Transcript Diapositiva 1

A brief introduction to the concept
of predictive control
1
Index
• Close Loop Control
• MPC without Constraints
• Optimal Control
• MPC with Constraints
• Introduction to Predictive
Control
• Example
• Predictive Control Elements
– Models
– Prediction
– Cost Function
– Optimization
2
Close Loop Control
• D (Disturbance) :
unknown and
uncontrollable inputs
[measurable or not]
• U (plant input) :
controllable inputs
[control action in closed
loop]
• Y (plant output) : system
output
3
Close Loop Control
• We want Y to track a certain signal (R).
• We want the system to behave like a
reference one
• We want to move from one point to
another minimizing a cost function.
4
Close Loop Control
• We want Y to track a certain signal (R).
5
Close Loop Control
• We want the system to behave like a
reference system does
6
Close Loop Control
• We want to move from one point to
another minimizing a cost function.
7
Close Loop Control
• R (Reference) : Signal
which the output should
follow.
controller
8
Index
• Close Loop Control
• MPC without Constraints
• Optimal Control
• MPC with Constraints
• Introduction to Predictive
Control
• Example
• Predictive Control Elements
– Models
– Prediction
– Cost Function
– Optimization
9
Discrete time Optimal
Control Formulation
System Model
x  k  1  Ax  k   Bu  k 
y  k   Cx  k   Du  k 
Cost Function
1 T
1  N 1 T

J  N   x  N   S  z  N    x  k   Q  z  k   u T  k   R  u  k 
2
2  k 0

10
Discrete time Optimal
Control Formulation
Control Law
u  k   K  x  k 
Problem : Determine K which minimizes the cost function.
Solution :
K  S  AT  S  A  AT  S  B  BT  S  B  R  BT  S  A  Q
1
11
Discrete time Optimal
Control Formulation
K
x  k  1  Ax  k   Bu  k 
y  k   Cx  k   Du  k 
•Static Formulation (fix controller).
•Disturbances are not taken into account.
•Inherent nonlinearities (ex. Saturations) are not taken into account.
•Exponential Convergence (infinite time).
•Linear cost function.
•Model uncertainty may be problematic.
12
Index
• Close Loop Control
• MPC without Constraints
• Optimal Control
• MPC with Constraints
• Introduction to Predictive
Control
• Example
• Predictive Control Elements
– Models
– Prediction
– Cost Function
– Optimization
13
MPC (Model Predictive Control)
• Optimal Control “practical” implementation
• Evolution : Practical Development (Industrial) ->
Theoretical Formulation (University)
• Very used in Process Industry.
–
–
–
–
Use of Empirical Information
Traditionally used in slow plants
Steady state Control
Simple to tune
14
MPC (Model Predictive Control)
• Can be used in a great variety of systems:
– Control action constrains
• Saturation
• Slew-rate
– Complex dynamics
• Non minimum phase
• Time delays
– Output constraints
– Multivariable systems
– “Nonlinear Systems”
• Can take profit from future behaviour knowledge
15
MPC (Model Predictive Control)
• Explicit use of models to forecast future
behaviors.
• Minimize a cost function.
• Different approaches differ on:
– Prediction Model
– Cost function
– Optimizer
16
MPC (Model Predictive Control)
• The key concept is to control the system in a
near future.
• Working in this way it is not necessary to take
into account the dynamic behaviour (ex. Inertia,
delays, …)
17
MPC (Model Predictive Control)
•
In order to taken into account this phenomena
it is necessary to recompute the “optimal
trajectory each time.
Due to noise and model uncertainty the
system does not moves over the
original optimal trajectory
18
Index
• Close Loop Control
• MPC without Constraints
• Optimal Control
• MPC with Constraints
• Introduction to Predictive
Control
• Example
• Predictive Control Elements
– Models
– Prediction
– Cost Function
– Optimization
19
MPC (Model Predictive Control)
• Components:
– Model to forecast the system
evolution
– Cost Function to define optimality
– Optimizer to obtain the optimal
trajectory
20
MPC (Model Predictive Control)
Reference
trajectory
Past Input/outputs
Model
Forecasted
outputs
Future
Inputs
Optimizer
Future Errors
Cost Functions
Constrains
Control Action
21
Model Predictive Control (Forecasting)
Modelo
Simulación
Modelo
Predicción a p pasos
22
Model Types used in MPC
• Impulsive Response (MAC,GPC,EPSAC)
• Step Response (DMC)
• Transfer functions (GPC, UPC, EPSAC,
EPSAC, EHAC, MUSMAR, MURHAC)
• State Space (PFC)
• Others (ex. Nonlinear, Fuzzy, Neural Nets
…)
23
Transfer function (GPC)
y  z  bm z m   b1 z  b0

n
u  z  an z   a1 z  a0
an 1
y k   
y  k  1 
an
bm
 u  k  m  n 
an
a0
a1
 y k 1 n  y k  n 
an
an
b0
b1
 u k 1 n  u k  n
an
an
24
Transfer function (GPC)
yˆ  k  1 k   
an 1
y k  
an
bm
 u  k  m  n  1 
an

a
a1
y  k  2  n   0 y  k  n  1 
an
an
b0
b1
 u  k  2  n   u  k  n  1
an
an
Per m=n
yˆ  k  1 k   
an 1
y k  
an
bm
 u  k  1 
an
a0
a1
 y  k  2  n   y  k  n  1 
an
an
b0
b1
 u  k  2  n   u  k  n  1
an
an
25
Transfer function (GPC)
yˆ  k  1 k   

an 1
y k  
an
bm
u  k  1 
an


a
a1
y  k  2  n   0 y  k  n  1 
an
an
b
b1
u  k  2  n   0 u  k  n  1
an
an
If you want to force system output to a certain value at k+1, then you can
Compute the necessary control action using the following equation:
an 1
a1


ˆ
y
k

1
k

y
k


y
k

2

n


 a  
 

a
an 
n
n

u  k  1 

bm1
b0
b1
bm  a0
u  k   u  k  2  n   u  k  n  1 
  y  k  n  1 
a
a
an
an
n
 n

26
Transfer function (GPC)
yˆ  k  2 k   

an 1
yˆ  k  1 k  
an

bm
b
uˆ  k  2   m1 uˆ  k  1 
an
an
a
a1
y  k  3  n   0 y  k  n  2 
an
an

b
b1
u  k  3  n  0 u  k  n  2
an
an
y k ,
, y k 1 n , y k  n ,
uˆ  k  2  , uˆ  k  1 ,
u k ,
, u k  n
In the generic case :
N 1
n
N
n
j 1
j 0
j 1
j 0
yˆ  k  N k    j yˆ  k  j k     j y  k  j    j uˆ (k  j )    ju (k  j )
yˆ  k  N k  ,
y k ,
, y k 1 n , y k  n ,
uˆ  k  N  ,
u k ,
, yˆ  k  1  N k  ,
, uˆ  k  1 ,
,u k  n
This can be computed as
a function of :
27
Transfer function (GPC)
Due to the fact that
can be computed in terms of
It is possible to write:
n
n
N
j 0
j 0
j 1
yˆ  k  N k     j y  k  j     j u (k  j )    j uˆ (k  j )
Respuesta libre
y k ,
, y k 1 n , y k  n ,
uˆ  k  N  ,
u k ,
Respuesta forzada
, uˆ  k  1 ,
, u k  n
Degrees of freedom
28
Model Predictive Control (Forecasting)
Reference trajectory
r(t+1)
Outputs
Previous control action effect
(forced response)
Future control action effect
(free response)
t
Control
action
t+1
t+2
t+3
time
Past control
actions
Future control actions
(degrees of freedom)
?
t
t+1
t+2
t+3
time
29
Disturbance Models
• Any Additional
information
introduced to improve
the forecast will
improve the system
performance.
• In some systems it is
possible to model the
disturbance behaviour
nˆ  k  N k   FN  z   n  k 
30
Index
• Close Loop Control
• MPC without Constraints
• Optimal Control
• MPC with Constraints
• Introduction to Predictive
Control
• Example
• Predictive Control Elements
– Models
– Prediction
– Cost Function
– Optimization
31
Cost Function
32
Cost Function
Goal :
The dynamic problem has been reformulated as a static one
Solution :
We are interested only in
uˆ  k 1
33
Cost Function
Usually Quadratic cost functions are used
Closed form solutions (Implicit optimization)
34
Model Predictive Control
setpoint trajectory
r(t+1)
CV
prediction horizon
Ny
t
t+1
t+2
t+3
...
t+Ny
time
MV past control
moves
control horizon
Nu
t
t+1
t+2
...
t+Nu-1
time
35
Cost Function with constrains
• Control Action:
u
y
Sys
– Saturation
– Derivative
• Output
– Saturation
umin  u  umax
u  umax
ymin  y  ymax
– Derivative
36
Optimization Problem
• No constraints:
– Linear system
• Quadratic Cost function
Closed form Solution
(Classical Control Theory can be applied)
• Generic Cost function
– Nonlinear system
• Constraints
Solver Needed
Solver Needed
Solve Needed
Solvers must work in Real Time
37
Example I
Measuring and avoiding the bullwhip effect: A control theoretic approach.
J. Dejonckheere, S.M. Disney, M.R. Lambrecht, D.R. Towill. European Journal
of Operational Research 147 (2003) 567–590
38
Example I
39
Example I
40
Example I
Overcome the production delay
41
Example I
42
43
Example I
44
Example I
45
Example II
46
Documentation
• Control Predictivo: Pasado, Presente y Futuro. Eduardo F.
Camacho & Calos Bordons. RIAI octubre 2004
• Predictive Control. A.W. Pike, M.J. Grimble, M.A. Johnson, A.W.
Ordys and S. Shakoor. Control Handbook (CRC Press 1996).
• Tutorial Overview of Model Predictive Control. James B.
Rawling. IEEE Control Systems Magazine. June 2000. pág. 3852.
• Application of Generalized Predictive Control to Industrial
Processes. David W. Clarke. IEEE Control Systems Magazine.
June 1998. pág. 52-55.
• A survey of industrial model predictive control technology. S.
Joe Qin, Thomas A. Badgwell. Control Engineering Practice 11
(2003) pàg. 733–764
47
Books
• Model Predictive Control in the Process Industry.
E.F. Camacho and C. Borbons. Springer-Verlag
(1995). ISBN : 3-540-19924-1.
• Adaptive Predictive Control : From the concepts
to plant optimization. Juan M. Martín Sánchez
and José Rodellar. Prentice Hall (1996). ISBN : 013-514861-8.
48