Estimation of basins of attraction for controlled systems with input

Estimation of basins of attraction for
controlled systems with input saturation and time-delays
Benjamin Biemond, Wim Michiels
Department of Computer Science, KU Leuven
0 – Outline
1
Introduction
2
Basin of attraction estimate for delay-free system
3
Time-delay system
4
Example
5
Conclusion
Benjamin Biemond
2/18
1 – Outline
1
Introduction
2
Basin of attraction estimate for delay-free system
3
Time-delay system
4
Example
5
Conclusion
Benjamin Biemond
3/18
1 – Introduction
4/18
Saturation
Plant
delay
delay
Controller
Sat(u)
Saturation:
u
piecewise affine
no global performance/ stabilisation of control system
⇒ only trajectories attracted from bounded basin of attraction
Benjamin Biemond
1 – Basin of attraction
5/18
Definition (Basin of attraction)
Subset in state space of all initial conditions with trajectories that converge
to the origin
Delay systems: initial functions xτ : [−τ, 0] → Rn
Approximated by sublevelsets of Lyapunov-Krasovskii functional
{xτ ∈ AC([−τ, 0], Rn )| V (xτ ) ≤ γ},
with
dV
dt
<0
Non-delayed polynomial systems: sublevelset of Lyapunov function
provides arbitrarily close estimate of basin of attraction [Giesl ’08]
Existing results for saturation and delays:
Polytopic overapproximation of saturation function [Fridman, Dambrine ’09;
Gomes da Silva et.al., ’11, Tarbouriech et.al, ’11] → linear techniques
Benjamin Biemond
1 – Model and approach
6/18
Sat(u)
x(t)
˙
= Ax(t) + Bsat(Kx(t − τ ))
u
In this presentation:
1
Find basin of attraction estimate for delay-free system
Lyapunov function Vnd : Rn → R≥0 , piecewise quadratic
→ exploits piecewise affine characteristic of differential equation
2
Benjamin Biemond
1 – Model and approach
6/18
Sat(u)
u
x(t)
˙
= Ax(t) + Bsat(Kx(t − τ ))
In this presentation:
1
Find basin of attraction estimate for delay-free system
Lyapunov function Vnd : Rn → R≥0 , piecewise quadratic
→ exploits piecewise affine characteristic of differential equation
2
Study delayed system
Lyapunov-Krasovskii functional V : AC([−τ, 0], Rn ) → R≥0 , with
Z
0
V (xτ ) =Vnd (xτ (0)) +
x˙ τ (s)T Qx˙ τ (s)ds
−τ
Z
0
Z
+
−τ
Benjamin Biemond
θ
0
x˙ τ (¯
s)K T B T RBK x˙ τ (¯
s)d¯
sdθ,
2 – Outline
1
Introduction
2
Basin of attraction estimate for delay-free system
3
Time-delay system
4
Example
5
Conclusion
Benjamin Biemond
7/18
2 – Basin of attraction estimate for delay-free system
x(t)
˙
= Ax(t) + Bsat(Kx(t))
sat(y) = sign(y) min(|y|, 1)
x ∈ Rn , K ∈ R1×n ,
A has eigenvalues in LHP
Continuous Lyapunov function

T

(x − ss ) Ps (x − ss ), Kx > 1
Vnd (x) = xT P0 x,
|Kx| ≤ 1


(x + ss )T Ps (x + ss ), Kx < −1
Free parameters ss , with Kss < 1, and P0 .
Basin of attraction estimate: {x ∈ Rn | Vnd (x) ≤ γ} with
γ = sup{¯
γ | Vnd (x) ≤ γ¯ ⇒ V˙ nd (x) ≤ 0}
Benjamin Biemond
8/18
2 – Basin of attraction estimate for delay-free system
x(t)
˙
= Ax(t) + Bsat(Kx(t))
sat(y) = sign(y) min(|y|, 1)
x ∈ Rn , K ∈ R1×n ,
A has eigenvalues in LHP
Continuous Lyapunov function

T

(x − ss ) Ps (x − ss ), Kx > 1
Vnd (x) = xT P0 x,
|Kx| ≤ 1


(x + ss )T Ps (x + ss ), Kx < −1
Free parameters ss , with Kss < 1, and P0 .
Basin of attraction estimate: {x ∈ Rn | Vnd (x) ≤ γ } with
Stronger requirement:
γ = sup{¯
γ | Vnd (x) ≤ γ¯ ⇒ V˙ nd (x) ≤ −Vnd (x)}
Benjamin Biemond
8/18
2 – Basin of attraction estimate for non-delayed system
γ = sup{¯
γ | Vnd (x) ≤ γ¯ ⇒ V˙ nd (x) ≤ −Vnd (x)}
Quadratically constraint quadratic problem
min
γ
=
x∈Rn
(x − ss )T Ps (x − ss )
Kx − 1 ≥ 0
(x − ss )T L(x − ss ) + 2(x − ss )T Ps (Ass + B) ≥ 0,
with L = Ps A + AT Ps + Ps .
s.t.
Non-convex constraint
solved using KKT-conditions → roots of polynomial equation
Benjamin Biemond
9/18
3 – Outline
1
Introduction
2
Basin of attraction estimate for delay-free system
3
Time-delay system
4
Example
5
Conclusion
Benjamin Biemond
10/18
3 – Time-delay system
11/18
x(t)
˙
= Ax(t) + Bsat(Kx(t)) +B (sat(Kx(t − τ )) − sat(Kx(t)))
|
{z
}
|
{z
}
ω
Fnd
Use Lyapunov-Krasovskii functional:
Z
0
V (xτ ) =Vnd (xτ (0)) +
x˙ τ (s)T Qx˙ τ (s)ds
−τ
Z
0
Z
+
−τ
For almost al t,
dVnd
dt
0
x˙ τ (¯
s)(BK)T RBK x˙ τ (¯
s)d¯
sdθ,
θ
= ∇Vnd Fnd +∇Vnd Bω.
| {z }
≤−Vnd
Now, use existing techniques to find expressions/upper bounds for terms
∇Vnd Bω and derivatives of quadratic terms in Lyapunov-Krasovskii
functional.
Benjamin Biemond
3 – Overapproximations
12/18
Z 0
Z 0Z 0
V (xτ ) = Vnd (xτ (0))+ x˙ τ (s)T Qx˙ τ (s)ds+
x˙ τ (¯
s)(BK)TRBK x˙ τ (¯
s)d¯
sdθ
−τ
−τ θ
|
{z
} |
{z
}
=:w
=:W
Let R1 ≺ 0 such that Pi R1−1 Pi ≺ δ1 Pi , i ∈ {0, s}, and [Fridman ’02]
Z
t
∇Vnd Bω ≤ τ δ1 Vnd +
t−τ
Benjamin Biemond
x(s)
˙ T (BK)T R1 (BK)x(s)ds
˙
3 – Overapproximations
12/18
Z 0
Z 0Z 0
V (xτ ) = Vnd (xτ (0))+ x˙ τ (s)T Qx˙ τ (s)ds+
x˙ τ (¯
s)(BK)TRBK x˙ τ (¯
s)d¯
sdθ
−τ
−τ θ
|
{z
} |
{z
}
=:w
=:W
Let R1 ≺ 0 such that Pi R1−1 Pi ≺ δ1 Pi , i ∈ {0, s}, and [Fridman ’02]
Z
t
∇Vnd Bω ≤ τ δ1 Vnd +
x(s)
˙ T (BK)T R1 (BK)x(s)ds
˙
t−τ
dw
=x(t)
˙ T Qx(t)
˙ − x(t
˙ − τ )T Qx(t
˙ − τ)
dt
+ x(t)
˙ T P3 (−x(t)
˙ + Ax(t) + Bsat(Kx(t − τ )))
Benjamin Biemond
3 – Overapproximations
12/18
Z 0
Z 0Z 0
T
V (xτ ) = Vnd (xτ (0))+ x˙ τ (s) Qx˙ τ (s)ds+
x˙ τ (¯
s)(BK)TRBK x˙ τ (¯
s)d¯
sdθ
−τ
−τ θ
|
{z
} |
{z
}
=:w
=:W
Let R1 ≺ 0 such that Pi R1−1 Pi ≺ δ1 Pi , i ∈ {0, s}, and [Fridman ’02]
Z
t
∇Vnd Bω ≤ τ δ1 Vnd +
x(s)
˙ T (BK)T R1 (BK)x(s)ds
˙
t−τ
dw
=x(t)
˙ T Qx(t)
˙ − x(t
˙ − τ )T Qx(t
˙ − τ)
dt
+ x(t)
˙ T P3 (−x(t)
˙ + Ax(t) + Bsat(Kx(t − τ )))
Use sat(Kx(t − τ )) = αKx(t − τ ), α ∈ [0, 1]
Benjamin Biemond
3 – Overapproximations
12/18
Z 0
Z 0Z 0
T
V (xτ ) = Vnd (xτ (0))+ x˙ τ (s) Qx˙ τ (s)ds+
x˙ τ (¯
s)(BK)TRBK x˙ τ (¯
s)d¯
sdθ
−τ
−τ θ
|
{z
} |
{z
}
=:w
=:W
Let R1 ≺ 0 such that Pi R1−1 Pi ≺ δ1 Pi , i ∈ {0, s}, and [Fridman ’02]
Z
t
∇Vnd Bω ≤ τ δ1 Vnd +
x(s)
˙ T (BK)T R1 (BK)x(s)ds
˙
t−τ
dw
=x(t)
˙ T Qx(t)
˙ − x(t
˙ − τ )T Qx(t
˙ − τ)
dt
+ x(t)
˙ T P3 (−x(t)
˙ + Ax(t) + Bsat(Kx(t − τ )))
≤x(t)(Q
˙
− P3 − τ2 P3 R2−1 P3 )x(t)
˙ − x(t
˙ − τ )T Qx(t
˙ − τ)
Z t
R2
+
x(s)
˙ T(BK)T
BK x(s)ds
˙
+ max x(t)
˙ TP3 (A + αBK)x(t)
2
α∈[0,1]
t−τ
Benjamin Biemond
3 – Overapproximations
12/18
Z 0
Z 0Z 0
V (xτ ) = Vnd (xτ (0))+ x˙ τ (s)T Qx˙ τ (s)ds+
x˙ τ (¯
s)(BK)TRBK x˙ τ (¯
s)d¯
sdθ
−τ
−τ θ
|
{z
} |
{z
}
=:w
=:W
with R = R1 + R22 :
Z t
dW
=−
x(s)
˙ T (BK)T RBK x(s)ds
˙
+ τ x(t)
˙ T (BK)TRBK x(t)
˙
dt
t−τ
Benjamin Biemond
3 – Overapproximations
12/18
Z 0
Z 0Z 0
V (xτ ) = Vnd (xτ (0))+ x˙ τ (s)T Qx˙ τ (s)ds+
x˙ τ (¯
s)(BK)TRBK x˙ τ (¯
s)d¯
sdθ
−τ
−τ θ
|
{z
} |
{z
}
=:w
=:W
with R = R1 + R22 :
Z t
dW
=−
x(s)
˙ T (BK)T RBK x(s)ds
˙
+ τ x(t)
˙ T (BK)TRBK x(t)
˙
dt
t−τ


0 P3 (A + αBK) 0
dV
Ψ
0  z(t)
≤(− + τ δ1 )Vnd + max z(t)T 0
dt
α∈[0,1]
0
0
−Q
with Ψ = −P3 + τ( 12 P3 R2−1 P3 + K T B T (R1 + 12 R2 )BK) and
z(t) = col(x(t), x(t),
˙
x(t
˙ − τ )).
Benjamin Biemond
3 – Sufficient conditions
13/18
Theorem
If there exist P3 , R1 , R2 , S ≺ 0 and scalar δ1 > 0 such that
1
(− + τ δ1 )S
2 P3 (A + αBK) ≺ 0,
1
T
Ψ
2 (A + αBK) P3
−δ1 Pi Pi
≺ 0,
S ≺ P0 ,
S ≺ 12 Ps .
Pi
−R1
with α ∈ {0, 1} and i ∈ {0, s}, then all trajectories with initial condition in
{xτ ∈ AC[−τ, 0]| V (xτ ) ≤ γ } are attracted towards the origin.
Benjamin Biemond
4 – Outline
1
Introduction
2
Basin of attraction estimate for delay-free system
3
Time-delay system
4
Example
5
Conclusion
Benjamin Biemond
14/18
4 – Example
15/18
T
0
1
0
−0.25
5.00 1.11
Let A =
,B=
,K=
, P0 =
−0.2 0.05
1
−0.2
1.11 10.74
T
We select = 0.05 and ss = 0.65 −0.05
4
3
2
x2
1
0
−1
−2
−3
−4
−6
−4
−2
0
2
4
x1
Solution to matrix inequality found for τ = 0.015
Benjamin Biemond
6
Black: Vnd , gray: xT P0 x
5 – Outline
1
Introduction
2
Basin of attraction estimate for delay-free system
3
Time-delay system
4
Example
5
Conclusion
Benjamin Biemond
16/18
5 – Conclusion
17/18
Presented novel Lyapunov-Krasovskii functional exploiting piecewise
affine nature of delay differential equation
Attains larger basin of attraction estimate for delay-free case
Sufficient condition for attraction for fixed delay
Fridman, E. and Dambrine, M. (2009) Control under quantization, saturation and delay: An LMI
approach Automatica, 45, 2258-2264.
Fridman, E. (2002). Effects of small delays on stability of singularly perturbed systems.
Automatica, 38(5), 897-902.
Giesl, P. (2008). Construction of a local and global Lyapunov function using radial basis functions.
IMA Journal of Applied Mathematics, 73(5), 782-802.
Tarbouriech, S., Garcia, G., Gomes da Silva Jr., J.M. and Queinnec, I. (2011). Stability and
stabilization of linear systems with saturating actuators. Springer- Verlag, London.
Benjamin Biemond
Thank you for your attention
Benjamin Biemond

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