Process Simulation using
MATLAB
Most chemical process models are nonlinear and
rarely have analytical solutions. However, numerical
solutions ca be easily obtained using MATLAB
Simulation of Nonlinear Models
• Example: Two interacting tank in series with outlet flowrate
being function of the square root of tank height.
F
F2
F1
F1 = R1 h1 - h2
F2 = R2 h2
– Parameter values
ft 2.5
R1 = 2.5
min
5 ft 2.5
R2 =
6 min
– Input variable F = 5 ft3/min
A1 = 5ft 2
A2 = 10 ft 2
• Modeling equations : material balance
d ( r A1h1 )
= r F - r F1
dt
Þ
d ( r A2 h2 )
= r F1 - r F2
dt
ù
é dh1 ù é F - R1 h - h
2
ú é f (h , h , F )ù
ê dt ú ê A1 A1 1
ú=ê 1 1 2
ê
ú=ê
ú
f
h
h
F
,
,
dh
R
R
ê
ú
(
)
ê 2ú
1
2
ë 2 1 2
û
h
h
h
1
2
2 ú
ê
êë dt úû ë A2
A2
û
• Steady-state height values : solve algebraic equations
F R1
A1 A1
R1
A2
h1 - h2 = 0
h1 - h2 -
R2
A2
h2 = 0
– Using MATLAB fsolve routine
• h1s = 10, h2s = 6
• Question
– What are the responses of tank height if the initial heights are
h1(0)=12 ft and h2(0)=7 ft ?
– Assume the system is at steady-state initially. What are the
responses of tank height if
• F changes from 5 to 7 ft3/min at t = 0
• F has periodic oscillation of F = 5 + sin(0.2t)
• F changes from 5 to 4 ft3/min at t = 20
• Solve differential equations using MATLAB ode45 routine or
Simulink dee
•
h1(0)=12 ft and h2(0)=7 ft
•
12
F = 5 + sin(0.2t)
12
h1
h2
11
11
10
height
height
10
9
9
h1
8
h2
8
7
7
6
6
0
10
20
30
40
50
60
70
80
90
5
100
0
50
100
time
•
150
time
F changes from 5 to 7 ft3/min at t = 0
•
F changes from 5 to 4 ft3/min at t = 20
20
11
18
10
h
h1
2
h2
16
1
h
9
height
height
8
14
12
7
6
10
5
8
6
4
0
20
40
60
80
100
time
120
140
160
180
200
3
0
20
40
60
80
time
100
120
140 150
Linear State-Space Model
• General dynamic models consist of a set of ODEs
x& = f ( x, u )
• A linear model is a subset of the general modeling equation.
The form of linear model is called as a state-space model.
x& = A x + Bu
– x : state vector (its elements are referred to as state variables)
– u : input vector of manipulated variables
– Matrices A and B are constant matrices
•
Example: Consider two tanks in series where the flow out of the
first tank enters the second tank. Assume the flow out of each
tank is a linear function of the tank height
F
F1 = R1 h1
Objective: develop a model to
describe how h2 change with
time, given the input flowrate F
F1
F2 = R2 h2
F2
– Material balance
dh1 F R1
=
- h1
dt
A1 A1
dh2 R1
R
=
h1 - 2 h2
dt
A2
A2
• Write the modeling equations in matrix form
é R1
ê
&
A1
é h1 ù
ê
=
ê& ú
ë h2 û ê R1
ê A
ë 2
ù
0 ú
é1ù
h
ú éê 1 ùú + ê A1 ú F
R2 ú ë h2 û ê ú
- ú
êë 0 úû
A2 û
x& = A x + B u
• Additional equation associated with a state-space model
y = C x + Du
– y : vector of output variables
– Output variables are typically a subset of x. They are variables
that can be measured or are of particular interest in a
simulation study
Ex: Both tank height are outputs
é1 0 ù é h1 ù
y=ê
= Cx
ê
ú
ú
ë0 1 û ë h2 û
( D = 0)
General form of State-Space Model
•
General form
dx1
= a11 x1 + a12 x2 + L + a1n xn + b11u1 + L + b1m u m
dt
M
M
dxn
= an1 x1 + an 2 x2 + L + ann xn + bn1u1 + L + bnm um
dt
y1 = c11 x1 + c12 x2 + L + c1n xn + d11u1 + L + d1m u m
M
n state variables (x)
m input variables (u)
r output variables (y)
M
yr = cr1 x1 + cr 2 x2 + L + crn xn + d r1u1 + L + d rm um
•
Matrix form
é x&1 ù é a11 a12
ê M ú=ê M
M
ê ú ê
êë x&n úû êë an1 an 2
é y1 ù éc11 c12
ê M ú=ê M
M
ê ú ê
êë yr úû êëcr1 cr 2
L a1n ù é x1 ù é b11 b12
O M úê M ú+ê M
M
úê ú ê
L ann úû êë xn úû êëbn1 bn 2
L c1n ù é x1 ù é d11 d12
O M úê M ú+ê M
M
úê ú ê
L crn úû êë xn úû êë d r1 d r 2
L b1m ù é u1 ù
O M úê M ú
úê ú
L bnm úû êëum úû
L d1m ù é u1 ù
O M úê M ú
úê ú
L d rm úû êëum úû
x& = A x + Bu
y = Cx + Du
Simulation of State-Space Models
• 除了一般的ODE解法器(ode45)之外，亦可使用MATLAB
提供之模擬線性模式的函數
Step 1: 建立線性狀態空間模式之物件 sys = ss(A, B, C, D)
Step 2: (the following functions assume a deviation variable form)
• [y,t,x] = initial(sys,x0)
– 計算線性模式 sys 的自由響應(輸入不改變之響應)，所使用
的起始條件指定於向量 x0 當中
• [y,t,x] = step(sys)
– 計算線性模式 sys 的單位步階響應
• [y,t,x] = impulse(sys)
– 計算線性模式 sys 的單位脈衝響應
• dimension of y : T x r x m
T = length of t
r = number of outputs
m = number of inputs
• [y,t,x] = lsim(sys,u,t,x0)
– 計算線性模式 sys 在時間 t 接受輸入向量 u 所得到的響應，
可指定起始條件 x0
Simulation Using Simulink
•
Simulation of state-space models using Simulink
(assume a deviation variable form)
選取拖曳 State-Space
(位於Continuous中)
• 輸入矩陣A, B, C, D 與起始條件
• 連結輸入與輸出
• 啟動模擬
• 若有多個輸入變數與輸出變數，則需使用Mux (輸入) 與
Demux (輸出) (位於Signal Routing中)
Mux:將多個訊號合成一個訊號
Demux:將一個訊號分成多個訊號
Exercise
•
State-space model, zero initial condition
é x&1 ù é -0.5 -0.8ù é x1 ù é1 -1ù é u1 ù
+ê
ê x& ú = ê 0.8
ê
ú
ú
ú êu ú
x
0
0
2
û ë 2û ë
û ë 2û
ë 2û ë
é y1 ù é 2 6 ù é x1 ù
ê y ú = ê1 -1ú ê x ú
û ë 2û
ë 2û ë
Find the responses for
– unit step input of u1
– unit step input of u2
– u1 = sin(t)
Using
– MATLAB functions
– Simulink