Optimal H Control Design under Model Information ...people.eng.unimelb.edu.au/ffarokhi/files/old_files/CDC-2013-Hinf.pdf · Optimal H 1Control Design under Model Information Limitations

Optimal H∞ Control Design under Model InformationLimitations and State Measurement Constraints

F. Farokhi, H. Sandberg, and K. H. Johansson

ACCESS Linnaeus Center, School of Electrical Engineering,KTH-Royal Institute of Technology, Stockholm, Sweden.

E-mails: {farokhi,hsan,kallej}@ee.kth.se

The 52nd IEEE Conference on Decision and Control (CDC 2013)

Friday December 13, 2013

Farokhi, Sandberg, and Johansson (KTH) Optimal H∞ Control Design ... Friday December 13, 2013 1 / 15

Decentralized Control

𝐶1

𝑃1

𝑃2

𝐶4

𝐶3

𝑃3

𝑃7

𝐶5

𝐶6

𝐶2 𝑃6

𝑃5

𝐶7

𝑃4

• Decentralized control extensively studied, e.g., [Witsenhausen ‘68 and‘71; Ho and Chu ‘72; Sandell and Athans ‘74; Anderson and Moore‘81; Siljak ‘89; Davison and Chang ‘90; Rotkowitz and Lall ‘06];

• Full model information {Pi} required.


Control Design with Limited Plant Model Information

𝐶1

𝑃1

𝑃2

𝐶4

𝐶3

𝑃3

𝑃7

𝐶5

𝐶6

𝐶2 𝑃6

𝑃5

𝐶7

𝑃4

• Design of Ci only depends on the model of surrounding Pj ;



𝐶1

𝑃1

𝑃2

𝐶4

𝐶3

𝑃3

𝑃7

𝐶5

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𝐶2 𝑃6

𝑃5

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𝑃4


Recent studies:

• Pi LTI and Ci static/dynamic/adaptive [Langbort & Delvenne;Farokhi, et al];

• Pi stochastically-varying parameters and Ci static [Farokhi &Johansson].



𝐶1

𝑃1

𝑃2

𝐶4

𝐶3

𝑃3

𝑃7

𝐶5

𝐶6

𝐶2 𝑃6

𝑃5

𝐶7

𝑃4


Recent studies:

• Pi LTI and Ci static/dynamic/adaptive [Langbort & Delvenne;Farokhi, et al];

• Pi stochastically-varying parameters and Ci static [Farokhi &Johansson].

Here: Numerical method for optimal control design overgeneral plant model model constraints


Networked Control Systems

P1 P2

K1 K2

x2x1u1 x2u2

P1

GK

P2

︸︷︷︸Architectural Constraints

1

GC

2

︸︷︷︸Model Information Constraints



P1 P2

K1 K2

x2x1u1 x2u2

P1

GK

P2


1

GC

2




P1 P2

K1 K2

x2x1u1 x2u2

1

GK

2


1

GC

2




P1 P2

K1 K2

x2x1u1 x2u2

1

GK

2


1

GC

2




P1 P2

K1 K2

x2x1u1 x2u2

1

GK

2


1

GC

2



Plant Model

Subsystem i is a parameter-dependent linear system

xi(t) =

N∑j=1

[Aij(αi)xj(t) + (Bw)ij(αi)wi(t) + (Bu)ij(αi)ui(t)

],

where

• xi(t) ∈ Rni , wi(t) ∈ Rmw,i , ui(t) ∈ Rmu,i are its state, exogenousinput, and control input;

• αi ∈ Rpi is its parameter vector.

Augmenting these subsystems results in

x(t) = A(α)x(t) +Bw(α)w(t) +Bu(α)u(t),

where α = (αi)Ni=1 belongs to the set of feasible parameters A.


Plant Model

Subsystem i is a parameter-dependent linear system

xi(t) =

N∑j=1

[Aij(αi)xj(t) + (Bw)ij(αi)wi(t) + (Bu)ij(αi)ui(t)

],

where

• xi(t) ∈ Rni , wi(t) ∈ Rmw,i , ui(t) ∈ Rmu,i are its state, exogenousinput, and control input;

• αi ∈ Rpi is its parameter vector.

Augmenting these subsystems results in

x(t) = A(α)x(t) +Bw(α)w(t) +Bu(α)u(t),

where α = (αi)Ni=1 belongs to the set of feasible parameters A.


Output Feedback Controller

Subsystem i observes

yi(t) =

N∑j=1

[(Cy)ij(αi)xj(t) + (Dyw)ij(αi)wj(t)

]∈ Roy,i ,

and uses static control law

ui(t) = Kiiyi(t).

Cy =

[(Cy)11 (Cy)12

0 (Cy)22

]1

GK

2Dyw =

[(Dyw)11 (Dyw)12

0 (Dyw)22

]




yi(t) =

N∑j=1


]∈ Roy,i ,


ui(t) = Kiiyi(t).

Cy =

[(Cy)11 (Cy)12

0 (Cy)22

]1

GK

2Dyw =

[(Dyw)11 (Dyw)12

0 (Dyw)22

]




yi(t) =

N∑j=1


]∈ Roy,i ,


ui(t) = Kiiyi(t).

Cy =

[(Cy)11 (Cy)12

0 (Cy)22

]1

GK

2Dyw =

[(Dyw)11 (Dyw)12

0 (Dyw)22

]


Control Design Strategy

Subsystem i uses control design strategy

Kii = Γii({αj |j → i in GC}).

1

GC

2 K11 = Γ11(α1)

K22 = Γ22(α2)

The set of all such control design strategies Γ is denoted by C.


Control Design Strategy

Subsystem i uses control design strategy

Kii = Γii({αj |j → i in GC}).

1

GC

2 K11 = Γ11(α1)

K22 = Γ22(α2)

The set of all such control design strategies Γ is denoted by C.


Performance Metric

Let us introduce the performance measure output vector

z(t) = Czx(t) +Dzww(t) +Dzuu(t) ∈ Roz ,

and define the closed-loop performance measure

J(Γ, α) = ‖Tzw(s; Γ, α)‖∞,

where

• Tzw(s; Γ, α): Closed-loop transfer function from w(t) to z(t).


Objective

Find the best control design strategy with limited information:

infΓ∈C

supα∈A

J(Γ, α) = infΓ∈C

supα∈A‖Tzw(s; Γ, α)‖∞

——————————————See (Ho and Chu, 1972; Sandell and Athans, 1974; Rotkowitz and Lall, 2006) for condi-tions to guarantee convexity and (Swigart and Lall, 2010, 2011; Shah and Parrilo, 2010)for explicitly calculating the optimal controller.


Objective


infΓ∈C

supα∈A



• For general GK, the outer problem is not necessarily convex and evenif convex, calculating the optimal controller might be difficult;

• The outer problem is an infinite-dimensional optimization problem;



Objective


infΓ∈C

supα∈A



• For general GK, the outer problem is not necessarily convex and evenif convex, calculating the optimal controller might be difficult;

• The outer problem is an infinite-dimensional optimization problem;

Find saddle point solutions of J using a numerical method



Assumptions

• A is a compact set;

• There exists a basis set (ξ`)L`=1 such that

A ∈ span((ξ`)L`=1)n×n,

Bw ∈ span((ξ`)L`=1)n×mw ,

Bu ∈ span((ξ`)L`=1)n×mu ,

Cy ∈ span((ξ`)L`=1)oy×n,

Dyw ∈ span((ξ`)L`=1)oy×mw ;

• There exists a basis set (η`′)L′

`′=1 such that

Γ ∈ span((η`′)L′

`′=1)mu×oy ;

• D>zuDzu = I and DywD>yw = I.


Saddle Point Solution

Find a saddle point of J(Γ, α) for Γ ∈ C ∩ span((η`′)L′`′=1) and α ∈ A.

Saddle Point

A pair (Γ∗, α∗) ∈ [C ∩ span((η`′)L′

`′=1)]×A is a saddle point of J(·, ·) if thereexist constants ε1, ε2 > 0 such that

J(Γ∗, α) ≤ J(Γ∗, α∗) ≤ J(Γ, α∗),

for any (Γ, α) ∈ [C ∩ span((η`′)L′

`′=1)]×A where ‖Γ− Γ∗‖≤ε1 and ‖α− α∗‖≤ε2.

• If GK is appropriately chosen (e.g., partially nested property) and(η`′)

L′`′=1 is not degenerate (i.e., @α s.t. all basis functions become

zero), then ε1 can be arbitrarily large (when considering dynamiccontrollers of high enough order);

• For special (ξ`)L`=1 (e.g., affine functions), ε2 may also be

arbitrarily large.




Saddle Point



J(Γ∗, α) ≤ J(Γ∗, α∗) ≤ J(Γ, α∗),







arbitrarily large.




Saddle Point



J(Γ∗, α) ≤ J(Γ∗, α∗) ≤ J(Γ, α∗),







arbitrarily large.


Numerical Procedure for Finding Saddle Points of J

1: Calculate ∂ΓJ(Γ, α) and ∂αJ(Γ, α); [Apkarian and Noll, ’06]

2: For each Γ(k), numerically extract α∗(Γ(k));

3: Then, update Γ(k+1) = Γ(k) + µkgk where gk ∈ ∂ΓJ(Γ(k), α∗(Γ(k))).

Theorem

Let {µk}∞k=0 be chosen such that limk→∞∑k

z=1 µz =∞ and

limk→∞∑k

z=1 µ2z <∞. Assume that the subgradients are uniformly

bounded for all iterations. If the numerical procedure converges, it gives asaddle point of J .


Numerical Procedure for Finding Saddle Points of J

1: Calculate ∂ΓJ(Γ, α) and ∂αJ(Γ, α); [Apkarian and Noll, ’06]

2: For each Γ(k), numerically extract α∗(Γ(k));

3: Then, update Γ(k+1) = Γ(k) + µkgk where gk ∈ ∂ΓJ(Γ(k), α∗(Γ(k))).

Theorem

Let {µk}∞k=0 be chosen such that limk→∞∑k

z=1 µz =∞ and

limk→∞∑k

z=1 µ2z <∞. Assume that the subgradients are uniformly

bounded for all iterations. If the numerical procedure converges, it gives asaddle point of J .


Numerical Example: Vehicle Platooning

• Regulating inter-vehicle distances d12 and d23v1(t)

d12(t)v2(t)

d23(t)v3(t)

=−%1/m1 0 0 0 0

1 0 −1 0 00 0 −%2/m2 0 00 0 1 0 −10 0 0 0 −%3/m3

v1(t)d12(t)v2(t)d23(t)v3(t)

+b1/m1 0 0

0 0 00 b2/m2 00 0 00 0 b3/m3

u1(t)u2(t)u3(t)

+w1(t)w2(t)w3(t)w4(t)w5(t)

%1 = %2 = %3 = 0.1 and b1 = b2 = b3 = 1.0

1

GK

2

3




d12(t)v2(t)

d23(t)v3(t)

=−%1/m1 0 0 0 0

1 0 −1 0 00 0 −%2/m2 0 00 0 1 0 −10 0 0 0 −%3/m3

v1(t)d12(t)v2(t)d23(t)v3(t)

+b1/m1 0 0

0 0 00 b2/m2 00 0 00 0 b3/m3

u1(t)u2(t)u3(t)

+w1(t)w2(t)w3(t)w4(t)w5(t)

%1 = %2 = %3 = 0.1 and b1 = b2 = b3 = 1.0

1

GK

2

3




d12(t)v2(t)

d23(t)v3(t)

=−%1/m1 0 0 0 0

1 0 −1 0 00 0 −%2/m2 0 00 0 1 0 −10 0 0 0 −%3/m3

v1(t)d12(t)v2(t)d23(t)v3(t)

+b1/m1 0 0

0 0 00 b2/m2 00 0 00 0 b3/m3

u1(t)u2(t)u3(t)

+w1(t)w2(t)w3(t)w4(t)w5(t)

%1 = %2 = %3 = 0.1 and b1 = b2 = b3 = 1.0

1

GK

2

3




d12(t)v2(t)

d23(t)v3(t)

=−%1/m1 0 0 0 0

1 0 −1 0 00 0 −%2/m2 0 00 0 1 0 −10 0 0 0 −%3/m3

v1(t)d12(t)v2(t)d23(t)v3(t)

+b1/m1 0 0

0 0 00 b2/m2 00 0 00 0 b3/m3

u1(t)u2(t)u3(t)

+w1(t)w2(t)w3(t)w4(t)w5(t)

z(t) =

[d12(t) d23(t) u1(t) u2(t) u3(t)

]>• Find a saddle point of J(Γ, α) = ‖Tzw (s; Γ, α)‖∞ when α = [m1m2m3]> ∈ [0.5, 1.0]3

and Γ belongs to the set of polynomials of αi, i = 1, 2, 3, up to the second order.




d12(t)v2(t)

d23(t)v3(t)

=−%1/m1 0 0 0 0

1 0 −1 0 00 0 −%2/m2 0 00 0 1 0 −10 0 0 0 −%3/m3

v1(t)d12(t)v2(t)d23(t)v3(t)

+b1/m1 0 0

0 0 00 b2/m2 00 0 00 0 b3/m3

u1(t)u2(t)u3(t)

+w1(t)w2(t)w3(t)w4(t)w5(t)

z(t) =

[d12(t) d23(t) u1(t) u2(t) u3(t)

]>• Find a saddle point of J(Γ, α) = ‖Tzw (s; Γ, α)‖∞ when α = [m1m2m3]> ∈ [0.5, 1.0]3

and Γ belongs to the set of polynomials of αi, i = 1, 2, 3, up to the second order.

1

GK

2

3



1

GC

2

3

Control Design with Local Model Information

maxα∈A∥∥Tzw (s; Γlocal, α

)∥∥∞ = 4.7905

1

GC

2

3

Control Design with Limited Model Information

maxα∈A∥∥Tzw (s; Γlimited, α

)∥∥∞ = 3.5533

1

GC

2

3

Control Design with Full Model Information

maxα∈A∥∥Tzw (s; Γfull, α

)∥∥∞ = 3.3596



25.8%

1

GC

2

3



)∥∥∞ = 4.7905

1

GC

2

3



)∥∥∞ = 3.5533

1

GC

2

3



)∥∥∞ = 3.3596



25.8%

5.4%

1

GC

2

3



)∥∥∞ = 4.7905

1

GC

2

3



)∥∥∞ = 3.5533

1

GC

2

3



)∥∥∞ = 3.3596


Conclusions and Future Work

Conclusions• Optimal control design for continuous-time linear parameter-

dependent systems under limited model information and partial statemeasurements;

• Expanded the control design strategy using basis functions;

• A numerical optimization method using the subgradients of theclosed-loop performance measure.

Future Work• Finding a good basis functions for expanding the control design

strategies.

For more information, please visit http://people.kth.se/~farakhi/


http://people.kth.se/~farakhi/

Conclusions and Future Work

Conclusions• Optimal control design for continuous-time linear parameter-

dependent systems under limited model information and partial statemeasurements;

• Expanded the control design strategy using basis functions;

• A numerical optimization method using the subgradients of theclosed-loop performance measure.

Future Work• Finding a good basis functions for expanding the control design

strategies.

For more information, please visit http://people.kth.se/~farakhi/


http://people.kth.se/~farakhi/

Documents

Optimal H Control Design under Model Information ...people.eng.unimelb.edu.au/ffarokhi/files/old_files/CDC-2013-Hinf.pdf · Optimal H 1Control Design under Model Information Limitations