Observability and Observer Designfor Switched Nonlinear Systems

Aneel Tanwani

aneel.tanwani@inria.frhttp://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).