Backstepping for Piecewise Affine Systems: A SOS Approach

OutlineIntroduction

PWA Controller SynthesisNumerical Example

Conclusion

Backstepping for Piecewise Affine SystemsAn SOS Approach

Behzad Samadi Luis Rodrigues

Department of Mechanical and Industrial EngineeringConcordia University

SMC 2007, Montreal

Samadi, Rodrigues Backstepping for Piecewise Affine Systems

OutlineIntroduction


Conclusion

Outline of Topics

Introduction

PWA Controller Synthesis

Numerical Example

Conclusion


OutlineIntroduction


Conclusion

Motivational example

Tunnel diode circuit


OutlineIntroduction


Conclusion


PWA characteristic


OutlineIntroduction


Conclusion


Piecewise affine (PWA) model:

x1 = −30x1 − 20x2 + 24 + 20u

x2 =

0.05x1 − 0.25x2, x2 < 0.20.05x1 + 0.1x2 − 0.07, 0.2 < x2 < 0.60.05x1 − 0.2x2 + 0.11, x2 > 0.6

Desired equilibrium point:

xcl =[

0.3714 0.6429]T


OutlineIntroduction


Conclusion

Piecewise Affine Systems

I A continuous-time PWA system is described as

x(t) = Aix(t) + ai + Biu(t), if x(t) ∈ Ri

I The polytopic cells, Ri , i ∈ I = {1, . . . ,M}, partition asubset of the state space X ⊂ Rn such that∪M

i=1Ri = X , Ri ∩Rj = ∅, i 6= j , where Ri denotes theclosure of Ri .

I Each cell is constructed as the intersection of a finite numberof half spaces

Ri = {x |Eix + ei � 0}


OutlineIntroduction


Conclusion







Ri = {x |Eix + ei � 0}


OutlineIntroduction


Conclusion







Ri = {x |Eix + ei � 0}


OutlineIntroduction


Conclusion


Practical examples:

I Mechanical systems with hard nonlinearities such assaturation, deadzone, Columb friction

I Contact dynamics

I Electrical circuits with diodes


OutlineIntroduction


Conclusion


I PWA systems are in general nonsmooth nonlinear systems.

I Controller synthesis methods for PWA systemsI Hassibi and Boyd (1998) - Quadratic stabilization and control

of piecewise linear systems - Limited to piecewise linearcontrollers for PWA systems with one variable in the domain ofnonlinearity

I Johansson and Rantzer (2000) - Piecewise linear quadraticoptimal control - No guarantee for stability

I Feng (2002) - Controller design and analysis of uncertainpiecewise linear systems - All local subsystems should be stable

I Rodrigues and How (2003) - Observer-based control ofpiecewise affine systems - Bilinear matrix inequality


OutlineIntroduction


Conclusion


I PWA systems are in general nonsmooth nonlinear systems.I Controller synthesis methods for PWA systems

I Hassibi and Boyd (1998) - Quadratic stabilization and controlof piecewise linear systems - Limited to piecewise linearcontrollers for PWA systems with one variable in the domain ofnonlinearity

I Johansson and Rantzer (2000) - Piecewise linear quadraticoptimal control - No guarantee for stability

I Feng (2002) - Controller design and analysis of uncertainpiecewise linear systems - All local subsystems should be stable

I Rodrigues and How (2003) - Observer-based control ofpiecewise affine systems - Bilinear matrix inequality


OutlineIntroduction


Conclusion

Objective

To propose a method for PWA controller synthesis using convexoptimization

I Convex optimization problems are numerically tractable.


OutlineIntroduction


Conclusion

Objective

To propose a method for PWA controller synthesis using convexoptimization

I Convex optimization problems are numerically tractable.


OutlineIntroduction


Conclusion

Sum of Squares Decomposition

I SOS decomposition for polynomials of degree d in n variables:

p(x) =m∑

i=1

f 2i (x)

for some polynomials fi

I SOS polynomials are non-negative.


OutlineIntroduction


Conclusion

Sum of Squares Decomposition

I SOS decomposition for polynomials of degree d in n variables:

p(x) =m∑

i=1

f 2i (x)

for some polynomials fiI SOS polynomials are non-negative.


OutlineIntroduction


Conclusion

Sum of Squares Programming

A sum of squares program is a convex optimization program of thefollowing form:

MinimizeJ∑

j=1

wjαj

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 aresome given real numbers, and the fi ,j are some given multivariatepolynomials.


OutlineIntroduction


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

]


OutlineIntroduction


Conclusion


I Piecewise polynomial Lyapunov functions for PWA systemswith continuous vector fields

I SOS Lyapunov functions for PWA systems withdiscontinuous vector fields


OutlineIntroduction


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 polynomialLyapunov function V (1)(x1) and a continuous PWA controllerx2 = 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,


OutlineIntroduction


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)i1γ

(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.


OutlineIntroduction


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.


OutlineIntroduction


Conclusion


The synthesis problem can be formulated as the following SOSprogram.

Find u = γ(2)i2

(X2), Γ(1)i1

(x1), Γ(2)i2

(X2), ci2j2(X2)

s.t. −∇x1V(2)i2.(A

(1)i1

x1 + a(1)i1

+ B(1)i1

x2)

−∇x2V(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,


OutlineIntroduction


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 isan arbitrary polynomial and

γ(2)i2

(X2) = K(2)i2

X2 + k(2)i2


OutlineIntroduction


Conclusion


I If the SOS program is feasible, a controller u = γ(2)i2

(X2) canbe found for the original PWA system (sufficient condition).

I The same procedure can be repeated for PWA systems instrict 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 = [xT1 . . . x

Tj ]T for j = 2, . . . , n.


OutlineIntroduction


Conclusion

Discontinuous PWA systems

I For discontinuous PWA systems, an SOS Lyapunov function isconstructed using affine controllers in each step.

I Since the controller in the last step will not be used in theconstruction of the Lyapunov function, the last controller canbe a PWA controller.


OutlineIntroduction


Conclusion

Tunnel Diode

Consider the tunnel diode PWA model:

x1 = −30x1 − 20x2 + 24 + 20u

x2 =

0.05x1 − 0.25x2, x2 < 0.20.05x1 + 0.1x2 − 0.07, 0.2 < x2 < 0.60.05x1 − 0.2x2 + 0.11, x2 > 0.6

Desired equilibrium point:

xcl =[

0.3714 0.6429]T


OutlineIntroduction


Conclusion

Tunnel Diode

First step:

I Consider the following system

x2 =

−0.25x2 + 0.05x1, x2 < 0.20.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6−0.2x2 + 0.11 + 0.05x1, x2 > 0.6

I Lyapunov function: V (x2) = 12(x2 − 0.6429)2

I The following control input can stabilize this system tox2 = 0.6429

x1 = γ(x2) = 0.3714− 4.8344(x2 − 0.6429)


OutlineIntroduction


Conclusion

Tunnel Diode

First step:


x2 =

−0.25x2 + 0.05x1, x2 < 0.20.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6−0.2x2 + 0.11 + 0.05x1, x2 > 0.6



x1 = γ(x2) = 0.3714− 4.8344(x2 − 0.6429)


OutlineIntroduction


Conclusion

Tunnel Diode

First step:


x2 =

−0.25x2 + 0.05x1, x2 < 0.20.1x2 − 0.07 + 0.05x1, 0.2 < x2 < 0.6−0.2x2 + 0.11 + 0.05x1, x2 > 0.6



x1 = γ(x2) = 0.3714− 4.8344(x2 − 0.6429)


OutlineIntroduction


Conclusion

Tunnel Diode

Second step:

I Construct the Lyapunov function

Vγ(x) =1

2(x2−0.6429)2+

1

2(x1−0.3714+4.8344(x2−0.6429))2

I Solve the SOS program to compute the controlinput(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


OutlineIntroduction


Conclusion

Tunnel Diode

Second step:

I Construct the Lyapunov function

Vγ(x) =1

2(x2−0.6429)2+

1

2(x1−0.3714+4.8344(x2−0.6429))2

I Solve the SOS program to compute the controlinput(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


OutlineIntroduction


Conclusion

Tunnel Diode

Simulation: x(0) = [0.5 0.1]T


OutlineIntroduction


Conclusion

Conclusion:

I A backstepping technique was developed for continuous anddiscontinuous PWA systems.

I The proposed technique consists of a series of convexproblems. Therefore, it is computationally efficient.

I A stabilizing controller was designed for the tunnel diodeexample by the proposed method.


OutlineIntroduction


Conclusion

Conclusion:





OutlineIntroduction


Conclusion

Conclusion:





Technology

Backstepping for Piecewise Affine Systems: A SOS Approach