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
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Practical Motivation
c Quanser
Memoryless Nonlinearities
Saturation Dead Zone Coulomb &
Viscous Friction
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Motivational example
Toycopter, a 2 DOF helicopter model
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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)
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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
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Piecewise Aﬃne Systems
PWA systems are in general nonsmooth nonlinear systems.
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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
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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
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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
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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.
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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}
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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
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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
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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
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