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Structured adaptive control,or how to solve LMIs with Simulink
Alexandru - Razvan LUZIDimitri PEAUCELLE
IEIIT-CNR Torino, october 2012
1/21
Introduction
Introduction
n Direct adaptive control:Adaptation of control gains done directly based on measurements.
s 6= Indirect adaptive control:Estimator of model parameters + scheduled control gain
n Feedback-loop stabilizing gains, MRAC not considered
n Lyapunov based stability proofs, not gradient approximation ‘MIT rule’
n Framework initiated by V.A. Yakubovich in the late 1960’s
l Contributions: new adaptive control law with asymptotic structure+ may solve LMIs
2/21
Plan
Plan
1 Passivity-based adaptive control
2 LMIs are strict-passifiable systems
3 Structured adaptive control
4 Numerical Example
3/21
Passivity-based adaptive control
Passivity-based adaptive control of LTI systems
Theorem
The following two conditions are equivalent:Ê There exists a static control u(t) = Fy(t) + w(t) for the system
x(t) = Ax(t) + Bu(t) , y(t) = Cx(t) , z(t) = y(t)
that makes the closed-loop strictly passive (with respect to w/z).
Ë For all Γ � 0 the following adaptive control
u(t) = K (t)y(t) + w(t) , K (t) = −y(t)yT (t)Γ
makes the closed-loop globally strictly-passive.
4/21
Passivity-based adaptive control
l Strict-passivity includes asymptotic stability of x = 0l Adaptive control converges to K (∞): strictly-passifying static gain
s Theorem for square systems - extensions exist for non-square systemss Not all stabilizable systems are strictly-passifiable
- modified adaptive laws exist for stabilizable systems
l Condition Ê also reads in terms of matrix inequalities as
∃Q � 0 : (A + BFC)TQ + Q(A + BFC) ≺ 0 , QB = CT
It happens to be an LMI constraint!
∃Q � 0 : ATQ + QA + CT (FT + F )C ≺ 0 , QB = CT
n Finding F solution to the LMI is equivalent to simulating the systemwith the adaptive control law and taking F = K (∞).
5/21
LMIs are strict-passifiable systems
All LMIs define strict-passifiable systems
l Let us consider an example: LMIs for an upper bound on the H∞ norm[ATP + PA + CTC PB + CTD
BTP + DTC −γ21 + DTD
]≺ 0 , P = PT � 0.
s All LMI constraints can be gathered in one ATP + PA + CTC PB + CTD 0BTP + DTC −γ21 + DTD 0
0 0 −P
≺ 0 , P = PT
6/21
LMIs are strict-passifiable systems
All LMIs define strict-passifiable systems
l Let us consider an example: LMIs for an upper bound on the H∞ norm[ATP + PA + CTC PB + CTD
BTP + DTC −γ21 + DTD
]≺ 0 , P = PT � 0.
s All LMI constraints can be gathered in one ATP + PA + CTC PB + CTD 0BTP + DTC −γ21 + DTD 0
0 0 −P
≺ 0 , P = PT
6/21
LMIs are strict-passifiable systems
s All LMI constraints can be gathered in one ATP + PA + CTC PB + CTD 0BTP + DTC −γ21 + DTD 0
0 0 −P
≺ 0 , P = PT
s Can be decomposed in a sum with elementary matrix variables
A + BTP
0 P 0P 0 00 0 −P
BP + BTγ (−γ21)Bγ ≺ 0 , P = PT
A =
CTC CTD 0DTC DTD 0
0 0 0
, BP =
A B 01 0 00 0 1
, Bγ =[
0 1 0]
7/21
LMIs are strict-passifiable systems
s All LMI constraints can be gathered in one ATP + PA + CTC PB + CTD 0BTP + DTC −γ21 + DTD 0
0 0 −P
≺ 0 , P = PT
s Can be decomposed in a sum with elementary matrix variables
A + BTP
0 P 0P 0 00 0 −P
BP + BTγ (−γ21)Bγ ≺ 0 , P = PT
A =
CTC CTD 0DTC DTD 0
0 0 0
, BP =
A B 01 0 00 0 1
, Bγ =[
0 1 0]
7/21
LMIs are strict-passifiable systems
s Can be decomposed in a sum with elementary matrix variables
A + BTP
0 P 0P 0 00 0 −P
BP + BTγ (−γ21)Bγ ≺ 0 , P = PT
s Or equivalently when gathering all variables in a block-diagonal matrix
A + BTFB ≺ 0 , B =[
BP Bγ
]with the structural equality constraints
F =
[FP 00 Fγ
], FP =
0 P 0P 0 00 0 −P
, P = PT , Fγ = −γ21
l The constraint A + BTFB ≺ 0 holds iff(A,B,C = BT ) is strictly-passifiable by F .
n LMI converted to strict-passification problem, with equality constraints.
8/21
LMIs are strict-passifiable systems
s Can be decomposed in a sum with elementary matrix variables
A + BTP
0 P 0P 0 00 0 −P
BP + BTγ (−γ21)Bγ ≺ 0 , P = PT
s Or equivalently when gathering all variables in a block-diagonal matrix
A + BTFB ≺ 0 , B =[
BP Bγ
]with the structural equality constraints
F =
[FP 00 Fγ
], FP =
0 P 0P 0 00 0 −P
, P = PT , Fγ = −γ21
l The constraint A + BTFB ≺ 0 holds iff(A,B,C = BT ) is strictly-passifiable by F .
n LMI converted to strict-passification problem, with equality constraints.
8/21
LMIs are strict-passifiable systems
s Can be decomposed in a sum with elementary matrix variables
A + BTP
0 P 0P 0 00 0 −P
BP + BTγ (−γ21)Bγ ≺ 0 , P = PT
s Or equivalently when gathering all variables in a block-diagonal matrix
A + BTFB ≺ 0 , B =[
BP Bγ
]with the structural equality constraints
F =
[FP 00 Fγ
], FP =
0 P 0P 0 00 0 −P
, P = PT , Fγ = −γ21
l The constraint A + BTFB ≺ 0 holds iff(A,B,C = BT ) is strictly-passifiable by F .
n LMI converted to strict-passification problem, with equality constraints.
8/21
LMIs are strict-passifiable systems
s Can be decomposed in a sum with elementary matrix variables
A + BTP
0 P 0P 0 00 0 −P
BP + BTγ (−γ21)Bγ ≺ 0 , P = PT
s Or equivalently when gathering all variables in a block-diagonal matrix
A + BTFB ≺ 0 , B =[
BP Bγ
]with the structural equality constraints
F =
[FP 00 Fγ
], FP =
0 P 0P 0 00 0 −P
, P = PT , Fγ = −γ21
l The constraint A + BTFB ≺ 0 holds iff(A,B,C = BT ) is strictly-passifiable by F .
n LMI converted to strict-passification problem, with equality constraints.
8/21
LMIs are strict-passifiable systems
n Procedure applies to any LMI: concludes with search of passifying
F =
F1 0. . .
0 FN
for a (symmetric) system (A,B,C = BT )
with additional structural equality constraints that can be compacted in
vec(Fi ) = Sixi ⇔ Uivec(Fi ) = 0
where vec(Fi ) is the vector composed of stacked columns of Fi ,xi are vectors of independent scalar decision variables and Ui = S⊥i .
l When starting from the canonical representation L0 +∑
j xjLj ≺ 0,then the structural constraints are all of the type
Fj =
[xj1rj1 0
0 −xj1rj2
]and rj1 + rj2 can be very large and Uj is huge.
l F and Ui s expected to be smaller when matrix representation.
9/21
LMIs are strict-passifiable systems
n Procedure applies to any LMI: concludes with search of passifying
F =
F1 0. . .
0 FN
for a (symmetric) system (A,B,C = BT )
with additional structural equality constraints that can be compacted in
vec(Fi ) = Sixi ⇔ Uivec(Fi ) = 0
where vec(Fi ) is the vector composed of stacked columns of Fi ,xi are vectors of independent scalar decision variables and Ui = S⊥i .
l When starting from the canonical representation L0 +∑
j xjLj ≺ 0,then the structural constraints are all of the type
Fj =
[xj1rj1 0
0 −xj1rj2
]and rj1 + rj2 can be very large and Uj is huge.
l F and Ui s expected to be smaller when matrix representation. 9/21
Structured adaptive control
Block-diagonal adaptive control with asymptotic structure
Theorem
The following two conditions are equivalent:Ê There exists a decentralized static control ui (t) = Fiyi (t) + wi (t)satisfying the structural constraints Uivec(Fi ) = 0 for the system
x(t) = Ax(t) +∑
Biui (t) , yi (t) = Cix(t) , z(t) = y(t)
that makes the closed-loop strictly passive (with respect to w/z).
Ë For all Γi � 0, αi > 0 the following adaptive control
ui (t) = Ki (t)yi (t) + wi (t) ,
Ki (t) = −yi (t)yTi (t)Γi − αi ·mat
(UT
i Ui · vec(Ki (t)))
Γi
makes the closed-loop globally strictly-passive.
‘mat’ is the function such that mat(vec(F )) = F .10/21
Structured adaptive control
Proof of Ê ⇒ Ë
l Ê reads as
∃F ,∃Q � 0 :(A + BFC)TQ + Q(A + BFC) < 0, QB = CT ,F = diag
[· · · Fi · · ·
], Ui · vec(Fi ) = 0
(1)
l Let the Lyapunov function for the non-linear system (with adaptive law)
V (x ,K ) =1
2
(xTQx +
∑i
Tr(
(Ki − Fi )Γ−1(Ki − Fi )T))
l After manipulations and using QB = CT , Ui · vec(Fi ) = 0, we get:
V (x ,K ) = xT (A + BFC)TQx + wT z −∑
i
αi (Ui · vec(Ki ))T (Ui · vec(Ki )).
11/21
Structured adaptive control
Proof of Ê ⇒ Ë
l Ê reads as
∃F ,∃Q � 0 :(A + BFC)TQ + Q(A + BFC) < 0, QB = CT ,F = diag
[· · · Fi · · ·
], Ui · vec(Fi ) = 0
(1)
l Let the Lyapunov function for the non-linear system (with adaptive law)
V (x ,K ) =1
2
(xTQx +
∑i
Tr(
(Ki − Fi )Γ−1(Ki − Fi )T))
l After manipulations and using QB = CT , Ui · vec(Fi ) = 0, we get:
V (x ,K ) = xT (A + BFC)TQx + wT z −∑
i
αi (Ui · vec(Ki ))T (Ui · vec(Ki )).
11/21
Structured adaptive control
Proof of Ê ⇒ Ë
l Ê reads as
∃F ,∃Q � 0 :(A + BFC)TQ + Q(A + BFC) < 0, QB = CT ,F = diag
[· · · Fi · · ·
], Ui · vec(Fi ) = 0
(1)
l Let the Lyapunov function for the non-linear system (with adaptive law)
V (x ,K ) =1
2
(xTQx +
∑i
Tr(
(Ki − Fi )Γ−1(Ki − Fi )T))
l After manipulations and using QB = CT , Ui · vec(Fi ) = 0, we get:
V (x ,K ) = xT (A + BFC)TQx + wT z −∑
i
αi (Ui · vec(Ki ))T (Ui · vec(Ki )).
11/21
Structured adaptive control
Proof of Ê ⇒ Ë (continued)
V (x ,K ) = xT (A + BFC)TQx + wT z −∑
i
αi (Ui · vec(Ki ))T (Ui · vec(Ki )).
s First term is strictly negative due to (1), until x = 0,s Last term is strictly negative, until Ui · vec(Ki ) = 0
n If no perturbations (w = 0) the system converges to the attractor
A = {(x ,K ) : x = 0 , Ui · vec(Ki ) = 0}
n On the attractor Ki = 0: the gains Ki (∞) are constant.
12/21
Structured adaptive control
Proof of Ê ⇒ Ë (continued)
V (x ,K ) = xT (A + BFC)TQx + wT z −∑
i
αi (Ui · vec(Ki ))T (Ui · vec(Ki )).
s First term is strictly negative due to (1), until x = 0,s Last term is strictly negative, until Ui · vec(Ki ) = 0
n If no perturbations (w = 0) the system converges to the attractor
A = {(x ,K ) : x = 0 , Ui · vec(Ki ) = 0}
n On the attractor Ki = 0: the gains Ki (∞) are constant.
12/21
Structured adaptive control
Proof of Ê ⇒ Ë (continued)
V (x ,K ) = xT (A + BFC)TQx + wT z −∑
i
αi (Ui · vec(Ki ))T (Ui · vec(Ki )).
n For initial conditions at equilibrium and nonzero perturbations
0 ≤ V (x(t),K (t)) =
∫ t
0V (x ,K )dτ <
∫ t
0wT z dτ
⇒ the system is strictly passive.
13/21
Structured adaptive control
Proof of Ê ⇒ Ë (continued)
V (x ,K ) = xT (A + BFC)TQx + wT z −∑
i
αi (Ui · vec(Ki ))T (Ui · vec(Ki )).
n For initial conditions at equilibrium and nonzero perturbations
0 ≤ V (x(t),K (t)) =
∫ t
0V (x ,K )dτ <
∫ t
0wT z dτ
⇒ the system is strictly passive.
13/21
Structured adaptive control
Proof of Ë ⇒ Ê
l The system with adaptive control is globally asymptotically stable,it converges to a asymptotically stable equilibrium:F i = Ki (∞) are stabilizing gains.
l Same reasoning holds for passivity.
14/21
Structured adaptive control
Summary
n All LMI problems are equivalent to static output-feedbackstrict-passification problems with structure constraints:
- gain F is block-diagonal- sub-blocks should satisfy Uivec(Fi ) = 0.
n If a structured strict-passification problem admits solutions,the block-diagonal adaptive law with asymptotic structure will converge toone of these.
l The LMIs can be solved by simulating the adaptive controlled systems.
s If the system converges Ki (∞) contain solutions of the LMIs.
s If does not converges the LMIs are infeasible.
15/21
Numerical Example
Numerical example
l Consider the transfer function:
G (s) =s2 + s + 1
s2 + s + 2
l Problem: compute the H∞ norm (or at least an upper bound).
s In Matlab: norm(G, Inf, 1e-4) = 1.3251
s LMI problem converted to adaptive passification
Ki = −yiyTi Γi − αi ·mat
(UT
i Ui · vec(Ki ))
Γi , y1 ∈ R6 , y2 ∈ R
with structural asymptotic constraints :
F1 =
0 P 0PT 0 00 0 −P
, P = PT ∈ R2×2 , F2 = −γ21 = −γ2.
16/21
Numerical Example
Numerical example
l Consider the transfer function:
G (s) =s2 + s + 1
s2 + s + 2
l Problem: compute the H∞ norm (or at least an upper bound).
s In Matlab: norm(G, Inf, 1e-4) = 1.3251
s LMI problem converted to adaptive passification
Ki = −yiyTi Γi − αi ·mat
(UT
i Ui · vec(Ki ))
Γi , y1 ∈ R6 , y2 ∈ R
with structural asymptotic constraints :
F1 =
0 P 0PT 0 00 0 −P
, P = PT ∈ R2×2 , F2 = −γ21 = −γ2.
16/21
Numerical Example
l Parameters for simulating the adaptive law (simulation in Simulink)s Initial conditions x = (1 . . . 1)T and Ki = 0s Γ1 = 1000 · 1, Γ2 = 10, α1 = α2 = 1
l Convergence to zero of the ‘outputs’ yi
17/21
Numerical Example
l Parameters for simulating the adaptive law (simulation in Simulink)s Initial conditions x = (1 . . . 1)T and Ki = 0s Γ1 = 1000 · 1, Γ2 = 10, α1 = α2 = 1l Convergence to zero of the ‘outputs’ yi
17/21
Numerical Example
s Convergence to structured values of the adapted gains Ki
K1(∞) =
266666640 0 4.6330 1.0671 0 00 0 1.0671 10.7960 0 0
4.6330 1.0671 0 0 0 01.0671 10.7960 0 0 0 0
0 0 0 0 −4.6330 −1.06710 0 0 0 −1.0671 −10.7960
37777775K2(∞) = −7.1307
18/21
Numerical Example
s Evolution of the (1 : 2, 3 : 4) elements of K1 that converge to P
s Solution of the LMIs
P =
[4.6330 1.06711.0671 10.7960
], γ = 2.6703 ≥ 1.3251 = γopt
19/21
Numerical Example
l Test for feasible / unfeasible casess Only K1 is adapated, γ is slowly linearly modified
s Unstable behavior when γ < 1.3251 = γopt .20/21
Conclusions
Conclusions et perspectives
n LMI feasibility problems can be solved by simulating systems
s Need for a parser to convert LMIs to adaptive control problems Simulation time is large - what is the best implementation ?s Is simulation time polynomial w.r.t. size of problem ?
n What about LMI optimization problems ?
s Decreasing parameters until system becomes unstable ?s Minimizing gap with dual LMI problem (it works).s Other ?
n Solving time-varying LMI problems ?
21/21