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Lyapunov Function Constructions for Slowly Time-Varying Systems - - PowerPoint PPT Presentation
Lyapunov Function Constructions for Slowly Time-Varying Systems - - PowerPoint PPT Presentation
Lyapunov Function Constructions for Slowly Time-Varying Systems MICHAEL MALISOFF Department of Mathematics Louisiana State University Joint with Fr ed eric Mazenc, Projet MERE INRIA-INRA Stability Regular Session Paper FrA15.3 45th IEEE
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REVIEW of MODEL and LITERATURE
Goals: For large constants α > 0, prove input-to-state stability (ISS) for ˙ x = f(x, t, t/α) + g(x, t, t/α)u, x(t0) = xo (Σ) and construct explicit corresponding ISS Lyapunov functions. Literature: Uses exponential-like stability of ˙ x = f(x, t, τ) aka (Σfro) for all relevant values of the scalar τ to show stability for u ≡ 0 but does not lead to explicit Lyapunov functions for (Σ) (Peuteman-Aeyels, Solo).
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REVIEW of MODEL and LITERATURE
Goals: For large constants α > 0, prove input-to-state stability (ISS) for ˙ x = f(x, t, t/α) + g(x, t, t/α)u, x(t0) = xo (Σ) and construct explicit corresponding ISS Lyapunov functions. Literature: Uses exponential-like stability of ˙ x = f(x, t, τ) aka (Σfro) for all relevant values of the scalar τ to show stability for u ≡ 0 but does not lead to explicit Lyapunov functions for (Σ) (Peuteman-Aeyels, Solo). Our Contributions: We explicitly construct Lyapunov functions for (Σ) in terms of given Lyapunov functions for (Σfro) without assuming any exponential-like stability of (Σfro) and we allow τ to be a vector.
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REVIEW of MODEL and LITERATURE
Goals: For large constants α > 0, prove input-to-state stability (ISS) for ˙ x = f(x, t, t/α) + g(x, t, t/α)u, x(t0) = xo (Σ) and construct explicit corresponding ISS Lyapunov functions. Literature: Uses exponential-like stability of ˙ x = f(x, t, τ) aka (Σfro) for all relevant values of the scalar τ to show stability for u ≡ 0 but does not lead to explicit Lyapunov functions for (Σ) (Peuteman-Aeyels, Solo). Our Contributions: We explicitly construct Lyapunov functions for (Σ) in terms of given Lyapunov functions for (Σfro) without assuming any exponential-like stability of (Σfro) and we allow τ to be a vector. Significance: Lyapunov functions for (Σfro) are often readily available. Explicit Lyapunov functions and slowly time-varying models are important in control engineering e.g. control of friction, pendulums, etc.
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MAIN ASSUMPTION and MAIN THEOREM
We first assume our (sufficiently regular) system (Σ) has the form ˙ x = f(x, t, p(t/α)) (Σp) where p : R → Rd is bounded and ¯ p := sup{|p′(r)| : r ∈ R} < ∞.
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MAIN ASSUMPTION and MAIN THEOREM
We first assume our (sufficiently regular) system (Σ) has the form ˙ x = f(x, t, p(t/α)) (Σp) where p : R → Rd is bounded and ¯ p := sup{|p′(r)| : r ∈ R} < ∞. Assume: ∃δ1, δ2 ∈ K∞; constants ca, cb, T > 0; a continuous function q : Rd → R; and a C1 V : Rn × [0, ∞) × Rd → [0, ∞) s.t. ∀ x ∈ Rn, t ≥ 0, and τ ∈ R(p) := {p(t) : t ∈ R}: (i) |Vτ(x, t, τ)| ≤ caV (x, t, τ), (ii) δ1(|x|) ≤ V (x, t, τ) ≤ δ2(|x|), (iii) t
t−T q(p(s))ds ≥ cb, and
(iv) Vt(x, t, τ) + Vx(x, t, τ)f(x, t, τ) ≤ −q(τ)V (x, t, τ) all hold.
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MAIN ASSUMPTION and MAIN THEOREM
We first assume our (sufficiently regular) system (Σ) has the form ˙ x = f(x, t, p(t/α)) (Σp) where p : R → Rd is bounded and ¯ p := sup{|p′(r)| : r ∈ R} < ∞. Assume: ∃δ1, δ2 ∈ K∞; constants ca, cb, T > 0; a continuous function q : Rd → R; and a C1 V : Rn × [0, ∞) × Rd → [0, ∞) s.t. ∀ x ∈ Rn, t ≥ 0, and τ ∈ R(p) := {p(t) : t ∈ R}: (i) |Vτ(x, t, τ)| ≤ caV (x, t, τ), (ii) δ1(|x|) ≤ V (x, t, τ) ≤ δ2(|x|), (iii) t
t−T q(p(s))ds ≥ cb, and
(iv) Vt(x, t, τ) + Vx(x, t, τ)f(x, t, τ) ≤ −q(τ)V (x, t, τ) all hold. Theorem A: For each constant α > 2Tca¯ p/cb, (Σp) is UGAS and V ♯
α(x, t) := e
α T
- t
α t α −T
- t
α
s
q(p(l))dl ds V (x, t, p(t/α)) is a Lyapunov function for (Σp). [UGAS: |φ(t; to, xo)| ≤ β(|xo|, t − to)]
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SKETCH of PROOF of THEOREM A
Step 1: Set ˆ V (x, t) := V (x, t, p(t/α)). Along trajectories of (Σp), d dt ˆ V ≤
- −q(p(t/α)) + ca¯
p α
- ˆ
V (x, t). (⋆) Important: The term involving α in (⋆) vanishes if Vτ ≡ 0.
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SKETCH of PROOF of THEOREM A
Step 1: Set ˆ V (x, t) := V (x, t, p(t/α)). Along trajectories of (Σp), d dt ˆ V ≤
- −q(p(t/α)) + ca¯
p α
- ˆ
V (x, t). (⋆) Important: The term involving α in (⋆) vanishes if Vτ ≡ 0. Step 2: Substitute (⋆) into ˙ V ♯
α
= E(t, α)
- d
dt ˆ
V +
- q(p(t/α)) − 1
T
- t
α t α −T
q(p(l))dl
- ˆ
V
- ≤
E(t, α) ca¯ p α − cb T
- ˆ
V (x, t), where V ♯
α(x, t) = E(t, α) ˆ
V (x, t) and E(t, α) := e
α T
- t
α t α −T
- t
α
s
q(p(l))dl
- ds
.
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EXAMPLE 1: STABILITY for ALL PARAMETER VALUES
The assumptions of Theorem A hold for ˙ x = f(x, t, cos2(t/α)) :=
x √ 1+x2
- 1 − 90 cos2 t
α
- V (x, t, τ) ≡ ¯
V (x) := e
√ 1+x2 − e.
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EXAMPLE 1: STABILITY for ALL PARAMETER VALUES
The assumptions of Theorem A hold for ˙ x = f(x, t, cos2(t/α)) :=
x √ 1+x2
- 1 − 90 cos2 t
α
- V (x, t, τ) ≡ ¯
V (x) := e
√ 1+x2 − e.
This follows from the estimates ∇ ¯ V (x)f(x, t, τ) ≤
- 2e
√ 2
e−1 − 45τ
- ¯
V (x) t
t−π
- 45 cos2(s) − 2e
√ 2
e−1
- ds = π
- 45
2 − 2e
√ 2
e−1
- > 0
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EXAMPLE 1: STABILITY for ALL PARAMETER VALUES
The assumptions of Theorem A hold for ˙ x = f(x, t, cos2(t/α)) :=
x √ 1+x2
- 1 − 90 cos2 t
α
- V (x, t, τ) ≡ ¯
V (x) := e
√ 1+x2 − e.
This follows from the estimates ∇ ¯ V (x)f(x, t, τ) ≤
- 2e
√ 2
e−1 − 45τ
- ¯
V (x) t
t−π
- 45 cos2(s) − 2e
√ 2
e−1
- ds = π
- 45
2 − 2e
√ 2
e−1
- > 0
so for all α > 0 we get UGAS and the Lyapunov function V ♯
α(x, t)
:= e
α π
- t
α t α −π
- t
α
s
- 45 cos2(l) − 2e
√ 2
e − 1
- dl
- ds
¯ V (x) = e
45 α
4
- sin( 2t
α )+π− 4πe √ 2 45(e−1)
- [e
√ 1+x2 − e]
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EXAMPLE 2: MECHANICAL SYSTEM with FRICTION
Model: Dynamics for x1=mass position and x2=velocity: ˙ x1 = x2 ˙ x2 = −σ1(t/α)x2 − k(t)x1 + u −
- σ2(t/α) + σ3(t/α)e−β1µ(x2)
sat(x2) (MSF) σi are positive friction-related coefficients; β1 is a positive constant corresponding to Stribeck effect; µ ∈ PD is related to Stribeck effect; k is a positive time-varying spring stiffness-related coefficient; and sat(x2) = tanh(β2x2), where β2 is a large positive constant. α > 1.
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EXAMPLE 2: MECHANICAL SYSTEM with FRICTION
Model: Dynamics for x1=mass position and x2=velocity: ˙ x1 = x2 ˙ x2 = −σ1(t/α)x2 − k(t)x1 + u −
- σ2(t/α) + σ3(t/α)e−β1µ(x2)
sat(x2) (MSF) σi are positive friction-related coefficients; β1 is a positive constant corresponding to Stribeck effect; µ ∈ PD is related to Stribeck effect; k is a positive time-varying spring stiffness-related coefficient; and sat(x2) = tanh(β2x2), where β2 is a large positive constant. α > 1. Assumptions: (a) σi ∈ C1,valued in (0, 1], σ′
i bounded; (b) ∃ constants
cb, T > 0 such that t
t−T σ1(r)dr ≥ cb ∀t ≥ 0; (c) k ∈ C1, k′ bounded,
∃ko, ¯ k > 0 s.t. ko ≤ k(t) ≤ ¯ k and k′(t) ≤ 0 ∀t ≥ 0.
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EXAMPLE 2: MECHANICAL SYSTEM with FRICTION
Model: Dynamics for x1=mass position and x2=velocity: ˙ x1 = x2 ˙ x2 = −σ1(t/α)x2 − k(t)x1 + u −
- σ2(t/α) + σ3(t/α)e−β1µ(x2)
sat(x2) (MSF) σi are positive friction-related coefficients; β1 is a positive constant corresponding to Stribeck effect; µ ∈ PD is related to Stribeck effect; k is a positive time-varying spring stiffness-related coefficient; and sat(x2) = tanh(β2x2), where β2 is a large positive constant. α > 1. Assumptions: (a) σi ∈ C1,valued in (0, 1], σ′
i bounded; (b) ∃ constants
cb, T > 0 such that t
t−T σ1(r)dr ≥ cb ∀t ≥ 0; (c) k ∈ C1, k′ bounded,
∃ko, ¯ k > 0 s.t. ko ≤ k(t) ≤ ¯ k and k′(t) ≤ 0 ∀t ≥ 0. We apply our theorem to (MSF) with p(t) = (σ1(t), σ2(t), σ3(t)).
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EXAMPLE 2: MECHANICAL SYSTEM with FRICTION (cont’d)
The function V (x, t, τ) = A(k(t)x2
1 + x2 2) + τ1x1x2,
A = 1 + ko 2 + (1 + 2β2)2 ko satisfies the following for the corresponding frozen dynamics:
1 2(x2 1 + x2 2) ≤ V (x, t, τ) ≤ A2¯
k(|x1| + |x2|)2 ≤ 2A2¯ k|x|2 Vt(x, t, τ) + Vx(x, t, τ)f(x, t, τ) ≤ − τ1ko
4A2¯ kV (x, t, τ)
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EXAMPLE 2: MECHANICAL SYSTEM with FRICTION (cont’d)
The function V (x, t, τ) = A(k(t)x2
1 + x2 2) + τ1x1x2,
A = 1 + ko 2 + (1 + 2β2)2 ko satisfies the following for the corresponding frozen dynamics:
1 2(x2 1 + x2 2) ≤ V (x, t, τ) ≤ A2¯
k(|x1| + |x2|)2 ≤ 2A2¯ k|x|2 Vt(x, t, τ) + Vx(x, t, τ)f(x, t, τ) ≤ − τ1ko
4A2¯ kV (x, t, τ)
Corollary: There exists a constant αo > 0 such that for all α > αo, (MSF) is UGAS and admits the Lyapunov function Vα(t, x) := V (x, t, p(t/α)) e
α¯ b T
- t
α t α −T
- t
α
s
σ1(l)dl ds where V is above, ¯ b = ko/(4A2¯ k), and p(t) = (σ1(t), σ2(t), σ3(t)).
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INPUT-TO-STATE STABILITY (ISS)
Additional Assumptions: To show ISS for ˙ x = F(x, t, u, α) := f(x, t, p(t/α)) + g(x, t, p(t/α))u (Σu) for large constants α > 0 and f, g, and p as before, we also assume: For all t ≥ 0, α > 0, and x ∈ Rn, (v) |Vx(x, t, p(t/α))| ≤ ca
- δ1(|x|),
and (vi) |g(x, t, p(t/α))| ≤ ca{1 +
4
- δ1(|x|)}.
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INPUT-TO-STATE STABILITY (ISS)
Additional Assumptions: To show ISS for ˙ x = F(x, t, u, α) := f(x, t, p(t/α)) + g(x, t, p(t/α))u (Σu) for large constants α > 0 and f, g, and p as before, we also assume: For all t ≥ 0, α > 0, and x ∈ Rn, (v) |Vx(x, t, p(t/α))| ≤ ca
- δ1(|x|),
and (vi) |g(x, t, p(t/α))| ≤ ca{1 +
4
- δ1(|x|)}.
ISS: ∃ β ∈ KL, γ ∈ K∞ s.t. |φ(t; to, xo, u)| ≤ β(|xo|, t − to) + γ(|u|∞). ISS-CLF: ∃µ1, µ2, χ ∈ K∞, µ3 ∈ PD s.t. µ1(|x|) ≤ W(x, t) ≤ µ2(|x|) and |u| ≤ χ(|x|) ⇒ Wt(x, t) + Wx(x, t)F(x, t, u, α) ≤ −µ3(|x|).
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INPUT-TO-STATE STABILITY (ISS)
Additional Assumptions: To show ISS for ˙ x = F(x, t, u, α) := f(x, t, p(t/α)) + g(x, t, p(t/α))u (Σu) for large constants α > 0 and f, g, and p as before, we also assume: For all t ≥ 0, α > 0, and x ∈ Rn, (v) |Vx(x, t, p(t/α))| ≤ ca
- δ1(|x|),
and (vi) |g(x, t, p(t/α))| ≤ ca{1 +
4
- δ1(|x|)}.
ISS: ∃ β ∈ KL, γ ∈ K∞ s.t. |φ(t; to, xo, u)| ≤ β(|xo|, t − to) + γ(|u|∞). ISS-CLF: ∃µ1, µ2, χ ∈ K∞, µ3 ∈ PD s.t. µ1(|x|) ≤ W(x, t) ≤ µ2(|x|) and |u| ≤ χ(|x|) ⇒ Wt(x, t) + Wx(x, t)F(x, t, u, α) ≤ −µ3(|x|). Theorem B: ∀ constants α > 4Tca¯ p/cb, the dynamics (Σu) are ISS and V ♯
α(x, t) := e
α T
- t
α t α −T
- t
α
s
q(p(l))dl ds V (x, t, p(t/α)) is an ISS-CLF for (Σu).
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ACKNOWLEDGEMENTS
- The authors thank the referees and Patrick De Leenheer, Marcio de
Queiroz, and Eduardo Sontag for their helpful comments on earlier presentations of this work.
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ACKNOWLEDGEMENTS
- The authors thank the referees and Patrick De Leenheer, Marcio de
Queiroz, and Eduardo Sontag for their helpful comments on earlier presentations of this work.
- A journal version will appear in Mathematics of Control, Signals,
and Systems. The authors thank J.H. van Schuppen for the
- pportunity to publish their work in his esteemed journal.
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ACKNOWLEDGEMENTS
- The authors thank the referees and Patrick De Leenheer, Marcio de
Queiroz, and Eduardo Sontag for their helpful comments on earlier presentations of this work.
- A journal version will appear in Mathematics of Control, Signals,
and Systems. The authors thank J.H. van Schuppen for the
- pportunity to publish their work in his esteemed journal.
- Part of this work was done while F. Mazenc visited the Louisiana
State University (LSU) Department of Mathematics. He thanks LSU for the kind hospitality he enjoyed during this period.
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ACKNOWLEDGEMENTS
- The authors thank the referees and Patrick De Leenheer, Marcio de
Queiroz, and Eduardo Sontag for their helpful comments on earlier presentations of this work.
- A journal version will appear in Mathematics of Control, Signals,
and Systems. The authors thank J.H. van Schuppen for the
- pportunity to publish their work in his esteemed journal.
- Part of this work was done while F. Mazenc visited the Louisiana
State University (LSU) Department of Mathematics. He thanks LSU for the kind hospitality he enjoyed during this period.
- Malisoff was supported by NSF Grant 0424011. He thanks Zvi
Artstein for illuminating discussions at the International Conference
- n Hybrid Systems and Applications in Lafayette, Louisiana.