Design of observers for lipschitz nonlinear systems using LMI

Copyright © IFAC Nonlinear Control Systems. Stuttgart, Germany, 2004

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DESIGN OF OBSERVERS FOR LIPSCHITZ NONLINEAR SYSTEMS USING LMI A. Alessandri·

• Institute of Intelligent Systems for Automation ISSIA-CNR National Research Council of Italy Via De Marini 6, 16149 Genova, Italy angelo~ge.issia.cnr.it

Abstract: The problem of constructing full-order state estimators for a class of systems with Lipschitz nonlillearities is addressed. By using Lyapunov functions and . ~unctionals, c?nditions have been found that guarantee the asymptotic stablhty of the estImation error. Such conditions can be conveniently expressed by means of Li~ear Matrix Inequalities (LMIs). Simulation results are reported to show the effectIveness of the proposed observer design methods. Copyright © 2004 [FAC Keywords: nonlinear state observer, Lyapunov function, Lyapunov functional, observer design, LMI.

vide a linear error dynamics in the transformed space (Krener and Isidori, 1983; Bestle and Zeitz, 1983; Krener and Respondek, 1985; Keller, 1987). This greatly simplifies the design of the observer but requires one to solve a set of partial differential equations to obtain the transformation (see also (Marino and Tomei, 1995)). A different observer structure was proposed in (Gauthier et al., 1992), where the nonlinearity in the error dynamics is compensated for by choosing a sufficiently high gain (Gauthier and Kupka, 1994). Thus, such estimators are usually called high-gain observers and used in output feedback control (see, for example, (Khalil, 1996)). A similar approach to highgain observer design was reported in (Ciccarella et al., 1994).

1. INTRODUCTION

An important subject in control theory is the design of state observers for nonlinear systems. A difficulty in dealing with such a problem concerns the construction of a Lyapunov function to ensure the stability of the estimation error. If an observer is found that is only locally stable, the above difficulties can be considerably reduced. For example, the popular extended Kalman filter (EKF, for short) has been proved, up to now, to be locally stable when it is used as an observer for discrete-time nonlinear systems (Boutayeb et al., 1997). Globally stable observers are preferable, even though it is well-known that local observers like the EKF perform quite well in many different applications.

The difficulties in proving the convergence of an estimator are due to the need of dominating the nonlinearities. One can take advantage of the specific types of nonlinearities (Arcak and Kokotovic, 2001; Fan and Arcak, 2003) or of the possible triangular system structure (Hou et al., 2000). In (Shim et al., 2003) passivity is used to construct observers for nonlinear systems

The literature on state observers is huge, hence, for the sake of brevity, we shall refer only to nonlinear continuous-time systems. The first attempts to construct observers for nonlinear systems were reported in (Thau, 1973; Kou et al., 1975). Later on, much attention was devoted to approaches that, by performing a state transformation, pro-

459

pair (A, C) is observable. In addition, we need to assume that f(', t) is locally Lipschitz, uniformly with respect to t, to ensure the uniqueness of the solution of the differential equation in (1). Finally, the solution has to be well-defined "It ~ O.

that are ISS (input-to-state stable) when the measurement disturbances are seen as the input and the estimation error as the output. In this context, the present work reports new results on the stability of the estimation error of observers for systems with Lipschitz nonlinearities. The convergence of the estimator has been proved by using both quadratic Lyapunov functions and Lyapunov functionals. The former are extensively used to prove the stability of the estimation error (see the previous references). The interest in the latter has arisen in the area of delay systems (see, for an introduction, (Boukas and Liu, 2002; Gu et al., 2003)), as well as in robust control (Balakrishnan, 2002). The construction of the proposed observers can be made by using LMIs (Boyd et al., 1994).

Remark 1. In the following, we shall refer to system equations (1); however, the proposed methods for observer design can be applied to a more general class of nonlinear observable systems that are diffeomorphic to (1). Specifically, in (Shim et al., 2001) necessary and sufficient conditions are provided to ensure the existence of a diffeomorphism that transforms quite a general nonlinear system into a system like (1) with

A = block diag (AI,"', Am)

The paper is organized as follows. Section 2 gives some basic definitions and introduces the notation. Section 3 is devoted to the description of the observer, stability analysis, and design. Section 4 illustrates the results obtained by simulations. Some conclusions and prospects for the future work are briefly discussed in Section 5.

010

A ~ --

0

T

<=?

T > 0, R - 8T

<=?

R > 0, T - 8 T R- I 8 > 0

8

C

= blockdiag (Cl"'"

Ci

= (1 0

... 0) E

Cm)

jRn,

L ni = n. Thus, an observer for (1) becomes

A full-order constant-gain observer for (1) is given by

i=Ax+f(x,t)+L(y-Cx)

t~O

(2)

where x(t) E Rn is the estimate of x(t) at time t and LE R nxm is a gain matrix. The dynamics of the estimation error e(t) ~ x(t) - x(t) is given by

e = (A -

>0

LC) e + f(x, t) - f(x, t)

t ~ 0 . (3)

In order to find a gain L that provides a convergent estimation error, most approaches rely on a quadratic Lyapunov function V = eT Pe, where P is a symmetric positive definite matrix. From (1) and (2), we obtain

v = eT

3. MAIN RESULTS

[( A - LC) T P

+ P (A -

+2 [f(x,t) - f(x,t)]T Pe

Consider a nonlinear system represented by t~O

m..

01 00

i=1

where R, 8, and T are matrices of appropriate dimensions.

x = A x + f(x, t) y=Cx

E m>n,xn,

a stable estimator in the original state space if one uses the inverse of the aforesaid diffeomorphism.

- 11· I1 is the Euclidean norm; - I denotes the identity matrix of appropriate dimension; - for a square matrix 8, 8 > 0 (8 < 0) means that this matrix is positive definite (negative definite); - Amin (8) and A max (8) are the minimum and maximum eigenvalues of the symmetric positive or negative definite matrix 8, respectively; - the Schur lemma can be expressed as follows: _I

)

. ..

m

and

The following notation and definitions will be used throughout the paper:

8) >

:::

[ 000 000

2. NOTATION AND DEFINITIONS

R ( 8T T

~~~

LC)] e (4)

Of course, the difficulty in proving that V is negative definite results from the second term on the right-hand side of (4). This is the key issue to resolve, and an interesting discussion of this question can be found in (Praly, 2001). If the function f is such that

(1)

where x(t) E X ~ Rn is the state vector, y(t) E ~ jRm is the output vector; moreover, the matrices A E jRnxn and C E jRmxn are such that the

Y

[j(x,t) - f(x,tW Pe:S 0

460

(5)

the stability analysis is simplified and, in order to ensure a stable error dynamics, LMI conditions can be derived for specific nonlinearities (Tibken, 2001; Arcak and Kokotovic, 2001; Howell and Hedrick, 2002; Fan and Arcak, 2003). In this paper, we shall refer to system with Lipschitz nonlinearities by assuming the following.

analysis via quadratic Lyapunov functions, it is easy to verify that the convergence of the estimation error to zero is exponential.

o By means of the change of variables L = p- 1Y, (6) reduces to the LMI in P, Y, and c

Assumption 1. The function f : X x [0,00) --+ jRn is Lipschitz in x, uniformly with respect to t, i.e., there exists k f > 0 such that

ATP - CTy (

+ PA

- YC

+ c k} I

P

P) 0 -cl < .

(8)

VXl,X2

This condition simplifies the design of the estimator, which otherwise need the solution of a complex algebraic Riccati equation (see, among others, (Raghavan and Hedrick, 1994; Raj amani, 1998; Rajamani and Cho, 1998; Alessandri, 2002)).

EX.

We have the following result. Theorem 1. Consider observer (2) for system (1) and let Assumption 1 hold; then, if there exist c > 0, a gain matrix L, and a symmetric positive definite matrix P such that

(

In the following, we shall derive a different result that is based on a Lyapunov functional. Theorem 2. Consider observer (2) for system (1) and let Assumption 1 hold; then, if there exist Q > 0, € > 0, a gain matrix L, and symmetric positive definite matrices P and Q such that

(A-LC)TP+P(A-LC)+Ck}1 P) 0 P -cl < ,

(6) then the estimation error converges exponentially to zero.

!.- > k}

Q

+ ~l

Q

Q

(9)

o Proof of Theorem 1. Using Assumption 1 and the Lyapunov function V = eT Pe, in (4) we can apply the Young inequality

(

(A-LC)TP+P(A-LC)+QP P) 0 P -cl < , (10)

then the estimation error converges asymptotically to zero.

2[f(x,t) - f(X,t)]T Pe

t)IIIIPell -::: cllf(x, t) 2 2 f(x, t)11 + ~ liP ell -::: ck} Ilx - xl1 c

-::: 21If(x, t) - f(x, -

o

2

+~IIPeI12=eT(Ck}I+~pp)e

Proof of Theorem 2. Let us consider the following Lyapunov functional

(7)

where c is any positive real constant, thus obtaining

J t

V = eT Pe

(Q) e- ot

+ Amin

Ilf(x(T), T)

o

V -::: eT [( A -

+~pp]e

LC) T P

+ P (A -

LC)

+ c kJI

(11) Consider the derivative of (11):

.

Therefore, the dynamics of the estimation error is stable if there exists P = pT > 0 such that (A - LC)T P

+ P (A -

LC)

+ c k} I +

1 - PP < c

J t

o.

-

Q

Amin

(Q) e- ot

2 Ilf(x(T), T) - f(X(T), T)11 dT

o

+ Amin (Q)

Using the Schur lemma in the last inequality, V turns out to be negative definite by imposing (6). Moreover, basing Oll. standard results of stability

e- ot IIf(x, t) - f(x, t)11

2

By using (3), (11), and the fact that e- ot < Vi ~ 0, the above equation yields

461

1,

V::::: eT [(A - LC)T P + P(A - LC)] e + eT P[f(x, t) - f(x, t)] + [f(x, t) - f(x, t)]T Pe - 0 V + 0 eT Pe + Amin (Q) Ilf(x, t) - f(x, t) 11 2 As V 2:

eT

(12)

observers for systems having Lipschitz nonlinearities, where the stability conditions are expressed by means of algebraic lliccati equations. Moreover, it is easy to verify that (9) and (15) imply (6), but the converse is not true. Finally, note that a necessary condition for (6) and (10) to hold is that the pair (A, C) is observable.

(13)

Remark 3. Assumption 1 may be restrictive as

Pe, from (9) we have

k2 V 2: eTpe 2: -.1.. (eTQe o

+clleIl 2 )

local Lipschitzianity is generally easier to satisfy. However, if the system admits an invariant set, the trajectories lie in a bounded set, which enables to construct a Lipschitz extension of the system to all JRn (Farza et al., 2004).

Assumption 1 provides k f

2:

Ilf(x, t) - f(x, t)11

-J.

IJell

and, as Amin (Q) we obtain

' e-;-

0

lIell : : : eTQe, 'ie E JR n , from

(13) 4. A NUMERICAL EXAMPLE

1(

T

,2

V 2: e Pe 2: -; Ilf(x, t) - f(x, t)11

+ c Ilf(x, t) 2:

~ (Amin o

- f(x, t)1I

(Q)

2

WQW eT

-0

Consider the model of a single link robot with a flexible joint rotating in a vertical plane (Marino and Tomei, 1995). The dynamics is described by

)

+ c) Ilf(x, t) - f(x, t) 11 2

After multiplying for gives

e

.

(14)

and a little algebra, (14)

Xl

= X2

.

k = - -JkX l + -X3 J

X2

l

Amin (Q) Ilf(x, t) - f(x, t)1I

- f(x, t) 11 2

2

-

0

X3 = X4 . k

V::::: -cllf(x, t)

X4

.

[(A - LC)T P

- c [f(x, t) - f(x, tW [J(x, t) - f(x, t)] Thus, V turns out to be negative definite if (10) is satisfied.

-X3 J2

1

+ -u J 2

We assumed to measure Xl and X3 (Le., y ~ (Xl,X3)T). The Lipschitz constant turned out to be equal to 0.3691.

o Note that (9) is an LMI and (10), for a fixed can be expressed by the LMI

Jl

where Xl and X2 are the link displacement and its velocity, respectively; X3 and X4 are the rotor displacement and its velocity, respectively. The parameter J l , J 2 , k, l, and 9 are the link inertia, the rotor inertia, the elastic constant, the position of the center of mass, and the gravity acceleration, respectively; they were chosen equal to 30 Kg/m 2 , 30 Kg/m 2 , 1 N/m, 1 m, and 10 m/s 2 , respectively.

+ P(A - LC) + op] e + eT P [f(x, t) - f(x, t)l + [f(x, t) - f(5;, tW Pe eT

k -

2

Using the above inequality in (12), we have

V:::::

= -J X l

mgl. -SIllXl

l

The system in in the block form of (1) and, using the procedure feasp of the Matlab LMI Toolbox (Gahinet et al., 1985), we solved (8):

0,

ATP_CTyT +PA-YC+oP P ) <0 (

P

-cl (15)

where c, Y, and P are the unknown and L p-ly.

c = 1.5755

=

P

Remark 2. Condition (8) and conditions (9) together with (15) are LMIs that may be difficult to satisfy for large values of kf. Such difficulties applies also to methods proposed in (Raghavan and Hedrick, 1994; Rajamani, 1998; Rajamani and Cho, 1998; Alessandri 2002) to construct

=

L =

(

0) 4.4045 3.9894 0 0 4.4045

(16)

03.9894

0.5453 -0.3701 0 0) -0.3701 0.5453 0 0 o 0 0.5453 -0.3701 ( o 0 -0.3701 0.5453

Once chosen 0 = 1, we solved the system of LMI given by (9) and (15), thus obtaining

462

8r---~--~--~--~----.

1.5 .. #1>;

6

4

-

state state estimate 1 state estimate 2

468

2

o

10

state state estimate 1

_L·~~:..::..~~~====s:;:ta=t=e=e=s=tim::::::a=te=2:::J

-1

_2l--~-...;...':::==:::::;:=:==::;:::==:::J

o

-

-0.5

2

time [sI

468

10

time [sI

10 .---~--~--~--~------, 1.5

/~

0.5 ,.

..

:

6

4

..

...

2/~"

,

-

.....:.,

..

-:-..._~

;-....~~- .;.;..;.

................

I

state state estimate 1 state estimate 2

4

6

8

-

-0.5

_2l--~-...;...':::==:::::;:=:==::;:::==:::J

2

,, ,

O!" ..

Of

o

" t'\

: .......•........: ""-~ .. ... . . . ~ . . .. ........•. / : : .

:

....

8

-1

time [sI

==:::;:=========::::=J

L'~ .. ~.'.:;.:.'.~ .. ~ .. ~ .. .:.;:. ... 2 4

o

10

state state estimate 1 state estimate 2

6

8

10

time [sI

Fig. 1. State variables and their estimates.

E

= 80.9686

p =

Q=

L

=

(

2.4058 0) 3.1259 0 0 2.4058

Lipschitz nonlinearities. The convergence of the estimation error of the proposed observer has been studied by means of both Lyapunov functions and functionals, and stability conditions have been found that can be expressed using LMIs. This enables one to accomplish the observer design in a simple and straightforward way, as shown by means of simulations. However, these conditions are difficult to be satisfied for larger values of the Lipschitz constant. This is a drawback as compared with the high-gain approach, which enables to design observers for systems with Lipschitz nonlinearities having an arbitrarily large Lipschitz constant (Gauthier et al., 1992; Gauthier and Kupka, 1994).

(17)

o 3.1259

6.6826 -23.4221 0 0) -23.4221 25.9034 0 0 o 0 78.4976 -23.4221 ( o 0 -23.4221 25.9034 68.1126 -17.8247 0 0) -17.8247 27.6065 0 0 o 0 68.1126 -17.8247 . ( o 0 -17.8247 27.6065

The results for a simulation run with x(O) = (l,l,l,l)T, x(O) (-1, -1, -1, _l)T, and u(t) = sin(t) are plotted in Fig. 1, where state estimates 1 and 2 refer to the choice of the gains in (16) and (17), respectively.

Lyapunov techniques have allowed to construct globally convergent observers for systems with Lipschitz nonlinearities without exploiting the special forms of the matrices A and C, which can be assumed to be in observable canonical form. In this case, as the subject of future investigations, one may take advantage of such special structures of A and C to derive less conservative stability

5. CONCLUDING REMARKS

The contribution of this paper lies in the construction of observers for a class of systems with

463

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conditions for the estimation error (for example, using backstepping, see (Krener and Kang, 2003)).

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