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
