Explicit Non-linear Optimal Control Law for Continuous Time Systems via Parametric Programming...
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Explicit Non-linear Optimal Control Law for Continuous Time Systems
via Parametric Programming
Vassilis Sakizlis,
Vivek Dua, Stratos Pistikopoulos
Centre for Process Systems Engineering
Department of Chemical Engineering
Imperial College, London.
Brachistrone Problem
wall- target
plane-obstacle
x
y
g
x=l
y=xtanθ+h
γ
Find closed-loop trajectory γ(x,y) of a gravity driven ball such that it will reach the opposite wall in minimum time
Outline
• Introduction
• Multi-parametric Dynamic Optimization
• Explicit Control Law
• Results
• Concluding Remarks
Introduction Model Predictive Control
Accounts for- Optimality- Constraints- Logical Decisions
Shortcomings-Demanding Computations-Demanding Computations-Applies to slow processes-Applies to slow processes-Uncertainty handling
Solve an optimization problem at each time interval
Application - Parametric Controllers (Parcos)
• Explicit Control law• Eliminate expensive, on-line computations
Optimization Problem
Parametric Solution
)( *xv
*xParametric Controller
v(t)=g(x*)
PLANT Process Outputs yInputDisturbances
w
Plant State x*Control v
Theory of PARCOS
•Complete mapping of optimal conditions in parameter space•Function c(x),vc (x),c
(x) •Critical regions CRc(x)0 c=1,Nc
What is Parametric Programming?
FeaturesFeatures
binary
continuous
parameters
)( s.t.
)(
::
δvx
xvg
xvfxv
:0,,
,,min)(,
)()(
xxv
x
Region CR1
Theory, Algorithms and Software Tools
for Multi-parametric Optimization Problems
Quadratic and convex nonlinear
Mixed integer linear, quadratic and nonlinear
Bilinear Applications
Process synthesis and planning
Design under Uncertainty
Reactive scheduling / Bilevel Programming
Stochastic Programming
Model based and hybrid control
Parametric Programming Developments
Model – based Control via Parametric Programming
0.15t ,10
,0
5.1),(0
""
01221.0992.0170.0
0.16954107.0269.1 ..
)100198.0
0084.00116.0(min)
,2,1||
*
,2,11,2
,2,11,1
24,2,1
1
0
2,2
2,1||
*
N
Nk
xxvxg
xx
vxxx
vxxxts
vxx
xxPxxφ(x
ktkttktk
t
ktktktkt
ktktktkt
ktktkt
N
kktkttN
TtN
vN
Formulate mp-QP (mp-LP)Obtain piecewise affine control lawPistikopoulos et al., (2002)Bemporad et al.,(2002)
c
o
cNc
Nc
XxCR
(x),φ(xv
,1
,0)(
),*
**
Objective
Discrete Model
Current States
Constraints
Parco / Explicit MPC Solution
c
cc
Nc
CRxCR
bxav
cc
,...1
0 if
law Control
2*1
*0
• Complex• Approximate
Multi-parametric Dynamic Optimizationmp-DO
),[
0)( ,0)()(
v(t)]dtv(t))()([min)(
*
221
T*
*
fo
o
f
t
t
Tf
Tf
v
ttt
xx
txGbtvDtxDg
vBxAx
RtxQtxPxxxf
• Feasible Set X*
For each x* X* there exists an optimizer v*(x*,t) such that the constraints g(v*,x*) are satisfied.
• Value Function (x*), x* X*
• Optimizer, states v*(x*,t), x(x*,t), x* X*
mp-DO Solution
Three methods
Complete discretization
Discrete state space model(Bemporad and Morari, 1999)
mp-(MI)QP (LP)mp-(MI)QP (LP) (Dua (Dua et al., et al., 2000,2001)2000,2001)
•Lagrange Polynomials for Parameterizing the Controls
(Vassiliadis et al., 1994)
•semi-infinite program - two stage decomposition .(similar
to Grossmann et al., 1983)
mp- (MI)DO (1)
mp- (MI)DO (2)mp- (MI)DO (2)• Euler – Lagrange conditions of OptimalityEuler – Lagrange conditions of Optimality• No state or control discretizationNo state or control discretization
Multi-parametric Dynamic Optimizationmp-DO
Optimality Conditions - Unconstrained problem(No inequality constraints)
)()(
)(
ConditionsBoundary
],[
Equations of System alDifferenti Augmented
*
1
ff
o
fo
T
T
tPxt
xtx
ttt
BRv
AQx
BvAxx
Two point boundary value problem
Multi-parametric Dynamic Optimizationmp-DO
Optimality Conditions - UnconstrainedUnconstrained problem
tfto
g(x,
v)
Constraint bound
Multi-parametric Dynamic Optimizationmp-DO
Optimality Conditions - ConstrainedConstrained problem
tfto
g(x,
v) -
co
nstr
aint
t1 t2
Boundary constrained arc
Unconstrained arc
UnknownsSwitching points
Multi-parametric Dynamic Optimizationmp-DO
Optimality Conditions - Constrained problem
],[0 ,0
)(
Equations of System alDifferenti Augmented
1
fo
iii
TT
TT
tttg
x
gBRv
x
gAQx
BvAxx
Complementarily Conditions
Multi-parametric Dynamic Optimizationmp-DO
Optimality Conditions - Constrained problem
TTTT
T
ff
o
gRvvQxxxH
tHtH
tHtH
tt
x
gtt
tPxt
txtx
txtx
xtx
)()(
)()(
)()(
)()(
)()(
)()(
)()(
)(
ConditionsBoundary
22
11
22
11
22
11
*
States - Continuity
Costates - Adjoints
Hamiltonian – Switching points
Multi-parametric Dynamic Optimizationmp-DO
Optimality Conditions - Constrained problem
• Solve analytically the dynamics, get time profiles of variables
• Substitute into Boundary Conditions Eliminate time
1111
2,1*
2,12,1
*2,12,1
,:
0)(,0))()()((
)()(
ofxwhere
ttbxtFtN
xtStM
Linear in Non Linear in t1,2
• Solve for ξ (sole unknown) and back-substitute into dynamics
• Get profiles of x(t,x*), v(t,x*), λ(t,x*), μ(t,x*)
Solution of mp-DO1. Fix a point in x-space
2. Solve DO and determine active constraints and boundary arcs
3. Determine optimal profiles for μ(t,x*),λ(t,x*),v(t,x*),t1(x*),t2(x*)
4. Determine region where profiles are valid:
0)},(~
{min)(0),(~
0)},({max)(0),(
*
],[
*2
*
*
],[
*1
*
21
txxGtx
txgxGtxg
ttt
tttfo
Optimality condition
Feasibility condition
Control Law
c
cc
Nc
xCRxxCR
xtbxxtAv
,,1
),()(0
if
),(),(ˆ
*2**1
***
Applied fort* t t*+Δt
OR
))(,())(,(lim)(ˆ **
0
* xttbxxttAxv kco
kc
t
Implement continuously
Continuous Control Law viamp-DO
* offunction nonlinear are , ,continuous xv
• Property 1:
• Property 2:
• Property 3:
• Property 4:
convex ,continuous is )( *x
NL continuous )(x t),(xt *kx
*kt
Feasible region: X* convex but each critical region non-convex
2 - state Exampleopen-loop unstable system
]5.1,0[
1001000]15.1[
0
1
01
63.063.1
]dtv100084.00099.0
0099.00116.0[min)(
*
24*
*
txx
vxg
vxx
xxPxxx
o
t
t
Tf
Tf
v
f
mp-DO Result
);43.113.0(0.21)93.009.1(4.12)( 2186.0
2176.10 xxexxetv tt
Region
)43.113.0
94.009.1ln(1.0 :where
04.2)43.113.0(22.0)94.009.1(4.1
21
21
2186.0
2177.10
xx
xxt
xxexxe tt
mp-DO Result
Results for constrained region:
mp-DO Result
Results for constrained region:
mp-DO Result
Complexity
mp-QP:
10
00
)21(10
i
Nnv
i ii
NqMax number of regions
mp-DO: 4211
00
i
nv
i ii
qMax number of regions
Reduced space of optimization variables and constraints
Constrained
Unconstrained
mp-DO Result - Simulations
mp-DO Result - Suboptimal
Feature: 25 regions correspond to the same active constraint over different time elements
Merge and get convex Hull
Compute feasible
Control lawIn Hull
v = -6.92x1-2.9x2-1.59
v = -6.58x1-3.02x2
mp-DO Result - Suboptimal
Brachistrone Problem
wall- target
plane-obstacle
x
y
g
x=l
y=xtanθ+h
γ
Find trajectory of a gravity driven ball such that it will reach the opposite wall in minimum time
Brachistrone Problem
),0[
)(
)(
1 ,)(
1,5.0)( tan,)tan(
)sin(2
)cos(2
min),( 00
f
oo
oo
f
f
tt
yty
xtx
lltx
hhxy
gyy
gyx
txy
Brachistrone Problem - Results
Brachistrone Problem - Results
Absence of disturbance: open=closed-loop profile
Brachistrone Problem - ResultsPresence of disturbance
Concluding Remarks
Issues
• Unexplored area of research
• Non-linearity in path constraints even if dynamics are linear
• Complexity of solution
Advantages
• Improved accuracy and feasibility over discrete time case
• Suitable for the case of model – based control
• Reduction in number of polyhedral regions
• Relate switching points to current state