Transcript Nonlinear Predictive Control for Fast Constrained Systems

Nonlinear Predictive Control
for Fast Constrained Systems
By
Ahmed Youssef
Introduction
What’s MBPC
N=N2-N1
CV
t
N1
N2
MV
Nu
CV: controlled variable
MV: manipulated variable
Introduction
Shortcomings of current industrial
nonlinear MBPC
•Computing the MBPC control law demands
significant on-line computation effort
•Inability to deal explicitly with the plant model
uncertainty.
Objective of research work
Reducing the computational complexity of nonlinear
MBPC & adding the robustness property whilst
preserving its good attributes to make it more effective
practical tool for controlling systems of fastconstrained dynamics.
State Dependent State-Space Models
Given the nonlinear dynamic model
xk 1  f ( xk , uk )
yt  g ( xt )
Reformulate into nonlinear state-dependent form
xt 1  A ( xt ) xt  B ( xt ) ut
yt  C ( xt ) xt
This is not a linearisation
Trivial example xk 1  sin(xk )  xk cos(xk ) u k
sin(xk )
A( xk ) 
; B( xk )  xk cos(xk )
xk
Hamilton-Jacobi-Bellman
x  f ( x)  g ( x) u



V ( x)  min  x T Q( x) x  u T R( x) u dt
u (t )
T
0
T
 V 
 V 
V
1  V 
1
T
 f ( x)  
 g ( x) R g ( x) 
  x T Q x  0
 
t
4 x 
x
x
NLQGPC Control Law
The NLQGPC quadratic infinite horizon cost function:
1 T
J  lim
 Jt
T  T  1 t 0
N
Jt  
j 0
y
t  j 1  rt  j 1

T 


T 
Qe yt  j 1  rt  j 1  ut  j Qu ut  j

The optimal control vector in terms of the states of the
system and reference model:
Ut,N  

 (t ,t  N ) A(t )   T H 1 A(t )
 
1
T
S
 (t ,t  N ) Qe
T
xˆt   T H 2  N  S(t ,t  N ) Qe Rt , N 

T
Qu  S(t ,t  N ) Qe S(t ,t  N )
 
  (t ) H  (t )
T
1

The Coupled Algebraic Riccati Equations

 
  S Q 
H 1j  AT  N Qe  N  H 1j 1 A  AT  N Qe S N  H 1j 1

T
T
 Qu  S N Qe S N  
T
T
H 1j 1
1
T
N
e


T 1


H j 1 A
N

H 2j   AT  N Qe  AT H 2j 1 N  AT T Qe S N  H 1j 1 
T

 Qu  S N Qe S N  
T
T
H 1j 1

 
1
 T H 2j 1N  S N T Qe


Dealing with Stability Issue
Control Lyapunov Function
A C1 function V(x): n   is said to be a
discrete CLF for the system:
if V(x) is positive definite, unbounded, and if
for all x  0
Stability via Satisficing
Satisficing is based on a point-wise cost / benefit
comparison of an action.
The benefits are given by the “Selectability”
function Ps(u,x), while the costs are given by the
“Rejectability” function Pr(u,x).
The “satisficing” set is those options for which
selectability exceeds rejectability:
CLF-Based Satisficing Technique
Start
f, B, P, 
uNLQGPC
Calculate 
implies

uS
No
implies
uaug = uNLQGPC - ( )(BTPB)-1BTPf
uS
 u NLQGPC B T P f
T
 
f T P B ( B T P B) 1 B T P f
f T P f  xT P x
 T
f P B ( B T P B) 1 B T P f
Satisficing generates the state
dependent set of controls that
render
the
closed-loop
system stable with respect to
a known CLF.
Dealing with Input Constraints
Examples of Actuator Constraints
• Magnitude Saturation
u max ,

u out  sat( u in )   u in ,
u ,
 min
• Rate-Limited Actuators
 max ,



u out  sat( u in )   u in ,

 min ,
• Actuator Dead-Zone
u in  u max ,
u min  u in  u max ,
u in  u min ,
u in   max ,
 min  u in   max ,
u in   min ,
Dd (u)  u  sat d (u)
Therefore the common term is the Saturation function
Approximation of Magnitude Saturation
the actuator range of operation is limited
0  u 
Limiting functions that
map the interval (-,)
onto (0, 1)
  u 
Limiting functions that
map the interval (-,)
onto (-1, 1)
Approximation of Magnitude Saturation
Sigmoid function S 01 (Black)
Error function
(Blue)
Tanh function
(Green)
Sigmoid function S 11 (Red)
Case Studies
F-8 fighter aircraft
F-16 fighter aircraft
Caltech Ducted Fan
Controlling of F-8 Fighter
0.5
Angle of Attack [rad]
0.4
Unconstrained NLQGPC
Constrained NLQGPC
CNLQGPC-Satisficing (Nominal)
CNLQGPC-Satisficing (Wind Gust)
0.3
0.2
0.1
0
-0.1
0
2
4
6
8
Time [s]
10
12
14
0.1
0
Pitch Rate [rad/s]
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
0
2
4
6
8
Time [s]
10
12
14
Elevator Deflection
0.01
0
-0.01
u [rad]
-0.02
-0.03
u  0.05236 rad
-0.04
-0.05
-0.06
-0.07
-0.08
0
2
4
6
8
Time [s]
10
12
14
Trajectories of Controlled System
0.8
0.6
0.4
x3
0.2
0
-0.2
-0.4
-1
-0.6
0
-0.8
0.5
1
0
-0.5
x1
x2
CONCLUSIONS
Properties of NLQGPC controller:
1. High performance
2. Less computational burden
3. Dealing with input constraints
4. Guaranteeing asymptotic stability to the
closed-loop system.
5. Possesses both performance robustness &
stability robustness