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Observability and Observer Designfor Switched Nonlinear Systems
Aneel Tanwani
[email protected]://www.inrialpes.fr/bipop/people/atanwani/
Collaborators: Hyungbo Shim, Daniel Liberzon, Stephan Trenn
SDH Reunion – Paris, June 4, 2012
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Switched Systems & Observability
Switched Jump Systems with Switching Signal σ(t)
x(t) = fσ(t)(x(t)) + gσ(t)(x(t))u(t)
x(tq) = pσ(t−q )(x(t−q ))
y(t) = hσ(t)(x(t)) Σ(σ, u) y
x??
Switching signal σ : R 7→ N; piecewise constant, right-continuous,ever increasing
Switching times {tq}, q ∈ N
Definition (Large-time observable on X ⊂ Rn)
∃T > t0 and u[t0,T ] s.t. x(T ) is determined uniquely from y[t0,T ], u[t0,T ],and σ[t0,T ] as long as x[t0,T ] ⊆ X .
Small-time observability: if, in addition, T > t0 is arbitrary.
Add ‘uniform’ when observability is uniform w.r.t. input u(t).
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Overview
This talk is about
Design of asymptotic observer for large-time observable system
Emphasis on the case when each mode is not observable (i.e., notsmall-time observable)Development of a sufficient condition for large-time observabilityObserver design strategy based on the sufficient condition
Related references:
[Herman ’77, Nijmeijer ’90, Isidori ’95]: local, instantaneousobservability for non-switched nonlinear systems
[Gauthier ’92 & ’94]: uniform observability w.r.t. inputs
[Hespanha ’05]: large-time observability of nonlinear systems
[Vidal ’03, Collins ’04, Babaali ’05]: recover discrete and continuousstate simultaneously, use of derivatives of output
[Balluchi ’03, Tanwani ’11]: large-time observability of switchedlinear systems and observer design
[Kang-Barbot ’09]: large-time observability of switched nonlinearsystems
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Motivating Example: Linear Case
Subsystem Γ1
x =
[0 00 0
]x
y =[1 0
]x
G1 :=
[1 00 0
]x2 unobservable
Subsystem Γ2
x =
[0 1−1 0
]x
y =[0 0
]x
G2 :=
[0 00 0
]x1, x2 unobservable
σ(·) : t
τ
τ
1
2
t1 t2
τ = π4 , y ≡ 0
x1
x2
kerG1 := Q11
kerG1 ∩ eA2τ ( kerG2 ∩ kerG1 := Q21) =: Q3
1
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Review of Linear Case in [T/Shim/Liberzon; HSCC’11]
Switched linear sys. w/ jump:
x(t) = Aqx(t)
x(tq) = Eqx(t−q )
y(t) = Cqx(t)
Kalman decomposition at eachmode:
ξ′q = F ′qξ′q + F ′′q ξq
ξq = Fqξq
y = Hqξq[ξ′q(tq)ξq(tq)
]= Rq
[ξ′q−1(t−q )ξq−1(t−q )
]
The coordinate change yields:
x(t−1 ) = M1ξ1(t−1 ) +N1ξ′1(t−1 )
x(t−2 ) = M2ξ2(t−2 ) +N2ξ′2(t−2 )
...
x(t−m) = Mmξm(t−m) +Nmξ′m(t−m)
= Ψmm−1
(Mm−1ξm−1(t−m−1)
+Nm−1ξ′m−1(t−m−1)
)= · · ·= Ψm
1
(M1ξ1(t−1 ) +N1ξ
′1(t−1 )
)where
Ψji := eAj(tj−tj−1)Ej−1 · · · eAi+1(ti+1−ti)Ei
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Review of Linear Case in [T/Shim/Liberzon; HSCC’11]
Pick the matrices Θj s.t.
span{Θj} = span{Ψmj Nj}⊥.
Then,Θm
...Θ2
Θ1
x(t−m) =
ΘmΨm
mMmξm(t−m) + ΘmΨmmNmξ
′m(t−m)
...Θ2Ψm
2 M2ξ2(t−2 ) + Θ2Ψm2 N2ξ
′2(t−2 )
Θ1Ψm1 M1ξ1(t−1 ) + Θ1Ψm
1 N1ξ′1(t−1 )
.It is shown in [HSCC’11] that
Left invertibility of
Θm
...Θ2
Θ1
⇐⇒ Switched system is large-time observable.
This approach applies to linear systems only.
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
A nonlinear example: Accumulating information
x = f1(x) =
0.1x3
x21 − x2
3 + 2x1
0.1(x1 + 1)
y = h1(x) = x2
x+ = p1(x) =
x1
2x2
x3
x = f2(x) =
x3
−(x21 − x2
3 + 2x1)x2
x1 + 1
y = h2(x) = x2
1 − x23 + 2x1
x+ = p2(x) = x
x = f3(x) =
x22
− 12x2
0
y = h3(x) = x1 + x2
2
X = {(x1, x2, x3) : x1 > 0, x3 > 0}
y = L2f1h1 = 0
y = Lf2h2 = 0 no one is observable
y = Lf3h3 = 0
Timeline: (t0)—(t−1 ) (t1)—(t−2 ) (t2)—
x2(t2) = x2(t−2 )
= exp
(∫ t−2
t1
−y(s)ds
)x2(t1)
= exp
(∫ t2
t1
−y(s)ds
)(2y(t−1 ))
x1(t2) = y(t2)− x22(t2)
x3(t2) = x3(t−2 )
= ±√
x21(t−2 ) + 2x1(t−2 )− y(t−2 )
= +
√x21(t2) + 2x1(t2)− y(t−2 )
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
A nonlinear example: Accumulating information
Lessons:
To recover x(t2), informationobtained at each individualmode is collected bytransporting through systemdynamics.
During the transportationthrough the continuousdynamics or the jump map, theobtained information is notcorrupted by unobservablequantities.
y = L2f1h1 = 0
y = Lf2h2 = 0 no one is observable
y = Lf3h3 = 0
Timeline: (t0)—(t−1 ) (t1)—(t−2 ) (t2)—
x2(t2) = x2(t−2 )
= exp
(∫ t−2
t1
−y(s)ds
)x2(t1)
= exp
(∫ t2
t1
−y(s)ds
)(0.1y(t−1 ))
x1(t2) = y(t2)− x22(t2)
x3(t2) = x3(t−2 )
= ±√
x21(t−2 ) + 2x1(t−2 )− y(t−2 )
= +
√x21(t2) + 2x1(t2)− y(t−2 )
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Observability Decomposition: Nonlinear Case
x = fq(x),
y = hq(x).
y = Lfqhq(x),
y = L2fqhq(x),
. . .
y(kq) = Lkqfqhq(x).
x(t0)
x(t1)
Observation space:
dOq := span{dλq,i(x) : 1 ≤ i ≤ kq},λq,i(x) ∈ {hq(x), Lfqhq(x), L2
fqhq(x), . . . }
Proposition
There exists a transformation:
ξ′q= Fq(ξ′q, ξq),
ξq = Fq(ξq),
y = Hq(ξq).
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
“Switched Observable Canonical Structure”
Suppose that each mode of thesystem is transformed as:
(Mode 1): [t0, t1)
ξ′1 = F ′1(ξ′1, ξ1) +G′1(ξ′1, ξ1)u
ξ1 = F1(ξ1) +G1(ξ1)u
y = H1(ξ1)
(Mode 2): [t1, t2)
ξ′2 = F ′2(ξ′2, z2, ξ2) +G′2(ξ′2, z2, ξ2)u
z2 = F ∗2 (z2, ξ2) +G∗2(z2, ξ2)u
ξ2 = F2(ξ2) +G2(ξ2)u
y = H2(ξ2)
z2(t1) = R2(ξ1(t−1 ))
= R2(ξ1(t−1 ), ξ2(t1))
(Mode 3): [t2, t3)
...
z3(t2) = R3(ξ2(t−2 ), z2(t−2 ), ξ3(t2))
...
(Mode m): [tm−1, tm)
there’s no ξ′mand dim(zm, ξm) = n.
IF ξq(t) is known for each [tq−1, tq),
THEN x(T ), tm−1 < T < tm, isdetermined by inversetransformation from(zm(T ), ξm(T )).
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Sufficient Condition for Large-time Observability local version
Without state jumps (∀pq(x) = x):
construct the observation space dOq = span{dλq,i(x) : 1 ≤ i ≤ kq},where λq,i(x) ∈ {hq(x), Lfqhq(x), Lgqhq(x), LgqLfqhq(x), . . . }let W0 := {0} and Wq := 〈Wq−1 + dOq|fq, gq〉
if ∃ m s.t. dimWm = n, then Σ admits Switched Obs. Canonical Form.
With state jumps:
introduce dO′q := span{d(λq,i ◦ pq−1) : 1 ≤ i ≤ kq} for each q ≥ 2
Wq := 〈Wq−1 + dOq|fq, gq〉 such that
(pq)∗(kerWq ∩ ker dO′q+1) ⊂ kerWq
if ∃ m s.t. dimWm = n, then Σ admits Switched Obs. Canonical Form.
Switched Obs. Canonical Form ⇒ Large-time Observability
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Example (continued)
Mode 1:ξ1,1 := λ1,1 = h1(x)
ξ1,2 := λ1,2 = Lf1h1(x)
W1 = dO1 =
row.span
{[0 1 0
x1 + 1 0 −x3
]}Mode 2:ξ2,1 := λ2,1 = h2(x) = Lf1h1(x)
W2 =W1
Mode 3:ξ3,1 := λ3,1 = h3(x)
W3 =W2 + dO3 = Rn =
row.span
0 1 0x1 + 1 0 −x3
1 2x2 0
Dynamics over [t0, t1):ξ1,1 = ξ1,2
ξ1,2 = 0
y = ξ1,1
Dynamics over [t1, t2):ξ2,1 = 0
y = ξ2,1
z2,1 = z2,1ξ2,1
z2,1(t1) = 2ξ1,1(t−1 )
Dynamics over [t2, t3):ξ3,1 = 0
y = ξ3,1
z3,1 = − 12z3,1
z3,2 = 2(ξ3,1 − z23,1 + 1)z23,1
z3,1(t2) = z2,1(t−2 ), z3,2(t2) = ξ2,1(t−2 )
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Observer Design
m = 2
Assumption
∃m s.t. the mode sequence 1→ 2→ · · · → m ensures large-timeuniform observability, and the sequence repeats.
Persistent switching and ∃D s.t. tq − tq−1 ≤ D, ∀q ∈ N.∀t ≥ t0, x(t) ∈ X : compact set, and |u(t)| ≤Mu.
Synchronous Observer (running parallel to the plant)
˙x(t) = fq(x(t)) + gq(x(t))u(t), t ∈ [tq−1, tq),
x(tq) =
{pq(x(t−q )), (q mod m) 6= 0,
pq(x†(t−q )), (q mod m) = 0,
x†(t−q ) is computed at time t−q from the stored y[tq−m,tq) and u[tq−m,tq).
fq(x) is globally Lipschitz s.t. fq(x) = fq(x) on X . Similar for others.
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Computing x†(t−m) from y and u on [t0, tm)
Plant:
(Mode 1): [t0, t1)
ξ1 = F1(ξ1) +G1(ξ1)u
y = H1(ξ1)
(Mode 2): [t1, t2)
ξ2 = F2(ξ2) +G2(ξ2)u
y = H2(ξ2)
z2 = F ∗2 (z2, ξ2) +G∗2(z2, ξ2)u
z2(t1) = R2(ξ1(t−1 ), ξ2(t1))
...
Observing method requires thatξq(t) ≈ ξq(t) for [tq−1, tq).
Observer:
(Mode 1): ξ1 = φ1(x(t0))
˙ξ1 = F1(ξ1)
+ G1(ξ1)u+ L1(ξ1, u, y) · (y − H1(ξ1))
(Mode 2): ξ2 = φ2(x(t1))
˙ξ2 = F2(ξ2)
+ G2(ξ2)u+L2(ξ2, u, y)·(y−H2(ξ2))
˙z2 = F ∗2 (z2, ξ2) + G∗2(z2, ξ2)u
z2(t1) = R2(ξ1(t−1 ), ξ2(t1))
ξ2(t) − ξ2(t)
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Idea of 2-pass Back-and-forth Observer
Forward observer: with ξf2 = φ2(x(t1)), on [t1, t2),
˙ξf2 = F2(ξf2 ) + G2(ξf2 )u+ Lf2 (ξf2 , u, y) · (y − H2(ξf2 ))
Backward observer: with ξb2(0) = ξf2 (t−2 ),
for s ∈ [0, t2 − t1),
˙ξb2 = −F2(ξb2)− G2(ξb2)u(t2 − s)
− Lb2(ξb2, u, y) · (y(t2 − s)− H2(ξb2))
Estimation error:
Convergence Result
At time t−m, set x†(t−m) = φ−1m (ξm(t−m), zm(t−m)). Then,
|x(t)− x(t)| ≤ Γ|x(t0)− x(t0)|, t0 ≤ t < tm
|x(tm)− x(tm)| ≤ γ|x(t0)− x(t0)|, 0 < γ < 1.
Repeating the process, it follows that limt→∞ |x(t)− x(t)| = 0.
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Example: Simulation result
0 1 2 3 4 5 6 7 8 9 10
1
2
3
Norm of estimation error
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 20
0.5
1
1.5Estimation of ξ -observer at mode 2
Introduction Review of Linear Case Geometric Conditions Observer Design Conclusion
Concluding Remarks
Summary:
Geometric conditions for observability in switched systems.
Hybrid observer design strategies based on geometric conditions.
Have dealt with linear and nonlinear ODEs, and linear DAEs.
References:
A. Tanwani, H. Shim, and D. Liberzon.Observability implies observer design for switched linear systems.Proc. of Conf. on Hyb. Sys: Comp. & Control, pg: 3-12, 2011.(Submitted to journal).
H. Shim, and A. Tanwani.On a Sufficient Condition for Observability of Switched NonlinearSystems and Observer Design Strategy.Proc. of American Control Conf., pg: 1206-1211, 2011.(Submitted to journal).