H∞ observer design for a class of nonlinear discrete systems

European Journal of Control (2009)2:157–165 # 2009 EUCA DOI:10.3166/EJC.15.157–165

H1 Observer Design for a Class of Nonlinear Discrete Systems Ignacio Pen˜arrocha1,, Roberto Sanchis1, and Pedro Albertos2, 1 2

Departament d’Enginyeria de Sistemes Industrials i Disseny, Universitat Jaume I, Campus de Riu Sec, 12071 Castelló, Spain; Departamento de Ingeniería de Sistemas y Automática, Universidad Politécnica de Valencia, 22012, E-46071, Valencia, Spain

In this paper, the problem of observing the state of a class of discrete nonlinear system is addressed. The design of the observer is dealt with using H1 performance techniques, taking into account disturbance and noise attenuation. The result is an LMI optimization problem that can be solved by standard optimization techniques. A design strategy is proposed based on the available disturbances information. Keywords: Lipschitz nonlineal systems; Linear matrix inequalities; H1 performance analysis; Discrete-time systems; Nonlinear observer.

1. Introduction The design of observers for non linear discrete systems has been studied by several authors in the literature. The idea behind is to obtain information about internal variables which are not directly available at the output, they are corrupted by noise or, in any practical situation, they are not accessible any time they are required. As usual in the nonlinear setting, there is no general solution for any nonlinearity. Observability conditions, as reviewed in [2], must be assumed. The simplest general assumption is to consider that the state and measurement functions satisfy some conic condition, [14]. Also, model uncertainties, noise and disturbances are assumed to be generally bounded. Most of the published works deal with a class of linear systems with

E-mail: [email protected] E-mail: [email protected] Correspondence to: P. Albertos, E-mail: [email protected]

additive nonlinearity characterized by a non linear term in the state and output equation, that are assumed to fulfill a Lipschitz condition. The use of linear matrix inequalities (LMI) has made possible to address the design of observers for that class of systems surpassing the drawbacks of previous approaches, where a high gain was needed to compensate for the non linear term, as initially proposed in [3]. Another alternative is the use of proportional/integral observers [1], that is, observers where the corrective action is proportional to the observation error and its integral, leading to a more complex observer dynamics. This approach has been applied for single output continuous time uniformly observed systems [4]. There are three ways of considering the Lipschitz condition in the additive nonlinearity in order to incorporate it in the LMI. The simplest one is the scalar form, consisting of k f ðx1 Þ  f ðx2 Þk  kx1  x2 k:

ð1Þ

This condition is taken into account, for example, in [15]. The drawback of this approach is that it can lead to very conservative results, or even to the non feasibility of the LMI for large . A more complex form of the Lipschitz condition consists of incorporating a matrix in the form k f ð1 Þ  f ðx2 Þk  kFðx1  x2 Þk:

ð2Þ

The advantage of this approach is that if the matrix F is adequately chosen, based on the form of the function f(x), the resulting LMI is less conservative and more likely to have a feasible solution. This approach is used, for example, in [10], [11] or [13]. Received 29 March 2008; Accepted 16 November 2008 Recommended by D. Normand-Cyrot and S. Monaco

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The most complex form of the Lipschitz condition (proposed in [16]) assumes that there are known upper and lower bounds on the elements of the Jacobian matrix of f (x), as aij 

@fi  bij : @xj

ð3Þ

This idea, and the use of the differential mean value theorem (DMVT), allows to express the condition as ! n X hij Mij ðx1  x2 Þ ð4Þ f ðx1 Þ  f ðx2 Þ ¼ i;j¼1

where Mij are empty matrices, except the element i, j that is 1, and where the terms hij are time varying but bounded by aij  hij  bij. Based on this condition, the error dynamics can be expressed as a Linear Parameter Varying system, with bounded parameters, that can be taken into account easily in the LMIs by considering the convex hull. The advantage of this approach is that less conservative results can be obtained. The important drawback is that the number of LMI to be solved simultaneously can grow exponentially with the system order (for a system of order n with p outputs the 2 number of LMI can be up to 2n þnp ). In the present paper, the design of observers for non linear discrete systems is addressed. The Lipschitz condition in the form of matrix F is considered, and the disturbance and noise attenuation are taken into account in order to minimize the norm of the estimation error. The use of Lipschitz matrix conditions is similar to the one developed in [10] and extended in [12] (where time delays and uncertainties are also considered) for continuous systems. In [13], the discrete observer design based on the matrix Lipschitz condition is studied, but the disturbances are not taken into account. In [15], on the other hand, the discrete case is also studied, but with the scalar Lipschitz condition and no disturbances consideration. With respect to the approach developed in [16], the presented work has the advantage of reaching a simple to solve LMI optimization problem, that consists of one single LMI, independently of the system order 2 (compared to the up to 2n þnp simultaneous LMI that must be solved in [16]). The drawback is that the result of the proposed approach is more conservative than the one presented in [16], and therefore, some problems solved by that approach could lead to an unfeasible LMI if the approach of the present paper is used. On the other hand, the proposed approach can be extended to the case when the outputs are measured scarcely and irregularly in time, increasing the LMIs to be solved to a number equal to the possible

measuring scenarios, while extending the approach of [16] to this case could lead to a really huge and almost unsolvable number of LMIs. In summary, the contribution of the paper is the design of a discrete observer for H1 disturbance attenuation, based on a matrix Lipschitz condition in the nonlinear terms, that has the advantage of a less conservative result if compared to previous works that use scalar Lipschitz conditions, and the advantage of a much simpler (lower computer cost) optimization problem to be solved compared to the approach in [16]. The outline of the paper is as follows: first, the problem is introduced, including the plant and observer equations, then, the prediction error dynamics is obtained, and the main result (the H1 design of the observer) is developed. Some examples illustrate the applicability of the proposed approach, compared with other works, and finally the main conclusions are summarized.

2. Problem Statement 2.1. Plant and Observer Consider a discrete nonlinear time-invariant MIMO system described by the equations x½t þ 1 ¼ Ax½t þ fðx½t; u½tÞ þ w½t;

ð5aÞ

y½t ¼ Cx½t þ hðx½tÞ þ v½t:

ð5bÞ

where x 2 Rn is the state, u 2 Rnu is the control input vector, y 2 Rny is the measured output variables, w[t] 2 Rn is the state disturbance and v[t] 2 Rny is the measurement noise. The pair (A, C) is assumed to be observable. The functions f (  ) : Rn þ nu ! Rn and h(  ) : Rn ! Rny are known nonlinear functions that are assumed to fulfill the Lipschitz condition, i.e., kfðx1 ; uÞ  fðx2 ; uÞkp  kF  ðx1  x2 Þkp ;

ð6Þ

khðx1 Þ  hðx2 Þkp  kH  ðx1  x2 Þkp ;

ð7Þ

for any vectorial norm p. In order to estimate the state from the output measurements, a model based observer is proposed. The state is initially estimated in open loop, leading to x½t  1 þ fð^ x½t  1; u½t  1Þ; x^½t  ¼ A^

ð8aÞ

This estimation is updated with the measurement as x½t ÞÞ: x^½t ¼ x^½t  þ Lðy½t  C x^½t   hð^ ð8bÞ where L is the gain matrix to be designed.

Discrete Nonlinear H1 Observer

159

The dynamic of the state observer depends on the gain matrix, L, that must be designed to assure predictor stability and a proper attenuation of the disturbances and sensor noises.

Lemma 1: [9] For any pair of vectors x, y 2 Rn and any positive definite matrix P 2 Rn  n, the following condition holds

2.2. Prediction Error

Lemma 2: Assume that x is a vector and A, B, P are matrices of proper dimensions, such that P is symmetric and positive definite (P ¼ PT > 0). Assume that y is a vector that satisfies

In order to design a predictor, that is, the predictor gain L, with these properties, the prediction error dynamic equation must be obtained, that is, an explicit relationship between prediction error at measurement instants t and previous one t  1 must be obtained. If the process equations are introduced in the state estimation equations, (8), the estimation error can be expressed as

kyk  kFxk;

ð13Þ

being F a matrix of proper dimensions. Then, for any ">0 ðAx þ ByÞ> PðAx þ ByÞ  x> Wx;

x~½t ¼ A~ x½t  1 þ fðx½t  1; u½t  1Þ  fð^ x½t  1; u½t  1Þ

ð14Þ

with

x½t ÞÞ  LðC~ x½t  þ hðx½tÞ  hð^ þ w½t  1  Lv½t

   1 W ¼ A> PþPB "IB> PB B> P Aþ" F> F:

where x~½t ¼ x½t  x^½t. As it is observed, due to the presence of f and h, it is not possible to explicitly write x~½t as a function of x~½t  1. In order to simplify the next mathematical developments, the following notation is introduced

ð15Þ Proof 1: Expanding the left expression in [14] one obtains ðAx þ ByÞ> PðAx þ ByÞ ¼ x> A> PAx þ 2x> A> PBy þ y> B> PBy:

x~½t  ¼ x½t  x^½t ; ~ ¼ fðx½t; u½tÞ  fð^ f½t x½t; u½tÞ;

Adding and subtracting "yT y on the right term it yields

~   ¼ hðx½tÞ  hð^ x½t Þ: h½t The predictor error dynamics at sampling instants can then be written as   ~   þ v½t ; x½t  þ h½t ð9Þ x~½t ¼ x~½t   L C~ where the open loop estimation error (~ x½t ) can be written as a function of the information at the previous control period as x½t  1 þ f~ ½t  1 þ w½t  1; x~½t  ¼ A~

2x> y  x> Px þ y> P1 y:

ð10Þ

~   fulfill and the functions f~½t and h½t k f~½tk  kF~ x½tk;

ð11Þ

~  k  kH~ kh½t x½t k:

ð12Þ

The design objective of the predictor is to find a gain L that stabilizes the observer and assures a proper attenuation of state disturbance and measurement noise. For the next section some previous results must be obtained.

ðAx þ ByÞ> PðAx þ ByÞ ¼ x> A> PAx   þ 2x> A> PBy  y> "I  B> PB y þ "y> y: Applying lemma 1, it leads to ðAx þ ByÞ> PðAx þ ByÞ  x> A> PAx  1 þ x> A> PB "I  B> PB B> PAx þ "y> y: Taking into account (13) it is easy to obtain ðAx þ ByÞ> PðAx þ ByÞ   1  x> A> PA þ A> PB "I  B> PB B> PA  þ"F> F x: Lemma 3: Assume x and u are vectors and B, P matrices of proper dimensions (such that P ¼ PT > 0). Then, for any G  0 ðx þ BuÞ> Pðx þ BuÞ  u> G2 u  x> Wx;

ð16Þ

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with

I. Pen˜arrocha et al.

 1 B> P: W ¼ P þ PB G2  B> PB

ð17Þ

Proof 2: Expanding the left expression in (16) it is easy to obtain

It is also demonstrated that if (18) holds, then the Lyapunov function of the state estimation error decreases, proving the convergence of the state estimation algorithm. Consider the index 1   X x~½t> x~½t v½t> G2v v½t  w½t> G2w w½t :

J¼

ðx þ BuÞ> Pðx þ BuÞ  u> G2 u ¼ x> Px   þ 2x> PBu  u> G2  B> PB u:

t¼0

Taking the Lyapunov function V½t ¼ Vð~ x½tÞ ¼ x~½t>P~ x½t and assuming null initial conditions, one can write

Applying lemma 1, it leads to ðx þ BuÞ> Pðx þ BuÞ  u> G2 u    1 B> P x:  x> P þ PB G2  B> PB

J

1  X x~½t  1> x~½t  1  v½t  1> G2v v½t  1 t¼1

  w½t> G2w w½t þ V½tjt¼1  V½tjt¼0 1  X x~½t  1> x~½t  1  v½t  1> G2v v½t  1 ¼

3. H1 Design Theorem 1: Consider the predictor algorithm ð8Þ applied to system ð5Þ. For some given Gw, Gv > 0, assume that there exist some matrices P ¼ PT 2 Rn  n, X 2 Rn  ny and some scalars "f, "h > 0 such that the next inequality fulfills 2 P PA  XCA P  XC P  XC   6 P  I  f F> F 6 > > h A H H h A> H> H 6 6 h A> H> HA 6 6 h H> H  f I  h H> H 6 6   G2w  h H> H 6 4        with Gw ¼ diagfw1 ; . . . ; wn g: Then, defining the predictor gain matrix as L ¼ P 1X, under null disturbances, the prediction error converges to zero asymptotically, and, under null initial conditions, the following condition holds v½tk22 þ kGw

  w½t> G2w w½t þ V½t ;

X

X

3

7 0 7 7 7 7 0 0 7  0; 7 0 0 7 7 h I 0 5  G2v 0

ð18Þ

where V½t ¼ V½t  V½t  1. Substituting V½t by

Gv ¼ diag fv1 ; . . . ; vny g

k~ x½tk22  kGv

t¼1

w½tk22 :

ð19Þ

Proof 3: In order to prove the theorem, a cost index including estimation error and disturbances is created. That index is bounded using the Lyapunov function of the state estimation error. Introducing the state estimation error dynamics in the index bound it is demonstrated that if LMI (18) holds, then the cost index is negative and therefore, condition (19) holds.

V½t ¼ x~½t> P~ x½t  x~½t  1> P~ x½t  1   ~    Lw½t > ¼ ðI  LCÞ~ x½t   Lh½t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ?

P ð?Þ  x~½t  1> P~ x½t  1; lemma 3 can be applied to eliminate the term v½t> G2v v½t, leading to J

1 X

x½t1v½t1> G2v v½t1 x~½t1> ðIPÞ~

t¼1

! >  ~   Pv ð?Þ : þ ðILCÞ~ x½t Lh½t |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ?

Discrete Nonlinear H1 Observer

with

161



Pv ¼ P þ PL G2v  L> PL

1

L> P:

ð20Þ

~   it yields Applying lemma 2 to eliminate the term h½t 1  X x~½t  1> ðI  PÞ~ x½t  1 J t¼1

w½t  with

1> G2w w½t

  1 þ x~½t  Ph x~½t  :  >



2 6 6 6 6 6 6 6 4

P  f F > F  I h A> H> HA

! h A> H> H

h H> HA

f I  h H> H

h H> HA

h H> H

0

0

1 X

3 A> Ph A þ f F> F  P þ I A> Ph A> Ph 4 Ph A Ph  f I Ph 5 0: Ph A Ph Ph  G2w

ð21Þ Substituting Ph as a function of Pv (using (21)) and applying Schur complements it leads to 3 2 > 3 2 > 3> h A> H> H 0 7 A ðI  LCÞ> A ðI  LCÞ> 7 7 6 ðI  LCÞ> 7 6 ðI  LCÞ> 7 7 6 7 7 6 6 7 Pv 6 7  0: h H> H 0 7 7 4 ðI  LCÞ> 5 4 ðI  LCÞ> 5 7 G2w  h H> H 0 5 L> L> 0 h I

Substituting the open loop prediction error by x~½t  ¼ A~ x½t  1 þ f~½t  1 þ w½t  1 and applying lemma 3 to eliminate the term w½t  1> G2w w½t  1 it yields J 

Substituting Pw as a function of Ph (using (22)) and applying Schur complements leads to 2

   1 Ph ¼ ðI  LCÞ> Pv þ Pv L h I  L> Pv L L> Pv ðI  LCÞ þ h H> H:

Substituting Pf as a function of Pw (using (24)) and applying Schur complements, the previous condition is equivalent to condition >

A Pw A  P þ I þ f F> F A> P w 0: Pw A Pw  f I

x½t  1 x~½t  1> ðI  PÞ~

t¼1

x½t  x~½t  1> P~ x½t  1; V½t ¼ x~½t> P~

! >

þ ðA~ x½t  1 þ f~ ½t  1Þ Pw ð?Þ ; |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ?

with  1 Ph : Pw ¼ Ph þ Ph G2w  Ph

Substituting Pv as a function of P using (20), applying Schur complements twice, and taking into account that PL ¼ X it finally leads to (18). Applying the same mathematical manipulations as before, the increment of the Lyapunov function, i.e.,

ð22Þ

will be negative if 2 3 P  PðI  LCÞA  PðI  LCÞ PL 6 7 P  f F> F 6 h A> H> H 0 7 > > 6 7  0; A H HA  h 6 7 > 4  f PI  h H H 0 5    h I

Applying now lemma 2 it yields that 1   X x~½t  1> ðI þ Pf  PÞ~ J x½t  1 ;

ð23Þ

t¼1

with

  1  Pf ¼ A> Pw þ Pw f I  Pw Pw A þ f F> F:

ð24Þ Condition (19) holds if J < 0, but this will always be true if I þ Pf  P 0:

holds. It must be noted that the matrix in inequality (25) can be formed taking the first, second, third and fifth blocks of rows and columns of matrix in LMI (18) with X ¼ PL and adding matrix diag{0, I, 0, 0}, that is a semi-definite matrix. This implies that if (18) holds, (25) holds, and then, the Lyapunov function of the state observed error decrease and the estimation error of algorithm (under null disturbances and noises) decreases exponentially to zero. Remark 1 (Design procedure): If the ‘2 norm of disturbance and noise measurement signals are considered

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to be known, the upper bound on k~ x½tk2 can be reduced to a minimum value by minimizing ny X

v2i kvi ½tk22 þ

n X

w2 i kwi ½tk22

i¼1

i¼1

along the variables  vi,  wi, P and X that satisfy the LMI ð18Þ. This convex minimization problem can be easily addressed using standard LMI solvers (such as Matlab LMI toolbox) that solve problems of the form Minimize

h> x

subject to

MðXÞ 0;

ð26Þ

where h is a constant vector, X denotes the matrix variables, x is a vector with all the components of X, and MðXÞ represents the matrices of the LMI problem. The computation cost of this minimization problem for one simple LMI, such as ð18Þ, is relatively low. First, G2v and G2w must be expressed as matricial variables and v21 ; . . . ; v2nv ; w2 1 ; . . . ; w2 nm must be written as the last components of vector x in ð26Þ. Then, the vector hT is defined as h> ¼ ½0 . . . 0 kv1 ½tk22 . . . kvny ½tk22 kw1 ½tk22 . . . kwn ½tk22 : The previous remark also applies if the RMS norms of the disturbances and measurement noise are known. In that case, the upper bound on k~ x½tkRMS can be minimized if v2 kv½tk2RMS þ w2 kw½tk2RMS

Then the prediction error at instant tk can be expressed as a function of the error at instant tk  1, using a variant observer matrix gain Lk that is a function of Nk. As a result, if there are s possible values of Nk, one can obtain s LMIs to be solved to obtain s matrix gains, LðNkÞ that are precalculated off line and applied as a function of the measurement characteristics, Nk. The authors are finishing the detailed development of the irregular measurement case. On the other hand the extension of the work presented in [16] to the scarce measurement case would lead to a huge and almost unsolvable number of LMIs.

4. Examples In this section some examples will illustrate the applicability of the proposed observer design technique, comparing it with other approaches.

4.1. Example 1 Consider the example of the flexible robot modeled in [5] and [8], being studied [6] and [16] (example 1). If an Euler approximation is applied, the dynamics of the robot can be described by equations 2 6 6 x½t ¼ ðI þ TAÞx½t  1 þ 6 4

is minimized along all variables  v,  w, P and X that satisfy the LMI (18). Remark 2: With respect the practical computation of matrices F and H, a simple general procedure may be to calculate the Jacobian of f and h, and then to obtain the maximum of the absolute values of each one of their elements, in the domain of validity of the involved variables for the specific problem. Those maximums would then form the elements of matrices F and H. For a specific problem, however, tighter matrix bounds F and H could perhaps be obtained through the exploitation of the structure of the particular functions f and h. Remark 3: The proposed approach can be extended to the case when the measurements are taken irregularly and scarcely in time (see [7] for details in the linear case). Assume that the outputs y[t] are measured only every Nk periods, where Nk can take values in a finite integer set of size s, and define tk as the instant when the k-th measurement is taken (hence Nk ¼ tk  tk  1).

0 0 0 3:33T sinðx3 Þ

3 7 7 7 5

þ w½t  1 y½t ¼ Cx½t þ v½t where T is the sampling period and 2

0

6 48:6 6 A¼6 4 0

1

0

1:25 0

48:6 0

19:5 C ¼ ½1 0 0

0 0

19:5

0

3

07 7 7; 15

0

The components of the state represent the angular position of the motor (x1), its angular velocity (x2), the angular position of the link (x3) and its velocity (x4). Vector w is the state disturbance whose components are null except the third one with an assumed norm of kw3 k2 ¼ 0:1T, and v is the measurement noise with an

Discrete Nonlinear H1 Observer

163

assumed norm kvk2 ¼ 0:1. The matrix F that fits the Lipschitz condition is in this case 2

0 60 F¼6 40 0

0 0 0 0

the solution of the LMI problem is unfeasible (applying LMIs on this work with matrix F ¼ 0.33TI), showing that the matrix Lipschitz condition is less conservative than the scalar one. The proposed method calculates the gain as a function of the available information of disturbances and measurement noises. To show this idea, assume now that the system has a smaller measurement noise of kvk2 ¼ 0:01. The resulting observer gain is then

3 0 07 7: 15 0

0 0 0 3:33T

If the method proposed in this work is applied, an observer gain L ¼ ½ 0:9996

16:5691

1:9360

h L ¼ 0:9998 24:0856

5:1833 >

3:3106

6:5132 T :

On the other hand, if the input disturbance is assumed to have an smaller value of kw3 k2 ¼ 0:01T, the resulting observer gain is h L ¼ 0:9996 12:5120 1:3297 4:1024 T :

is obtained. This gain cannot be compared with the one obtained in [16] because here a discrete-time observer is considered, but in Fig. 1 the evolution of the state and its estimate is shown to be similar to the behavior reached in [16]. If the scalar Lipschitz condition is applied, that is:

This shows that the proposed design strategy fits the observer gain to minimize the observation error taking into account the available information of the disturbances and noises.

kfðx½tÞ  fð^ x½tÞk  0:33Tk~ x½tk

2.5 5

5

2 0

x2(t)

x1(t)

1.5 −5

1 −10 0.5

0

0

0.5

1

1.5 time(s)

2

2.5

−15

3

0

0.5

1

1.5 time(s)

2

2.5

3

0

0.5

1

1.5 time(s)

2

2.5

3

2

3

2

1

1 x4(t)

x3(t)

0 0

−1 −1 −2

−2 −3

0

0.5

1

1.5 time(s)

2

Fig. 1. States (solid lines) and estimations (dotted lines).

2.5

3

−3

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4.2. Example 2 Consider the MIMO discrete time non linear system defined by 2 3 0:15 0:2 0:01 6 7 A ¼ 4 0:1 0:9 0:1 5 0:02 0:26 0:8 3 2 0:6 0:4

0:5 1 0:5 7 6 C¼ B¼4 1 0 5 0 1:5 1 0:15 0:9

and assume that the disturbances w and v are vectors of independent white noises of variances 0.2, 0.3, 0.1 and 0.1, 0.1 respectively. In this case the matrices that define the bounds on the Lipschitz conditions of f and h are easy to obtain 2 3 0:05 0:1 0:01 6 7 F ¼ 4 0:02 0:01 0:01 5 0:03 0:01 0:1 0:05 H¼ 0:05 0:1

2

3 0:05 sinðx1 Þ cosð2x2 Þ  0:01 sinðx3 Þ 5; fðx; uÞ ¼ Bu þ 4 0:01 sinðx2 Þ cosð2x1 Þ sinðx3 Þ 2 0:15 sin ð3x3 Þ þ 0:01 sinðx2 Þ cosð3x1 Þ

2

4

1.5

3

0:9

0:1 0:1



0:05 sinð2x1 Þ cosðx2 Þ sinð2x3 Þ hðxÞ ¼ 0:05 cos2 ðx3 Þ þ 0:05 sinð2x2 Þ cosðx1 Þ



2

1

1

0.5

x2[t]

x1[t]

0 0

−0.5

−2

−1

−3

−1.5

−4

−2 −2.5

−1

−5 0

5

10

15

20

25

−6

30

8

0

5

10

15

20

25

30

0

5

10

15 sample

20

25

30

10

6 5 4

y1[t], y2[t]

x3[t]

2 0 −2 −4

0

−5

−10

−6 −8

0

5

10

15 sample

20

25

30

Fig. 2. States, outputs (solid lines) and estimations (dotted lines).

−15

Discrete Nonlinear H1 Observer

Applying the proposed design method, the following observer matrix gain is obtained: 2 3 0:1048 0:01135 L ¼ 4 0:5409 0:2762 5; 0:8134 0:5439 and its implementation leads to the state estimations shown in Fig. 2. On the other hand, if a scalar Lipschitz condition is taken into account instead of the matrix one, the value of the bounding constant would be  ¼ 0.9. In this case, the resulting LMI is unfeasible, and hence, no solution can be found. Finally, in order to apply the method described in [16], a huge number of 215 ¼ 32768 LMIs should be solved simultaneously, because in this case n ¼ 3, p ¼ 3, q ¼ 2, implying a huge computational effort.

5. Conclusions In this paper, the design of observers for non linear discrete systems has been addressed. The non linear terms in the state and output equation has been assumed to fulfill a matrix Lipschitz condition, leading to a less conservative result than the assumption of a scalar Lipschitz condition. The proposed design strategy takes into account the attenuation of disturbances and measurement noise. The final design procedure is based on the solution of a convex minimization problem subject to a linear matrix inequality, that can be solved by means of standard LMI solvers. The problem is formulated in terms of the available knowledge about the norms of the disturbances, leading to a solution that minimizes the bound on the state estimation error norm. The proposed approach is suitable to be extended to the case when the measurements are taken scarcely and irregularly in time. Those alternative approaches already discussed would also lead to a huge number of LMIs to be solved.

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