This document proposes a stability analysis method for sampled-data piecewise affine (PWA) systems using convex optimization and a time-delay approach. It formulates the stability analysis of sampled-data PWA slab systems as a convex optimization problem. The analysis uses a Lyapunov-Krasovskii functional and proves that all trajectories converge to an invariant set if certain constraints are satisfied. Future work includes formulating controller synthesis for sampled-data PWA slab systems also as a convex optimization problem.

1. Sampled-Data Piecewise Affine Slab Systems:
A Time-Delay Approach
Behzad Samadi Luis Rodrigues
Department of Mechanical and Industrial Engineering
Concordia University
ACC 2008, Seattle, WA

2. Outline of Topics

3. Practical Motivation
c Quanser
Memoryless Nonlinearities
Saturation Dead Zone Coulomb &
Viscous Friction

4. Motivational example
Toycopter, a 2 DOF helicopter model

5. Motivational example
Pitch model of the experimental helicopter:
˙x1 =x2
˙x2 =
1
Iyy
(−mheli lcgx g cos(x1) − mheli lcgz g sin(x1) − FkM sgn(x2)
− FvMx2 + u)
where x1 is the pitch angle and x2 is the pitch rate.
Nonlinear part:
f (x1) = −mheli lcgx g cos(x1) − mheli lcgz g sin(x1)
PWA part:
f (x2) = −FkM sgn(x2)

6. Sampled-Data PWA Systems: A Time-Delay Approach
x1
f(x1) f (x1)
ˆf (x1)
-3.1416 -1.885 -0.6283 0.6283 1.885 3.1416
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
PWA approximation - Helicopter model

7. Objective
To propose a stability analysis method for sampled-data PWA
systems using
convex optimization
time-delay approach
Continuous−time
PWA systems
PWA controller
Hold

8. Piecewise Affine Systems
PWA systems are in general nonsmooth nonlinear systems.

9. 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

10. Sampled-Data PWA Systems: A Time-Delay Approach
PWA slab system
˙x = Ai x + ai + Bu, for x ∈ Ri
with the region Ri defined as
Ri = {x | σi < CRx < σi+1},
where CR ∈ R1×n and σi for i = 1, . . . , M + 1 are scalars such
that
σ1 < σ2 < . . . < σM+1

11. Sampled-Data PWA Systems: A Time-Delay Approach
PWA slab system
˙x = Ai x + ai + Bu, for x ∈ Ri
with the region Ri defined as
Ri = {x | σi < CRx < σi+1},
where CR ∈ R1×n and σi for i = 1, . . . , M + 1 are scalars such
that
σ1 < σ2 < . . . < σM+1
Continuous-time PWA controller
u(t) = Ki x(t) + ki , x(t) ∈ Ri

12. Sampled-Data PWA Systems: A Time-Delay Approach
Lyapunov-Krasovskii functional:
V (xs, ρ) := V1(x) + V2(xs , ρ) + V3(xs , ρ)
where
xs(t) :=
x(t)
x(tk )
, tk ≤ t < tk+1
V1(x) := xT
Px
V2(xs , ρ) :=
0
−τM
t
t+r
˙xT
(s)R ˙x(s)dsdr
V3(xs , ρ) := (τM − ρ)(x(t) − x(tk))T
X(x(t) − x(tk ))
and P, R and X are positive definite matrices.

13. Sampled-Data PWA Systems: A Time-Delay Approach
The closed-loop system can be rewritten as
˙x(t) = Ai x(t) + ai + B(Ki x(tk) + ki ) + Bw,
for x(t) ∈ Ri and x(tk ) ∈ Rj where
w(t) = (Kj − Ki )x(tk ) + (kj − ki ), x(t) ∈ Ri , x(tk ) ∈ Rj
The input w(t) is a result of the fact that x(t) and x(tk ) are
not necessarily in the same region.

14. Sampled-Data PWA Systems: A Time-Delay Approach
Theorem (1)
For the sampled-data PWA system, assume there exist symmetric
positive matrices P, R, X and matrices Ni for i = 1, . . . , M such
that the conditions are satisfied and let there be constants ∆K and
∆k such that
w ≤ ∆K x(tk ) + ∆k
Then, all the trajectories of the sampled-data PWA system in X
converge to the following invariant set
Ω = {xs | V (xs, ρ) ≤ σaµ2
θ + σb}

15. Sampled-Data PWA Systems: A Time-Delay Approach
for all i ∈ I(0),
Ωi + τMM1i + τMM2i < 0

 Ωi + τMM1i τM
Ni
0
τM NT
i 0 −τMR

 < 0
for all i /∈ I(0), ¯Λi ≻ 0,
Ωi + τMM1i + τMM2i < 0




Ωi + τMM1i τM


Ni
0
0


τM NT
i 0 0 −τMR



 < 0

16. Sampled-Data PWA Systems: A Time-Delay Approach
Solving an optimization problem to maximize τM subject to the
constraints of the main theorem and η > γ > 1 leads to
τ⋆
M = 0.2193

17. Sampled-Data PWA Systems: A Time-Delay Approach
x1
x2
-3 -2 -1 0 1 2 3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Sampled data PWA controller for Ts = 0.2193

18. Sampled-Data PWA Systems: A Time-Delay Approach
x1
x2
-3 -2 -1 0 1 2 3
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
Continuous time PWA controller

19. Summary of the contributions:
Formulating stability analysis of sampled-data PWA slab
systems as a convex optimization problem
Future work:
Formulating controller synthesis for sampled-data PWA slab
systems as a convex optimization problem