Sampled-Data Piecewise Affine Slab Systems: A Time-Delay Approach

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

Outline of Topics

Practical Motivation

c©Quanser

Memoryless Nonlinearities

Saturation Dead Zone Coulomb &Viscous Friction

Motivational example

Toycopter, a 2 DOF helicopter model

Motivational example

Pitch model of the experimental helicopter:

x1 =x2

x2 =1

Iyy(−mheli lcgxg cos(x1)−mheli lcgzg sin(x1)− FkM sgn(x2)

− FvMx2 + u)

where x1 is the pitch angle and x2 is the pitch rate.

Nonlinear part:

f (x1) = −mheli lcgxg cos(x1)−mheli lcgzg sin(x1)

PWA part:f (x2) = −FkM sgn(x2)

Sampled-Data PWA Systems: A Time-Delay Approach

x1

f(x

1)

f (x1)

f (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

Objective

To propose a stability analysis method for sampled-data PWAsystems using

convex optimization

time-delay approach

Continuous−time

PWA systems

PWA controller

Hold

Piecewise Affine Systems

PWA systems are in general nonsmooth nonlinear systems.

Piecewise Affine Systems

PWA systems are in general nonsmooth nonlinear systems.

Controller synthesis methods for PWA systems

Hassibi and Boyd (1998) - Quadratic stabilization and controlof piecewise linear systems - Limited to piecewise linearcontrollers for PWA systems with one variable in the domain ofnonlinearityJohansson and Rantzer (2000) - Piecewise linear quadraticoptimal control - No guarantee for stabilityFeng (2002) - Controller design and analysis of uncertainpiecewise linear systems - All local subsystems should be stableRodrigues and How (2003) - Observer-based control ofpiecewise affine systems - Bilinear matrix inequality

Sampled-Data PWA Systems: A Time-Delay Approach

PWA slab system

x = Aix + 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

thatσ1 < σ2 < . . . < σM+1

Sampled-Data PWA Systems: A Time-Delay Approach

PWA slab system

x = Aix + 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

thatσ1 < σ2 < . . . < σM+1

Continuous-time PWA controller

u(t) = Kix(t) + ki , x(t) ∈ Ri

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) := xTPx

V2(xs , ρ) :=

∫ 0

−τM

∫ t

t+r

xT(s)Rx(s)dsdr

V3(xs , ρ) := (τM − ρ)(x(t)− x(tk))TX (x(t)− x(tk))

and P , R and X are positive definite matrices.

Sampled-Data PWA Systems: A Time-Delay Approach

The closed-loop system can be rewritten as

x(t) = Aix(t) + ai + B(Kix(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) arenot necessarily in the same region.

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 Xconverge to the following invariant set

Ω = xs | V (xs , ρ) ≤ σaµ2θ + σb

Sampled-Data PWA Systems: A Time-Delay Approach

for all i ∈ I(0),

Ωi + τMM1i + τMM2i < 0

Ωi + τMM1i τM

[

Ni

0

]

τM[

NTi 0

]

−τMR

< 0

for all i /∈ I(0), Λi ≻ 0,

Ωi + τMM1i + τMM2i < 0

Ωi + τMM1i τM

Ni

00

τM[

NTi 0 0

]

−τMR

< 0

Sampled-Data PWA Systems: A Time-Delay Approach

Solving an optimization problem to maximize τM subject to theconstraints of the main theorem and η > γ > 1 leads to

τ⋆M = 0.2193

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

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

Summary of the contributions:

Formulating stability analysis of sampled-data PWA slabsystems as a convex optimization problem

Future work:

Formulating controller synthesis for sampled-data PWA slabsystems as a convex optimization problem