Backstepping for Piecewise Affine Systems: A SOS Approach

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

Outline
Introduction
Numerical Example
Conclusion
Outline of Topics
Introduction
Numerical Example
Conclusion

Outline
Introduction
Numerical Example
Conclusion
Motivational example
Tunnel diode circuit

Outline
Introduction
Numerical Example
Conclusion
PWA characteristic

Outline
Introduction
Numerical Example
Conclusion
Piecewise aﬃne (PWA) model:
˙x1 = −30x1 − 20x2 + 24 + 20u
˙x2 =



0.05x1 − 0.25x2, x2 < 0.2
0.05x1 + 0.1x2 − 0.07, 0.2 < x2 < 0.6
0.05x1 − 0.2x2 + 0.11, x2 > 0.6
Desired equilibrium point:
xcl = 0.3714 0.6429
T

Outline
Introduction
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

Outline
Introduction
Numerical Example
Conclusion
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 .

Outline
Introduction
Numerical Example
Conclusion
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}

Outline
Introduction
Numerical Example
Conclusion
Practical examples:
Mechanical systems with hard nonlinearities such as
saturation, deadzone, Columb friction
Contact dynamics
Electrical circuits with diodes

Outline
Introduction
Numerical Example
Conclusion
PWA systems are in general nonsmooth nonlinear systems.

Outline
Introduction
Numerical Example
Conclusion
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

Outline
Introduction
Numerical Example
Conclusion
Objective
To propose a method for PWA controller synthesis using convex
optimization

Outline
Introduction
Numerical Example
Conclusion
Objective
To propose a method for PWA controller synthesis using convex
optimization
Convex optimization problems are numerically tractable.

Outline
Introduction
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

Outline
Introduction
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.

Outline
Introduction
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.

Outline
Introduction
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

Outline
Introduction
Numerical Example
Conclusion
Piecewise polynomial Lyapunov functions for PWA systems
with continuous vector ﬁelds
SOS Lyapunov functions for PWA systems with
discontinuous vector ﬁelds

Outline
Introduction
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,

Outline
Introduction
Numerical Example
Conclusion
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.

Outline
Introduction
Numerical Example
Conclusion
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.

Outline
Introduction
Numerical Example
Conclusion
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,

Outline
Introduction
Numerical Example
Conclusion
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

Outline
Introduction
Numerical Example
Conclusion
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.

Outline
Introduction
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.

Outline
Introduction
Numerical Example
Conclusion
Tunnel Diode
Consider the tunnel diode PWA model:
˙x1 = −30x1 − 20x2 + 24 + 20u
˙x2 =



0.05x1 − 0.25x2, x2 < 0.2
0.05x1 + 0.1x2 − 0.07, 0.2 < x2 < 0.6
0.05x1 − 0.2x2 + 0.11, x2 > 0.6
Desired equilibrium point:
xcl = 0.3714 0.6429
T

Outline
Introduction
Numerical Example
Conclusion
Tunnel Diode
First step:
Consider the following system
˙x2 =



−0.25x2 + 0.05x1, x2 < 0.2
0.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6
−0.2x2 + 0.11 + 0.05x1, x2 > 0.6

Outline
Introduction
Numerical Example
Conclusion
Tunnel Diode
First step:
˙x2 =



−0.25x2 + 0.05x1, x2 < 0.2
0.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6
−0.2x2 + 0.11 + 0.05x1, x2 > 0.6
Lyapunov function: V (x2) = 1
2(x2 − 0.6429)2

Outline
Introduction
Numerical Example
Conclusion
Tunnel Diode
First step:
˙x2 =



−0.25x2 + 0.05x1, x2 < 0.2
0.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6
−0.2x2 + 0.11 + 0.05x1, x2 > 0.6
Lyapunov function: V (x2) = 1
2(x2 − 0.6429)2
The following control input can stabilize this system to
x2 = 0.6429
x1 = γ(x2) = 0.3714 − 4.8344(x2 − 0.6429)

Outline
Introduction
Numerical Example
Conclusion
Tunnel Diode
Second step:
Construct the Lyapunov function
Vγ(x) =
1
2
(x2−0.6429)2
+
1
2
(x1−0.3714+4.8344(x2−0.6429))2

Outline
Introduction
Numerical Example
Conclusion
Tunnel Diode
Second step:
Construct the Lyapunov function
Vγ(x) =
1
2
(x2−0.6429)2
+
1
2
(x1−0.3714+4.8344(x2−0.6429))2
Solve the SOS program to compute the control
input(α = 0.25):
u =



−0.35009 + 1.2572x1 − 0.1216x2, x2 < 0.2
−0.34175 + 1.2603x1 − 0.20165x2, 0.2 < x2 < 0.6
−0.3784 + 1.2567x1 − 0.13739x2, x2 > 0.6

Outline
Introduction
Numerical Example
Conclusion
Tunnel Diode
Simulation: x(0) = [0.5 0.1]T

Outline
Introduction
Numerical Example
Conclusion
Conclusion:
A backstepping technique was developed for continuous and
discontinuous PWA systems.

Outline
Introduction
Numerical Example
Conclusion
Conclusion:
The proposed technique consists of a series of convex
problems. Therefore, it is computationally eﬃcient.

Outline
Introduction
Numerical Example
Conclusion
Conclusion:
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.

Backstepping for Piecewise Affine Systems: A SOS Approach

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Backstepping for Piecewise Affine Systems: A SOS Approach