CHAPTER 12: Controller Design, Tuning, & Troubleshooting Anis Atikah Ahmad anisatikah@unimap.edu.my

Steps in Process Control Formulate the control objectives. Select controlled, manipulated, and measured variables. Choose the control strategy and the control structure Specify controller settings.

Outline Performance Criteria for Closed-Loop Systems PID Controller Design: Model-Based Design Method Direct Synthesis Method Internal Model Control (IMC) Controller Tuning Relations IMC Tuning Relations Tuning Relations Based on Integral Error Criteria Miscellaneous Tuning Method Controllers with Two Degrees o Freedom On-Line Controller Tuning Continuous Cycling Method Guidelines for Common Control Loops Troubleshooting Control Loops

Performance Criteria for Closed-Loop Systems Which of the following provides the best response???? Unit-step disturbance responses of FOPTD model & PI controller

Performance Criteria for Closed-Loop Systems The function of a feedback control system is to ensure that the closed loop system has desirable dynamic and steady-state response characteristics. Ideally, we would like the closed-loop system to satisfy the following performance criteria: The closed-loop system must be stable. The effects of disturbances are minimized, providing good disturbance rejection Rapid, smooth responses to set-point changes are obtained, that is, good set-point tracking.

Performance Criteria for Closed-Loop Systems Steady-state error (offset) is eliminated. Excessive control action is avoided The control system is robust, that is, insensitive to changes in process conditions and to inaccuracies in the process model.

PID Controller Settings PID controller settings can be determined by a number of alternative techniques: Direct Synthesis (DS) method Internal Model Control (IMC) method Controller tuning relations Frequency response techniques Computer simulation On-line tuning after the control system is installed.

1. Direct Synthesis (DS) method In the Direct Synthesis (DS) method, the controller design is based on a process model and a desired closed-loop transfer function. Consider the block diagram of a feedback control system in Figure 12.2. The closed-loop transfer function for set-point changes is: Eq. 12-1

1. Direct Synthesis (DS) method [cont.] For simplicity, let and assume that Gm = Km. Then Eq. 12-1 reduces to: Rearranging and solving for Gc gives an expression for the feedback controller: Eq. 12-2 Eq. 12-3a

1. Direct Synthesis (DS) method [cont.] Equation 12-3a cannot be used for controller design because the closed-loop transfer function Y/Ysp is not known. Also, it is useful to distinguish between the actual process G and the model, that provides an approximation of the process behavior. A practical design equation can be derived by replacing the unknown G by , and Y/Ysp by a desired closed-loop transfer function, (Y/Ysp)d: Eq. 12-3b

1. Direct Synthesis (DS) method [cont.] Note that the controller transfer function in (12-3b) contains the inverse of the process model. For processes without time delays, the first-order model in Eq. 12-4 is a reasonable choice, Where τc is the desired closed loop time constant. Because the steady-state gain is one, no offset occurs for set-point changes. Eq. 12-3b Eq. 12-4

1. Direct Synthesis (DS) method [cont.] By substituting (12-4) into (12-3b) : Solving for Gc , the controller design equation becomes: The term provides integral control action and thus eliminates offset. Design parameter provides a convenient controller tuning parameter that can be used to make the controller more aggressive (small ) or less aggressive (large ). Eq. 12-5

Direct Synthesis (DS) method [cont.] If the process transfer function contains a known time delay θ, a reasonable choice for the desired closed-loop transfer function is: The time-delay term in (12-6) is essential because it is physically impossible for the controlled variable to respond to a set-point change at t = 0. Combining Eqs. 12-6 and 12-3b : Eq. 12-6

Direct Synthesis (DS) method [cont.] Combining Eqs. 12-6 and 12-3b Gives: The following derivation is based on approximating the time-delay term in the denominator of (12-7) with a truncated Taylor series expansion: Eq. 12-7 Eq. 12-8

1. Direct Synthesis (DS) method [cont.] Substituting (12-8) into the denominator of Eq. 12-7 and rearranging gives Note that this controller also contains integral control action. Eq. 12-9

1. Direct Synthesis (DS) method [cont.] 1. First-Order-plus-Time-Delay (FOPTD) Model Consider the standard FOPTD model, Substituting Eq. 12-10 into Eq. 12-9 and rearranging gives a PI controller; with the following controller settings: Eq. 12-10 Eq. 12-11

Direct Synthesis (DS) method [cont.] 2. Second-Order-plus-Time-Delay (FOPTD) Model Consider a second-order-plus-time-delay model, Substitution into Eq. 12-9 and rearranging gives a PID controller; with the following controller settings: Eq. 12-12 Eq. 12-13 Eq. 12-14

Example 12.1 Use the DS design method to calculate PID controller settings for the process: Consider three values of the desired closed-loop time constant: . Evaluate the controllers for unit step changes in both the set point and the disturbance, assuming that Gd = G. Repeat the evaluation for two cases: a. The process model is perfect ( = G). The model gain is incorrect, = 0.9, instead of the actual value, K = 2.

Example 12.1- Solution Use the DS design method to calculate PID controller settings for the for two cases: Comparing with standard PID controller; Thus; comparing

Example 12.1- Solution The controller settings are as follows: (a)For K =2 (b)For K =0.9 The values of Kc decrease as τc increases, but the values of τI and τD and do not change

Example 12.1- Solution Simulation results for (a) ( = G), As τc increases, the responses become more sluggish Increasing τc

Example 12.1- Solution Simulation results for (b) ( = 0.9). ( = 0.9). Increasing τc As τc increases, the responses become more sluggish

Example 12.1- Solution Which one is better???WHY?

2. Internal Model Control (IMC) The Internal Model Control (IMC), similar to DS method, is based on an assumed process model and leads to analytical expressions for the controller settings. The IMC method is based on the simplified block diagram shown in Fig. 12.6b. The model response is subtracted from the actual response Y, and the difference, is used as the input signal to the IMC controller, Fig. 12.6

Internal Model Control (IMC) The two block diagrams are identical if controllers Gc and Gc* satisfy the relation: Eq. 12.16 Any IMC controller is equivalent to a standard feedback controller Gc, and vice versa. Fig. 12.6

2. Internal Model Control (IMC) The following closed-loop relation for IMC can be derived as follows: For the special case of a perfect model, Eq. 12.17 reduces to Eq. 12.17 Eq. 12.18

2. Internal Model Control (IMC) The IMC controller is designed in two steps: The process model is factored as where contains any time delay and right half plane zeros. 2. The controller is specified as: Where f is a low-pass filter with steady state gain of one and: Eq. 12.19 Eq. 12.20 Eq. 12.21

2. Internal Model Control (IMC) For an ideal case where the process is perfect ( ), substituting the IMC controller transfer function in closed loop relation, gives Thus, the closed-loop transfer function for set-point changes (D=0) is: Eq. 12.22 Eq. 12.23

Example 12.2 Use the IMC design method to design a FOPTD model. Assume that f is specified by eq 12.21 with r=1, and consider 1/1 Padé approximation for the time delay term FOPTD Model: According to 1/1 Padé approximation, Substituting into FOPTD Model; Eq. 12.24a Eq. 12.25

Example 12.2 cont. Using 2 steps in IMC Controller design; STEP 1: Factor this model as Model: Thus, Using 2 steps in IMC Controller design; contains any time delay and right half plane zeros Eq. 12.26 Eq. 12.27

Example 12.2 cont. Using 2 steps in IMC Controller design; STEP 2: Specify the controller into Thus, substituting and , Thus, Using 2 steps in IMC Controller design; Eq. 12.28

Example 12.2 cont. Using 2 steps in IMC Controller design; STEP 2: Specify the controller into Thus, substituting and , Thus, Using 2 steps in IMC Controller design; Eq. 12.28

Example 12.2 cont. The equivalent controller Gc can be obtained from eq 12.16; And rearranged into PID controller; Eq. 12.29 Eq. 12.30 *Type of controller (PI or PID) depends on time-delay approximation. Repeating this derivation for Taylor series approximation gives a standard PI controller.

3. Controller Tuning Relations 3.1. IMC Tuning Relations Table 12.1

3. Controller Tuning Relations 3.1. IMC Tuning Relations Lag-dominant models (θ/τ<<1) First- or second-order models with relatively small time delays (θ/τ<<1 ) are referred to as lag-dominant models. The IMC and DS methods provide satisfactory set-point responses, but very slow disturbance responses, because the value of τI is very large. Fortunately, this problem can be solved in three different ways. Approximate the lag-dominant model by integrator-plus-time delay model. Then apply IMC tuning relation in Table 12.1. (Case M or N)

3. Controller Tuning Relations 3.1. IMC Tuning Relations Lag-dominant models (θ/τ<<1) 2. Limit the value of τI For lag-dominant models, the standard IMC controllers for first-order and second-order models provide sluggish disturbance responses because τI is very large. As a remedy, Skogestad (2003) has proposed limiting the value of :

3. Controller Tuning Relations 3.1. IMC Tuning Relations Lag-dominant models (θ/τ<<1) 3. Design the controller for disturbance rejection, rather than set-point tracking. For example, develop an extension of the DS approach based on closed-loop transfer function for disturbance. (Y/D)d, rather than (Y/Ysp)d

Example 12.4 Consider a lag-dominant model withθ/τ =0.01: Design four PI controller; IMC (τc=1) IMC (τc=2) based on the integrator approximation IMC (τc=1) with Skogestad’s modification Direct synthesis method for disturbance rejection. The controller settings are Kc=0.551 and τI=4.91 Evaluate the four controllers by comparing their performance for unit step changes in both set point and disturbance.

Example 12.4-solution The PI controller settings are: Kc τI IMC 0.5 100 Integrator approximation 0.556 5 Skogestad 8 DS (disturbance) 0.551 4.91

Example 12.4-solution Set-point response IMC controller provides an excellent set-point response, while the other three controllers have significant overshoots and longer settling times. Set-point response However, the IMC controller produces an unacceptably slow disturbance response owing to its large τI value. Disturbance response

3. Controller Tuning Relations 3.2 Tuning Relations Based on Integral Error Criteria Controller tuning relations have been develop that optimize the closed-loop response for a simple process model & a specified disturbance or set-point change. The optimum settings minimize an integral error criterion. Three popular integral error criteria are: Integral of the absolute value of the error (IAE) Integral of squared error (ISE) Integral of the time-weighted absolute error (ITAE)

3. Controller Tuning Relations 3.2 Tuning Relations Based on Integral Error Criteria IAE value Graphical interpretation of IAE

3. Controller Tuning Relations 3.2 Tuning Relations Based on Integral Error Criteria

3. Controller Tuning Relations 3. 3 Miscellaneous Tuning Relations Hägglund and Åström tuning relations Skogestad tuning relations G(s) Kc τI Condition Kc τI τD

3. Controller Tuning Relations Important Notes on Controller Design & Tuning Relations 1. Kc  1/K where K = KvKpKm . 2. q /t ↑  Kc ↓ (more time delay → poorer performance) 3. q /t ↑  tI and tD ↑ ( tD / tI = 0.1 -0.3 ) 4. When integral control action is added to a proportional-only controller,  Kc ↓ The addition of derivative action  Kc ↑ (stability margin increased)

Controllers with two degree of freedom The specification of controller settings for standard PID controller typically requires a tradeoff between set-point tracking and disturbance rejection. Fortunately, two simple strategies can be used to adjust the set-point and disturbance responses independently. These strategies referred to as controllers with two degrees of freedom. First strategy: set point changes are introduced gradually rather than as abrupt step changes . Eg: the set-point can be ramped or “filtered” by passing it through first order transfer function:.

Controllers with two degree of freedom Second strategy: adjusting the set-point response, based on a simple modification of the PID control law: Set-point weighting factor, 0<β<1 As β increases, the set point response become faster but exhibits more overshoot. When β =1, the modified control law reduces to standard PID control law.

Example 12.6 For the first-order- plus delay model of Example 12.4, the PI controller with DS- d settings provided the best disturbance response. Can set- point weighting significantly reduce the overshoot without adversely affecting the settling time?

Example 12.6 Set-point weighting with ß =0.5 provides a significant improvement, because the overshoot is greatly reduced and the settling time is significantly decreased.

4. On-line Controller Tuning 4.1 Continuous Cycling Method Step 1. After the process has reached steady state (at least approximately), eliminate the integral and derivative control action by setting τD to zero and τI to the largest possible value. Step 2. Set Kc equal to a small value (e.g., 0.5) and place the controller in the automatic mode. Step 3. Introduce a small, momentary set-point change so that the controlled variable moves away from the set point. Gradually increase Kc in small increments until continuous cycling occurs. The term continuous cycling refers to a sustained oscillation with a constant amplitude. The numerical value of Kc that produces continuous cycling (for proportional-only control) is called the ultimate gain, Kcu. The period of the corresponding sustained oscillation is referred to as the ultimate period, Pu.

4. On-line Controller Tuning 4.1 Continuous Cycling Method Step 4. Calculate the PID controller settings using the Ziegler-Nichols (Z-N) tuning relations in Table 12.6. Step 5. Evaluate the Z-N controller settings by introducing a small set-point change and observing the closed-loop response. Fine-tune the settings, if necessary. 52

4. On-line Controller Tuning 4.1 Continuous Cycling Method Figure 12.12 Experimental determination of the ultimate gain Kcu. 53

4. On-line Controller Tuning 4.1 Continuous Cycling Method Table 12.6. Controller settings based on the continuous cycling method

Guidelines for Common Control Loops 1. Flow Rate Flow control loops are characterized by fast response with essentially no time delay For flow control loops, PI control is generally used. The presence of recurring high-frequency noise discourages the use of derivative action because it amplifies the noise. 2. Gas Pressure PI controllers are normally used with only a small amount of integral control action (тI is large) Derivative action is normally not needed because the process response times are usually quite small compared to those of other process operations.

Guidelines for Common Control Loops 3. Temperature General guidelines are difficult to state because of the wide variety of processes and equipment involving heat transfer and different time scales. PID controller are commonly employed to provide more rapid responses than can be obtained with PI controllers. 4. Liquid Level Standard P or PI controllers are commonly used for level control. Because offset is not important in averaging level control, it is reasonable to use a proportional-only controller. 56

Troubleshooting Control Loops If a control loop is not performing satisfactorily, then troubleshooting is necessary to identify the source of the problem. Based on experience in the chemical industry, Buckley (1973) has observed that a control loop that once operated satisfactorily can become either unstable or excessively sluggish for a variety of reasons that include: a. Changing process conditions, usually changes in throughput rate. b. Sticking control valve stem c. Plugged line in a pressure or differential pressure transmitter. d. Fouled heat exchangers, especially reboilers for distillation columns. e. Cavitating pumps (usually caused by a suction pressure that is too low).

Troubleshooting Control Loops The starting point for troubleshooting is to obtain enough background information to clearly define the problem. Many questions need to be answered: 1.What is the process being controlled? 2.What is the controlled variable? 3.What are the control objectives? 4.Are closed-loop response data available? 5.Is the controller in the manual or automatic mode? Is it reverse or direct acting? 6.If the process is cycling, what is the cycling frequency? 7. What control algorithm is used? What are the controller settings? 8. Is the process open-loop stable? 9. What additional documentation is available, such as control loop summary sheets, piping and instrumentation diagrams, etc.? After acquiring this background information, the next step is to check out each component in the control loop.