Chapter 2 Mathematical Background Professor Shi-Shang Jang Chemical Engineering Department National Tsing-Hua University Taiwan March,

2-1 The Essence of Process Dynamics 2  As the plant is not operated at its steady state as designed, the process is in a dynamic state.  In many cases, small deviations of process steady state (noise) is allowed, but keep monitoring on the system is very important.  In case of strong deviations (trend), process control becomes a must.

Example: Thermal Process Inputs: f(t), T i (t),T s (t) Output: T(t) TsTs CV: T(t) MV: f(t) DV: T i (t), T s (t)

Noise v.s. Trend 4 Noise Trend T time

Conception of deviation variable 5 Plant Manipulative variables Controlled variables Disturbances H= h deviation = h - h steady-state tank Valve liquid level h control stream FcFc Wild stream FwFw

The Essence of Process Dynamics - Continued 6 The feedback/feedforward process control needs to understand the relationships between ◦ CVs and MVs ◦ DVs and CVs the relationships are called process models. For the ease of mathematical analyses, the process modeling implements in Laplace transform instead of direct use of time domain process model. Implementation of deviation variables is needed.

Conception of deviation variable (  =T- 70  ) 7 Noise Trend  time

2-2 Linear Systems 8

Consider a process system with a CV, say y(t), and a MV, say m(t). In process industries, it is traditional to assume the system is lumped and linear. A lumped process system is such that: A lumped system is linear if 9

Linear Systems-Continued A lumped process system is autonomous if An autonomous lumped system is said to be stable at the origin if 10

Linear Systems-Continued An autonomous lumped system is called asymptotic stable if Theory 2-1: A linear system is stable if the roots of the characteristic equation all have negative real part. Corollary 2-1: A linear system is asymptotic stable if it is stable. 11

Linear Systems-Examples 12 Note that no matter what the initial conditions is, the solution indicates : The system is hence asymptotic stable.

Linear Systems-Examples 13 Note that the above system is unstable since: Note that the above system is also unstable since:

Linear Systems-Continued Theory 2-2: A forced linear system is stable if the inputs of the system is bounded. A process control system is basically assuming that the controlled variable (CV) is to be controlled to a certain set point (zero), thus it is assumed that the input (MV) is the forcing function to a linear system with a dependent variable of CV. 14

Linear Systems-Continued A forced system is of the following form, for example: y”+a 1 y’+a 0 y=bm(t) In case of m(t) is in special functions such sint the above equation can be solved by implementing particular solutions i,e., y(t)=y h +y p However, in case of feedback process control, m(t) is most likely as a function of y(t), i.e. 15

Linear Systems-Continued In most analysis of a feedback control systems, it is our tradition to assume that the system performance is in its steady state, i.e. y”+a 1 y’+a 0 y=bm(t), y(0)=0; y’(0)=0 For the convenience of the analysis, a Laplace transform expression of the above linear system becomes necessary. 16

2-3 Laplace Transform: Definitions and Properties 17 Definition 2-1: Consider a function of time f(t), the Laplace transform of f(t) is denoted by Property 2-1: If f(t)=1,  t  0, f(t)=0,  t<0, this is called a step function as shown below (to describe an abrupt event to cause the change of operation), then: time 0

Scenario 1: Step function (shift) 18 The process endures an abrupt long term change, such a PM on a tool of manufacturing plant.

Definitions and Properties- Continued 19 Corollary 2-1: Given the following step function: Then: Property 2-2: Given a ramp function: Then: time 00

Scenario 2: Ramp function (drift) 20 The process endures slow change on the process such as tool aging of a manufacturing plant.

Definitions and Properties - Continued 21 Property 2-3: delta function  (t) Then: Note that, a pulse function can also represent an abrupt event in plant operation, but the event can be vanished very soon unlike step function. Definition 2-2: A pulse function is denoted by: Definition 2-3: A delta function  (t)=lim c  0 f(x) time 0c 1/c

Scenario 3: Pulse function (Excursion) 22 The process endures a sudden short change but returns to its original state.

Definitions and Properties- Continued 23 Property 2-4: Exponential function Corollary 2-2 Then:

Scenario 4: Exponential function (Excursion) 24 The process endures a sudden abrupt change, but return to its original state gradually..

Definitions and Properties- Continued 25 Sinusoildal FunctionsProperty 2-5: Then: Property 2-6: Then: Sinusoildal Functions

Scenario 5: Sinusoidal function 26 The process endures a sinusoidal wave input to force it response another periodical wave.

Theory 27 Theory 2-1: Linearity Consider two functions f(t) and g(t) with their Laplace transform F(s) and G(s) exists, and let a and b be two constants, then Theory 2-2: Derivative Let f(t) be differentiable, and its Laplace transform F(s) exist, then

Theory - Continued 28 Remark 2-1: Deviation variable (perturbation variable) Let y(t) has a steady state y s, then y d is called a deviation variable if y d (t)=y(t)-y s Remark 2-2: In the study of control theory, we all assume that the process is originally at its steady state, and then a change of the system starts. Therefore, y(0)= y s, or y d (0)=0. Corollary 2-3: Derivative of a deviation variable

Theory - Continued 29 Theory 2-3: Final Value Theorem Consider a function f(t) with its Laplace Transform F(s), then: Theory 2-4: Dead time (Time Delay, Translation in Time) Consider a function f(t) with its Laplace Transform F(s), then:

General Procedure 30 Time domain Laplace domain Step 1 Take Laplace Transform ODE Initial conditions Step 2 Solve for Step 3 Factor D(s) perform partial fraction expansion Step 4 Take inverse Laplace transform Solution y(t)

Solution of a Linear System Example: Solve the differentiation equation Sol: Take the Laplace transform on both sides of the equation or, By Laplace Transform

Solution of a Linear System By Laplace Transform – Partial Fraction Expansion

2-4 Transfer Functions – Example: Thermal Process Inputs: f(t), T i (t),T s (t) Output: T(t) TsTs

2-4 Transfer Functions – Cont. Let f be a constant  V= constant, C v =C p

Transfer Functions – Cont. where, G p (s) is call the transfer function of the process, in block diagram: G p (s) M(s) Y(s)

Scenario Simulation (Step Change) 36

Homework and Reading Assignments 37 Homework – Due 3/19 Text p57 2-1,2-2, 2-6 (a), (c) Reading Assignments ： Laplace Transform(p11-26)

2-5 Linearization of a Function X0X0 X 0 - △ X0+△X0+△ - △ 0 △ F(X) X aX+b

Linearization of a Function (single variable) Example: Linearize the Arrhenius equation, at T=300  C, k(300)=100, E=22,000kcal/kmol.

Linearization of a Function (two variables)

Linearization of Differential Equations Definition 2-4: A linear function L(x) is such that for x  R n, for all scalars a and b, L(ax 1 +bx 2 )= aL(x 1 )+bL(x 2 ). Definition 2-5: A Linear system is denoted as the following: Provided that L 1,…,L n are linear functions

Linearization - Continued Consider a function f(x 1,…,x n ), the linearization of such a function to a point is defined by the first order Tylor expansion at that point, i.e.

Linearization of a single input single output system

Example – Level Process h V f0f0 f Cross-sectional=A A=5m 2 f 0 =1m 3 /min Abrupt change of f 0 from 1 to 1.2m 3 /min

Example – Level Process - Continued Taking Laplace Transform on both sides: Now, F 0 (s)=0.2/s

[A B C D]=linmod('example_model'); [b,a]=ss2tf(A,B,C,D); example_model Linearization

47 Level Time(min.)

Conclusive Remarks for Laplace Transform Laplace transform is convenient tool to represent the dynamics of a linear system. Laplace transform is very easy to use to special inputs, especially in case of abrupt change of system inputs and time delays. Linearization is a frequent approach for a nonlinear system, and it is a good approximation in small change of the inputs.

Homework – Due 3/19 Text p ,2-23 Linearization of Function(p50-56)