Simple conditions for L 2 stability and stabilization of networked - - PowerPoint PPT Presentation
Introduction Problem statement Stability Analysis Stabilization Conclusion Simple conditions for L 2 stability and stabilization of networked control systems Y. Ariba, C. Briat, K.H. Johansson KTH, Stockholm, Sweden IFAC World Congres 2011,
Introduction Problem statement Stability Analysis Stabilization Conclusion
KTH, Stockholm, Sweden IFAC World Congres 2011, Milano, Italy
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Introduction Problem statement Stability Analysis Stabilization Conclusion
◮ Introduction ◮ Problem statement ◮ Stability analysis ◮ Control ◮ Conclusion and Future Works
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Introduction Problem statement Stability Analysis Stabilization Conclusion
Controller
Network
System Hold
x(t) tk
xk
^
u(t) v(t) y(t)
Exogenous input performance
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Introduction Problem statement Stability Analysis Stabilization Conclusion
◮ Time-delay systems [Yu et al.], [Fridman et al.] ◮ Impulsive systems [Naghshtabrizi et al.], [Seuret] ◮ Sampled-data systems [Mirkin] ◮ Robust techniques [Fujioka], [Oishi et al.] ◮ Functional-based approaches [Seuret]
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Introduction Problem statement Stability Analysis Stabilization Conclusion
◮ Process model
˙ x(t) = Ax(t) + Bu(t) + Ev(t) y(t) = Cx(t) + Du(t) + Fv(t) x(0) = x0 (1)
◮ Control-law model
u(t) = Kx(tk) t ∈ [tk, tk+1) tk+1 − tk ≤ (1 + m)Tmax + τk+1 τk ∈ [0, τmax] (2)
◮ tk: arrival instants of a new control input ◮ τk = τ(tk), k ∈ N ◮ m is the number of consecutive dropouts ◮ Varying sampling period → Actual (varying) sampling period+data loss+varying
propagation delays
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Introduction Problem statement Stability Analysis Stabilization Conclusion
◮ Find a sampled-data state-feedback control law such that the closed-loop system
is asymptotically (exponentially) stable and
L2 disturbance attenuation constraints; or
constraint. SMATI := {(m, τ, T) ∈ N × R+ × R++ : (1 + m)T + τ ≤ MATI} . (3)
◮ m: number of consecutive dropouts ◮ T: sampling period ◮ τ: propagation delay
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Introduction Problem statement Stability Analysis Stabilization Conclusion
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Introduction Problem statement Stability Analysis Stabilization Conclusion
Theorem The interconnected system above is well-posed if there exists a symmetric matrix Θ satisfying the conditions E −A T
⊥ Θ
E −A
⊥ ≻ 0
(4) and
PT ∇
PT ∇
(5) for all u ∈ L2e and all T > 0.
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Introduction Problem statement Stability Analysis Stabilization Conclusion
x(t) δ(t) v(t)
= I1n ∆sh1n ∆γ
˙ x(t) ˙ x(t) y(t)
, (6) 1 −1 1 1
˙ x(t) ˙ x(t) y(t)
= A + BK −BK E C + DK −DK F
x(t) δ(t) v(t)
(7)
◮ I: integral operator, ◮ ∆sh : θ →
t
tk
θ(s)ds, t ≤ tk+1 ⇒ δ(t) = x(t) − x(tk).
◮ ∆γ: virtual operator characterizing the L2 gain of the transfer v → y.
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Introduction Problem statement Stability Analysis Stabilization Conclusion
Lemma The integration operator I is characterized by the IQC:
I1n
−P 1n I1n
for all x ∈ Ln
2e and for any matrix P ∈ Sn ++.
◮ Lyapunov condition for stability ◮ Frequency domain I → s−1
s−11n ∗ −P −P 1n s−11n
Pre- and post-multiply by s∗X(s)∗ and sX(s) −(s + s∗)X(s)∗PX(s) 0, ∀ℜ[s] ≥ 0 (8) Characterization of all ℜ[s] ≥ 0: pick any P = P T ≻ 0
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Introduction Problem statement Stability Analysis Stabilization Conclusion
Lemma The operator ∆sh can be characterized by the IQC:
∆sh1n
π2 µ2S1 −S2 −S2 S1 1n ∆sh1n
for all x ∈ Ln
2e and for any matrices S1, S2 ∈ Sn ++.
◮ S1: Bound on the L2-gain of 2µ/π, tk+1 − tk ≤ µ, k ∈ N [Mirkin] ◮ S2: Passivity of ∆sh [Fujioka]
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Introduction Problem statement Stability Analysis Stabilization Conclusion
Lemma The operator ∆γ, which has an L2-induced norm equal to γ−1, is characterized by the IQC:
∆γ
1r 1q ∆γ
for all x ∈ Lq
2e.
◮ Upper bound γ on L2-gain of v → y
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Introduction Problem statement Stability Analysis Stabilization Conclusion
Theorem The NCS system is asymptotically stable for all (m, τ, T) ∈ Sµ if there exist matrices P, S1, S2 ∈ Sn
++ and a scalar η > 0 such that the LMI
E −A T
⊥ Θ
E −A
⊥ ≺ 0
(9) holds where E, A are defined in (7) and Θ = −P − 4 π2 µ2S1 −S2 −η1q ∗ S1 1r . (10) Moreover, the closed-loop system satisfies ||y||L2 ≤
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Introduction Problem statement Stability Analysis Stabilization Conclusion
˙ x(t) = 1 1
1 −1.006 −1.006 x(tk). (11)
◮ Maximal constant sampling period: 5.8117.
MATI
for n = 2 [Yu, 04] unfeasible 4 n(n+1)
2
12 [Yue, 04] 0.970 2 n(n+1)
2
+ 6n2 30 [Tan, 08] 0.995 4 n(n+1)
2
+ 16n2 76 [Naghshtabrizi, 06](without delay) 1.272 7 n(n+1)
2
+ 16n2 85 Proposed result 1.561 3 n(n+1)
2
+ 1 10
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Introduction Problem statement Stability Analysis Stabilization Conclusion
Theorem There exists a matrix K ∈ Rm×n such that the NCS is asymptotically stable for all (m, τ, T) ∈ Sµ if there exist matrices P, S1 ∈ Sn
++, X ∈ Rn×n, U ∈ Rm×n and a
scalar γ > 0 such that the LMI −(X + XT ) P + A′
cl
−BU E X µ π 2 S1 ⋆ −P C′
cl T
⋆ ⋆ −S1 −(DU)T ⋆ ⋆ ⋆ −γI F T ⋆ ⋆ ⋆ ⋆ −γI ⋆ ⋆ ⋆ ⋆ ⋆ −P −µ π 2 S1 ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ −S1 ≺ 0 (12) holds with A′
cl = AX + BU and C′ cl = CX + DU. Furthermore, the closed-loop
system controlled with gain K = UX−1 satisfies ||y||L2 ≤ γ||v||L2.
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Introduction Problem statement Stability Analysis Stabilization Conclusion
◮ Let us consider the open-loop system
˙ x(t) = −0.8 −0.01 1 0.1
0.4 0.1
(13)
◮ [Yu, 04]: system stabilizable for µ ≤ 0.6011. ◮ Proposed result: system stabilizable for µ ≤ 3.64826 with the controller gain
K = −0.3482 −0.3097 .
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Introduction Problem statement Stability Analysis Stabilization Conclusion
◮ Approach based on well-posedness and IQCs ◮ LMI results for both stability and stabilization ◮ Tradeoff: L2-gain minimization vs. MATI maximization ◮ Low numerical complexity ◮ Can be extended to robust stability and stabilization
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