New SDH-04-06-2012 *0.5cmObserver based stabilization of linear … · 2012. 6. 14. · Introduction Stability results Observerbased control Simulation results Strange attractors

Introduction Stability results Observer based control Simulation results Strange attractors Conclusion & Perspectives

SDH-04-06-2012

Observer based stabilization of linear system under sparse

measurement

J-P Barbot

J-P Barbot Observer based stabilization under sparse measurement


Road Map

◮ Introduction

◮ A Stability Results

◮ Stable unmeasured subspace

◮ Unstable unmeasured subspace

◮ Example

◮ Example of base



Questions

(a)

(b)

Figure : (a) The traditional paradigm to capture information ; (b) TheCS paradigm to capture information, where Φ is a random sensing matrix.



Questions

How use Compressive Sensing in Control System theory ?This question generates several important questions :

◮ a- How to pass from signal to system ?

◮ b- What is the appropriate base for system ?

◮ c- How to verify the RIP condition for system ?

◮ d- How to bypass the optimization algorithm in order to dealwith real time algorithms ?



Answers

Answer to question a :

◮ Remember Kalman-Bucy filter

◮ Signal is generated by a system

◮ Cluster, Bayesian approach,... can be see as the first step of amodelization

Answer to question b :

◮ Bases in CS are Frequency, Wavelet, Curvelet,...

◮ Oscillations are generated by system

◮ Some normal forms may be use as base ?

◮ Some other assumption than s < m << N are generally donein CS (approach signal processing). In system theory forexample diagnostic, only few systems are selected.

An example of base will be given at the end of this presentation.J-P Barbot Observer based stabilization under sparse measurement


Answers

Answer to question c :In Compressive Sensing

y = Φv

with y ∈ Rm and v ∈ Rn. The RIP condition is : There existsδ ∈]0, 1[ such that

(1− δ)||v ||22 < ||Φv ||22 < (1 + δ)||v ||22

With respect to linear system undersampling this condition isclosed to a random sampling in order to preserve the systemobservability.



Answers

Answer to question d :In this presentation an answer to question d is given, but adding astability study of the observer based control.In the next, it is firstly recalled and introduced some stabilityresults before to present our main results.



References

• Some references in Signal Processing

◮ E. Candes, The restricted isometry property and itsimplications for compressed sensing, Compte Rendus del’Academie des Sciences, Series I, vol. 346, pp. 910, 2008.

◮ E. Candes and M. B. Walkin, An introduction to compressivesampling, IEEE Signal Processing Magazine, vol. 21, no. 2,pp. 2130, 2008.

◮ D. Donoho, I. Drori, V. Stoden, Y. Tsaig, and M. Shahram,Sparselab, http ://sparselab.stanford.edu.

◮ J. Starck and J. L. Bodin, Astronomical data analysis andsparcity : from wavelelets to compressed sensing, Processingof the IEEE, vol. 98, pp. 10211030, 2010.



References

• Some references in Signal Processing

◮ A. Woiselle, J. Starck, and M. Fadili, 3d data denoising andinpainting with the fast curvelet transform, J. of MathematicalImaging and Vision (JMIV), vol. 39, no. 2, pp. 121139, 2011.

◮ L. Berec, A multi-model method to fault detection anddiagnosis : Bayesian solution. an introductory treatise,International Journal of Adaptive Control and SignalProcessing, vol. 12, p. 8192, 1998.

◮ L. Yu, H. Sun, J-P. Barbot, and G. Zeng, Bayesiancompressive sensing for cluster structured sparse signals,Signal Processing, vol. 92, no. 1, pp. 259269, 2012.

◮ ...



References

• Some references in Control Systems Theory

◮ B. Anderson, T. Brinsmead, F. De Bruyne, J. Hespanha, D.Liberzon, and A. Morse, Multiple model adaptive control...,George Zames Special Issue, International Journal of Robustand Non- linear Control, vol. 10, pp. 909929, 2000.

◮ S. Bhattachara and T. Basar, Sparcity based feedback design :a new paradigm in opportunistic sensing, IEEE ACC, 2011.

◮ Y. Khaled, J.-P. Barbot, D. Benmerzouk, and K. Busawon, Anew type of impulsive observer for hyperchaotic system, inIFAC 3th Conference on analysis and control of chaoticsystems, Cancum, Mxico, June 20 th- 22th 2012 2012.



References

• Some references in Control Systems Theory

◮ H. Nijmeijer and I. Mareels, An observer looks atsynchronization, IEEE Transactions on Circuits andSystems-1 : Fundamental theory and Applications, vol. 44, no.10, pp. 882891, 1997.

◮ W. Kang and A. J. Krener, Extended quadratic controllernormal form and dynamic state feedback linearization ofnonlinear systems, SIAM J. Control and Optimization, Vol 30,pp 1319-1337, 1992.

◮ A. Yang, M. Gastpar, R. Bajcsy, and S. Sastry, Distributedsensor perception via sparse representation, Proceddings ofIEEE, vol. 98, no. 6, pp. 10771088, 2010.

◮ ...



Nonlinear case

Let us consider the folowing system :

x1(t) = f1(x1(t), x2(t)); t 6= tk

x2(t) = f2(x2(t)); t 6= tk

x1(t+k ) = Rx1(tk)

x2(t+k ) = x2(tk)

(1)

where x1(t) ∈ Rp, x2(t) ∈ R

n−p, f1 : Rn → R

p andf2 : R

n−p → Rn−p.

Assumption

f1 is at least locally lipschitz, where l1 and l2 are respectively theLipschitz constants with respect to x1 and x2.



Nonlinear case

Assumption

The sampling sequence tk ∈ T = {ti : i ∈ N} ⊂ R verifies thatthere exists τmax and τmin with τmax > τmin > 0 such that ∀i > 0

ti+1 ≥ ti + τmin and ti+1 ≤ ti + τmax (2)

Moreover we define :

θk , tk+1 − tk

x(t+k ) , limh→0

x(tk + h)

x(t−k ) , limh→0

x(tk − h) = x(tk)



Nonlinear case

TheoremIf the system (1) verifies the previous assumptions and satisfies thefollowing conditions :

1. there exists a strictly positive definite functionV2 : R

n−p 7→ R+ V2 ∈ C1, with V2(0) = 0 such that ∀x2 6= 0

V2(x2(t)) < 0 and V2(0) = 0 .

2. ‖R‖1el1θk < 1 for all k > 0.

then, ∀ǫ∗ > 0, the ball 1 B2ǫ∗ is globally asymptotically stableconsidering the sequence of time t+k (i.e. the time just after thereset instants).

1. with respect to norm one



Nonlinear case

Proposition

Assume that the conditions of the theorem hold for system (1),then for any ǫ∗ > 0 (of theorem 1) the ball Bβ is globallyasymptotically stable with β = ǫ∗

‖R‖1+ ǫ∗, if ‖R‖1 6= 0, and the

system (1) converges globally asymptotically to zero if ‖R‖1 = 0.



Linear case

Let us consider the following class of linear system :

x1(t) = A11x1(t) + A12x2(t); t 6= tk

x2(t) = A22x2(t)

x1(t+k ) = Rx1(t)

x2(t+k ) = x2(t)

(3)

Where x(t) ∈ Rn, A11 ∈ R

p×p, A12 ∈ Rp×(n−p),

A22 ∈ R(n−p)×(n−p) and R ∈ R

p×p.



Linear case

TheoremIf the system (3) verifies the following condition :

1. A22 is Hurwitz continuous

2. ‖ReA11θk‖2 < 1 ∀k > 0

Then, ∀ǫ > 0, the state of system (3) converges to a ball of radiusepsilon.



Stable unmeasured subspace

Now, we consider the following class of system :

{

x(t) = Ax(t) + Bu(t)

y(tk) = Cx(tk)(4)

Where x(t) ∈ Rn is the state, u(t) ∈ R

m is the input andy(tk) ∈ R

p is the output. The matrixes A, B and C are constantand of appropriate dimension. Moreover the system is assumeddetectable and stabilizable.




Assumption

There exist a regular matrix T which transform the system 4 intothe system 5 described below (with x = Tx).

˙x1(t) = A11x1(t) + A12x2(t) + B1u(t)

˙x2(t) = A22x2(t) + B2u(t)

y(tk) = x1(tk)

(5)

Where x(t) = (xT1 , xT2 )T with A11 ∈ Rp×p, A12 ∈ R

p×(n−p),A22 ∈ R

(n−p)×(n−p), B1 ∈ Rp×m, B2 ∈ R

(n−p)×m and A22 Hurwitzcontinuous.




For the system (5) the proposed observer base control is :a- The control

u(t) = −Kx , (−K1,−K2)

(

x1x2

)

(6)

b- the Impulsive observer

˙x1(t) = A11x1(t) + A12x2(t) + B1u(t)˙x2(t) = A22x2(t) + B2u(t)

x1(t+k ) = Rx1(tk) + (Ip − R)x1(tk)

y(tk) = x1(tk)

(7)

Where K ∈ Rm×n is a pole placement matrix and R ∈ R

m×n is thereset matrix.




TheoremIf the system (4) verifies the previous assumptions, then it ispossible, for considered sampling sequence (2), to find an observerbased control (6)-(7) which stabilize practically asymptotically thesystem.



Unstable unmeasured subspace

We consider again the system 4, but with :

Assumption

There exist a regular matrix T which transform the system 4 intothe system 8 described below (with x = Tx).

˙x1(t) = A11x1(t) + A12x2(t) + B1u(t)

˙x2(t) = A22x2(t) + B2u(t)

y(tk) = x1(tk)

(8)

Where x(t) = (xT1 , xT2 )T with A11 ∈ Rp×p, A12 ∈ R

p×(n−p),A22 ∈ R

(n−p)×(n−p), B1 ∈ Rp×m, B2 ∈ R

(n−p)×m and A22 isunstable.




The observer based control for system (8) is quite different in itsobserver part than the one proposed in the previous section andhas the following form :a- Control Law :

u(t) = −Kx , (K1,K2)

(

x1x2

)

(9)




b- Generalized impulsive observer

˙x1(t) = A11x1(t) + A12x2(t) + B1u(t)˙x2(t) = A22x2(t) +M(x2(t)− x2(t)) + B2u(t)

˙x1(t) = A11x1(t) + A12x2(t) + L1(x1(t)− x1(t)) + B1u(t)

˙x2(t) = A22x2(t) + L2(x1(t)− x1(t)) + B2u(t)

x1(t+k ) = Rn1 x1(tk) + (Id − R)x1(tk)

(10)Where K ∈ R

m×n, with Rn1 = diag{r1, .., rn1} and −1 < ri < 1, fori ∈ [1, n1]

⋂

N and R , W , M, L1 et L2 are matrixes of appropriatedimensions.




Figure : The block diagram of state feedback with generalizedimpulsive observer




TheoremThe system (4), which verified previous assumptions, is practicallystabilized, under the sampling 2 by the observer based control(9)-(10), if the following conditions are verified :

1. ‖ReA11θk‖ < 1.

2. the matrixe

A22 −M 0 M

0 A11 − L1 A12

0 −L2 A22

is Hurwitz.

3. The matrix

(

A11 − B1K1 A12 − B1K2

−B2K1 A22 − B2K2

)

is Hurwitz.

4. ‖x(0)‖ < KM , ‖x(0)‖ < KM , ‖x(0)‖ < KM and ∀k > 0θk < θm(KM)



Simulation results

Consider the following linear triangular system :

x1 = 2x1 + x2 + 3x3 + u

x2 = x2 + x3

x3 = −x3 − u

y(tk) = x1(tk)

(11)



Simulation results

The goal is to make the system (11) stable. For this, let synthesizethe proposed observer base control (10). By choosing K1 = 48,

K2 = (30, 55), R = 0, M =

(

5 00 0

)

, L1 = 6 and L2 =

(

60

)

.

The parameters of observer control satisfy the conditions oftheorem 3, with corresponding matrix of observer is

−4 1 0 5 00 −1 0 0 00 0 −4 1 30 0 −6 1 10 0 0 0 −1



Simulation results

and the corresponding matrix of control is

−48 31 581 1 0

−30 −56 −48



Simulation results

0 5 10 15 20 25 30−50

0

50

0 5 10 15 20 25 30−50

0

50

0 5 10 15 20 25 30−50

0

50

TIME(S)

x1

x1obs

x2

x2obs

x3

x3obs

Figure : controlled state



Simulation results

Now, consider the following Chaotic system :

x1(t) = a(x2(t)− x1(t))

x2(t) = bx1(t) + cx2(t)− x1(t)x3(t) + x4(t)

x3(t) = −dx3(t) + x1(t)x2(t)

x4(t) = −kx1(t) + x4(t)

y(tk) = x2(tk)

(12)

Avec : a = 35, b = 7, c = 12, d = 3, k = 5 et(x10, x20, x30, x40) = (0.05, 0.01, 0.05, 0.5). Les exposants deLyapunov sont : λ1 = 16.472, λ2 = 1.2729, λ3 = −3 etλ4 = −39.7443. The system (12) is hyperchaotic.



Simulation results

−15

−10

−5

0

5

10

15

−15

−10

−5

0

5

10

1510

15

20

25

30

35

40

x1

x2

x 3

Figure : Phase portraitJ-P Barbot Observer based stabilization under sparse measurement


Simulation results

Impulsionnel observer

˙x1(t) = a(x2(t)− x1(t))˙x2(t) = bx1(t) + cx2(t)− x1(t)x3(t) + x4(t)˙x3(t) = −dx3(t) + x1(t)x2(t)˙x4(t) = −kx1(t) + x4(t) +M(z4(t)− x4(t))

x2(t+k ) = x2(tk) + R(x2(tk)− x2(tk))

(13)



Simulation results

Observer correction

z4(t) = zd2(t)− bx1(t)− cz2(t) + x1(t)x3(t)

z2(t) = zd2(t) + λ1|z2(t)− x2(t)|12 sign(z2(t)− x2(t))

zd2(t) = λ2sign(z2(t)− x2(t))



Simulation results

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−40

−20

0

20

40

Temps(s)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

10

20

30

40

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20

−10

0

10

20

30

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−20

−10

0

10

20

30

x1

x1obs

x4

x4obs

x2

x2obs

x3

x3obs

Figure : observer and system states for θ = 0.01J-P Barbot Observer based stabilization under sparse measurement


Strange attractors identification and state observation

Let us consider the following network of chaotic systems :1-Lorenz System :

ΣLorenz =

x1 = 13.5x1 + 10.5x2 − x2x3

x2 = 22.5(x1 − x2)

x3 = −17

6x3 + x1x2

(14)

2- Lu system :

ΣLu =

x1 = 22.2x1 − x2x3

x2 = 30(x1 − x2)

x3 = −8.8

3x3 + x1x2

(15)



Strange attractor identification and state observation

3- Chen system :

ΣChen =

x1 = 28x1 + 7x2 − x2x3

x2 = 35(x1 − x2)

x3 = −3x3 + x1x2

(16)

4- Qi system :

ΣQi =

x1 = 24(x1 + x2)− x2x3

x2 = 42.5(x1 − x2) + x1x3

x3 = −13x3 + x1x2

(17)




Figure : The block diagram of multi-observer baseJ-P Barbot Observer based stabilization under sparse measurement



−60−40

−200

2040

60

0

10

20

30

40

50

60−40

−30

−20

−10

0

10

20

30

40

Figure : Phases Portrait for Strange attractor




0 20 40 60 80 100 120 140 160 180 200−1

0

1

e 1(Lor

enz)

0 20 40 60 80 100 120 140 160 180 200−1

0

1

e 1(Lu)

0 20 40 60 80 100 120 140 160 180 200−1

0

1

e 1(Che

n)

0 20 40 60 80 100 120 140 160 180 200−1

0

1

Times(s)

e 1(Qi)

Figure : Observation error



Conclusion & Perspectives

◮ Some preliminaries results and relations between CS andControl Theory were highlighted.

◮ This work was done in collaboration with : Y. Khaled, L. Yu,G. Zheng, D. Benmerzouk, H. Sun, D. Boutat, K. Busawon.

Open Problems

◮ RIP and ’observability’ (new definition)

◮ Observability normal form with respect to sparse measurement

◮ Observer and observer based control proof in nonlinear case

◮ ...


Documents

New SDH-04-06-2012 *0.5cmObserver based stabilization of linear … · 2012. 6. 14. · Introduction Stability results Observerbased control Simulation results Strange attractors