data informativity for system analysis and control part 1...

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Data informativity for system analysis andcontrol part 1: analysis

Jaap Eising, Henk J. van Waarde, Harry L. Trentelman, and M. Kanat Camlibel

University of Groningen

39th Benelux Meeting on Systems and Control

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Motivation

Model-based vs Data-driven.Control and Analysis of unknown systems:

Data Model Controller/Analysis

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Motivation


Model-based control


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Motivation


Model-based controlSystem identification


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Motivation




Persistency of excitation

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Motivation




Persistency of excitation

Data-driven control

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Motivation

The main observation:

Consider the discrete-time, linear time-invariant system:

x(t+ 1) = Asx(t) +Bsu(t).

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Motivation




Suppose that As and Bs are unknown, but that we have access to measurementson this system. We want to resolve whether the system (As, Bs) is stabilizable.

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Motivation




Suppose that As and Bs are unknown, but that we have access to measurementson this system. We want to resolve whether the system (As, Bs) is stabilizable.We can only conclude that the system is stabilizable if all systems compatiblewith the measurements are stabilizable.

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Motivation

Contents

1. Motivation

2. Informativity Framework

3. Results on informativity for analysis

4. Conclusions

Not in this talk: Data-driven control, controller design, disturbances, noise.

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Informativity Framework


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The informativity framework for analysis.

Three important parts.

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Model Class ΣContains the ‘true’system.

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Data DGenerated by the ‘true’system.Set of all systems thatare compatible: ΣD.

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Property PAny system-theoreticproperty.Set of all systems thathave this property: ΣP .

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We say that the data D are informative for property P if ΣD ⊆ ΣP .

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We say that the data D are informative for property P if ΣD ⊆ ΣP .

Informativity problemProvide necessary and sufficient conditions on D under which the data areinformative for property P .

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Informativity for stabilizability.

Consider the class of discrete-time, linear time-invariant input/state systems,containing ‘true’ system: x(t+ 1) = Asx(t) +Bsu(t).

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Consider the class of discrete-time, linear time-invariant input/state systems,containing ‘true’ system: x(t+ 1) = Asx(t) +Bsu(t).Suppose that we measure the state and input for time 0 up to T .

U− :=[u(0) u(1) · · · u(T − 1)

], X :=

[x(0) x(1) · · · x(T )

],

X− :=[x(0) x(1) · · · x(T − 1)

], X+ :=

[x(1) x(2) · · · x(T )

].

Then the set of all systems compatible with the data is equal to:

ΣD = {(A,B) | X+ = AX− +BU−}

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Consider the class of discrete-time, linear time-invariant input/state systems,containing ‘true’ system: x(t+ 1) = Asx(t) +Bsu(t).Suppose that we measure the state and input for time 0 up to T .

U− :=[u(0) u(1) · · · u(T − 1)

], X :=

[x(0) x(1) · · · x(T )

],

X− :=[x(0) x(1) · · · x(T − 1)

], X+ :=

[x(1) x(2) · · · x(T )

].

Then the set of all systems compatible with the data is equal to:

ΣD = {(A,B) | X+ = AX− +BU−}

Consider stabilizability, and let:

ΣP = {(A,B) | (A,B) is stabilizable}

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With data (U−, X) as before and sets:

ΣD = {(A,B) | X+ = AX− +BU−}ΣP = {(A,B) | (A,B) is stabilizable.}

Informativity for stabilizability

The data (U−, X) are informative for stabilizability if ΣD ⊆ ΣP .

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The informativity framework for control.

Let the ‘true’ system be (As, Bs) and data (U−, X) as before.

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Suppose now that we are interested in stabilization by static state feedback: FindK such that As +BsK is stable.

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We say the data D is informative for property P(K) if there exists a K such thatΣD ⊆ ΣP(K).

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We say the data D is informative for property P(K) if there exists a K such thatΣD ⊆ ΣP(K).

Informativity problem for controlProvide necessary and sufficient conditions on D under which there exists a Ksuch that the data are informative for property P(K).

Control design problem

If D are informative for P(K), find K such that ΣD ⊆ ΣP(K).

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Results on informativity for analysis


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Data-driven Hautus test.Consider the class of discrete-time, linear time-invariant input/state systems,containing ‘true’ system: x(t+ 1) = Asx(t) +Bsu(t).

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Data-driven Hautus test.Consider the class of discrete-time, linear time-invariant input/state systems,containing ‘true’ system: x(t+ 1) = Asx(t) +Bsu(t).With data (U−, X) as before and sets:

Σi/s = {(A,B) | X+ = AX− +BU−},Σstab = {(A,B) | (A,B) is stabilizable},Σcont = {(A,B) | (A,B) is controllable}.

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Data-driven Hautus test.Consider the class of discrete-time, linear time-invariant input/state systems,containing ‘true’ system: x(t+ 1) = Asx(t) +Bsu(t).

Theorem (Data-driven Hautus tests)

The data (U−, X) are informative for controllability if and only if

rank(X+ − λX−) = n ∀λ ∈ C.

Similarly, the data (U−, X) are informative for stabilizability if and only if

rank(X+ − λX−) = n ∀λ ∈ C with |λ| > 1.

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Example

Theorem (Data-driven Hautus test)


rank(X+ − λX−) = n ∀λ ∈ C.

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Example



rank(X+ − λX−) = n ∀λ ∈ C.

Let n = 2 and

x0 =

[00

], x1 =

[10

], x2 =

[01

], u0 = 1, u1 = 0.

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Example



rank(X+ − λX−) = n ∀λ ∈ C.

Let n = 2 and

x0 =

[00

], x1 =

[10

], x2 =

[01

], u0 = 1, u1 = 0.

This implies that

X+ =

[1 00 1

], X− =

[0 10 0

], =⇒ X+ − λX− =

[1 −λ0 1

]

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Example



rank(X+ − λX−) = n ∀λ ∈ C.

Let n = 2 and

x0 =

[00

], x1 =

[10

], x2 =

[01

], u0 = 1, u1 = 0.

Indeed:

Σi/s =

{([0 a11 a2

],

[10

])| a1, a2 ∈ R

}

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Observability

Given C, consider the class of autonomous discrete-time, linear time-invariantsystems, containing ‘true’ system: x(t+ 1) = Asx(t), y(t) = Cx(t).

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Observability

Given C, consider the class of autonomous discrete-time, linear time-invariantsystems, containing ‘true’ system: x(t+ 1) = Asx(t), y(t) = Cx(t).With state measurements X as before we have sets:

Σs = {A | X+ = AX−},Σstab = {A | (C,A) is observable},Σcont = {A | (C,A) is detectable}.

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Observability

Given C, consider the class of autonomous discrete-time, linear time-invariantsystems, containing ‘true’ system: x(t+ 1) = Asx(t), y(t) = Cx(t).


The data X are informative for observability with C if and only if kerC ⊆ imX−and for all λ ∈ C and v ∈ Cn we have(

X+ − λX−CX−

)v =⇒ X−v = 0.

The data X are informative for detectability with C if and only if kerC ⊆ imX−and for all λ ∈ C with |λ| > 1 and v ∈ Cn we have(

X+ − λX−CX−

)v =⇒ X−v = 0.

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Example

Let C =(1 0 0

)and suppose that X− =

0 00 11 0

, X+ =

0 11 00 0

.

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Example

Let C =(1 0 0


0 00 11 0

, X+ =

0 11 00 0

.


The data X are informative for observability with C if and only if kerC ⊆ imX−

and for all λ ∈ C and v ∈ Cn we have

(X+ − λX−CX−

)v =⇒ X−v = 0.

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Example

Let C =(1 0 0


0 00 11 0

, X+ =

0 11 00 0

.


The data X are informative for observability with C if and only if kerC ⊆ imX−

and for all λ ∈ C and v ∈ Cn we have

(X+ − λX−CX−

)v =⇒ X−v = 0.

Indeed kerC ⊆ imX− and

(X+ − λX−CX−

)=

0 11 −λ−λ 00 0

.

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Conclusions

Conclusions

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Conclusions

Conclusions

> We have introduced a framework for dealing with informativity problems.

> We resolved analysis problems regarding controllability, stabilizability,observability and detectability.

> Some examples illustrated these results.

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Conclusions

Conclusions

> We have introduced a framework for dealing with informativity problems.

> We resolved analysis problems regarding controllability, stabilizability,observability and detectability.

> Some examples illustrated these results.

Based on: H. J. Van Waarde, J. Eising, H. L. Trentelman, and M. K. Camlibel,“Data informativity: a new perspective on data-driven analysis and control,”IEEE Transactions on Automatic Control.

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Conclusions

Extensions

> Data-driven control.

> Different model classes:

• Nonlinear systems.

• Disturbances.

• Parametrization (e.g. network structure).

> Different data:

• Noise.

• Input/output measurements

> Different properties.

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Conclusions

Questions?

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