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Backstepping for Piecewise Affine Systems: A SOS Approach
1. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Backstepping for Piecewise Affine Systems
An SOS Approach
Behzad Samadi Luis Rodrigues
Department of Mechanical and Industrial Engineering
Concordia University
SMC 2007, Montreal
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
6. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Piecewise Affine Systems
A continuous-time PWA system is described as
˙x(t) = Ai x(t) + ai + Bi u(t), if x(t) ∈ Ri
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
7. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Piecewise Affine Systems
A continuous-time PWA system is described as
˙x(t) = Ai x(t) + ai + Bi u(t), if x(t) ∈ Ri
The polytopic cells, Ri , i ∈ I = {1, . . . , M}, partition a
subset of the state space X ⊂ Rn such that
∪M
i=1Ri = X, Ri ∩ Rj = ∅, i = j, where Ri denotes the
closure of Ri .
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
8. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Piecewise Affine Systems
A continuous-time PWA system is described as
˙x(t) = Ai x(t) + ai + Bi u(t), if x(t) ∈ Ri
The polytopic cells, Ri , i ∈ I = {1, . . . , M}, partition a
subset of the state space X ⊂ Rn such that
∪M
i=1Ri = X, Ri ∩ Rj = ∅, i = j, where Ri denotes the
closure of Ri .
Each cell is constructed as the intersection of a finite number
of half spaces
Ri = {x|Ei x + ei 0}
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
9. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Piecewise Affine Systems
Practical examples:
Mechanical systems with hard nonlinearities such as
saturation, deadzone, Columb friction
Contact dynamics
Electrical circuits with diodes
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
11. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Piecewise Affine Systems
PWA systems are in general nonsmooth nonlinear systems.
Controller synthesis methods for PWA systems
Hassibi and Boyd (1998) - Quadratic stabilization and control
of piecewise linear systems - Limited to piecewise linear
controllers for PWA systems with one variable in the domain of
nonlinearity
Johansson and Rantzer (2000) - Piecewise linear quadratic
optimal control - No guarantee for stability
Feng (2002) - Controller design and analysis of uncertain
piecewise linear systems - All local subsystems should be stable
Rodrigues and How (2003) - Observer-based control of
piecewise affine systems - Bilinear matrix inequality
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
13. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Objective
To propose a method for PWA controller synthesis using convex
optimization
Convex optimization problems are numerically tractable.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
14. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Sum of Squares Decomposition
SOS decomposition for polynomials of degree d in n variables:
p(x) =
m
i=1
f 2
i (x)
for some polynomials fi
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
15. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Sum of Squares Decomposition
SOS decomposition for polynomials of degree d in n variables:
p(x) =
m
i=1
f 2
i (x)
for some polynomials fi
SOS polynomials are non-negative.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
16. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Sum of Squares Programming
A sum of squares program is a convex optimization program of the
following form:
Minimize
J
j=1
wj αj
subject to fi,0 +
J
j=1
αj fi,j (x) is SOS, for i = 1, . . . , I
where the αj ’s are the scalar real decision variables, the wj ’s are
some given real numbers, and the fi,j are some given multivariate
polynomials.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
17. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Backstepping for PWA systems
Consider the following PWA system
˙x1 = A
(1)
i1
x1 + a
(1)
i1
+ B
(1)
i1
x2, for E
(1)
i1
x1 + e
(1)
i1
> 0
˙x2 = A
(2)
i2
X2 + a
(2)
i2
+ B
(2)
i2
u, for E
(2)
i2
X2 + e
(2)
i2
> 0
where ij = 1, . . . , Mj for j = 1, 2 and
X2 =
x1
x2
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
18. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Backstepping for PWA systems
Piecewise polynomial Lyapunov functions for PWA systems
with continuous vector fields
SOS Lyapunov functions for PWA systems with
discontinuous vector fields
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
19. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Continuous PWA systems
It is assumed that for the following subsystem
˙x1 = A
(1)
i1
x1 + a
(1)
i1
+ B
(1)
i1
x2, for E
(1)
i1
x1 + e
(1)
i1
> 0,
with i1 = 1, . . . , M1 there exist a continuous piecewise polynomial
Lyapunov function V (1)(x1) and a continuous PWA controller
x2 = γ(1)(x1) with
V (1)(x1) = V
(1)
i1
(x1)
γ(1)(x1) = K
(1)
i1
(x1) + k
(1)
i1
, for E
(1)
i1
x1 + e
(1)
i1
> 0,
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
20. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Continuous PWA systems
In addition, the continuous piecewise polynomial
V (1)
(x1) = V
(1)
i1
(x1), x1 ∈ Ri1
is a Lyapunov function for the closed loop system satisfying
− V
(1)
i1
.(A
(1)
i1
x1 + a
(1)
i1
+ B
(1)
i1
γ
(1)
i1
(x1))
−Γ
(1)
i1
(x1).(E
(1)
i1
x1 + e
(1)
i1
) − αV
(1)
i is SOS
where α > 0, Γ
(1)
i1
(x1) is an SOS function.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
21. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Continuous PWA systems
Consider now the following candidate Lyapunov function
V (2)
(X2) = V (1)
(x1) +
1
2
(x2 − γ(1)
(x1)).(x2 − γ(1)
(x1))
Note that V (2)(X2) is a continuous piecewise polynomial function.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
22. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Continuous PWA systems
The synthesis problem can be formulated as the following SOS
program.
Find u = γ
(2)
i2
(X2), Γ
(1)
i1
(x1), Γ
(2)
i2
(X2), ci2j2 (X2)
s.t. − x1 V
(2)
i2
.(A
(1)
i1
x1 + a
(1)
i1
+ B
(1)
i1
x2)
− x2 V
(2)
i2
.(A
(2)
i2
X2 + a
(2)
i2
+ B
(2)
i2
u)
−Γ
(1)
i1
(x1).(E
(1)
i1
x1 + e
(1)
i1
)
−Γ
(2)
i2
(X2).(E
(2)
i2
X2 + e
(2)
i2
) − αV
(2)
i2
is SOS,
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
23. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Continuous PWA systems
Γ
(1)
i1
(x1) and Γ
(2)
i2
(X2) are SOS
γ
(2)
i2
(X2) − γ
(2)
j2
(X2) = ci2j2 (X2)(E
(2)
i2j2
X2 + e
(2)
i2j2
)
where i1 = 1, . . . , M1, i2 = 1, . . . , M2, R
(2)
i2
and R
(2)
j2
are level-1
neighboring cells, E
(2)
i2j2
X2 + e
(2)
i2j2
= 0 contains their boundary, ci2j2 is
an arbitrary polynomial and
γ
(2)
i2
(X2) = K
(2)
i2
X2 + k
(2)
i2
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
24. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Backstepping for PWA systems
If the SOS program is feasible, a controller u = γ
(2)
i2
(X2) can
be found for the original PWA system (sufficient condition).
The same procedure can be repeated for PWA systems in
strict feedback form
˙x1 = A
(1)
i1
x1 + a
(1)
i1
+ B
(1)
i1
x2, for E
(1)
i1
x1 + e
(1)
i1
> 0
˙x2 = A
(2)
i2
X2 + a
(2)
i2
+ B
(2)
i2
x3, for E
(2)
i2
X2 + e
(2)
i2
> 0
...
˙xn = A
(n)
in
Xn + a
(n)
in
+ B
(n)
in
u, for E
(n)
in
Xn + e
(n)
in
> 0
where ij = 1, . . . , Mj and Xj = [xT
1 . . . xT
j ]T
for j = 2, . . . , n.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
25. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Discontinuous PWA systems
For discontinuous PWA systems, an SOS Lyapunov function is
constructed using affine controllers in each step.
Since the controller in the last step will not be used in the
construction of the Lyapunov function, the last controller can
be a PWA controller.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
33. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Conclusion:
A backstepping technique was developed for continuous and
discontinuous PWA systems.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
34. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Conclusion:
A backstepping technique was developed for continuous and
discontinuous PWA systems.
The proposed technique consists of a series of convex
problems. Therefore, it is computationally efficient.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
35. Outline
Introduction
PWA Controller Synthesis
Numerical Example
Conclusion
Conclusion:
A backstepping technique was developed for continuous and
discontinuous PWA systems.
The proposed technique consists of a series of convex
problems. Therefore, it is computationally efficient.
A stabilizing controller was designed for the tunnel diode
example by the proposed method.
Samadi, Rodrigues Backstepping for Piecewise Affine Systems
