Optimal input design for nonlinear dynamical systems: a graph-theory approach · 2015-01-15 ·...
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Optimal input design for nonlinear dynamicalsystems: a graph-theory approach
Patricio E. Valenzuela
Department of Automatic Control and ACCESS Linnaeus Centre
KTH Royal Institute of Technology, Stockholm, Sweden
Seminar Uppsala university
January 16, 2015
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System identification
ytSystemut
vmt
vt︷ ︸︸ ︷
System identification: Modeling based on input-output data.
Basic entities: data set, model structure, identification method.
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System identification
The maximum likelihood method:
θnseq = arg maxθ∈Θ
lθ(y1:nseq)
wherelθ(y1:nseq) := log pθ(y1:nseq)
As nseq → ∞, we have:
θnseq → θ0
√nseq
(θnseq − θ0
)→ Normal with zero mean and
covariance {IeF }−1
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System identification
√nseq
(θnseq − θ0
)→ Normal with zero mean and
covariance {IeF }−1
where
IeF := E
{∂
∂θlθ(y1:nseq)
∣∣∣∣θ=θ0
∂
∂θ⊤lθ(y1:nseq)
∣∣∣∣θ=θ0
∣∣∣∣∣u1:nseq
}
Different u1:nseqs ⇒ different IeF s.
Covariance matrix of√nseq
(θnseq − θ0
)affected by u1:nseq !
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Input design for dynamic systems
Input design: Maximize information from an experiment.
Existing methods: focused on linear systems.
Recent developments for nonlinear systems (Hjalmarsson 2007,Larsson 2010, Gopaluni 2011, De Cock-Gevers-Schoukens 2013,Forgione-Bombois-Van den Hof-Hjalmarsson 2014).
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Input design for dynamic systems
Challenges for nonlinear systems:
Problem complexity (usually non-convex).
Model restrictions.
Input restrictions.
How could we overcome these limitations?
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Summary
1. We present a method for input design for dynamic systems.
2. The method is also suitable for nonlinear systems.
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Outline
Problem formulation for output-error models
Input design based on graph theory
Extension to nonlinear SSM
Closed-loop application oriented input design
Conclusions and future work
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Problem formulation for output-error models
Input design based on graph theory
Extension to nonlinear SSM
Closed-loop application oriented input design
Conclusions and future work
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Problem formulation for output-error models
et
G0ut yt
G0 is a system, where:
-et: white noise (variance λe)-ut: input-yt: system output
Goal: Design
u1:nseq := (u1, . . . , unseq)
as a realization of a stationaryprocess maximizing IF .
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Problem formulation for output-error models
Here,
IF :=1
λeE
{nseq∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤
}
=1
λe
∫ nseq∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤ dP (u1:nseq)
ψθ0t (ut) := ∇θyt(ut)|θ=θ0
yt(ut) := G(ut; θ)
Design u1:nseq ∈ Rnseq ⇔ Design P (u1:nseq) ∈ P.
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Problem formulation for output-error models
Here,
IF :=1
λeE
{nseq∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤
}
=1
λe
∫ nseq∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤ dP (u1:nseq)
Assumption
ut ∈ C (C finite set)
IF =1
λe
∑
u1:nseq ∈Cnseq
nseq∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤ p(u1:nseq)
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Problem formulation for output-error models
Characterizing p(u1:nseq) ∈ PC :
p nonnegative,∑p(x) = 1,
p is shift invariant.
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Problem formulation for output-error models
Problem
Design uopt1:nseq
∈ Cnseq as a realization from popt(u1:nseq), where
popt(u1:nseq) := arg maxp∈PC
h(IF (p))
where h : Rnθ×nθ → R is a matrix concave function, and
IF (p) =1
λe
∑
u1:nseq ∈Cnseq
nseq∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤ p(u1:nseq)
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Problem formulation for output-error models
Input design based on graph theory
Extension to nonlinear SSM
Closed-loop application oriented input design
Conclusions and future work
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Input design problem
Problem
Design uopt1:nseq
∈ Cnseq as a realization from popt(u1:nseq), where
popt(u1:nseq) := arg maxp∈PC
h(IF (p))
where h : Rnθ×nθ → R is a matrix concave function, and
IF (p) =1
λe
∑
u1:nseq ∈Cnseq
nseq∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤ p(u1:nseq)
Issues:
1. IF (p) requires a sum of nseq-dimensional terms (nseq large).
2. How could we represent an element in PC?
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Input design problem
Solving the issues:
1. IF (p) requires a sum of nseq-dimensional terms (nseq large).
Assumption
u1:nseq is a realization of a stationary process with memory nm
(nm < nseq).
⇒ IF (p) requires a sum of nm-dimensional terms.
Minimum nm: related with the memory of the system.
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Input design problem
PC
Solving the issues:
2. How could we represent an element in PC?
PC is a polyhedron.
VPC: Set of extreme points of PC .
⇒ PC can be described as a convex combination of VPC.
The elements in VPCcan be found by using Graph theory!
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Graph theory in input design
Example: de Bruijn graph, C := {0, 1}, nm := 2.
(ut−1, ut)(1, 1)
(ut−1, ut)(0, 0)
(ut−1, ut)(1, 0)
(ut−1, ut)(0, 1)
Elements in VPC⇔ Prime cycles in GCnm
Prime cycles in GCnm ⇔ Elementary cycles in GC(nm−1)
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Graph theory in input design
Example: de Bruijn graph, C := {0, 1}, nm := 2.
(ut−1, ut)(0, 0)
(ut−1, ut)(1, 0)
(ut−1, ut)(0, 1)
(ut−1, ut)(1, 1)
Elements in VPC⇔ Prime cycles in GCnm
Prime cycles in GCnm ⇔ Elementary cycles in GC(nm−1)
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Graph theory in input design
Example: de Bruijn graph, C := {0, 1}, nm := 2.
(ut−1, ut)(0, 0)
(ut−1, ut)(1, 0)
(ut−1, ut)(0, 1)
(ut−1, ut)(1, 1)
Elements in VPC⇔ Prime cycles in GCnm
Prime cycles in GCnm ⇔ Elementary cycles in GC(nm−1)
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Graph theory in input design
Example: de Bruijn graph, C := {0, 1}, nm := 2.
(ut−1, ut)(0, 0)
(ut−1, ut)(1, 0)
(ut−1, ut)(0, 1)
(ut−1, ut)(1, 1)
Elements in VPC⇔ Prime cycles in GCnm
Prime cycles in GCnm ⇔ Elementary cycles in GC(nm−1)
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Graph theory in input design
Example: de Bruijn graph, C := {0, 1}, nm := 2.
(ut−1, ut)(0, 0)
(ut−1, ut)(1, 0)
(ut−1, ut)(0, 1)
(ut−1, ut)(1, 1)
Elements in VPC⇔ Prime cycles in GCnm
Prime cycles in GCnm ⇔ Elementary cycles in GC(nm−1)
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Graph theory in input design
Example: de Bruijn graph, C := {0, 1}, nm := 2.
(ut−1, ut)(1, 1)
(ut−1, ut)(0, 0)
(ut−1, ut)(1, 0)
(ut−1, ut)(0, 1)
There are algorithms to find elementary cycles (Johnson 1975,Tarjan 1972).
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Graph theory in input design
Once vi ∈ VPCis known
⇒ The distribution for each vi is known.
⇒ An input signal {uit}t=N
t=0 can be drawn from vi.
Therefore,
I(i)F :=
1
λe
∑
u1:nm ∈Cnm
nm∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤ vi(u1:nm)
≈ 1
λe N
N∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤
for all vi ∈ VPC.
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Graph theory in input design
Therefore,
I(i)F :=
1
λe
∑
u1:nm ∈Cnm
nm∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤ vi(u1:nm)
≈ 1
λe N
N∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤
for all vi ∈ VPC.
The sum is approximated by Monte-Carlo!
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Input design based on graph theory
To design an experiment in Cnm:
1. Compute all the prime cycles of GCnm .
2. Generate the input signals {uit}t=N
t=0 from the prime cycles ofGCnm , for each i ∈ {1, . . . , nV}.
3. For each i ∈ {1, . . . , nV}, approximate I(i)F by using
I(i)F ≈ 1
λeN
N∑
t=1
ψθ0t (ut)ψ
θ0t (ut)
⊤
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Input design based on graph theory
To design an experiment in Cnm:
4. Define γ := {α1, . . . , αnV} ∈ R
nV .Solve
γopt := arg maxγ∈R
nV
h(IappF (γ))
where
IappF (γ) :=
nV∑
i=1
αi I(i)F
nV∑
i=1
αi = 1
αi ≥ 0 , for all i ∈ {1, . . . , nV}
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Input design based on graph theory
To design an experiment in Cnm:
5. The optimal pmf popt is given by
popt =nV∑
i=1
αopti vi
6. Sample u1:nseq from popt using Markov chains.
IappF (γ) linear in the decision variables ⇒ The problem is convex!
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Example I
et
G0ut yt
G(ut; θ) =
xt+1 =1
θ1 + x2t
+ ut
yt = θ2 x2t + et
x1 = 0
with θ =[θ1 θ2
]⊤= θ0 =
[0.8 2
]⊤.
et: white noise, Gaussian, zero mean, variance λe = 1.
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Example I
et
G0ut yt
G(ut; θ) =
xt+1 =1
θ1 + x2t
+ ut
yt = θ2 x2t + et
x1 = 0
with θ =[θ1 θ2
]⊤= θ0 =
[0.8 2
]⊤.
We consider h(·) = log det(·), and nseq = 104.
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Example I
G(ut; θ) =
xt+1 =1
θ1 + x2t
+ ut
yt = θ2 x2t + et
x1 = 0
Results:
h(IF ) Case 1 Case 2 Case 3 Binary
log{det(IF )} 3.82 4.50 4.48 3.47
Case 1: nm = 2, C = {−1, 0, 1}Case 2: nm = 1, C = {−1, −1/3, 1/3, 1}Case 3: nm = 1, C = {−1, −0.5, 0, 0.5, 1}
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Example I
Results (95 % confidence ellipsoids):
0.79 0.8 0.81
1.98
1.985
1.99
1.995
2
2.005
2.01
2.015
2.02
θ1
θ2
Red: Case 2; Blue: Binary input. 33
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Problem formulation for output-error models
Input design based on graph theory
Extension to nonlinear SSM
Closed-loop application oriented input design
Conclusions and future work
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Extension to nonlinear SSM
Nonlinear state space model:
x0 ∼ µ(x0)
xt|xt−1 ∼ fθ(xt|xt−1, ut−1)
yt|xt ∼ gθ(yt|xt, ut)
where θ ∈ Θ.
-fθ, gθ, µ: pdfs-xt: states-ut: input-yt: system output
Goal: Design
u1:nseq := (u1, . . . , unseq)
as a realization of a stationaryprocess maximizing IF .
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Extension to nonlinear SSM
Here,
IF = E
{S(θ0)S⊤(θ0)
}
S(θ0) = ∇θ log pθ(y1:nseq |u1:nseq)∣∣θ=θ0
Design u1:nseq ∈ Rnseq ⇔ Design P (u1:nseq) ∈ P.
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Extension to nonlinear SSM
Fisher’s identity:
∇θ log pθ(y1:T |u1:T ) = E{∇θ log pθ(x1:T , y1:T |u1:T )
∣∣y1:T , u1:T}
∇θ log pθ(y1:T |u1:T ) =T∑
t=1
∫
X 2ξθ(xt−1:t, ut)pθ(xt−1:t|y1:T )dxt−1:t
with
ξθ(xt−1:t, ut) = ∇θ
[log fθ(xt|xt−1, ut−1) + log gθ(yt|xt, ut)
]
Assumption
ut ∈ C (C finite set)
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Extension to nonlinear SSM
Recall PC :
p nonnegative,∑p(x) = 1,
p is shift invariant.
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Extension to nonlinear SSM
Problem
Design uopt1:nseq
∈ Cnseq as a realization from popt(u1:nseq), where
popt(u1:nseq) := arg maxp∈PC
h(IF (p))
where h : Rnθ×nθ → R is a matrix concave function, and
IF (p) = E
{S(θ0)S⊤(θ0)
}
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Input design problem for nonlinear SSM
Problem
Design uopt1:nseq
∈ Cnseq as a realization from popt(u1:nseq), where
popt(u1:nseq) := arg maxp∈PC
h(IF (p))
where h : Rnθ×nθ → R is a matrix concave function, and
IF (p) = E
{S(θ0)S⊤(θ0)
}
Issues:
1. How could we represent an element in PC? ⇒ Use thegraph theory approach!
2. How could we compute IF (p)?
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Input design based on graph theory (revisited)
To design an experiment in Cnm:
1. Compute all the prime cycles of GCnm .
2. Generate the input signals {uit}t=N
t=0 from the prime cycles ofGCnm , for each i ∈ {1, . . . , nV}.
3. For each i ∈ {1, . . . , nV}, approximate I(i)F by using
I(i)F := Evi(u1:nm )
{S(θ0)S⊤(θ0)
}
≈ (new expression required!)
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Estimating IF
Approximate IF as
IF :=1
M
M∑
m=1
Sm(θ0)S⊤m(θ0)
Difficulty: Sm(θ0) is not available.
Solution: Estimate Sm(θ0) using particle methods!
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Particle methods to estimate Sm(θ0)
Goal: Approximate {pθ(x1:t|y1:t)}t≥1.
{x(i)1:t, w
(i)t }N
i=1: Particle system.
Approach: Auxiliary particle filter + Fixed-lag smoother.
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Particle methods to estimate Sm(θ0)
Estimate Sm(θ0) as
Sm(θ0) :=T∑
t=1
N∑
i=1
w(i)κtξθ0(x
a(i)κt,t−1
t−1 , xa
(i)κt,t
t , ut)
where
ξθ(xt−1:t, ut) = ∇θ
[log fθ(xt|xt−1, ut−1) + log gθ(yt|xt, ut)
]
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Input design based on graph theory (revisited)
To design an experiment in Cnm:
1. Compute all the prime cycles of GCnm .
2. Generate the input signals {uit}t=N
t=0 from the prime cycles ofGCnm , for each i ∈ {1, . . . , nV}.
3. For each i ∈ {1, . . . , nV}, approximate I(i)F by using
I(i)F := Evi(u1:nm )
{S(θ0)S⊤(θ0)
}
≈ 1
M
M∑
m=1
Sm(θ0)S⊤m(θ0)
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Input design based on graph theory (revisited)
To design an experiment in Cnm:
4. Define γ := {α1, . . . , αnV} ∈ R
nV .For k ∈ {1, . . . , K}, solve
γopt,k := arg maxγ∈R
nV
h(Iapp,kF (γk))
where
Iapp,kF (γk) :=
nV∑
i=1
αi,k I(i),kF
nV∑
i=1
αi,k = 1
αi,k ≥ 0 , for all i ∈ {1, . . . , nV}
Compute γ as the sample mean of {γopt,k}Kk=1.
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Input design based on graph theory (revisited)
To design an experiment in Cnm:
5. The optimal pmf popt is given by
popt =nV∑
i=1
αopti vi
6. Sample u1:nseq from popt using Markov chains.
IappF (γ) linear in the decision variables ⇒ The problem is convex!
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Example II
Nonlinear state space model:
xt+1 = θ1xt +xt
θ2 + x2t
+ ut + vt, vt ∼ N (0, 0.12)
yt =1
2xt +
2
5x2
t + et, et ∼ N (0, 0.12)
where θ =[θ1 θ2
]⊤, θ0 =
[0.7 0.6
]⊤.
Input design: nseq = 5 · 103, nm = 2, C = {−1, 0, 1}, andh(·) = log det(·).
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Example II
Input sequence:
0 20 40 60 80 100−1.5
−1
−0.5
0
0.5
1
1.5
t
ut
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Example II
Results:
Input / h(IF ) log det(IF )
Optimal 25.34
Binary 24.75Uniform 24.38
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Problem formulation for output-error models
Input design based on graph theory
Extension to nonlinear SSM
Closed-loop application oriented input design
Conclusions and future work
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Closed-loop application oriented input design
et
System
Ky,t(·)
H
− yd (setpoint)
(ext. excitation) rt ut yt
System (θ0 ∈ Θ):
xt+1 = Aθ0xt +Bθ0ut
yt = Cθ0xt + νt
νt = H(q; θ0)et
{et}: white noise, known distribution.Feedback: ut = rt +Ky,t(yt − yd)
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Closed-loop application oriented input design
Model:
xt+1 = A(θ)xt +B(θ)ut
yt = C(θ)xt + νt
νt = H(q; θ)et
θ ∈ Θ.
Goal: Perform an experiment to obtain θnseq .⇒ design r1:nseq !
Requirements:
1. yt, ut should not be perturbed excessively.
2. θnseq must satisfy quality constraints.
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Closed-loop application oriented input design
Minimize control objective:
J = E
{nseq∑
t=1
∥∥∥yt − yd∥∥∥
2
Q+ ‖ut − ut−1‖2
R
}
Requirements:
1. yt, ut should not be perturbed excessively.
Probabilistic bounds:
P{|yt − yd| ≤ ymax} > 1 − ǫy
P{|ut| ≤ umax} > 1 − ǫx
for t = 1, . . . , nseq
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Closed-loop application oriented input design
ΘSI(α)
Requirements:
2. θnseq must satisfy quality constraints.
Maximum likelihood: As nseq → ∞, we have:
√nseq
(θnseq − θ0
)∈ AsN (0, {Ie
F }−1)
⇒ Identification set:
ΘSI(α) ={θ : (θ − θ0)⊤ Ie
F (θ − θ0) ≤ χ2α(nθ)
}
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Closed-loop application oriented input design
Θapp(γ)
Requirements:
2. θnseq must satisfy quality constraints.
Quality constraint: Application set
Θ(γ) =
{θ : Vapp(θ) ≤ 1
γ
}
Relaxation: Application ellipsoid
Θapp(γ) :=
{θ : (θ − θ0)⊤ ∇2
θVapp(θ)∣∣∣θ=θ0
(θ − θ0) ≤ 2
γ
}
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Closed-loop application oriented input design
Θapp(γ)
ΘSI(α)
Requirements:
2. θnseq must satisfy quality constraints.
Quality constraint:ΘSI(α) ⊆ Θapp(γ)
achieved by
1
χ2α(nθ)
IeF � γ
2∇2
θVapp(θ)∣∣∣θ=θ0
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Closed-loop application oriented input design
Optimization problem:
min{rt}
nseqt=1
J = E
{nseq∑
t=1
‖yt − yd‖2Q + ‖∆ut‖2
R
}
s. t. System constraints
P{|yt − yd| ≤ ymax} > 1 − ǫy
P{|ut| ≤ umax} > 1 − ǫx
IF � γχ2α(nθ)
2∇2
θVapp(θ)
Difficulty: P (and IeF ) hard to optimize.
Solution: Use the graph-theory approach!
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Closed-loop application oriented input design
Graph theory approach:r1:nseq realization from p(r1:nm) with alphabet C.
To design an experiment in Cnm:
1. Compute all the prime cycles of GCnm .
2. Generate the signals {rit}t=N
t=0 from the prime cycles of GCnm ,for each i ∈ {1, . . . , nV}.
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Closed-loop application oriented input design
Graph theory approach:r1:nseq realization from p(r1:nm) with alphabet C.
Given e1:Nsim , r(j)1:Nsim
, approximate
J (j) ≈ 1
Nsim
Nsim∑
t=1
∥∥∥y(j)t − yd
∥∥∥2
Q+
∥∥∥u(j)t − u
(j)t−1
∥∥∥2
R
Pet,r
(j)t
{|u(j)t | ≤ umax} ≈ Monte Carlo
Pet,r
(j)t
{|y(j)t − yd| ≤ ymax} ≈ Monte Carlo
I(j)F computed as in previous parts.
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Closed-loop application oriented input design
Optimization problem (graph-theory):
min{β1, ..., βnv }
nv∑
j=1
βjJ(j)
s. t. System constraints
Constraints on {βj}nv
j=1nv∑
j=1
βjPet,r
(j)t
{|u(j)t | ≤ umax} > 1 − ǫx
nv∑
j=1
βjPet,r
(j)t
{|y(j)t − yd| ≤ ymax} > 1 − ǫy
nv∑
j=1
βj I(j)F � γχ2
α(n)
2nseq∇2
θVapp(θ)
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Closed-loop application oriented input design
Optimal pmf:
popt :=nv∑
j=1
βoptj pj
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Example III
Consider the open-loop, SISO state space system
xt+1 = θ02 xt + ut
yt = θ01 xt + et
[θ0
1 θ02
]⊤=
[0.6 0.9
]⊤.
Input:ut = rt − ky yt
ky = 0.5 known.
Goal: Estimate[θ1 θ2
]⊤using indirect identification.
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Example III
Design {rt}500t=1, nm = 2, rt ∈ C = {−0.5, −0.25, 0, 0.25, 0.5}.
Performance degradation:
Vapp(θ) =1
500
500∑
t=1
‖yt(θ0) − yt(θ)‖22
yd = 0
ǫy = ǫx = 0.07
ymax = 2, umax = 1
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Example III
Reference:
0 20 40 60 80 100−0.5
0
0.5r t
0 20 40 60 80 100−0.5
0
0.5
r t
0 20 40 60 80 100−0.5
0
0.5
t
r t
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Example III
Input:
0 20 40 60 80 100
−1
1ut
0 20 40 60 80 100
−1
1
ut
0 20 40 60 80 100
−1
1
t
ut
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Example III
Output:
0 20 40 60 80 100
−2
0
2yt
0 20 40 60 80 100
−2
0
2
yt
0 20 40 60 80 100
−2
0
2
t
yt
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Example III
Application and identification ellipsoids: (98% confidence level)
0.4 0.6 0.8
0.7
0.8
0.9
1
1.1
θ1
θ2
Θapp(γ)
ΘbinarySI (α)
ΘoptSI (α)
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Problem formulation for output-error models
Input design based on graph theory
Extension to nonlinear SSM
Closed-loop application oriented input design
Conclusions and future work
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Conclusions
A new method for input design was introduced.
The method can be used for nonlinear systems.
Convex problem even for nonlinear systems.
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Future work
Reducible Markov chains.
Computational complexity.
Robust input design.
Application oriented input design for MPC.
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Thanks for your attention.
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Optimal input design for nonlinear dynamicalsystems: a graph-theory approach
Patricio E. Valenzuela
Department of Automatic Control and ACCESS Linnaeus Centre
KTH Royal Institute of Technology, Stockholm, Sweden
Seminar Uppsala university
January 16, 2015
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Outline
Problem formulation for output-error models
Input design based on graph theory
Extension to nonlinear SSM
Closed-loop application oriented input design
Conclusions and future work
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Appendix I: Graph theory in input design
Example: Elementary cycles for a de Bruijn graph, C := {0, 1},nm := 1.
ut = 0 ut = 1
One elementary cycle: z = (0, 1, 0).
⇒ Prime cycles for a de Bruijn Graph, C := {0, 1}, nm := 2:v1 = ((0, 1), (1, 0), (0, 1)).
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Appendix II: Graph theory in input design
Example: Generation of input signal from a prime cycle.
Consider a de Bruijn graph, C := {0, 1}, nm := 2.
v1 = ((0, 1), (1, 0), (0, 1))
{u1t }t=N
t=0 : Take last element of each node.
Finally,
{uit}t=N
t=0 = {1, 0, 1, 0, . . . , ((−1)N + 1)/2}
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Appendix III: Building A
For i ∈ Cnm , define
Ai := {j ∈ Cnm : (j, i) ∈ E} .
(the set of ancestors of i).
For each i ∈ Cnm , let
Aij =
P{i}∑k∈Ai
P{k}, if j ∈ Ai and
∑k∈Ai
P{k} 6= 0
1#Ai
, if j ∈ Ai and∑
k∈Ai
P{k} = 0
0 , otherwise.
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Apendix IV: Example nonlinear case
et
G0ut yt
G0(ut) = G1(q, θ)ut +G2(q, θ)u2t
where
G1(q, θ) = θ1 + θ2 q−1
G2(q, θ) = θ3 + θ4 q−1
et: Gaussian white noise, zero mean, variance λe = 1.
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Apendix IV: Example nonlinear case
et
G0ut yt
G0(ut) = G1(q, θ)ut +G2(q, θ)u2t
where
G1(q, θ) = θ1 + θ2 q−1
G2(q, θ) = θ3 + θ4 q−1
We consider h(·) = det(·), cseq = 3, nm = 2, C := {−1, 0, 1},and N = 5 · 103.
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Apendix IV: Example nonlinear case
Stationary probabilities:
−1 0 1
−1
0
1
ut
ut−
1
det(IappF ) = 0.1796.
Results consistent with previous contributions (Larsson et al.2010).
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Appendix V: Particle methods to estimate Sm(θ0)
Auxiliary particle filter:
pθ(x1:t|y1:t) :=N∑
i=1
w(i)t∑N
k=1w(k)t
δ(x1:t − x(i)1:t)
{x(i)1:t, w
(i)t }N
i=1: Particle system.
Two step procedure to compute {x(i)1:t, w
(i)t }N
i=1:
1. Sampling/propagation.
2. Weighting.
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Appendix V: Particle methods to estimate Sm(θ0)
Two step procedure to compute {x(i)1:t, w
(i)t }N
i=1:
1. Sampling/propagation:
{a(i)t , x
(i)t } ∼ wat
t−1∑N
k=1w(k)t−1
Rθ,t(xt|xat
t−1, ut−1)
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Appendix V: Particle methods to estimate Sm(θ0)
Two step procedure to compute {x(i)1:t, w
(i)t }N
i=1:
2. Weighting:
w(i)t :=
gθ(yt|x(i)t , ut) fθ(xt|x(i)
t−1, ut−1)
Rθ,t(xt|x(i)t−1, ut−1)
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Appendix V: Particle methods to estimate Sm(θ0)
Difficulty: Auxiliary particle filter suffers particle degeneracy.
Solution: Use fixed-lag smoother.
Main idea FL-smoother:
pθ(xt|y1:T , u1:T ) ≈ pθ(xt|y1:κt, u1:κt)
for κt = min(t+ ∆, T ), for some fixed-lag ∆ > 0.
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Apendix VI: Equivalence time and frequency domain
Frequency Time
Design variable Φu(ω) p(u1:nm)
RestrictionsΦu(ω) ≥ 0
1
2π
∫ π−π Φu(ω)dω ≤ 1
p ≥ 0∑u1:nm
p(u1:nm) = 1
p stationary
Information matrix IF (Φu) IF (p)
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