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Observer design for systems with nonsmall and unknown time-varying delay

Alexandre Seuret, Thierry Floquet, Jean-Pierre Richard, Sarah Spurgeon

To cite this version:

Alexandre Seuret, Thierry Floquet, Jean-Pierre Richard, Sarah Spurgeon. Observer design for systems

with nonsmall and unknown time-varying delay. TDS’07, IFAC Workshop on Time Delay Systems,

IFAC, Sep 2007, Nantes, France. �inria-00179798�

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OBSERVER DESIGN FOR SYSTEMS WITH NON SMALL AND UNKNOWN

TIME-VARYING DELAY Alexandre Seuret Thierry Floquet∗∗

Jean-Pierre Richard∗∗ Sarah K. Spurgeon

Control and Instrumentation Research Group, University of Leicester, University Road, Leicester LE1 7RH, UK.

as389@leicester.ac.uk, eon@le.ac.uk

∗∗LAGIS, CNRS UMR 8146, Ecole Centrale de Lille, BP 48, 59651 Villeneuve d’Ascq Cedex, France.

Équipe Projet ALIEN, Centre de Recherche INRIA Futurs.

thierry.floquet@ec-lille.fr, jean-pierre.richard@ec-lille.fr

Abstract: This paper deals with the design of observers for linear systems with unknown, time-varying, but bounded delays (on the state and on the input). In this work, the problem is solved for a class of systems by combining the unknown input observer approach with an adequate choice of a Lyapunov-Krasovskii functional for non small delay systems. This result provides workable conditions in terms of rank assumptions and LMI conditions. The dynamic properties of the observer are also analyzed. A 4th-order example is used to demonstrate the feasibility of the proposed solution.

Keywords: Sliding Mode Observer, Time-Delay Systems, Unknown Delay, Non Small Delay, Linear Matrix Inequalities.

1. INTRODUCTION

State observation is an important issue for both linear and nonlinear systems. This work considers the observation problem for the case of linear systems with non small and unknown delay. Sev- eral authors proposed observers for delay systems (see, e.g., (Sename, 2001; Richard, 2003)). Most of the literature, as pointed out in (Richard, 2003), considers that the value of the mainly constant delay can be used in the observer realization.

This means that the delay is known or measured.

Likewise, what is defined as “observers without internal delay” (Darouach, 2001; Darouach et

1 Partially supported by EPSRC Platform reference , EP/D029937/1 : entitled “Control of Complex Systems”

al., 1999; Fairmaret al., 1999) involves the output knowledge at the present and delayed instants.

There are presently very few results in which the observer does not assume knowledge of the delay (Choi and Chung, 1997; de Souza et al., 1999;

Fridmanet al., 2003b; Seuretet al., 2007; Wanget al., 1999). These interesting approaches consider linear systems and guarantee anHperformance.

They are based on stability techniques indepen- dent of the delay and lead to the minimization of the state observation error. It is interesting to reduce the probable conservatism of such results by taking into account information on a delay upper-bound and derive an asymptotically stable observer.

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In (Seuret et al., 2007), the authors design an observer using a computational delay which can be assimilated to an estimation of the delayˆh. The conditions which guarantee the convergence of the error dynamics developed for this observer do not take into account the value ofˆh. This means that the estimation of the state is guaranteed whatever the delay estimation hˆ is. The property is not related to the conservatism of the conditions.

The errors between the real and computational delays are controlled by the discontinuous sliding function. The greater the error between the real and computational delays, the greater the gain of the discontinuous function will be. It is thus straightforward to conclude that a worse estimate of the delay can lead to a high gain in the sliding injection.

In this paper, another method is proposed to solve the problem of the observation of linear systems with unknown time-varying delays which are as- sumed to be “non small” i.e. the delay function lies in an interval excluding0. The result is based on a combination of some results on sliding mode observers (see, e.g., (Barbotet al., 1996; Edwards and Spurgeon, 1998; Floquet et al., 2004; Perru- quetti and Barbot, 2002)) with an appropriate choice of a Lyapunov-Krasovskii functional. The dynamical properties of the observer will also be discussed. For the sake of simplicity, the unknown time delayτ(t)is assumed to be the same for the state and the input. In order to reduce the conser- vatism of the developed conditions, the existence of known real numbers d, τ1 and τ2 is assumed such that∀t∈IR+:

τ1≤τ(t)≤τ2

˙

τ(t)≤d <1. (1) Here the delay used in the observer is the average of the delay (τ21)/2. Then the design of the observer does not require the definition or the computation of a delay estimate and the stability conditions only depend on the parameters of the studied system.

Throughout the article, the notation P > 0 for P ∈ IRn×n means that P is a symmetric and positive definite matrix.[A1|A2|...|An]is the con- catenated matrix with matricesAi. In represents then×nidentity matrix.

2. PROBLEM STATEMENT

Consider the linear time-invariant system with state and input delay:





˙

x(t) = Ax(t) +Aτx(t−τ(t))

+Bu(t) +Bτu(t−τ(t)) +Dζ(t) y(t) = Cx(t)

x(s) = φ(s), ∀s∈[−τ2,0]

(2)

wherex∈IRn, u∈IRmandy∈IRqare the state vector, the input vector and the measurement vector, respectively. ζ ∈ IRr is an unknown and bounded perturbation that satisfies:

kζ(t)k ≤α1(t, y, u), (3) where α1 is a known scalar function. φ ∈ C0([−τ2,0],IRn)is the vector of initial conditions.

It is assumed that A, Aτ, B, Bτ, C and D are constant known matrices of appropriate dimen- sions. The following structural assumptions are required for the design of the observer:

A1. rank(C[Aτ|Bτ|D]) = rank([Aτ|Bτ|D]),p, A2. p < q≤n,

A3. The invariant zeros of (A,[Aτ|Bτ|D], C) lie inC.

Under these assumptions and using the same linear change of coordinates as in (Edwards and Spurgeon, 1998), Chapter 6, the system can be transformed into:









˙

x1(t) = A11x1(t) +A12x2(t) +B1u(t),

˙

x2(t) = A21x1(t) +A22x2(t) +B2u(t) +G1x1(t−τ(t)) +G2x2(t−τ(t)) +Guu(t−τ(t)) +D1ζ(t),

y(t) = T x2(t),

(4) where x1 ∈ IRnq, x2 ∈ IRq and where G1, G2, Gu,D1 andA21 are defined by:

G1=

· 0 G¯1

¸ , G2=

· 0 G¯2

¸ , Gu=

· 0 G¯u

¸ , D1=

· 0 D¯1

¸

, A21=

·A211

A212

¸ ,

with G¯1∈ IRp×(nq), G¯2∈ IRp×q, G¯u ∈IRp×m, D¯1 ∈ IRp×r, A211 ∈ IR(qp)×(nq) , A212 ∈ IRp×(nq) and T an orthogonal matrix involved in the change of coordinates given in (Edwards and Spurgeon, 1998).

Under these conditions, the system can be de- composed into two subsystems. A1 implies that the unmeasurable state x1 is not affected by the delayed terms and the perturbations.A3 ensures that the pair(A11, A211)is at least detectable.

In this article, the following lemma will be used:

Lemma 1. (Hu et al., 2004) For any matrices A, P0>0 andP1>0, the inequality

ATP1A−P0<0,

is equivalent to the existence of a matrix Y such that:

· −P0 ATYT Y A −Y −YT +P1

¸

<0.

3. OBSERVER DESIGN Define the following sliding mode observer:

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



















˙ˆ

x1(t) = A111(t) +A12x2(t) +B1u(t) +(LTTGlT−A11L)(x2(t)−xˆ2(t)) +LTTν(t),

˙ˆ

x2(t) = A211(t) +A22x2(t) +B2u(t) +G11(t−h) +G2x2(t−h) +Guu(t−h)−TTν(t),

−(A21L+TTGlT)(x2(t)−xˆ2(t)) ˆ

y(t) = Txˆ2(t),

(5) where the linear gainGlis a Hurwitz matrix andL has the form£L¯ 0¤

withL¯ ∈IR(nq)×(qp). The computed delayh= (τ21)/2is an implemented value that is chosen according to the delay defi- nition. It corresponds to the delay average. The discontinuous injection termν is given by:

ν(t) =













−ρ(t, y, u) Py(y(t)−y(t))ˆ kPy(y(t)−y(t))ˆ k if y(t)−y(t)ˆ 6= 0,

0

otherwise.

(6)

where Py > 0, Py ∈ IRp×p and where ρ is a nonlinear positive gain yet to be defined. Note that the non delayed terms depending on x2 are known becausex2(t) =TTy(t). Defineµ= (τ2− τ1)/2.

Remark 1. Compared to (Seuretet al., 2007), this observer does not required an artificial delayh. Itˆ only needs to have an average value of the delay.

Contrary to the observer proposed in (Seuret et al., 2007), the implemented delay will appear in the conditions which guarantee stability.

Defining the state estimation errors as e1 = x1(t)−ˆx1(t)ande2=x2(t)−xˆ2(t), one obtains:









˙

e1(t) =A11e1(t)−LTT(GlT e2(t) +ν(t)) +A11Le2(t),

˙

e2(t) =A21e1(t) +G1e1(t−τ(t)) +(TTGlT +A21L)e2(t) +TTν+ξ(t) +D1ζ(t),

(7) whereξ0:IR 7−→IRp is given by:

ξ(t) =G1(ˆx1(t−τ(t))−xˆ1(t−h)) +G2(x2(t−τ(t))−x2(t−h)) +Gu(u(t−τ(t))−u(t−h)).

which can be rewritten as:

ξ(t) = [G1 G2 Gu]

Z tτ(t) th

·x˙ˆ1(s)

˙ x2(s)

˙ u(s)

¸ ds.

The function ξ only depends on the known vari- ablesxˆ1,x2anduand on the unknown delayτ(t).

One can then assume that there exists a known scalar functionα2such that:

kξ(t)k ≤α2(t,xˆ1, x2, u). (8)

Let us define an expression for ρin (6) by using results introduced in the case of control law design (Fridman et al., 2003a). Defineγ, a real positive number and ρsuch that:

ρ(t, y, u) =kD11(t, y, u) +α2(t,xˆ1, x2, u) +γ, (9) Introduce the change of coordinates

·¯e1

¯ e2

¸

=

TL

·e1

e2

¸

with TL =

·Inq L

0 T

¸

. Using the fact thatLG1=LG2=LGu=LD1= 0, one obtains:





˙¯

e1(t) = (A11+LA21)¯e1(t),

˙¯

e2(t) = T A211(t) +T G11(t−τ(t)) +Gl¯e2(t)−T G1L¯e2(t−τ(t)) +T ξ(t) +T D1ζ(t) +ν,

(10)

Theorem 1. Under assumptions A1−A3 and (8) and for all Hurwitz matrices Gl, system (10) is asymptotically stable for any delay τ(t) in [τ1 τ2] if there exist symmetric definite positive matricesP1 andR1∈IR(nq)×(nq),P2∈IRq×q, a symmetric matrix Z2 ∈ IRq×q and a matrix W ∈IR(nq)×(qp) such that the following linear matrix inequalities hold:

Ψ0 Ψ1 Ψ1 Ψ2 0

−2P1+hR1 0 0 0

−2P1+hR1a 0 0

Ψ3 P2T G1

−P1

<0.

(11) where

Ψ0= AT11P1+P1A11+AT211WT+W A211, Ψ1= AT11P1+AT211WT,

Ψ2= (A21+G1)TTTP2,

Ψ3= GTlP2+P2Gl+hZ2+ 2µZ2a+R2,

and h

(1d)R2 [W 0]T [W 0] P1

i<0, h R1 (T G1)TP2

P2T G1 Z2

i≥0, h R1a (T G1)TP2

P2T G1 Z2a

i≥0.

(12)

The gainL¯ is given byL¯ =P11W.

Proof. Consider the Lyapunov-Krasovskii func- tional:

V(t) = ¯eT1(t)P1¯e1(t) + Z 0

h

Z t t+θ

˙¯

eT1(s)R1e˙¯1(s)dsdθ +

Z µ

µ

Z t t+θ

˙¯

eT1(s)R1ae˙¯1(s)dsdθ +¯eT2(t)P22(t) +

Z t tτ(t)

¯

eT2(s)R22(s)ds.

(13) The functionalV can be divided into three parts.

The first line of (13) is designed to control the errorse1(t) subject to the constant delayh. The second line presents a functional which takes into

(5)

account the delay variation around the average delay h. The last part which appears in the last line of (13) controls the errore2(t).

Using the following transformatione¯i(t−τ(t)) =

¯

ei(t)−Rt

the˙¯i(s)ds−Rth

tτ(t)e˙¯i(s)ds, one has:

V˙(t) =

¯

eT1(t)[(A11+LA21)TP1+P1(A11+LA21)]¯e1(t) +2¯eT2(t)P2T(A21+G1)¯e1(t) + ¯eT2(t)[GTlP2

+P2Gl+R2]¯e2(t)−2¯eT2(t)P2T G1L¯e2(t−τ(t))

−(1−τ(t))¯˙ eT2(t−τ(t))R22(t−τ(t)) +he˙¯T1(t)R1e˙¯1(t)−

Z t th

e˙¯T1(s)R1e˙¯1(s)ds +2µe˙¯T1(t)R1ae˙¯1(t)−

Z tτ2

tτ1

e˙¯T1(s)R1ae˙¯1(s)ds +η1(t) +η2(t) +η3(t)−2ρ(t, y, u)kP22(t)k, where

η1(t) =−2¯eT2(t)P2T G1

Z t th

˙¯

e1(s)ds, η2(t) =−2¯eT2(t)P2T G1

Z th tτ(t)

e˙¯1(s)ds, η3(t) = 2¯eT2(t)P2[T D1ζ(t) +T ξ(t)].

The LMI condition (12) implies that for any vectorX:

XTh

R1 (T G1)TP2

P2T G1 Z2

iX ≥0.

Developing this relation for X =

·e˙¯1(s)

¯ e2(t)

¸ , the following inequality holds:

−2¯e2(t)P2G1e˙¯1(s)≤e¯2(t)TZ22(t) + ˙¯eT1(s)R1e˙¯1(s).

Then, an integration with respect to s of the previous inequality leads to an upper bound of η1(t):

η1(t)≤ Z t

t−h

¯

eT2(t)Z2¯e2(t)ds +

Z t th

˙¯

eT1(s)R1e˙¯1(s)ds, η1(t)≤ h¯eT2(t)Z2¯e2(t)

+ Z t

th

˙¯

eT1(s)R1e˙¯1(s)ds.

(14)

By using the same techniques, an upper bound of η2 is found:

η2(t)≤ 2µ¯eT2(t)Z2a2(t) +

Z th+µ thµ

˙¯

eT1(s)R1ae˙¯1(s)ds. (15) From (9) and from the orthogonality of the matrix T, the following inequality holds:

η3(t)−2ρ(t, y, u)kP22(t)k ≤ −2γkP22(t)k. (16) Taking into account (14), (15), (16) and the fact thate˙¯1(t) = (A11+ ¯LA211)¯e1(t),V˙ can be upper- bounded as follows:

V˙(t)≤ e¯T1(t)(P111+ ˜AT11P1+hA˜T11R111

+2µA˜T11R1a11)¯e1(t)−2γkP2e2(t)k +¯eT2(t)(P2Gl+GTl P2+R2+hZ2

+2µZ2a)¯e2(t) + 2¯eT2(t)P2(A21+G1)¯e1(t)

−(1−τ(t))¯˙ eT2(t−τ(t))R22(t−τ(t))

−2¯eT2(t)P2T G1L¯e2(t−τ(t))

whereA˜11= (A11+ ¯LA211).

Then, the last term of this inequality can be upperbounded by noting that:

−2¯eT2(t)P2T G1L¯e2(t−τ(t))

≤ ¯eT2(t)P2T G1P1−1(T G1)TP22(t) +¯eT2(t−τ(t))LTP1L¯e2(t−τ(t))

≤ ¯eT2(t)P2T G1P1−1(T G1)TP22(t)

+¯eT2(t−τ(t))(P1L)TP1−1(P1L)¯e2(t−τ(t)), which leads to the following upperbound:

V˙(t)≤ h¯e

1(t)

¯ e2(t)

iT

Ψhe¯

1(t)

¯ e2(t)

i−2γkP2¯e2(t)k +¯eT2(t−τ(t))ψ3¯e2(t−τ(t)),

(17)

whereΨ =hψ

10 Ψ2

ψ20

iand :

ψ10= (A11+ ¯LA211)TP1+P1(A11+ ¯LA211) +h(A11+ ¯LA211)TR1(A11+ ¯LA211) +2µ(A11+ ¯LA211)TR1a(A11+ ¯LA211), ψ20=GTl P2+P2Gl+R2+hZ2+ 2µZ2a

+P2T G1P1−1(T G1)TP2, ψ30= (1−d)R2+ (P1

£L¯ 0¤

)TP11(P1

£L¯ 0¤ ).

(18) This matrix inequality is not an LMI because of the multiplication of matrix variables. Con- sidering ψ20 and ψ30, the Schur complement can remove these nonlinearities but forψ10 Lemma 1 is required. As ψ10 must be negative definite to have a solution to the problem (17), the use of Lemma 1 is possible. Applying it twice to ψ10, the nonlinear condition can be expressed as:

Ψ0 A¯T11YT A¯T11YaT Ψ2 0

ψ2 0 0 0

ψ3 0 0

Ψ3 −P2T G1

P1

<0. (19)

·

(1d)R2

h¯

LTP1

0

i

P1

¸

<0. (20) where

11= A11+ ¯LA211

ψ2= −Y −YT+hR1

ψ3= −Ya−YaT +hR1a

Choosing Y = P1, Ya = P1 and defining W = P1L, the LMI condition from the Theorem ap-¯ pears. Then, if (11) and (12) are satisfied, (19) and (20) are also satisfied. Finally the error dy- namics are asymptotically stable and converge to the solutione(t) = 0.

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4. DYNAMIC PROPERTIES OF THE OBSERVER

Corollary 1. With the observer design in Theo- rem 1, an ideal sliding motion takes place on S0={¯e2= 0} in finite time.

Proof.Consider the Lyapunov function:

V2(t) = ¯eT2(t)P2¯e2(t) (21) Differentiating (21) along the trajectories of (10) yields:

2(t) = ¯eT2(t)(GTl P2+P2Gl)¯e2(t) +2¯eT2(t)P2

TTν+A211(t) +G11(t−τ(t)) +G1L¯e2(t−τ(t))

+D1ζ(t) +ξ(t)].

Noting that Gl is Hurwitz and (6), the following inequality holds:

2(t)≤ 2kP2¯e2(t)k[kA211(t) +G11(t−τ(t)) +G1L¯e2(t−τ(t))k −γ].

From Theorem 1, the errors e¯1 and e¯2 are asymptotically stable. There thus exists an in- stant t0 and a real positive number δ such that

∀t≥t0, kA21¯e1(t) +G1¯e1(t−τ(t)) +G1L¯e2(t− τ(t))k ≤γ−δ. This leads to:

∀t≥t0, V˙2(t) ≤ −2δkP2¯e2(t)k

≤ −2δp

λmin(P2)p

V2(t). (22) where λmin(P2) is the lowest eigenvalue of P2. Integrating the previous inequality shows that a sliding motion takes place on the manifold S0 in finite time.

5. EXAMPLE

Consider the system with time-varying delay (4) and:

A11=

· 0 0 0 1

¸

, A12=

·

1 0 0 0.1

¸

, A21=

· 2 3 2 1

¸

, A22=

·

1 0 0 1

¸

, G1=

· 0 0 0.1 0.21

¸

, G2=

· 0 0 0.2 1

¸

, T =

· 1 0 0 1

¸

, Gu=

· 0 1

¸

, D1=B1=B2=

· 0 0

¸

,

The delay is chosen as τ(t) = τ01sin(ω1t)), with τ0 = 0.225s, τ1 = 0.075 and ω1 = 0.5s−1. The control law is

u(t) =u0sin(ω2t)

with u0 = 2and ω2 = 3and the Hurwitz matrix Glis

·−5 0 0 −3

¸ .

Using Theorem 1, the following observer gain is obtained:

L¯ =

·−0.1144 0.0280

¸

Since the system (4) is open loop stable, its dynamics are bounded. Thus the functionα2(t,xˆ1, x2, u)can be chosen as a constantK= 4.

The simulation results are given in the following figures. Figure 1 shows the observation errors.

Figures2and3show the comparison between the real and observed states.

0 2 4 6 8 10

−14

−12

−10

−8

−6

−4

−2 0 2 4 6

time

ex 1

(t) ex

2

(t) ey

1

(t) ey

2

(t)

Fig. 1. Observation errors for τ0 = 0.225 and τ1= 0.075

0 5 10 15 20

−10

−5 0 5 10 15

time

x11(t)

^x11(t) x12(t)

^x12(t)

Fig. 2.x1 andxˆ1

Figure 1 shows the system enters a sliding motion at time t = 2.8s. The unmeasured variables converge asymptotically to0.

6. CONCLUSION

The problem of designing observers for linear sys- tems with non small and unknown variable delay on both the input and the state has been solved in this article. Delay-dependent LMI conditions have been found to guarantee asymptotic stability of the dynamical error system. The conditions only

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0 2 4 6 8 10

−20

−15

−10

−5 0 5 10 15 20 25 30

time

y11(t) y12(t)

^y11(t)

^y12(t)

Fig. 3.x2andxˆ2

depend on the real delay definition and do not require any estimated or computational delay. In addition, the dynamic properties of the proposed observer can be characterized.

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