Table of Contents PART I: INTRODUCTION Chapter1: Introduction

Table of Contents PART I: INTRODUCTION Chapter1: Introduction
Chapter: 2 Control Loop Hardware PART II: PROCESS DYNAMICS Chapter 3: Dynamic Modeling Chapter 4: Laplace Transforms Chapter 5: Transfer Functions Chapter 6: Dynamic Behavior of Ideal Systems

PART III: PID CONTROL Chapter 7: PID Control Chapter 8: PID Controller Tuning Chapter 9: PID Controller Tuning Chapter 10: Troubleshooting Control Loops Chapter 11: Frequency Response Analysis PART IV: ADVANCED PID CONTROL Chapter 12: Cascade, Ratio, and Feed forward Control

Chapter 13: PID Enhancements
Chapter 14: PID Implementation Issues PART V: CONTROL OF MIMO PROCESSES Chapter 15: PID Controllers Applied to MIMO Systems Chapter 16: Model Predictive Controller Chapter 17: Multi-Unit Controller Design Chapter 18: Case Studies

Chemical and Bio-Process Control
James B. Riggs M. Nazmul Karim

Chapter 1 Introduction

A Career in Process Control
Requires that engineers use all of their chemical engineering training (i.e., provides an excellent technical profession that can last an entire career) Can become a technical “Top Gun” Allows engineers to work on projects that can result in significant savings for their companies (i.e., provides good visibility within a company)

A Career in Process Control
Provides professional mobility. There is a shortage of experienced process control engineers. Is a well paid technical profession for chemical engineers.

Chemical Process Industries (CPI)
Hydrocarbon fuels Chemical products Pulp and paper products Agrochemicals Man-made fibers

Bio-Process Industries
Use micro-organisms to produce useful products Pharmaceutical industry Ethanol from grain industry

Importance of Process Control for the CPI
PC directly affects the safety and reliability of a process. PC determines the quality of the products produced by a process. PC can affect how efficient a process is operated. Bottom Line: PC has a major impact on the profitability of a company in the CPI.

Safety and Reliability
The control system must provide safe operation Alarms, safety constraint control, start-up and shutdown. A control system must be able to “absorb” a variety of disturbances and keep the process in a good operating region: Thunderstorms, feed composition upsets, temporary loss of utilities (e.g., steam supply), day to night variation in the ambient conditions

Benefits of Improved Control
Old Controller

Old Controller New Controller

Better Control Means Products with Reduced Variability
For many cases, reduced variability products are in high demand and have high value added (e.g., feedstocks for polymers). Product certification procedures (e.g., ISO 9000) are used to guarantee product quality and place a large emphasis on process control.

Old Controller New Controller Improved Performance

Maximizing the Profit of a Plant
Many times involves controlling against constraints. The closer that you are able to operate to these constraints, the more profit you can make. For example, maximizing the product production rate usually involving controlling the process against one or more process constraints.

Constraint Control Example
Consider a reactor temperature control example for which at excessively high temperatures the reactor will experience a temperature runaway and explode. But the higher the temperature the greater the product yield. Therefore, better reactor temperature control allows safe operation at a higher reactor temperature and thus more profit.

Importance of Process Control for the Bio-Process Industries
Improved product quality. Faster and less expensive process validation. Increased production rates.

Driving a Car: An Everyday Example of Process Control
Control Objective (Setpoint): Maintain car in proper lane. Controlled variable- Location on the road Manipulated variable- Orientation of the front wheels Actuator- Driver’s arms/steering wheel Sensor- Driver’s eyes Controller- Driver Disturbance- Curve in road

Logic Flow Diagram for a Feedback Control Loop

Temperature Control for a Heat Exchanger: ChE Control Example

Heat Exchanger Control
Controlled variable- Outlet temperature of product stream Manipulated variable- Steam flow Actuator- Control valve on steam line Sensor- Thermocouple on product stream Disturbance- Changes in the inlet feed temperature

DO Control in a Bio-Reactor

DO Control Controlled variable- the measured dissolved O2 concentration Manipulated variable- air flow rate to the bio-reactor Actuator- variable speed air compressor Sensor- ion-specific electrode in contact with the broth in the bio-reactor Disturbance- Changes in the metabolism of the microorganisms in the bio-reactor

Logic Flow Diagram for a Feedback Control Loop

Comparison of Driving a Car and Control of a Heat Exchanger
Actuator: Driver’s arm and steering wheel vs. Control valve Controller: the driver vs. an electronic controller Sensor: the driver’s eyes vs. thermocouple Controlled variable: car’s position on the road vs. temperature of outlet stream

The key feature of all feedback control loops is that the measured value of the controlled variable is compared with the setpoint and this difference is used to determine the control action taken.

In-Class Exercise Consider a person skiing down a mountain. Identify the controller, the actuator, the process, the sensor and the controlled variable. Also, indicate the setpoint and potential disturbances. Remember that the process is affected by the actuator to change the value of the controlled variable.

Types of Feedback Controllers
On-Off Control- e.g., room thermostat Manual Control- Used by operators and based on more or less open loop responses PID control- Most commonly used controller. Control action based on error from setpoint (Chaps 6-8). Advanced PID- Enhancements of PID: ratio, cascade, feedforward (Chaps 9-11). Model-based Control- Uses model of the process directly for control (Chap 13).

Duties of a Control Engineer
Tuning controllers for performance and reliability (Chap 7) Selecting the proper PID mode and/or advanced PID options (Chap 6, 10-12) Control loop troubleshooting (Chap 2 & 8) Multi-unit controller design (Chap 14) Documentation of process control changes

Characteristics of Effective Process Control Engineers
Use their knowledge of the process to guide their process control applications. They are “process” control engineers. Have a fundamentally sound picture of process dynamics and feedback control. Work effectively with the operators.

Operator Acceptance A good relationship with the operators is a NECESSARY condition for the success of a control engineer. Build a relationship with the operators based on mutual respect. Operators are a valuable source of plant experience. A successful control project should make the operators job easier, not harder.

Process Control and Optimization
Control and optimization are terms that are many times erroneously interchanged. Control has to do with adjusting flow rates to maintain the controlled variables of the process at specified setpoints. Optimization chooses the values for key setpoints such that the process operates at the “best” economic conditions.

Optimization and Control of a CSTR

Optimization Example

Economic Objective Function
VB > VC, VA, or VAF At low T, little formation of B At high T, too much of B reacts to form C Therefore, the exits an optimum reactor temperature, T*

Optimization Algorithm
1. Select initial guess for reactor temperature 2. Evaluate CA, CB, and CC 3. Evaluate  4. Choose new reactor temperature and return to 2 until T* identified.

Graphical Solution of Optimum Reactor Temperature, T*

Process Optimization Typical optimization objective function, :  = Product values-Feed costs-Utility costs The steady-state solution of process models is usually used to determine process operating conditions which yields flow rates of products, feed, and utilities. Unit costs of feed and sale price of products are combined with flows to yield  Optimization variables are adjusted until  is maximized (optimization solution).

Generalized Optimization Procedure

Optimization and Control of a CSTR

In-Class Exercise Identify an example for which you use optimization in your everyday life. List the degrees of freedom (the things that you are free to choose) and clearly define the process and how you determine the objective function.

Overview of Course Material
Control loop hardware (Chap 2) Dynamic modeling (Chap 3) Transfer functions and idealized dynamic behavior (Chap 4-6) PID controls (Chap 7-10) Advanced PID controls (Chap 12-14) Control of MIMO processes (Chap 15-18)

Fundamental Understanding and Industrially Relevant Skills
Laplace tranforms and transfer functions (Ch 4-5) Idealized dynamic behavior (Ch 6) Frequency response analysis (Ch 11) Industrially Relevant Skills- Control hardware and troubleshooting (Ch 2&10) Controller Implementation and tuning (Ch 7-9) Advanced PID techniques (Ch 12-14) MIMO control (Ch 15-18)

Process Control Terminology
Important to be able to communicate with operators, peers, and boss. New terminology appears in bold in the text New terminology is summarized at the end of each chapter. Review the terminology regularly in order to keep up with it.

Overall Course Objectives
Develop the skills necessary to function as an industrial process control engineer. Skills Tuning loops Control loop design Control loop troubleshooting Command of the terminology Fundamental understanding Process dynamics Feedback control

Overview All feedback control loops have a controller, an actuator, a process, and a sensor where the controller chooses control action based upon the error from setpoint. Control has to do with adjusting flow rates to maintain controlled variables at their setpoints while for optimization the setpoints for certain controllers are adjusted to optimize the economic performance of the plant.

Chapter 3 Dynamic Modeling

Process Dynamics Chemical Engineering courses are generally taught from a steady-state point-of-view. Dynamics is the time varying behavior of processes. Chemical processes are dynamically changing continuously. Steady-state change indicates where the process is going and the dynamic characteristics of a system indicates what dynamic path it will take.

Uses of Dynamic Process Models
Evaluation of process control configurations For analysis of difficult control systems for both existing facilities and new projects Process design of batch processes Operator Training Start-up/shut-down strategy development

Classification of Models
Lumped parameter models- assume that the dependent variable does not change with spatial location within the process, e.g., a perfectly well mixed vessel. Distributed parameter models- consider that the dependent variable changes with spatial location within the process.

Example of a Lumped Parameter Process

Example of a Distributed Parameter Process

Modeling Approaches Lumped parameter processes- Macroscopic balances are typically applied for conservation of mass, moles, or energy and result in ODE’s. Distributed parameter processes- Microscopic balances are typically applied yielding differential equations for conservation of mass, moles, or energy for a single point in the process which result in PDE’s.

Conservation Equations: Mass, Moles, or Energy Balances

Mass Balance Equation

Accumulation Term

Other Terms in Mass Balance Eq.

Mole Balance Equation

Accumulation Term

Other Terms in Mass Balance Eq.

Thermal Energy Balance Equation

Accumulation Term

Other Terms in Energy Balance Eq.

Constitutive Relationships
Usually in the form of algebraic equations. Used with the balance equations to model chemical engineering processes. Examples include: Reaction kinetic expressions Equations of state Heat transfer correlation functions Vapor/liquid equilibrium relationships

Degree of Freedom Analysis
The number of degrees of freedom (DOF) is equal to the number of unknowns minus the number of equations. When DOF is zero, the equations are exactly specified. When DOF is negative, the system is overspecified. When DOF is positive, it is underspecified.

Different Types of Modeling Terms
Dependent variables are calculated from the solution of the model equations. Independent variables require specification by the user or by an optimization algorithm and represent extra degrees of freedom. Parameters, such as densities or rate constants, are constants used in the model equations.

Dynamic Models of Control Systems
Control systems affect the process through the actuator system which has its own dynamics. The process responds dynamically to the change in the manipulated variable. The response of the process is measured by sensor system which has its own dynamics. There are many control systems for which the dynamics of the actuator and sensor systems are important.

Dynamic Modeling Approach for Process Control Systems

Dynamic Model for Actuators
These equations assume that the actuator behaves as a first order process. The dynamic behavior of the actuator is described by the time constant since the gain is unity

Heat addition as a Manipulated Variable
Consider a steam heated reboiler as an example. A flow control loop makes an increase in the flow rate of steam to the reboiler. The temperature of the metal tubes in the reboiler increases in a lagged manner. The flow rate of vapor leaving the reboiler begins to increase. The entire process is lumped together into one first order dynamic model.

Dynamic Response of an Actuator (First Order System)

Dynamic Model for Sensors
These equations assume that the sensors behave as a first order system. The dynamic behavior of the sensor is described by the time constant since the gain is unity T and L are the actual temperature and level.

Dynamic Model for an Analyzer
This equation assumes that the analyzer behaves as a pure deadtime element. The dynamic behavior of the sensor is described by the analyzer deadtime since the gain is unity

Dynamic Comparison of the Actual and Measured Composition

Model for Product Composition for CSTR with a Series Reaction

Model for Cell Growth in a Fed-Batch Reactor

Class Exercise: Dynamic Model of a Level in a Tank
Model equation is based on dynamic conservation of mass, i.e., accumulation of mass in the tank is equal to the mass flow rate into the tank minus the mass flow rate out.

Class Exercise Solution: Dynamic Model for Tank Level
Actuator on flow out of the tank. Process model Level sensor since the level sensor is much faster than the process and the actuator

Sensor Noise Noise is the variation in a measurement of a process variable that does not reflect real changes in the process variable. Noise is caused by electrical interference, mechanical vibrations, or fluctuations within the process. Noise affects the measured value of the controlled variable; therefore, it should be included when modeling process dynamics.

Modeling Sensor Noise Select standard deviation () of noise.  is equal to 50% of repeatability. Generate random number. Use random number in a correlation for the Gaussian distribution which uses This result is the noise on the measurement. Add the noise to the noise free measurement of the controlled variable.

Numerical Integration of ODE’s
Accuracy and stability are key issues. Reducing integration step size improves accuracy and stability of explicit integrators The ODE’s that represent the dynamic behavior of control systems in the CPI are not usually very stiff. As a result, a Euler integrator is usually the easiest and most effective integrator to use.

Development of Dynamic Process Models for Process Control Analysis
It is expensive, time consuming, and requires a specific expertise. It is typically only used in special cases for particularly difficult and important processes.

Overview Dynamic modeling for process control analysis should consider the dynamics of the actuator, the process, and the sensor as well as sensor noise.

Chapter 4 Laplace Transforms

Laplace Transforms Provide valuable insight into process dynamics and the dynamics of feedback systems. Provide a major portion of the terminology of the process control profession.

Laplace Transforms Useful for solving linear differential equations.
Approach is to apply Laplace transform to differential equation. Then algebraically solve for Y(s). Finally, apply inverse Laplace transform to directly determine y(t). Tables of Laplace transforms are available.

Method for Solving Linear ODE’s using Laplace Transforms

Some Commonly Used Laplace Transforms

Final Value Theorem Allows one to use the Laplace transform of a function to determine the steady-state resting value of the function. A good consistency check.

Initial-Value Theorem
Allows one to use the Laplace transform of a function to determine the initial conditions of the function. A good consistency check

Apply Initial- and Final-Value Theorems to this Example
Laplace transform of the function. Apply final-value theorem Apply initial- value theorem

Partial Fraction Expansions
Expand into a term for each factor in the denominator. Recombine RHS Equate terms in s and constant terms. Solve. Each term is in a form so that inverse Laplace transforms can be applied.

Heaviside Method Individual Poles

Heaviside Method Repeated Poles

Heaviside Method Example with Repeated Poles

Example of Solution of an ODE
ODE w/initial conditions Apply Laplace transform to each term Solve for Y(s) Apply partial fraction expansions w/Heaviside Apply inverse Laplace transform to each term

Overview Laplace transforms are an effective way to solve linear ODEs.

Transfer Functions and State Space Models
Chapter 5 Transfer Functions and State Space Models

Transfer Functions Provide valuable insight into process dynamics and the dynamics of feedback systems. Provide a major portion of the terminology of the process control profession.

Transfer Functions Defined as G(s) = Y(s)/U(s)
Represents a normalized model of a process, i.e., can be used with any input. Y(s) and U(s) are both written in deviation variable form. The form of the transfer function indicates the dynamic behavior of the process.

Derivation of a Transfer Function
Dynamic model of CST thermal mixer Apply deviation variables Equation in terms of deviation variables.

Derivation of a Transfer Function
Apply Laplace transform to each term considering that only inlet and outlet temperatures change. Determine the transfer function for the effect of inlet temperature changes on the outlet temperature. Note that the response is first order.

In-Class Exercise: Derive a Transfer Function

Poles of the Transfer Function Indicate the Dynamic Response
For a, b, c, and d positive constants, transfer function indicates exponential decay, oscillatory response, and exponential growth, respectively.

Poles on a Complex Plane

Exponential Decay

Damped Sinusoidal

Exponentially Growing Sinusoidal Behavior (Unstable)

What Kind of Dynamic Behavior?

Unstable Behavior If the output of a process grows without bound for a bounded input, the process is referred to a unstable. If the real portion of any pole of a transfer function is positive, the process corresponding to the transfer function is unstable. If any pole is located in the right half plane, the process is unstable.

Routh Stability Criterion

Routh Array for a 3rd Order System

Routh Stability Analysis Example

In-Class Exercise

Zeros of a Transfer Function
The zeros of a transfer functions are the value of s that render N(s)=0. If any of the zeros are positive, an inverse response is indicated. If all the zeros are negative, overshoot can occur in certain situations

Combining Transfer Functions
Consider the CST thermal mixer in which a heater is used to change the inlet temperature of stream 1 and a temperature sensor is used to measure the outlet temperature. Assume that heater behaves as a first order process with a known time constant.

Transfer function for the actuator Transfer function for the process Transfer function for the sensor

In-Class Exercise: Overall Transfer Function For a Self-Regulating Level

Block Diagram Algebra Series of transfer functions
Summation and subtraction Divider

Block Diagram Algebra

In-Class Exercise: Block Diagram Algebra

Solution of In-Class Exercise

What if the Process Model is Nonlinear
Before transforming to the deviation variables, linearize the nonlinear equation. Transform to the deviation variables. Apply Laplace transform to each term in the equation. Collect terms and form the desired transfer functions. Or instead, use Equation

Transfer Function for a Nonlinear Process

Advantages of Equation 5.7.3
Equation was derived based on linearing the nonlinear ODE, applying deviation variables, applying Laplace transforms and solving for Y(s)/U(s). Therefore, Equation is much easier to use than deriving the transfer function for both linear and nonlinear first-order ODEs.

Application of Equation 5.7.3

In-Class Exercise

State Space Models State space models are a system of linear ODEs that approximate a system of nonlinear ODEs at an operating point. Similar to Equation 5.7.3, state space models can be conveniently generated using the definitions of the terms in the coefficient matrices and the nonlinear ODEs.

State Space Models

Poles of a State Space Model

Overview The transfer function of a process shows the characteristics of its dynamic behavior assuming a linear representation of the process.

Dynamic Behavior of Ideal Systems
Chapter 6 Dynamic Behavior of Ideal Systems

Ideal Dynamic Behavior
Idealized dynamic behavior can be effectively used to qualitatively describe the behavior of industrial processes. Certain aspects of second order dynamics (e.g., decay ratio, settling time) are used as criteria for tuning feedback control loops.

Inputs

First Order Process Differential equation Transfer function
Note that gain and time constant define the behavior of a first order process.

First Order Process

Determine the Process Gain and Process Time Constant from Gp(s)

Estimate of First-Order Model from Process Response

In-Class Exercise By observing a process, an operator indicates that an increase of 1,000 lb/h of feed (input) to a tank produces a 8% increase in a self-regulating tank level (output). In addition, when a change in the feed rate is made, it takes approximately 20 minutes for the full effect on the tank to be observed. Using this process information, develop a first-order model for this process.

Second Order Process Differential equation Transfer function
Note that the gain, time constant, and the damping factor define the dynamic behavior of 2nd order process.

Underdamped vs Overdamped

Effect of  on Overdamped Response

Effect of  on Underdamped Response

Characteristics of an Underdamped Response
Rise time Overshoot (B) Decay ratio (C/B) Settling or response time Period (T)

Example of a 2nd Order Process
The closed loop performance of a process with a PI controller can behave as a second order process. When the aggressiveness of the controller is very low, the response will be overdamped. As the aggressiveness of the controller is increased, the response will become underdamped.

Determining the Parameters of a 2nd Order System from its Gp(s)

Second-Order Model Parameters from Process Response

High Order Processes The larger n, the more sluggish the process response (i.e., the larger the effective deadtime) Transfer function:

Example of Overdamped Process
Distillation columns are made-up of a large number of trays stacked on top of each other. The order of the process is approximately equal to the number of trays in the column

Integrating Processes
In flow and out flow are set independent of level Non-self-regulating process Example: Level in a tank. Transfer function:

Deadtime Transport delay from reactor to analyzer: Transfer function:

FOPDT Model High order processes are well represented by FOPDT models. As a result, FOPDT models do a better job of approximating industrial processes than other idealized dynamic models.

Determining FOPDT Parameters
Determine time to one-third of total change and time to two-thirds of total change after an input change. FOPDT parameters:

Determination of t1/3 and t2/3

In-Class Exercise Determine a FOPDT model for the data given in Problem 5.51 page 208 of the text.

Inverse Acting Processes
Results from competing factors. Example: Thermometer Example of two first order factors:

Lead-Lag Element

Recycle Processes Recycle processes recycle mass and/or energy.
Recycle results in larger time constants and larger process gains. Recycles (process integration) are used more today in order to improve the economics of process designs.

Mass Recycle Example

Overview It is important to understand terms such as:
Overdamped and underdamped response Decay ratio and settling time Rectangular pulse and ramp input FOPDT model Inverse acting process Lead-Lag element Process integration and recycle processes

Chapter 7 PID Control

PID Controls Most common controller in the CPI.
Came into use in 1930’s with the introduction of pneumatic controllers. Extremely flexible and powerful control algorithm when applied properly.

General Feedback Control Loop

Closed Loop Transfer Functions
From the general feedback control loop and using the properties of transfer functions, the following expressions can be derived:

Characteristic Equation
Since setpoint tracking and disturbance rejection have the same denominator for their closed loop transfer functions, this indicates that both setpoint tracking and disturbance rejection have the same general dynamic behavior. The roots of the denominator determine the dynamic characteristics of the closed loop process. The characteristic equation is given by:

Feedback Control Analysis
The loop gain (KcKaKpKs) should be positive for stable feedback control. An open-loop unstable process can be made stable by applying the proper level of feedback control.

Characteristic Equation Example
Consider the dynamic behavior of a P-only controller applied to a CST thermal mixer (Kp=1; p=60 sec) where the temperature sensor has a s=20 sec and a is assumed small. Note that Gc(s)=Kc.

Example Continued- Analysis of the Closed Loop Poles
When Kc =0, poles are and which correspond to the inverse of p and s. As Kc is increased from zero, the values of the poles begin to approach one another. Critically damped behavior occurs when the poles are equal. Underdamped behavior results when Kc is increased further due to the imaginary components in the poles.

In-Class Exercise Determine the dynamic behavior of a P- only controller with Kc equal to 1 applied to a first-order process in which the process gain is equal to 2 and the time constant is equal to 22. Assume that Gs(s) is equal to one and Ga(s) behaves as a first-order process with a time constant of 5.

PID Control Algorithm

Definition of Terms e(t)- the error from setpoint [e(t)=ysp-ys].
Kc- the controller gain is a tuning parameter and largely determines the controller aggressiveness. I- the reset time is a tuning parameter and determines the amount of integral action. D- the derivative time is a tuning parameter and determines the amount of derivative action.

Transfer Function for a PID Controller

Example for a First Order Process with a PI Controller

Example of a PI Controller Applied to a Second Order Process

Properties of Proportional Action
Closed loop transfer function base on a P-only controller applied to a first order process. Properties of P control Does not change order of process Closed loop time constant is smaller than open loop p Does not eliminate offset.

Offset Resulting from P-only Control

Proportional Action for the Response of a PI Controller

Proportional Action The primary benefit of proportional action is that it speedup the response of the process.

Properties of Integral Action
Based on applying an I- only controller to a first order process Properties of I control Offset is eliminated Increases the order by 1 As integral action is increased, the process becomes faster, but at the expense of more sustained oscillations

Integral Action for the Response of a PI Controller

Integral Action The primary benefit of integral action is that it removes offset from setpoint. In addition, for a PI controller all the steady-state change in the controller output results from integral action.

Properties of Derivative Action
Closed loop transfer function for derivative-only control applied to a second order process. Properties of derivative control: Does not change the order of the process Does not eliminate offset Reduces the oscillatory nature of the feedback response

Derivative Action for the Response of a PID Controller

Derivative Action The primary benefit of derivative action is that it reduces the oscillatory nature of the closed-loop response.

Position Form of the PID Algorithm

Proportional Band Another way to express the controller gain.
Kc in this formula is dimensionless. That is, the controller output is scaled 0-100% and the error from setpoint is scaled 0-100%. In more frequent use years ago, but it still appears as an option on DCS’s.

Conversion from PB to Kc
Proportional band is equal to 200%. The range of the error from setpoint is 200 psi. The controller output range is 0 to 100%.

Conversion from Kc to PB
Controller gain is equal to 15 %/ºF The range of the error from setpoint is 25 ºF. The controller output range is 0 to 100%.

Digital Equivalent of PID Controller
The trapezoidal approximation of the integral. Backward difference approximation of the first derivative

Digital Version of PID Control Algorithm

Derivation of the Velocity Form of the PID Control Algorithm

Velocity Form of PID Controller
Note the difference in proportional, integral, and derivative terms from the position form. Velocity form is the form implemented on DCSs.

Correction for Derivative Kick
Derivative kick occurs when a setpoint change is applied that causes a spike in the derivative of the error from setpoint. Derivative kick can be eliminated by replacing the approximation of the derivative based on the error from setpoint with the negative of the approximation of the derivative based on the measured value of the controlled variable, i.e.,

Correction for Aggressive Setpoint Tracking
For certain process, tuning the controller for good disturbance rejection performance results in excessively aggressive action for setpoint changes. This problem can be corrected by removing the setpoint from the proportional term. Then setpoint tracking is accomplished by integral action only.

The Three Versions of the PID Algorithm Offered on DCS’s
(1) The original form in which the proportional, integral, and derivative terms are based on the error from setpoint

The Three Versions of the PID Algorithm Offered on DCSs
(2) The form in which the proportional and integral terms are based on the error from setpoint while the derivative-on- measurement is used for the derivative term.

The Three Versions of the PID Algorithm Offered on DCS’s
(3) The form in which the proportional and derivative terms are based on the process measurement and the integral is based on the error from setpoint.

Guidelines for Selecting Direct and Reverse Acting PID’s
Consider a direct acting final control element to be positive and reverse to be negative. If the sign of the product of the final control element and the process gain is positive, use the reverse acting PID algorithm. If the sign of the product is negative, use the direct acting PID algorithm If control signal goes to a control valve with a valve positioner, the actuator is considered direct acting.

Level Control Example Process gain is positive because when flow in is increased, the level increases. If the final control element is direct acting, use reverse acting PID. For reverse acting final control element, use direct acting PID.

Level Control Example Process gain is negative because when flow out is increased, the level decreases. If the final control element is direct acting, use direct acting PID. For reverse acting final control element, use reverse acting PID.

In-Class Exercise Write the position form of the PID algorithm for Example 3.4, and assume that the control valve on the feed line to the mixer has an air-to-close actuator. Use the form that is not susceptible to derivative kick. Specify whether the controller is a direct-acting or reverse-acting controller.

In-Class Exercise Write the velocity form of the PID algorithm for Example 3.1, and assume that the control valve on the feed line to the mixer has an air-to-open actuator. Use the form that is not susceptible to derivative kick or proportional kick. Specify whether the controller is a direct-acting or reverse- acting controller.

Filtering the Process Measurement
Filtering reduces the effect of sensor noise by approximating a running average. Filtering adds lag when the filtered measurement is used for control. Normally, use the minimum amount of filtering necessary. f- filter factor (0-1)

Feedback Loop with Sensor Filtering

Effect of Filtering on Closed Loop Dynamics

Analysis of Example f is equal to t (1/f-1) as f becomes small, f becomes large. As f is increased, p’ will increase. Critical issue is relative magnitude of f compare to p.

Effect of the Amount of Filtering on the Open Loop Response

Effect of a Noisy Sensor on Controlled Variable without Filtering

Effect of a Noisy Sensor on Controlled Variable with Filtering

An Example of Too Much and Too Little Filtering

Relationship between Filter Factor (f), the Resulting Repeatability Reduction Ratio (R) and the Filter Time Constant (f)

Key Issues for Sensor Filtering
To reduce the effect of noise (i.e., R is increased), f must be reduced, which increases the value of f. Filtering slows the closed-loop response significantly as f becomes larger than 10% of p. The effect of filtering on the closed-loop response can be reduced by increasing the frequency with which the filter is applied, i.e., reducing f.

PID Controller Design Issues
Over 90% of control loops use PI controller. P-only: used for fast responding processes that do not require offset free operation (e.g., certain level and pressure controllers) PI: used for fast responding processes that require offset free operation (e.g., certain flow, level, pressure, temperature, and composition controllers)

Integrating Processes
For integrating processes, P-only control provides offset-free operation. In fact, if as integral action is added to such a case, the control performance degrades. Therefore, for integrating processes, P-only control is all that is usually required.

PID Controller Design Issues
PID: use for sluggish processes (i.e., a process with large deadtime to time constant ratios) or processes that exhibit severe ringing for PI controllers. PID controllers are applied to certain temperature and composition control loops. Use derivative action when:

Comparison between PI and PID for a Low p/p Ratio

Comparison between PI and PID for a High p/p Ratio

Analysis of Several Commonly Encountered Control Loops
Flow control loops Level control loops Pressure control loops Temperature control loops Composition control loops DO control loop Biomass controller

Flow Control Loop Since the flow sensor and the process (changes in flow rate for a change in the valve position) are so fast, the dynamics of the flow control loop is controlled by the dynamics of the control valve. Almost always use PI controller.

Deadband of a Control Valve
Deadband of industrial valves is between ±10%- ±25%. As a result, small changes in the air pressure applied to the valve do not change the flow rate.

Deadband of Flow Control Loop
A control valve (deadband of ±10-25%) in a flow control loop or with a positioner typically has a deadband for the average flow rate of less than ±0.5% due to the high frequency opening and closing of the valve around the specified flow rate.

Level Control Loop Dynamics of the sensor and actuator are fast compared to the process. Use P-only controller if it is an integrating process.

Pressure Control Process
The sensor is generally faster than the actuator, which is faster than the process. Use P-only controller if it is an integrating process otherwise use a PI controller.

Temperature Control Loop
The dynamics of the process and sensor are usually slower than the actuator. Use a PI controller unless the process is sufficiently sluggish to warrant a PID controller.

Analysis of PI Controller Applied to Typical Temperature Loop

Further Analysis of Dynamic of a Typical Temperature Control Loop
Note that as the controller gain is increased, i.e., KcKp increase, the closed loop time constant becomes smaller. Also, note that as the controller gain is increased, the value of  decreases.

Composition Control Loop
The process is usually the slowest element followed by the sensor with the actuator being the fastest. Use a PI controller unless the process is sufficiently sluggish to warrant a PID controller.

DO Control Loop The process and the sensor have approximately the same dynamic response. This is a fast responding process for which offset- free operation is desired. Therefore, PI controller should be used.

Biomass Controller The process for this system is the slowest element.
Because the process is a high-order sluggish process, a PID controller is required.

Overview The characteristic equation determines the dynamic behavior of a closed loop system Proportional, integral, and derivative action each have unique characteristics. There are a number of different ways to apply a PID controller. Use a PI controller unless offset is not important or if the process is sluggish. When analyzing the dynamics of a loop, consider the dynamics of the actuator, the process, and the sensor separately.

Performance of P-only, PI and PID Controllers
Chapter 8 Performance of P-only, PI and PID Controllers

P-only Control For an open loop overdamped process as Kc is increased the process dynamics goes through the following sequence of behavior overdamped critically damped oscillatory ringing sustained oscillations unstable oscillations

Dynamic Changes as Kc is Increased for a FOPDT Process

Root Locus Diagram (Kc increases a to g)

Effect of Kc on Closed-Loop 

Effect of Kc on Closed-Loop p

P-only Controller Applied to First-Order Process without Deadtime
Without deadtime, the system will not become unstable regardless of how large Kc is. First-order process model does not consider combined actuator/process/sensor system. Therefore, first-order process model without deadtime is not a realistic model of a process under feedback control.

PI Control As Kc is increased or I is decreased (i.e., more aggressive control), the closed loop dynamics goes through the same sequence of changes as the P-only controller: overdamped, critically damped, oscillatory, ringing, sustained oscillations, and unstable oscillations.

Effect of Variations in Kc
Effect of Variations in tI

Analysis of the Effect of Kc and I
When there is too little proportional action or too little integral action, it is easy to identify. But it is difficult to differentiate between too much proportional action and too much integral action because both lead to ringing.

Response of a Properly Tuned PI Controller

Response of a PI Controller with Too Much Proportional Action

Response of a PI Controller with Too Much Integral Action

PID Control Kc and I have the same general effect as observed for PI control. Derivative action tends to reduce the oscillatory nature of the response and results in faster settling for systems with larger deadtime to time constant ratios.

Comparison between PI and PID for a Low p/p Ratio

Comparison between PI and PID for a Higher qp/tp Ratio

An Example of Too Much Derivative Action

Effect of D on Closed-Loop 

Demonstration: Visual Basic Simulator
Effect of Kc, I, and D

Overview As the controller aggressiveness is increased (i.e., Kc is increased or I is decreased), the response goes from overdamped to critically damped to oscillatory to ringing to sustained oscillations to unstable. Too little proportional or integral action are easy to identify while too much proportional or integral results in ringing. Differentiating between too much integral or proportional action requires comparing the lag between the controller output and the CV.

Chapter 9 PID Tuning Methods

Controller Tuning Involves selection of the proper values of Kc, I, and D. Affects control performance. Affects controller reliability Therefore, controller tuning is, in many cases, a compromise between performance and reliability.

Tuning Criteria Specific criteria General criteria Decay ratio
Minimize settling time General criteria Minimize variability Remain stable for the worst disturbance upset (i.e., reliability) Avoid excessive variation in the manipulated variable

Decay Ratio for Non-Symmetric Oscillations

Performance Assessment
Performance statistics (IAE, ISE, etc.) which can be used in simulation studies. Standard deviation from setpoint which is a measure of the variability in the controlled variable. SPC charts which plot product composition analysis along with its upper and lower limits.

Example of an SPC Chart

Classical Tuning Methods
Examples: Cohen and Coon method, Ziegler- Nichols tuning, Cianione and Marlin tuning, and many others. Usually based on having a model of the process (e.g., a FOPDT model) and in most cases in the time that it takes to develop the model, the controller could have been tuned several times over using other techniques. Also, they are based on a preset tuning criterion (e.g., QAD)

Controller Tuning by Pole Placement
Based on model of the process Select the closed-loop dynamic response and calculate the corresponding tuning parameters. Application of pole placement shows that the closed-loop damping factor and time constant are not independent. Therefore, the decay ratio is a reasonable tuning criterion.

Controller Design by Pole Placement
A generalized controller (i.e., not PID) can be derived by using pole placement. Generalized controllers are not generally used in industry because Process models are not usually available PID control is a standard function built into DCSs.

IMC-Based Tuning A process model is required (Table 9.4 contain the PID settings for several types of models based on IMC tuning). Although a process model is required, IMC tuning allows for adjusting the aggressiveness of the controller online using a single tuning parameter, f.

Controller Reliability
The ability of a controller to remain in stable operation with acceptable performance in the face of the worst disturbances that the controller is expected to handle.

Analysis of the closed loop transfer function for a disturbance shows that the type of dynamic response (i.e., decay ratio) is unaffected by the magnitude to the disturbance.

We know from industrial experience that certain large magnitude disturbance can cause control loops to become unstable. The explanation of this apparent contradiction is that disturbances can cause significant changes in Kp, p, and p which a linear analysis does not consider.

Controller Reliability Example: CSTR with CA0 Upsets

Is determined by the combination of the following factors Process nonlinearity Disturbance type Disturbance magnitude and duration If process nonlinearity is high but disturbance magnitude is low, reliability is good. If disturbance magnitude is high but process nonlinearity is low, reliability is good.

Tuning Criterion Selection

Tuning Criterion Selection Procedure
First, based on overall process objectives, evaluate controller performance for the loop in question. If the control loop should be detuned based on the overall process objectives, the tuning criterion is set. If the control loop should be tuned aggressively based on the overall process objectives, the tuning criterion is selected based on a compromise between performance and reliability.

Selecting the Tuning Criterion based on a Compromise between Performance and Reliability
Select the tuning criterion (typically from critically damped to 1/6 decay ratio) based on the process characteristics: Process nonlinearity Disturbance types and magnitudes

Effect of Tuning Criterion on Control Performance
The more aggressive the control criterion, the better the control performance, but the more likely the controller can go unstable.

Filtering the Sensor Reading
For most sensor readings, a filter time constant of 3 to 5 s is more than adequate and does not slow down the closed-loop dynamics. For a noisy sensor, sensor filtering usually slows the closed-loop dynamics. To evaluate compare the filter time constant with the time constants for the acutator, process and sensor.

Recommended Tuning Approach
Select the tuning criterion for the control loop. Apply filtering to the sensor reading Determine if the control system is fast or slow responding. For fast responding, field tune (trail-and-error) For slow responding, apply ATV-based tuning

Field Tuning Approach Turn off integral and derivative action.
Make initial estimate of Kc based on process knowledge. Using setpoint changes, increase Kc until tuning criterion is met

Field Tuning Approach Decrease Kc by 10%.
Make initial estimate of I (i.e.,I=5p). Reduce I until offset is eliminated Check that proper amount of Kc and I are used.

An Example of Inadequate Integral Action
Oscillations not centered about setpoint and slow offset removal indicate inadequate integral action.

Field Tuning Example

ATV Identification and Online Tuning
Perform ATV test and determine ultimate gain and ultimate period. Select tuning method (i.e., ZN or TL settings). Adjust tuning factor, FT, to meet tuning criterion online using setpoint changes or observing process performance: Kc=KcZN/FT I=ZN×FT

ATV Test Select h so that process is not unduly upset but an accurate a results. Controller output is switched when ys crosses y0 It usually take 3-4 cycles before standing is established and a and Pu can be measured.

Applying the ATV Results
Calculate Ku from ATV results. ZN settings TL settings

Comparison of ZN and TL Settings
ZN settings are too aggressive in many cases while TL settings tend to be too conservative. TL settings use much less integral action compared to the proportional action than ZN settings. As a result, in certain cases when using TL settings, additional integral action is required to remove offset in a timely fashion.

Advantages of ATV Identification
Much faster than open loop test. As a result, it is less susceptible to disturbances Does not unduly upset the process.

Online Tuning Provides simple one-dimensional tuning which can be applied using setpoint changes or observing controller performance over a period of time.

ATV Test Applied to Composition Mixer

CST Composition Mixer Example
Calculate Ku Calculate ZN settings Apply online tuning

Online Tuning for CST Composition Mixer Example
FT=0.75 FT=0.5

Control Performance for Tuned Controller

Critically Damped Tuning for CST Composition Mixer

Comparison Between 1/6 Decay Ratio and Critically Damped Tuning

ATV based tuning

PID Tuning Procedure Tune PI controller using field tuning or ATV identification with online tuning. Increase D until minimum response time is obtained. Initially set D=Pu/8. Increase D and Kc by the same factor until desired response is obtained. Check response to ensure that proper amount of integral action is being used.

Comparison between PI and PID for the Heat Exchanger Model

Comparison of PI and PID
The derivative action allows for larger Kc which in turn results in better disturbance rejection for certain processes.

PID Tuning Example

Initial Settings for Level Controllers for P-only Control
Based on critically damped response. FMAX is largest expected change in feed rate. LMAX is the desired level change under feedback control. Useful as initial estimates for slow responding level control systems.

Initial Settings for Level Controllers for PI Control
Ac is cross-sectional area to tank and r is liquid density. FMAX is largest expected change in feed rate. LMAX is the desired level change under feedback control. Useful as initial estimates for slow responding level control systems.

Initial Settings for Level Controllers
Use online tuning adjusting Kc and I with FT to obtain final tuning. Remember that Kc is expressed as (flow rate/%); therefore, height difference between 0% and 100% is required to calculate I.

In-Class Example Calculate the initial PI controller settings for a level controller with a critically damped response for a 10 ft diameter tank (i.e., a cylinder placed on its end) with a measured height of 10 ft that normally handles a feed rate of 1000 lb/h. Assume that it is desired to have a maximum level change of 5% for a 20% feed rate change and that the liquid has a density corresponding to that of water.

Control Interval, t t is usually 0.5 to 1.0 seconds for regulatory loops and 30 to 120 seconds for supervisory loops for DCS’s. In order to adequately approach continuous performance, select the control interval such that: t < 0.05(p+p) For certain processes, t is set by the timing of analyzer updates and the previous formula can be used to assess the effect on control performance

Effect of Control Interval on Control Performance
When the controller settings for continuous control are used with t=0.5, instability results. Results shown here are based on retuning the system for t=0.5 resulting in a 60% reduction in Kc.

Overview Controller tuning is many times a compromise between performance and reliability. Reliability is determined by process nonlinearity and the disturbance type and magnitude. The controller tuning criterion should be based on controller reliability and the process objectives.

Overview Classical tuning methods, pole placement and IMC tuning are not recommended because they are based on a preset tuning criterion and they usually require a process model. Tune fast loops should be tuned using field tuning and slow loops using ATV identification with online tuning.

Chapter 10 Control Loop Troubleshooting

Objectives for Control Loop Troubleshooting
Be able to implement an overall troubleshooting methodology Be able to determine whether or not the actuator, process, sensor, and controller are functioning properly. Recall the major failure modes for each of the elements of a control loop.

Troubleshooting Loops in the CPI

Control Diagram of a Typical Control Loop

Components and Signals of a Typical Control Loop

What is Control Loop Troubleshooting?
Control loop is suspected of not functioning properly. Poor overall control performance Erratic behavior Control loop was removed from service. Identify the source of the problem. Correct the problem. Retune the controller and monitor.

Overall Approach to Troubleshooting Control Loops
Check subsystem separately. Actuator system Controller Sensor Process Then check performance of the entire control loop What’s been changed lately?

Checking the Actuator System
Apply block sine wave input changes to the setpoint for the flow controller. Determine the deadband of the flow control loop from a block sine wave test. Also, estimate the time constant for the flow control loop from the block sine wave test. If the time constant is less than 2 seconds and the deadband is less than 0.5%, there is no need to evaluate the actuator system further

Block Sine Wave Test

Common Problems with the Actuator System
Excessive valve deadband Improperly sized control valve Valve packing is tightened too much Improperly tuned valve positioner

Check the Sensor System
Evaluate the repeatability of the sensor during steady-state operation. Evaluate the sensor dynamics. This may require an independent measurement of the controlled variable. Or check the elements that could contribute to a slow responding sensor.

Most Common Sensor Failures
Transmitter Improperly calibrated Excessive signal filtering Temperature sensor Off calibration Improperly located thermowell Buildup of material on the thermowell Pressure Plugged line to pressure sensor

Most Common Sensor Failures
Sampling system for GC Plugged line in sampling system Flow indicator Plugged line to differential pressure sensor Level indicator

Check the Controller Check the filtering on the measured value of the controlled variable. Check the cycle time for the controller. Check the tuning on the controller.

Factors that Affect the Closed-Loop Performance of a Control Loop
The type and magnitude of disturbances Primarily affects variability in CV Can affect nonlinear behavior The lag associated with the components of the feedback control loop (actuator, process, and sensor) Results in slower disturbance rejection which affects variability Precision of the feedback components Directly affects variability

Testing the Entire Control Loop
Closed-loop block sine wave test Variability of the controlled variable over a period of a week or more.

Closed-Loop Block Sine Wave Test

Closed-Loop Block Sine Wave Test
Closed-loop deadband Indication of the effect of actuator deadband, sensor noise, and resolution of A/D and D/A converters Closed-loop settling time Indication of the combined lags of the control loop components A means of determining if all the major problems with in a control loop have been corrected.

SPC Chart A Method for Evaluating the Long-Term Performance of a Controller

Long-Term Measurement of Variability
Direct measure of control performance in terms that relate to economic objectives Takes longer to develop than closed-loop block sine wave test

Troubleshooting Example
Symptom- The variability in the impurity level in the overhead product of a distillation column is greater than the specified limit. Step 1 Check the actuator system By applying a series of block sine wave tests, it was determined that the deadband and time constant of the flow control loop were 0.3% and 1.5 second which indicates that the actuator system is functioning properly.

Troubleshooting Example (cont)
Step 2 Check the controller The filtering on the product analyzer reading was found to be excessive The controller was retuned The control performance was improved but at times it was still not meeting the product variability specifications

Troubleshooting Example (cont)
Check the product analyzer The repeatability was determined by observing steady- state periods and was found to be well within the product variability specifications. The cycle time of the controller was found to be appropriate. Excessive transport delay in the sample system was identified and a new sample pump installed The composition controller was retuned and control performance met specifications.

Troubleshooting Exercise
Students pair up into groups of two. One student represents the “process” and the other, who is acting as the control engineer, performs the troubleshooting. The process student must choose a loop fault and the control engineer requests the results of certain tests from the process. After the engineer identifies the problem and fixes it, the students switch roles and repeat the exercise.

Overview of CPI Troubleshooting
In order to ensure that a control loop is functioning properly, the control engineer must have a thorough knowledge of the proper design and operation (Table 2.3) of the various components that comprise the control loop.

Troubleshooting in the Bio-Tech Industries

Overall Approach For the CPI, troubleshooting usually involves evaluation of one control loop at a time. For the bio-tech industries, it usually involves evaluating the operation of a bio-reactor. For the bio-tech industries, poor operation of a bio-reactor can involve poorly performing control loops or poorly performing sensors. Therefore, troubleshooting is a global problem.

Expert Systems Expert systems for troubleshooting a bio- reactor are based on distilling the experience of experts into a set of “if-then- else” rules that guide the operator to the root problem(s). Expert systems can identify batches that can be returned to a normal operating window. Otherwise, the batch can result in off- specification products that are useless.

Actuator Systems Block sine wave tests can be used to determine the deadband and time constant for the actuator system. See Table 2.3 for desired performance levels.

Sensor Systems Coriolis flow meters- require periodic calibration.
Ion-specific electrodes (DO, pH and Redox)- require regular replacement and proper location is important. DO- membrane should be replaced regularly. pH- calibration drift a problem requiring calibration Redox- regular calibration a problem

Sensor Systems Turbidity sensor- cell can accumulated in the measurement cell. Mass spec- highly reliable due to regular calibration with air samples. HPLC- use “guard” columns to reduce fouling of the HPLC column. FIA- malfunctioning valves a major problem.

Overview of Bio-Tech Troubleshooting
Expert systems are used to guide the troubleshooting activity. Troubleshooting bio-reactors is a global problem requiring a complete understanding of the entire system. Effective troubleshooting of bio-reactors can greatly reduce the frequency of “bad” batches and is, therefore, economically important.

Frequency Response Analysis
Chapter 11 Frequency Response Analysis

Frequency Response Analysis
Is the response of a process to a sinusoidal input Considers the effect of the time scale of the input. Important for understanding the propagation of variability through a process. Important for terminology of the process control field. But it is NOT normally used for tuning or design of industrial controllers.

Process Exposed to a Sinusoidal Input

Key Components of Frequency Response Analysis

Effect of Frequency on Ar and 

Bode Plot: A Convenient Means of Presenting Ar and  versus 

Ways to Generate Bode Plot
Direct excitation of process. Combine transfer function of the process with sinusoidal input. Substitute s=i into Gp(s) and convert into real and imaginary components which yield Ar() and (). Apply a pulse test.

Developing a Bode Plot from the Transfer Function

Derivation of the Bode Plot for a First Order Process

Properties of Bode Plots

Bode Plot of Complex Transfer Functions
Break transfer function into a product of simple transfer functions. Identify Ar() and  of each simple transfer function from Table 8.1. Combine to get Ar() and () for complex transfer function according to properties. Plot results as a function of 

Example of a Bode Plot of a Complex Transfer Function

Example Continued

Bode Stability Criterion

Bode Stability Criterion
A system is stable if Ar is less than 1.0 at the critical frequency (i.e.,  that corresponds to=-180º) Closed loop stability of a system can be analyzed by applying the Bode Stability Criterion to the product of the transfer functions of the controller and the process, i.e., Gc(s)Gp(s).

Gain Margin

Gain Margin Gain Margin = 1/Ar*
Where Ar* is the amplitude ratio at the critical frequency. Controllers are typically designed with gain margins in the range of 1.4 to 1.8 which implies that Ar at the critical frequency varies between 0.7 and 0.55, respectively.

Phase Margin

Phase Margin PM = 
Where * is  at the crossover frequency. Controllers are typically designed with a PM between 30º to 45º.

Tuning a Control from the Gain Margin

Tuning a Control from the Phase Margin
Phase margin determines the phase angle at the crossover frequency. The amplitude ratio at the crossover frequency is one; therefore, the controller gain can be calculated from the equation for the amplitude ratio.

Example of a Pulse Test

Developing a Process Transfer Function from a Pulse Test

Limitations of Transfer Functions Developed from Pulse Tests
They require an open loop time constant to complete. Disturbances can corrupt the results. Bode plots developed from pulse tests tend to be noisy near the crossover frequency which affects GM and PM calculations.

Nyquist Diagram

Nyquist Diagram (Complex Plane Plot)

Nyquist Stabilty Criterion

Closed Loop Frequency Response

Example of a Closed Loop Bode Plot

Analysis of Closed Loop Bode Plot
At low frequencies, the controller has time to reject the disturbances, i.e., Ar is small. At high frequencies, the process filters (averages) out the variations and Ar is small. At intermediate frequencies, the controlled system is most sensitive to disturbances.

Peak Frequency of a Controller
The peak frequency indicates the frequency for which a controller is most sensitive.

Overview Understanding how the frequency of inputs affects control performance and control loop stability is important. The analytical aspects of frequency response analysis are rarely used industrially.

Cascade, Ratio, and Feedforward Control
Chapter 12 Cascade, Ratio, and Feedforward Control

Cascade, Ratio, and Feedforward Control
Each of these techniques offers advantages with respect to disturbance rejection: Cascade reduces the effect of specific types of disturbances. Ratio reduces the effect of feed flow rates changes Feedforward control is a general methodology for compensating for measured disturbances.

Compensating for Disturbances Reduces Deviations from Setpoint and Settling Time

Level Controller on a Tank With and Without Cascade Control

Analysis of Cascade Example
Without a cascade level controller, changes in downstream pressure will disturb the tank level. With cascade level controller, changes in downstream pressure will be absorbed by the flow controller before they can significantly affect tank level because the flow controller responds faster to this disturbance than the tank level process.

Key Features for Cascade Control to be Successful
Secondary loop should reduce the effect of one or more disturbances. Secondary loop must be at least 3 times faster than master loop. The CV for the secondary loop should have a direct effect on the CV for the primary loop. The secondary loop should be tuned tightly.

Cascade Reactor Temperature Control

Analysis of Example Without cascade, changes in the cooling water temperature will create a significant upset for the reactor temperature. With cascade, changes in the cooling water temperature will be absorbed by the slave loop before they can significantly affect the reactor temperature.

Multiple Cascade Example
This approach works because the flow control loop is much faster than the temperature control loop which is much faster than the composition control loop.

Example Draw schematic: A temperature controller on the outlet stream is cascaded to a pressure controller on the steam which is cascaded to a control valve on the condensate.

Solution

Ratio Control Useful when the manipulated variable scales directly with the feed rate to the process. Dynamic compensation is required when the controlled variable responds dynamically different to feed rate changes than it does to a changes in the manipulated variable.

Typical Performance Improvements using Ratio Control

Ratio Control for Wastewater Neutralization

Analysis of Ratio Control Example
The flow rate of base scales directly with the flow rate of the acidic wastewater. The output of the pH controller is the ratio of NaOH flow rate to acid wastewater flow rate; therefore, the product of the controller output and the measured acid wastewater flow rate become the setpoint for the flow controller on the NaOH addition.

Ratio Control Applied for Vent Composition Control

Ratio Control Requiring Dynamic Compensation

Example Draw schematic: For a control system that adjusts the ratio of fuel flow to the flow rate of the process fluid to control the outlet temperature of the process fluid. Use a flow controller on the fuel.

Solution

Feedforward and Feedback Level Control

Analysis of Feedforward and Feedback Level Control
Feedback-only must absorb the variations in steam usage by feedback action only. Feedforward-only handle variation in steam usage but small errors in metering will eventually empty or fill the tank. Combined feedforward and feedback has best features of both controllers.

Derivation of FF Controller

Lead/Lag Element for Implementing FF Control

Effect of Lead/Lag Ratio

Static Feedforward Controller
A static feedforward controller make a correction that is directly proportional to the disturbance change. A static feedforward controller is used when the process responds in a similar fashion to a change in the disturbance and the manipulated variable.

Feedforward When p«d

Example of Feedforward Control for d<p

Static Feedforward Results
When the inlet temperature drops by 20ºC, Q is immediately increased by 20 kW. Deviations from setpoint result from dynamic mismatch

Perfect Feedforward Control
FF correction is mirror image of disturbance effect. Net effect is no change in controlled variable.

Required Dynamic Compensation
Since the Q affects the process slower than Ti , initially overcompensation in Q is required followed by cutting back on Q to 20 kW.

Results with Dynamic Compensation

Feedforward Control Action

Effect of Lead/Lag Ratio

Tuning a FF Controller Make initial estimates of lead/lag parameters based on process knowledge. Under open loop conditions, adjust Kff until steady- state deviation from setpoint is minimized.

Tuning a FF Controller Analyzing the dynamic mismatch, adjust ff.

Tuning a FF Controller Finally, adjust (ld - lg) until approximately equal areas above and below the setpoint result.

Tuning a FF Controller

Feedback Control Can effectively eliminate disturbances for fast responding processes. But it waits until the disturbance upsets the process before taking corrective action. Can become unstable due to nonlinearity and disturbance upsets.

Feedforward Control Compensates for d’s before process is affected
Most effective for slow processes and for processes with significant deadtime. Can improve reliability of the feedback controller by reducing the deviation from setpoint. Since it is a linear controller, its performance will deteriorate with nonlinearity.

Combined FF and FB Control

Combined FF and FB for the CSTR

Results for CSTR

Analysis of Results for CSTR
FB-only returns to setpoint quickly but has large deviation from setpoint. FF-only reduces the deviation from setpoint but is slow to return to setpoint. FF+FB reduces deviation from setpoint and provides fast return to setpoint.

Example Draw schematic: For a combined feedforward and feedback controller in which the inlet feed temperature is the feedforward variable and the outlet temperature is the feedback variable. The combined controller output is the setpoint for a steam pressure controller.

Solution

Overview Cascade can effectively remove certain disturbances if the slave loop is at least 3 times faster than the master loop. Ratio control is effective for processes that scale with the feed rate. Feedforward can be effective for measured disturbances for slow responding processes as long as the process nonlinearity is not too great.

Chapter 13 PID Enhancements

Limitations of Convential PID Controllers
The performance of PID controllers can be substantially limited by: Process nonlinearity Measurement deadtime Process constraints This chapter will consider approaches for PID controllers to handle each of these problems.

Inferential Control Uses easily measure process variables (T, P, F) to infer more difficult to measure quantities such as compositions and molecular weight. Can substantially reduce analyzer delay. Can be much less expensive in terms of capital and operating costs. Can provide measurements that are not available any other way.

Effect of Deadtime on Control Performance

Inferential Temperature Control for Distillation Columns

Choosing a Proper Tray Temperature Location
A tray temperature used for inferential control should show strong sensitivity.

Feed Composition Affects Composition/Temperature Correlation

Feedback Correction for Feed Composition Changes

Inferential Reactor Conversion Control

Molecular Weight of a Polymer

Soft Sensors Based on Neural Networks
Neural network (NN) provides nonlinear correlation. Weights are adjusted until NN agrees with plant data NN-based soft sensors are used to infer NOx levels in the flue gas from power plants.

Heat Exchangers are Nonlinear with Respect of Flow Rate Changes

Effect of Scheduling Controller Tuning
Shows the results for a nonscheduled controller that was tuned for v=7 ft/sec after the feed rate is changed to v=4 ft/sec and the results for a scheduled controller for the same upset.

Scheduling Controller Tuning
Can be effective when either a measured disturbance or the controlled variable correlates with nonlinear process changes. Tune the controller at different levels of the scheduling parameter and combine the results so that the controller tuning parameters vary over the full operating range.

Override/Select Controls
Process are many times operated at the safety or equipment limits in order to maximize process throughput. During upset periods, it is essential that safety limits are enforced.

Furnace Tube Temperature Constraint Control

Analysis of Tube Temperature Constraint Controller
Under normal operation, the controller adjusts the furnace firing rate to maintain process stream at the setpoint temperature. At higher feed rates, excessive tube temperatures can result greatly reducing the useful life of the furnace tubes. The LS controller reduces the firing rate to ensure that the furnace tubes are not damaged.

Column Flooding Constraint Control

Controlling Multiple Constraints

Hot Spot Temperature Control

Cross-Limiting Firing Controls

Firing Rate Increase

Firing Rate Decrease

Analysis of Cross-Limiting Firing Controls
It is critical that excess oxygen is maintained during firing rate increases or decreases or CO will form. When the firing rate is increased, the air flow rate will lead the fuel flow rate. When the firing rate is decreased, the fuel flow rate will lead the air flow rate. Air flow rate controller is based on equivalent fuel flow rate.

Override Control

Override/Select Control
Override/Select control uses LS and HS action to change which controller is applied to the manipulated variable. Override/Select control uses select action to switch between manipulated variables using the same control objective.

Computed Manipulated Variable Control
Used when the desired manipulated variable is not directly controllable. Reduces the effect of certain types of disturbances.

Computed Reboiler Duty Control

Internal Reflux Control

Overview Where applicable, inferential control reduces deadtime at a very effective price. When process nonlinearity becomes excessive, consider scheduling controller tuning. Use override/select controls to satisfy safety and operational constraints. Computed manipulated variable control can effective reduce the effects of certain disturbances.

PID Implementation Issues
Chapter 14 PID Implementation Issues

Reset Windup for PID Controllers
Windup results when the manipulated variable is not able to control to the setpoint resulting in sustained offset causing the integral of the error from setpoint to accumulate. When control returns, accumulated error causes an upset. Windup can occur when a control valve saturates or when a control loop is not being used (e.g., select control).

Reset Windup Note that controller output saturates causing area “A” to accumulate by the integral action. After the disturbance returns to its normal level, the controller output remains saturated for a period of time causing an upset in y.

Anti-Reset Windup When the manipulated variable saturates, the integral is not allowed to accumulate. When control returns, the controller takes immediate action and the process returns smoothly to the setpoint.

Methods for Anti-Reset Windup
Turn off the integral when a valve saturates or a control loop is not in use. Clamp the controller output to be greater than 0% and less than 100%. Apply internal reset feedback Apply external reset feedback

Industrial Approach External reset feedback Controller output clamping
Digitally turn-off integral calculation

Internal Reset Feedback

Conventional PI Controller
Therefore, internal reset feedback is equivalent to a conventional PI controller. It still has windup, but controller output can be clamped.

External Reset Feedback
An extension of internal reset feedback, therefore, it is equivalent to a conventional PI controller. When u saturates, windup will cease preventing windup. Less windup than clamping, but requires umeas.

Bumpless Transfer When a control loop is turned on without bumpless transfer, the process can become unduly upset. With bumpless transfer, an internal setpoint is used for the controller and the internal setpoint is ramped at a slow rate from the initial conditions to the actual desired setpoint to order to provide a smooth startup of a control loop.

Comparison of True and Internal Setpoints

Control Performance With and Without Bumpless Transfer

Split Range Flow Control
In certain applications, a single flow control loop cannot provide accurate flow metering over the full range of operation. Split range flow control uses two flow controllers (one with a small control valve and one with a large control valve) in parallel. At low flow rates, the large valve is closed and the small valve provides accurate flow control. At large flow rates, both valve are open.

Split Range Flow Controller

Coordination of Control Valves for Split Range Flow Control

Example for Split Range Flow Control

Titration Curve for a Strong Acid-Strong Base System
Therefore, for accurate pH control for a wide range of flow rates for acid wastewater, a split range flow controller for the NaOH is required.

Other Split-Range Flow Control Examples
When the controlled flow rate has a turn down ratio greater than 9 See value sizing examples in Chapter 2

Split Range Temperature Control

Overview All controllers that employ integral action should have anti-reset windup applied. Bumpless transfer provides a means for smooth startup of a control loop. When accurate metering of a flow over a very wide flow rate range is called for, use split range flow control.

PID Controllers Applied to MIMO Processes
Chapter 15 PID Controllers Applied to MIMO Processes

2×2 Example of a MIMO Process

Example of a 2×2 MIMO Process
Two inputs: Setpoints for flow controller on steam and reflux. Two outputs: Composition of products B and D

Configuration Selection (Choosing the u/y Pairings)
That is, which manipulated variable is to be used to control which controlled variable. Choosing an inferior configuration can dramatically reduce control performance. For many processes, configuration selection is a difficult and challenging process (e.g., dual composition control for distillation).

Single Loop Controllers Applied to a 2×2 MIMO Process

Example of Single Loop PID Controllers Applied to 2×2 Process
L is adjusted by PID controller to maintain composition of D at its setpoint. Steam flow is adjusted by PID controller to maintain composition of B at its setpoint.

Coupling Effect of Loop 2 on y1

Example of Coupling L is adjusted to maintain the composition of D which causes changes in the composition of B. The bottom loop changes the flow rate of steam to correct for the effect of the reflux changes which causes changes in the composition of D.

The Three Factors that Affect Configuration Selection
Coupling Dynamic response Sensitivity to Disturbances

Steady-State Coupling

Relative Gain Array When 11 is equal to unity, no coupling is present. When 11 is greater than unity, coupling works in the opposite direction as the primary effect. When 11 is less than unity, coupling works in the same direction as the primary effect.

Numerator of 11

Denominator of 11

RGA Example

c1 = 1.0 y2 = K21 c1 = 0.05

c2 = -y2/K22 = -0.05/2 =-0.025

(y1)coup = c2 K12 = (0.1) =-.0025

Calculation of RGA

RGA Calculation for 2×2 System

RGA Analysis RGA is a good measure of the coupling effect of a configuration if all the input/output relationships have the same general dynamic behavior. Otherwise, it can be misleading.

Example Showing Dynamic Factors

Dynamic Example Note that the off-diagonal terms possess dynamics that are 10 times faster than the diagonal terms. As a result, adjustments in c1 to correct y1 result in changes in y2 long before y1 can be corrected. Then the other control loop makes adjustments in c2 to correct y2, but y1 changes long before y2. Thus adjustments in c1 cause changes in y1 from the coupling long before the direct effect.

Direct Pairing (Thin Line) and Reverse Pairing (Thick Line)

Dynamic RGA

Dynamic RGA for Direct (a) and Reverse (b) Pairings
Consider the frequency, , corresponding to desired closed loop response which indicates b better than a

Overall Dynamic Considerations
Pairings of manipulate and controlled variables should be done so that each controlled variable responds as quickly as possible to changes in its manipulated variable.

Sensitivity to Disturbances
In general, each configuration has a different sensitivity to a disturbance. Note that thick and thin line represent the results for different configurations

Configuration Selection
It is the combined effect of coupling, dynamic response, and sensitivity to disturbances that determines the control performance for a particular control configuration for a MIMO process.

Configuration Selection for a C3 Splitter

(L,V) Configuration Applied to the C3 Splitter

Reflux Ratio Applied to the Overhead of the C3 Splitter

Configuration Selection Example
L, L/D, and V are the least sensitive to feed composition disturbances. L and V have the most immediate effect on the product compositions followed by L/D and V/B with D and B yielding the slowest response.

Control Performance

Analysis of Configuration Selection Example
Note that (L,V) is the worst configuration in spite of the fact that it is the least susceptible to disturbances and the fastest acting configuration, but it is the most coupled. Even though (D,V) had an RGA of 0.06, it had decent control performance. (L,B) is best since it has good decoupling and the overhead product is most important.

Tuning Decentralized Controllers
When a particular loop is 3 times or more faster than the rest of the loops, tune it first. When tuning two or more loop with similar dynamics, use ATV identification with online tuning

One-Way Decoupler

Overview The combined effect of coupling, sensitivity to disturbances, and dynamic response determine the performance of a configuration Implement tuning of fast loops first and use a single tuning factor when several loops are tuned together. One-way decoupling can be effective when the most important controlled variable suffers from significant coupling.

Model Predictive Control
Chapter 16 Model Predictive Control

Single Loop Controllers

MPC Controller

Model Predictive Control
Most popular form of multivariable control. Effectively handles complex sets of constraints. Has an LP on top of it so that it controls against the most profitable set of constraints. Several types of industrial MPC but DMC is the most widely used form.

DMC is based on Step Response Models
Allow the development of empirical input/output process models. The coefficients correspond to the step response behavior of the process

Example of a SRM t i u y(t) ai 0 0 1 0 0 1 1 0 0 0 2 2 0 0.63 0.63

Open-Loop Step Test for Thermal Mixer

SRM for the Thermal Mixer (u=0.05)
ti yi ai 5 50.00 0.0 1 10 49.77 -0.23 -4.68 2 15 49.35 -0.65 -13.0 3 20 49.08 -0.92 -18.4 4 25 48.94 -1.06 -21.2 30 48.87 -1.13 -22.6 6 35 48.84 -1.16 -23.2 7 40 48.83 -1.17 -23.5 8 45 48.82 -1.18 -23.7 9 50

Example of SRM Applied to a Process with Complex Dynamics

Complex Dynamics An integrating process (e.g., a level) is modeled the same except that the Tss is based on attaining a constant slope (i.e., a constant ramp rate). An integrating variable is referred to as a “ramp variable”.

SRM for Ramp Function

Using SRM to Calculate y(t)
Given: SRM: [ ] y0= u= Ts=20 sec Solve for y(t): y = y0 + u SRM y1=5 + 5×0 = 5; y2 = 5 + 5×0.4 = 7; y3 = 5 +5×0.8 = 9; y4 = 5 + 5×.9 = 9.5 y5 = 5 + 5×1 = or in vector form y = [ …] y(20 s)=5; y(40 s)=7; y(60 s)=9; y(80 s)=9.5 y(100 s)=10; y(120 s)=10; y(140 s)=

Class Exercise Given: SRM: [0 0.4 0.8 0.9 1.0] Solve for y(t):
y0= u= Ts=20 sec Solve for y(t):

Solution to Class Exercise
Given: SRM: [ ] y0= u= Ts=20 sec Solve for y(t): y = y0 + u SRM y1=5 + -2×0 = 5; y2 = ×0.4 = 4.2; y3 = 5 +-2×0.8 = 3.4; y4 = ×.9 =3.2 y5 = ×1 = or in vector form y = [ …]

Calculating SRM from Step Response Data
Remember previous equation: y = y0 + u SRM Solving for SRM yields: SRM = (y-y0)/u

Example of the Calculation of SRM from Step Response Data

Class Exercise for the Calculation of the SRM from Step Response Data

Simulation Demonstration
A thermal mixing process mixes two streams with different temperatures (25ºC and 75ºC) to produce a product (about 50ºC). Consider a 10% increase (0.05 kg/s) increase in the flow rate of colder stream. Develop SRM from step response.

Step Test Result and SRM

Comparison for the Same u (+10%)

Comparison for the Same u (-10%)

Demonstration Conclusions
MPC model agrees closely for the same step input change, but shows significant mismatch for a different size of input change due to process nonlinearity.

Nonlinear Effects This SRM will not accurately represent negative changes in the MV or different magnitude MV changes due to process nonlinearity. Therefore, a better way to develop a SRM is to make a number of MV changes with different magnitude and directions and average the results to get the SRM.

SRM’s for Different Size u’s

Another Approach Use the average ai’s

Using Average ai’s for u=+10%

Using Average ai’s for u=-10%

Using Average ai’s for u=+5%

Relation to MPC Model Identification
Developing SRM’s that represent an “average” response is the basis of good MPC models.

Class Exercise Using the composition mixer simulator (CMIXER), use a +5% step input change (0.025 kg/min) to generate a SRM for this process First, implement the step input change Next, determine the model interval Then, calculate the SRM coefficients Finally, compare simulator result to MPC model prediction for u (+10% to -10%)

SRM Summary SRM’s represent the dynamic behavior of a dependent variable (output variable) for a process for a unit step change in an input. SRM’s are empirical models that are flexible enough to model complex dynamics SRM’s are linear approximations of process behavior and can be estimated using step responses for the process.

Application of SRM for a Series of Input Changes
SISO MPC Controller Application of SRM for a Series of Input Changes

A Series of Input Changes
Definition of the SRM is based on a single step input change. But feedback control requires a large number of changes to the input variables. To use a SRM to predict the output behavior for a series of input changes, linear superposition is assumed, i.e., the process response is equal to the sum of the individual step responses.

Superposition of Two Step Input Changes

Equation Form for Two Sequential Input Changes
Consider the the response of a process to two step input changes separately

Combined Response for Two Input Changes

Combined Response for a Series of 4 Input Changes

Matrix Form for Combined Response for a Series of 4 Input Changes

The Dynamic Matrix The Dynamic Matrix determines the dynamic response of the process to a series of input changes, nc, for a prediction horizon of np steps into the future. The dynamic Matrix is constructed from the coefficients of the SRM A is np×nc; SRM is m (this case has 4 inputs and 7 output predictions)

Zero’s in Dynamic Matrix
Where do they come from? Consider the first row: y1 is cannot be affected by u1, u2, and u3 because y1 is measured before u1, u2, and u3 are applied. Likewise, y2 is cannot be affected by u2 and u3

The Matrix Form of the Generalized Model Equation

Dynamic Matrix Example

Class Exercise

Solution to Class Exercise

SISO MPC Controller Prediction Vector

Previous Inputs Affect Future Response of the Process

Prediction Vector The prediction vector, yP, contains the combined effect of the previous input changes on the future values of the output variable. The prediction vector is calculated from the product of the previous input changes and the prediction matrix, which is constructed using the coefficients of the SRM.

Process Model The future values of the output variable are equal to the sum of the contribution from previous inputs (prediction vector) and the contribution from future change in the input variable:

Process Model In order to correct for the mismatch between the predicted value of y at the current time and the measured value, a correction vector, , is added to the process model equation.

Correction for Mismatch in Ramp Variable
The error between the predicted value of y(t0) and the measured value is applied as before: = y(t0) - yP(t0) An unmeasured disturbance can affect the slope of the modeled SRM Therefore, an additional tuning factor for ramp variables is the “rotation factor (RF)”, which is the fraction of the mismatch  that is used to change the slope of the SRM of the ramp variable. Slopenew=Slopeold+RF×

SISO MPC Controller DMC Controller

Development of DMC Control Law

DMC Control Law

Analysis of DMC Controller
(ATA)-1AT is a constant matrix that is determined explicitly from the coefficients of the SRM. If you double the number of coefficients in the SRM, you will increase the number of terms in the dynamic matrix by a factor of 4 E takes into account changes in the setpoint, the influence of previous inputs, and the correction for model mismatch. A full set of future moves are determine by this control law at each control interval.

Moving Horizon Controller

Moving Horizon Controller
Even though the DMC controller calculates a series of future moves, only the first of the calculated moves is actually implemented. The next time the controller is called (Ts later), a new series of moves is calculated, but only the first is applied. If the SRM were perfect, the subsequent set of first moves would be equal to the first series of moves calculated by the DMC controller.

Implementation Details
SISO MPC Controller Implementation Details

Implementation Details
Choose Ts based on when new information is available on the CV. E.g., analyzer update. For temperature sensor, use Ts=10-15 s. Model horizon: m= Tss / Ts (normally set m=60-90 coefficients. Always set m>30. Control horizon: nc=½m Prediction horizon: np=m+nc

Controller Implementation Example
Consider a SISO DMC temperature control loop that has an open loop time to steady- state equal to 20 minutes. Applying the previous equations:

Class Example Consider a SISO DMC controller applied to the overhead product of a C3 splitter. The overhead composition analyzer update every 10 min and the time to steady-state for the overhead composition for a reflux flow rate change is 7 hours. Determine m, np and nc for this controller.

Solution

SISO MPC Controller Tuning

DMC Controller Tuning Previous DMC control law results in aggressive control because it is based on minimizing the error from setpoint without regard to changes in the MV. (ATA)-1 is usually ill-conditioned due to normal levels of process/model mismatch. These problems can be overcome by adding a diagonal matrix, Q, to the Dynamic Matrix, A.

DMC Controller with Move Suppression

The Dynamic Matrix with the Move Suppression Factor Added

Move Suppression

DMC Control Law

Effect of Move Suppression Factor
The larger q, the greater the penalty for MV moves. The smaller q, the greater the penalty for errors from setpoint.

Tmixer DMC Tuning Example Q=300

Controller Tuning Reliability versus performance.
Meeting the overall process objectives.

Checking the Controller Tuning for a Setpoint Changes in the Opposite Direction Q=30

Class Exercise SISO DMC Tuning
Tune the DMC controller for the CMixer

SISO MPC Controller Testing

Controller Testing Use disturbance upsets to test controller performance

Controller Disturbance Rejection Q=60

Class Exercise Testing the Controller Tuning
Using disturbance upset tests, test the performance of your tuned DMC controller.

Model Identification: The Least Squares Solution

Controller Disturbance Rejection Ts=7, Q=30

What if the SRM become worse? Ts=11; Q=30

What if the SRM become worse? Ts=15; Q=30

SRM Model Mismatch The controller still works in this case, but the performance is penalized.

MIMO DMC Control

MIMO DMC Control Extension of SISO to MIMO Constraint control
Economic LP Large-scale applications

Extension of DMC to MIMO processes
MIMO DMC Control Extension of DMC to MIMO processes

Dynamic Matrix for a MIMO Process

Recall The SISO Dynamic Matrix

Example of a Partitioned Dynamic Matrix

Dynamic Matrix for a MIMO Process

A Partitioned Control Vector

An Example of a Partitioned Control Vector

Control Law for MIMO DMC Controller

Relative Weighting Factor
The relative weighting factor, wi , allows the application control engineer to quantitatively rank the importance of each CV and constraint. For each of the constraints and CV’s, the control engineer must determine how large a violation of the constraint or deviation from setpoint requires drastic action to maintain reliable operation (Equal Concern Errors).

Equal Concern Errors ECE’s are based on process experience.
Consider a distillation column 0.5 psi above the differential pressure limit of 2 psi will cause drastic action to prevent flooding. 1% impurity levels above the specified impurity levels in the products will require rerunning the product. Reboiler temperatures greater than 10ºF above the reboiler temperature constraint will result in excessive fouling of the reboiler.

Feedforward Variables
Measured disturbances should be modeled as inputs for the MPC controller to reduce the effect of those disturbances on the CV. FF variables are treated as inputs that are not manipulated.

MIMO DMC Control Constraint Control

Constraint Control With DMC, constraints are treated as CV’s.
As different combinations of constraints become active, different weighting factors, wi, are used which are based on the equal concern errors. Therefore, as different combinations of active constraints are encountered, the wi’s automatically determine the relative importance of each. The DMC controller determines the optimal control action based on considering that the wi’s change along the entire path to the setpoints. Therefore, the entire control law must be solved at each control interval.

MIMO DMC Control Law

MPC Model Identification
MIMO DMC Control MPC Model Identification

MPC Model ID

MIMO Model Identification
For DMC, impulse models are used for identification so that bad data can be “sliced” out of the training data. Sources of bad data: analyzer failure, data not recorded, process not operated in normal fashion (by-pass opened, atypical feed to the process, saturated regulatory controls, etc.), and large unmeasured disturbance to the process.

Model Identification SRM’s have the flexibility to model a full range of process dynamics (e.g., recycle systems). MPC controllers that use preset functional forms for models (e.g., state-space models or a set of preset transfer function models) can be inferior to SRM based models in certain cases.

MIMO DMC Control Economic LP

Economic LP The LP combines process optimization with the DMC controller. The LP is based on economic parameters (e.g., product values, energy costs, etc.) and the steady-state gains of the process. The last term in each SRM is used to provide the necessary gain information; therefore, the LP is consistent with the MPC controller.

Example of an LP

Economic LP and PID Control Performance

Economic LP and MPC Control Performance

MPC Control Performance (i.e., size of control circle)
Depends on accuracy of models (e.g., process nonlinearity, type and size of unmeasured disturbances, changes in the process and operating conditions). Depends on the performance of regulatory controls and sensors. Can also depend on the control MPC technology used.

Economic LP Consider the case with 5 MV’s and 10 control objectives (i.e., upper and lower limits on CV’s, upper and lower limits on MV’s, and upper rate of change limit on MV’s): because there are 5 degrees- of-freedom (i.e., one for each MV) the LP will determine which 5 constraints to simultaneously operate against. The 5 constraints, in this case, become the setpoints for the MPC controller. The LP is applied each control interval along with the MPC controller.

The LP and the MPC Controller
Both use the steady-state gains from the SRM. The LP determines the setpoints for the controller from the economic parameters and the process gains. The controller drives the process to the most profitable set of constraints, i.e., keeps the process making the most profit even though the most profitable set of process constraints changes with time parameters. Balanced ramp variables are constraints for the LP.

Large-Scale Applications
MIMO DMC Control Large-Scale Applications

Size of a Typical Large MPC Application
MV’s: 40 CV’s: 60 Constraints: There are some companies that have even larger applications.

MPC Project Organization

Understand the process Set the scope of the MPC controller Choose the control configuration Design the Plant Test Conduct the Pretest Conduct the Plant Test and collect the data Analyze data and determine MPC model Tune the controller Commission the controller Post audit

Understand the process

Understand the Process
Process understanding is the single most important issue for a successful MPC application. Stated another way, if you do not fully understand the process and its preferred operation, it is highly unlikely that you will be able to develop a successful MPC application.

How to Develop Process Understanding
Study PFD’s Study P&ID’s Talk to the operators (how it really works!) Talk to plant engineers (how they want it to work) Interview plant economic planners (how much its worth). If available, run steady-state process simulator Read the plant operating manuals Spend time at the process on graveyards.

Be able to answer the following questions
What is the purpose of the plant? Where does the feed come from? Where do the product go? How much flexibility is there for feed supply and product demand? How do the seasons affect the operation? Also, feed supply and product demand?

Be able to answer the following questions
What are the 3 or 4 most important constraints that operators worry about? Where is energy used and how expensive is it? What are the product specifications? Are there environmental or tax issues that affect plant operations?

Set the scope of the MPC controller

Set the Scope of the Project (Very Important Step)
What part of the plant should be included in the controller? That is, how much of the plant should be included in the MPC controller to meet the project objectives? To answer this question, you must discuss the objectives of the project with the operations personnel, the technical staff for that portion of the plant, and the scheduling people. Only then can you be sure that you are solving the “correct” problem.

Select the control configuration

MPC Configuration Selection
Which PID loops do you open (i.e., turn over to MPC controller) and which do you leave closed. That is, if you leave a PID loop closed, the MPC controller sets the PID loop setpoint (e.g., flow controller). What are the MV’s for the distillation columns in the process? Should the MPC controller be responsible for accumulator or reboiler level control?

MPC Configuration Selection
From an overall point of view, what are the MV’s and the CV’s for the MPC controller? A challenging problem that requires process knowledge and control experience while keeping focused on the overall process objectives.

Reasons for Leaving a PID Loop Closed
The PID loop may be more effective in eliminating certain disturbance due to the higher frequency of application. For example, flow control loops. In addition, certain pressure, level, and temperature PID loops should be left closed.

Situations for which a PID loop should be opened
Loops with very long dynamic and/or large deadtime Process lines with two control valves in series. Certain levels for which allowing the MPC controller control the level adds important flexibility to better meet the primary objectives of the process. For example, it can provide more effective decoupling.

Use Good Control Engineering
Ensure that the MV’s have a direct and immediate affect on the CV’s. Use computed MV’s to reject certain disturbances, e.g., use internal reflux control to reduce the effect of rain storms or computed heat input for a pumparound. Use inferential measurements to reduce the effect of deadtime (e.g., inferential temp control) Use CV transformations that linearize the overall response of the process.

Variable Transformations
For a control valve, P=K×Flow2 For high purity distillation columns, use log transformed compositions: x’=log(x) Use different linear models depending on the operating range.

Design the Plant Test

Plant Test Guidelines Make from 5 to 15 step tests for each MV
The MV changes should be as random as possible to prevent correlated data. MV moves should be as large as possible but not so large that it upsets the process (e.g., 1-10%). For each MV, make a change after 1p, 2p, 3p, 4p, and 5p, where p is Tss/4. Above all, remember the product specs and the process constraints

Develop the Testing Plan
Identify all the MV’s you plan to move. Tabulate the following data for each MV: Tag number of the MV and physical description Nominal value Range of move sizes Make a proposed testing sequence and indicate when each MV is moved and by how much. Discuss the MV move sizes with operations. Remember the spec and constraints.

Choose Sampling Period
Small sample period: high time resolution but increased size of data set. In general, sample as new information become available. For fast responding processes, sample every few seconds. For slow processes, sample every 5 min Look at the smallest expected Tss. Estimate sample period as Tss/50 Compare with sensor dynamics

Make MV Changes one MV at a Time to Generate Uncorrelated Results

Develop a Roughed-out Gain Matrix
Using your process knowledge, determine whether the steady-state process gain is positive, negative, or zero for each input (MV & DV)/ output (CV) pair. Using + for positive, - for negative, or 0 for zero, construct the roughed-out gain matrix. Inputs are listed vertically while outputs are listed horizontally.

Roughed-out Gain Matrix Ex.

Ex. Roughed-out Gain Matrix

Conduct the Pretest

Pretest Check all the sensors, control valves and regulatory control loops for proper operation. Ensure that the process equipment is in proper repair. Check material and energy balances on the unit. Make sure that all the major pieces of equipment are in operation. Check the feed to the unit. Then, apply step input changes for each MV and compare to your roughed out gain matrix. Reconcile any deviations from what you expected (e.g., roughed out gain matrix, dynamic response, etc.). This will enhance your process understanding.

Conduct the Plant Test and collect the data

Testing in Optimal Operating Mode
Because you will want your plant to operate effectively at its optimum operating conditions, you should, if possible, test you plant near the optimum operating conditions to reduce the effects of process/model mismatch. Therefore, you will need to determine the optimum operating conditions for your plant, e.g., discuss with an experienced plant engineer or use a nonlinear process optimizer.

Plant Test Overview The plant test is the most important step in an MPC project because if the MPC models do not agree with the process, controller reliability and performance will be poor. If the process changes (e.g., different equipment configuration, different feed, different regulatory controller tuning, etc.), mismatch between the MPC model and the process will result, affecting controller performance.

Plant Test Check List Meet (5-10 min) with the operators at the start of each shift and explain what you are doing. Make sure that each MV move is made on time, in the proper direction, and by the correct amount. Make sure that a technical person is observing entire testing period (24-7).

Plant Test Check List Make real-time plots of the data and explain all the behavior that you see. Identify abnormal periods and make notes: significant unmeasured disturbances, instrument failures, utility outages. The MPC model should not be trained using data collected during these periods. Talk to the operators about the process. Buy pizza, donuts, ice cream and barbeque for the operators

Using Roughed-out Gain Matrix
During the testing phase, make sure that the plant results agree with your roughed-out gain matrix. If not, you will need to explain the difference. That is, your process knowledge needs to be improved a bit or something is not working properly (e.g., an analyzer is off-line)

Unmeasured Disturbances
Training an MPC controller on data collected during periods with unmeasured disturbances will reduce the accuracy of your model and reduce the effectiveness of the overall project. Some unmeasured disturbances are inevitable, but you must strive to keep them to an absolute minimum.

Examples of Unmeasured Disturbances to be Avoided
Back flushing a condensor Changes in by-pass streams Product grade changes Changes in the controller settings for the regulatory controllers. Any change in the process operating conditions not modeled in the controller or process equipment changes

Identified Periods with Unmeasured Disturbances
Slice out the data from those periods so that this data is not used to train the MPC model; therefore, the identified model will not be corrupted.

Analyze the data and develop the MPC model

Obtain the MPC Process Models
Use the accepted plant test data with the MPC model identification software to develop each of the input/output SRM models.

Evaluate Process Models
For each input/output pair, plot the SRM on a small graph. Arrange the SRM models into a matrix similar to the roughed-out gain matrix. Review the results to ensure that you have models that make sense. Use the statistical tools from the MPC model ID software to evaluate each SRM

Evaluate Process Models
Examine the difference between the plant test data and your models (residual). Plot the residual for positive and negative MV changes. This will help identify the degree of process nonlinearity. Also, it will help evaluate CV transformations that behave more linearly.

Plant Economics The LP will require incremental costs (feed costs, utility costs) and incremental revenues (product values) For previous distillation example, you will need the incremental cost of steam ($/lb), the incremental cost of feed ($/lb), and the incremental value of both products ($/lb).

Tune the controller

General Approach to Tuning
Use the SRM’s of the process (MPC model) off-line as the process to develop the preliminary settings for the controller. Three areas that you need to get right: Economics Constraints Dynamics

Testing Economics Run plant/controller simulations (off-line) with different combinations of constraints to ensure that the controller pushes the plant in the correct direction. Run test cases to see which LP constraints are operative.

Constraints Check the equal concern errors for each constraint to ensure that the proper relative weighting factors for each combination of active constraints is used. That is, when constraint 1 and 5 are active, what is the relative importance of the CV’s and constraints. Also, use the plant/controller simulation (off-line) to test the relative weighting.

Dynamics Select the move suppression factors for each MV using off-line simulations of the plant. Run the simulations a number of times for setpoint changes and unmeasured disturbances to get the MSF’s right. Operators are a good source for letting you know how much the MV’s should move in certain situations and how fast the process should move to CV’s, but remember that they are most familiar with how the plant used to operate.

Operator Training In some states, it is a legal requirement.
Typically operators receive one to two hours of training with MPC. Nevertheless, better operator training increases your chance for success.

Watchdog Timer The controller should run every minute 24-7
Otherwise, a DCS alarm should sound. The most common approach is to use a watchdog timer (a small program running on the DCS) to communicate with the MPC controller. If the watchdog fails to receive a healthy response from the controller after an appropriate period of time, the watchdog forces each MV into a fallback (shed) position and writes an alarm to the operator.

Commission the controller

General Approach Now it is time to install the controller in the closed loop in the process. Since the controller will directly affect the plant operation, it is essential to proceed methodically and with caution. Use a checklist and be observant

Commissioning Checklist
Make sure that the operators have been trained Install control software, database, and other real-time components. Check controller database Control CV’s, DV’s, and MV’s match raw DCS inputs exactly Controller models, tuning parameters, etc. match off-line versions exactly

Commissioning Checklist (cont.)
Turning controller on the prediction mode: Each MV configured correctly in the DCS? Correct shed mode for each MV? MV’s out of cascade (controller cannot change)? Controller turned on the prediction mode? The size of the calculated MV moves make sense? The CV predictions make sense?

First-time MV check Use conservative controller settings Reduce difference between upper and lower limits on MV (or rate limits) Make sure controller is running Put MV’s into cascade one at a time. Ensure that the calculated MV change is applied to DCS setpoint.

Close the loop Ensure that the controller is running Turn ON the controller master switch With MV’s clamped, put one or two MV’s into cascade Relax the MV limits somewhat and check for good controller behavior Add more MV’s gradually, ensuring good control performance.

Final Checks Ensure that the standard deviations of the CV’s decrease and that the MV’s are not moving around too much. Ensure that the LP is driving the process in the correct direction.

Post audit

Post audit Collect post commissioning data for the key CV’s and compare to pre-project data. Ensure that the standard deviations of the key CV’s have decreased. Evaluate whether the upper and lower limits on the MV’s are still set correctly. Ensure that the controller is running the process at the economically best set of constraints most of the time.

Advantages of MPC Combines multivariable constraint control with process optimization. A generic approach that can be applied to a wide range of processes. Allows for more systematic controller maintenance. Depends on the particular process

MPC Offers the Most Significant Advantages
For high volume processes, such as refineries and high volume chemical intermediate plants. For processes with unusual process dynamics. For processes with significant economic benefit to operated closer to optimal constraints. For processes that have different active constraints depending on product grade, changes in product values, summer/winter operation, or day/night operation. For processes that it is important to have smooth transitions to new operating targets.

Application Example C3 splitter in an ethylene plant (reboiler duty is free; therefore, run at maximum reboiler duty and apply single-ended control.) DCS % of cost % of inc. benefit AdPID 10% of cost % of inc. benefit MPC % of cost % of inc. benefit RTO % of cost % of inc. benefit

Application Example Reformer. Control is easy since it only involve inlet temperature controllers. DCS % of cost % of inc. benefit AdPID 10% of cost % of inc. benefit MPC % of cost % of inc. benefit RTO % of cost % of inc. benefit

Application Examples FCC Unit. Optimal constraints change with economics, operating conditions, etc. Finding the optimal riser temperature requires RTO (yield model and nonlinear problem). DCS % of cost % of inc. benefit AdPID 10% of cost % of inc. benefit MPC % of cost % of inc. benefit RTO % of cost % of inc. benefit

Limitations of MPC It is a linear method; therefore, it is not recommended for highly nonlinear processes (e.g., pH control). It does not adapt to process changes. If the process changes significantly, the MPC model must be re-identified. For certain cases, the economic LP may not be accurate enough. For these cases, it may be necessary to add nonlinear optimization on top of the LP.

Different Forms of MPC

Primary Commercial MPC Software
ABB Aspentech DMCplus Fisher-Rosemount Foxboro Honeywell RMPCT Yokogawa

MPC software based on models with specified dynamic characteristics can be inferior to SRM based approaches.

Conclusions

Conclusions The challenges of this course:
A very difficult area with considerable mathematics and industrial practice. Only two days Wide variety of backgrounds of the students.

Conclusions You should now be familiar with MPC terminology and technology and the major issues associated with the commercial application of MPC. Are you an MPC expert? Not yet! But with more study and a lot more experience, you can become one.

Conclusions “This is not the end.
It is not even the beginning of the end. But it is, perhaps, the end of the beginning” (Sir Winston Churchill, Nov. 10, 1942)

Chapter 17 Multiunit Controller Design

Approach to Multiunit Controller Design
1. Identify process objectives. 2. Identify the process constraints. 3. Identify significant disturbances. 4. Determine the type and location of sensors. 5. Determine the location of control valves. 6. Apply a degree-of-freedom analysis. 7. Implement energy management.

Approach (Continued) 8. Control process production rate.
9. Select the manipulated variables that meet the control objectives. 10. Address how disturbances will be handled. 11. Develop a constraint handling strategy. 12. Control inventories. 13. Check component balances. 14. Control individual unit operations. 15. Apply process optimization.

C2 Splitter Case Study 1. Identify process objectives.
2. Identify the process constraints. 3. Identify significant disturbances. 4. Determine the type and location of sensors. 5. Determine the location of control valves.

C2 Splitter with Sensors and Valves

C2 Splitter Case Study 6. Apply a degree-of-freedom analysis.
7. Implement energy management. 8. Control process production rate. 9. Select the manipulated variables that meet the control objectives. 10. Address how disturbances will be handled.

C2 Splitter with Control Configuration

C2 Splitter Case Study 11. Develop a constraint handling strategy.
12. Control inventories. 13. Check component balances. 14. Control individual unit operations. 15. Apply process optimization.

C2 Splitter with Constraint Controls

Recycle Reactor Case Study
1. Identify process objectives. 2. Identify the process constraints. 3. Identify significant disturbances. 4. Determine the type and location of sensors. 5. Determine the location of control valves.

Recycle Reactor with Sensors and Valves

6. Apply a degree-of-freedom analysis. 7. Implement energy management. 8. Control process production rate. 9. Select the manipulated variables that meet the control objectives. 10. Address how disturbances will be handled.

Recycle Reactor with Control Configuration

11. Develop a constraint handling strategy. 12. Control inventories. 13. Check component balances. 14. Control individual unit operations. 15. Apply process optimization.

Recycle Reactor with Constraint Controls

Chapter 18 Control Case Studies

Control Systems Considered
Temperature control for a heat exchanger Temperature control of a CSTR Composition control of a distillation column pH control

Temperature Control for Heat Exchangers

Heat Exchangers Exhibit process deadtime and process nonlinearity.
Deadtime and gain both increase as tubeside flow decreases. Major disturbances are feed flow and enthalpy changes and changes in the enthalpy of the heating or cooling medium.

Inferior Configuration for a Steam Heated Heat Exchanger

Analysis of Inferior Configuration
This configuration must wait until the outlet product temperature changes before taking any corrective action for the disturbances listed.

Preferred Configuration for a Steam Heated Heat Exchanger

Analysis of Preferred Configuration
For the changes in the steam enthalpy and changes in the feed flow or feed enthalpy, they will cause a change in the heat transfer rate which will in turn change the steam pressure and the steam pressure controller will take corrective action. There this configuration will respond to the major process disturbances before their effect shows up in the product temperature.

Modfication to Perferred Configuration

Analysis Modfication to Perferred Configuration
A smaller less expensive valve can be used for this approach, i.e., less capital to implement. This configuration should be slower responding than the previous one since the MV depends on changing the level inside the heat exchanger in order to affect the process.

Scheduling of PI Controller Settings

Inferior Configuration for a Liquid/Liquid Heat Exchanger

Preferred Configuration for a Liquid/Liquid Heat Exchanger

Comparison of Configurations for Liquid/Liquid Heat Exchangers
For the inferior configuration, the process responds slowly to MV changes with significant process deadtime. Moreover, process gain and deadtime change significantly with the process feed rate. For the preferred configuration, the system responds quickly with very small process deadtime. Process deadtime and gain changes appear as disturbances.

Temperature Control for CSTRs

CSTR Temperature Control
Severe nonlinearity with variations in temperature. Effective gain and time constant vary with temperature. Disturbances include feed flow, composition, and enthalpy upsets, changes in the enthalpy of the heating or cooling mediums, and fouling of the heat transfer surfaces.

Preferred Configuration for Endothermic CSTR

Exothermic CSTR’s Open loop unstable
Minimum and maximum controller gain for stability Normal levels of integral action lead to unstable operation PD controller required Must keep p/p less than 0.1

Deadtime for an Exothermic CSTR
mix- Vr divided by feed flow rate, pumping rate of agitator, and recirculation rate. ht- MCp/UA coolant- Vcoolant divided by coolant recirculation rate s- sensor system time constant (6-20 s)

Exothermic CSTR Temperature Control

Maximizing Production Rate

Using Boiling Coolant

Distillation Control

Distillation Control Distillation control affects-
Product quality Process production rate Utility usage Bottom line- Distillation control is economically important

The Challenges Associated with Distillation Control
Process nonlinearity Coupling Severe disturbances Nonstationary behavior

Material Balance Effects

Effect of D/F and Energy Input on Product Purities [Thin line larger V]

Combined Material and Energy Balance Effects
Energy input to a column generally determines the degree of separation that is afforded by the column while the material balance (i.e., D/F) determines how the separation will be allocated between the two products.

Vapor and Liquid Dynamics
Boilup rate changes reach the overhead in a few seconds. Reflux changes take several minutes to reach the reboiler. This difference in dynamic response can cause interesting composition dynamics.

Effect of Liquid and Vapor Dynamics [(D,V) configuration]
Consider +V L/V decrease causes impurity to increase initially After V reaches accumulator, L will increase which will reduce the impurity level. Result: inverse action

Disturbances Feed composition upsets Feed flow rate upsets
Feed enthalpy upsets Subcooled reflux Loss of reboiler steam pressure Column pressure swings

Regulatory Control Flow controllers. Standard flow controllers on all controlled flow rates. Level controllers. Standard level controllers applied to reboiler, accumulators, and internal accumulators Pressure controllers. Examples follow

Minimum Pressure Operation

Manipulating Refrigerant Flow

Flooded Condenser

Venting for Pressure Control

Venting/Inert Injection

Inferential Temperature Control
Use pressure corrected temperature Use CAD model to ID best tray temperature to use

Single Composition Control - y
L is fast responding and least sensitive to z. No coupling present. Manipulate L to control y with V fixed.

Single Composition Control - x
V is fast responding and least sensitive to z. No coupling present. Manipulate V to control x with L fixed

Dual Composition Control Low L/D Columns
For columns with L/D < 5, use energy balance configurations: (L,V) (L,V/B) (L/D,V) (L/D,V/D)

Dual Composition Control High L/D Columns
For columns with L/D > 8, use material balance configurations: (D,B) (D,V) (D,V/B) (L,B) (L/D,B

When One Product is More Important than the Other
When x is important, use V as manipulated variable. When y is important, use L as manipulated variable. When L/D is low, use L, L/D, V, or V/B to control the less important product. When L/D is high, use D, L/D, B, or V/B to control the less important product

Configuration Selection Examples
Consider C3 splitter: high L/D and overhead propylene product is most important: Use (L,B) or (L,V/B) Consider low L/D column where the bottoms product is most important: Use (L,V) or (L/D,V).

When One Product is More Important than the Other
Tune the less important composition control loop loosely (e.g., critically damped) first. Then tune the important composition control loop tightly (i.e., 1/6 decay ratio) Provides dynamic decoupling

Typical Column Constraints
Maximum reboiler duty Maximum condenser duty Flooding Weeping Maximum reboiler temperature

Max T Constraint - y Important

Max T Constraint - x Important

Keys to Effective Distillation Control
Ensure that regulatory controls are functioning properly. Check analyzer deadtime, accuracy, and reliability. For inferential temperature control use RTD, pressure compensation, correct tray. Use internal reflux control. Ratio L, D, V, B to F. Choose a good control configuration. Implement proper tuning.

pH Control

pH Control pH control is important to any process involving aqueous solutions, e.g., wastewater neutralization and pH control for a bio-reactor. pH control can be highly nonlinear and highly nonstationary. Titration curves are useful because they indicate the change in process gain with changes in the system pH or base-to-acid ratio.

Strong Acid and Weak Acid Titration Cures for a Weak Base Which is an easier control problem?

Effect of pKa on the Titration Curves for a Strong and Weak Base

Titration Curves The shape of a titration curve is determined from the pKa and pKb of the acid and the base, respectively.

Degree of Difficulty for pH Control Problems
Easiest: relatively uniform feed rate, influent concentration and influent titration curve with a low to moderate process gain at neutrality. (Fixed gain PI controller or manual control) Relatively easy: variable feed rate with relatively uniform influent concentration and influent titration curve. (PI ratio control)

Degree of Difficulty for pH Control Problems
More Difficult: variable feed rate and influent concentration, but relatively uniform titration curve. (A ratio controller that allows the user to enter the titration curve) MOST DIFFICULT: variable feed rate, influent concentration and titration curve. Truly a challenging problem. (An adaptive controller, see text for discussion of inline pH controllers).

Table of Contents PART I: INTRODUCTION Chapter1: Introduction

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