Download - 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.
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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
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Outline
• Introduction
• Multi-parametric Dynamic Optimization
• Explicit Control Law
• Results
• Concluding Remarks
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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
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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
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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
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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
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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
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Parco / Explicit MPC Solution
c
cc
Nc
CRxCR
bxav
cc
,...1
0 if
law Control
2*1
*0
• Complex• Approximate
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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*
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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
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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
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Multi-parametric Dynamic Optimizationmp-DO
Optimality Conditions - UnconstrainedUnconstrained problem
tfto
g(x,
v)
Constraint bound
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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
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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
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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
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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*)
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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
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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
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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
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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
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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
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mp-DO Result
Results for constrained region:
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mp-DO Result
Results for constrained region:
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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
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Constrained
Unconstrained
mp-DO Result - Simulations
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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
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mp-DO Result - Suboptimal
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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
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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
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Brachistrone Problem - Results
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Brachistrone Problem - Results
Absence of disturbance: open=closed-loop profile
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Brachistrone Problem - ResultsPresence of disturbance
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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