Dynamic equations on time-scale: application to stability analysis - - PowerPoint PPT Presentation
Introduction Problem Statement Stability analysis Stabilization Dynamic equations on time-scale: application to stability analysis and stabilization of aperiodic sampled-data systems C. Briat and U. T. Jnsson KTH, Stockholm, Sweden IFAC
Introduction Problem Statement Stability analysis Stabilization
KTH, Stockholm, Sweden IFAC World Congres 2011, Milano, Italy
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Introduction Problem Statement Stability analysis Stabilization
◮ Introduction ◮ Problem statement and Preliminaries ◮ Stability analysis ◮ Stabilization ◮ Conclusion and Future Works
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Introduction Problem Statement Stability analysis Stabilization
◮ Discrete-time systems with varying sampling period ◮ Several frameworks ◮ Time-delay systems [Yu et al.], [Fridman et al.] ◮ Impulsive systems [Naghshtabrizi et al.], [Seuret] ◮ Sampled-data systems [Mirkin] ◮ Robust techniques [Fujioka], [Hetel et al.], [Oishi et al.], [Ariba et al.] ◮ Functional-based approaches [Seuret]
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Introduction Problem Statement Stability analysis Stabilization
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Introduction Problem Statement Stability analysis Stabilization
◮ Continuous-time LTI system
˙ x(t) = Ax(t) + Bu(t) x(0) = x0 (1) with state x and control input u.
◮ Sampled-data control law
u(t) = Kx(tk), t ∈ [tk, tk+1) (2) where Tk := tk+1 − tk ≤ T, k ∈ N.
◮ Stability analysis problem: given K, find the set T of admissible T > 0 for which
stability still holds.
◮ Find controller gain K maximizing the maximal sampling period.
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Introduction Problem Statement Stability analysis Stabilization
◮ Unification/generalization of continuous-time and discrete-time systems ◮ Examples: T = R+, T = Z+, T = {0} ∪ {1/k}k∈N, T =
[t2k, t2k+1]
◮ Forward jump operator: σ(t) = {inf s ∈ T : t < s} ◮ Graininess: µ(t) = σ(t) − t (distance) ◮ Dynamical system on time-scale:
x∆(t) = Ax(t) + Bu(t) x(0) = x0 (3)
◮ ∆ operator [Goodwin]:
f∆(t) := lim
s→t,s∈T
f(t) − f(s) t − s if µ(t) = 0 f(σ(t)) − f(t) µ(t) if µ(t) > 0.
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Introduction Problem Statement Stability analysis Stabilization
◮ Linear systems → V (x) = xT Px ◮ Stability condition
AT P + PA + µAT PA ≺ 0, P = P T ≻ 0 (4)
◮ µ fixed: equivalent to a DT criterion ◮ µ → 0: equivalent to a CT criterion ◮ Spectrum condition: λ(A) ⊂ D(−1/µ, 1/µ) ◮ µ → 0: D(−1/µ, 1/µ) → C− ◮ µ = µ0 = 0: D(1/µ0, 1/µ0) analogous to the unit disc.
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Introduction Problem Statement Stability analysis Stabilization
◮ DT system:
x(t) =
t
tk
eA(t−s)dsBK(tk)
(5)
◮ System on TS:
z∆(tk) = A∆(µ(tk))z(tk) (6) where the new state z coincide with x at sampling instants and
A∆(µ(tk)) = µ(tk)−1 eAµ(tk) + Φ(µ(tk))BK(tk) − I
= tk eA(µ(tk)−s)ds (7)
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Introduction Problem Statement Stability analysis Stabilization
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Introduction Problem Statement Stability analysis Stabilization
Theorem The dynamical system z∆(tk) = A∆(tk)z(tk), z(t0) = z0, (t0, tk) ∈ T2, tk ≥ t0 is robustly exponentially stable for µ(tk) ∈ µ if the following statements hold:
and µ(tk) ∈ µ.
++ verifying θ1I P(tk) θ2I for some
0 < θ1 < θ2 < +∞ and β ∈ (0, 1/ sup{µ}) such that Mµ(tk)(P(σ(tk)), A∆(tk), P ∆(tk) + βP(tk)) 0 (8) holds for all µ(tk) ∈ µ and all tk ∈ T.
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Introduction Problem Statement Stability analysis Stabilization
Theorem The dynamical system z∆(tk) = A∆(µ(tk))z(tk), z(t0) = z0, (t0, tk) ∈ T2, tk ≥ t0 is robustly exponentially stable for µ(tk) ∈ µ, inf{µ} > 0, if the following statements hold:
++ such that
Mµ(P(µn), A∆(µc), S(µc, µn)) ≺ 0 (9) holds for all (µn, µc) ∈ µ2 and where S(µc, µn)) = µ−1
c
(P(µn) − P(µc)).
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Introduction Problem Statement Stability analysis Stabilization
◮ Scalar system ([Mirkin], [Fridman], [Fujioka])
˙ x(t) = −2x(t) + x(tk). (10)
◮ TS formalism
A∆(µ) = 3 2µ
(11)
◮ Lyapunov condition
3 µ
1 + 3 4
◮ True for µ > 0 ◮ When µ → 0, the LHS tends to -6.
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Introduction Problem Statement Stability analysis Stabilization
Let us consider the system A = 1 −0.1
BK =
−1.15
Ref. Maximal varying sampling period [Fridman,04] 0.8696 [Yue,05] 0.8871 [Ariba,07 1.009 [Naghshtabrizi,08] 1.1137 [Mirkin,07] 1.3659 [Seuret,09b] 1.6894 [Oishi,09] 1.7294 Proposed result 1.72941 Theoretical 1.7294194 (constant case)
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Introduction Problem Statement Stability analysis Stabilization
A =
−2 0.1
BK = 1
◮ Constant sampling period ◮ P (µ) = P0 → µ = [0.5004, 1.9203] ◮ P (µ) = P0 + P1µ → µ = [0.2013, 2.0204] ◮ Other disjoint intervals
µ ∈ {[2.4706, 3.6963], [5.4307, 6.3447], [7.0277, 7.7249], [10.3916, 10.7559], [11.4973, 11.7179]} .
◮ Aperiodic case:
µ ∈ {[0.2187, 1.0031], [0.500, 1.9256], [2.47, 3.6], [2.77, 3.6963], [5.4584, 5.8004], [5.8172, 6.3447], [7.0339, 7.5009], [7.5000, 7.7070]} . (13)
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Introduction Problem Statement Stability analysis Stabilization
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Introduction Problem Statement Stability analysis Stabilization
Theorem There exists a quadratically stabilizing switching sampling-period-dependent state-feedback control law if there exist X ∈ Sn
++ and a bounded continuous matrix function U : µ → Rn×m such that the LMI
Ξ11(µ) Ξ12(µ) ⋆ −µ−1X
(14)
holds for all µ ∈ µ with
Ξ12(µ) = µ−1[Ae(µ)X + Φ(µ)BU(µ)]T Ξ11(µ) = Ξ12(µ) + Ξ12(µ)T Ae(µ) = exp(Aµ) − I (15)
In such a case, the controller matrix is given by K(µ) = U(µ)X−1.
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Introduction Problem Statement Stability analysis Stabilization
Theorem There exists a robustly stabilizing sampling-period-dependent state-feedback control law if there exist a matrix Z ∈ Rn×n, bounded continuous matrix functions P : µ → Sn
++, U : µ → Rn×m and a sufficiently large positive scalar function
ǫ : µ2 → R++ such that the matrix inequality
Ξ11(µc, µn) Ξ12(µc, µn) Z ⋆ Ξ22(µc, µn) ⋆ ⋆ Ξ33(µc, µn) ≺ 0 (16)
holds for all µ ∈ µ, inf{µ} > 0, with S(µc, µn) = µ−1
c
(P(µn) − P(µc)) and
Ξ11(µc, µn) = −Z − ZT + µcP (µn) Ξ12(µc, µn) = µ−1
c
[Ae(µc)X + Φ(µc)BU(µc)] + P (µn) Ξ22(µc, µn) = −ǫ(µc, µn)P (µn) + S(µc, µn) Ξ33(µc, µn) = −P (µn)/ǫ(µc, µn) Ae(µc) = exp(Aµc) − I (17)
In such a case, the controller matrix is given by K(µc) = U(µc)Z−1.
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Introduction Problem Statement Stability analysis Stabilization
◮ System 1:
A =
−2 0.1
1
(18)
◮ System 2:
A = 7 4 5 11
1 1
(19)
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Introduction Problem Statement Stability analysis Stabilization
degree of K(µ) µ+ for System 1 µ+ for System 2 1.8938 0.1566 1 2.2072 0.3299 2 2.2121 0.5020 3 2.2170 0.6717 4 2.2182 0.8353 5 2.2206 0.9817 6 2.2206 1.0196
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Introduction Problem Statement Stability analysis Stabilization
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Introduction Problem Statement Stability analysis Stabilization
◮ Time-scale approach for stability analysis of sampled-data systems ◮ Stability analysis via sampling-period dependent Lyapunov Functions ◮ Stabilization via sampling-period dependent controllers ◮ Extension to dynamic output feedback possible ◮ Future works will be devoted to the theory of systems on time-scales
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Introduction Problem Statement Stability analysis Stabilization
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