High Gain Observer Design for Some Networked Control Systems

New results on high gain observer design for networked control systems via an emulation-like approach are presented. By using a general framework and a Lyapunov approach, we derive some explicit conditions on the maximum allowable transmission interval that ensure an exponential convergence of the observation error for a large class of network protocols.


I. INTRODUCTION
The control and observation of systems called Networked control systems (NCSs) are currently attracting a lot of attention in the control community. In many applications, the interest for NCSs is motivated by many advantages they offer such as the ease of maintenance and installation, the greater flexibility and the low cost. For these reasons, many industrial control applications use a serial communication channel to connect sensors and controllers. In NCSs, the serial communication channel has many nodes (sensors and actuators) but the signals of these nodes cannot be transmitted at the same time. The rule that selects which node will use the network to transmit its data, is T. Ahmed-Ali is with GREYC UMR CNRS 6072, Caen Cedex 14050, France (e-mail: tarek.ahmed-ali@greyc.ensicaen.fr). called scheduling network protocol. In the present letter, only communication constraints on sensors are considered. It is well known that the stability of these systems is largely determined by the transmission protocol used and by the so-called maximum allowable transfer interval (MATI), i.e., the maximum allowable time between any two successive transmissions in the network. There exist several families of protocols in the literature, but in this study, only Uniformly Globally Exponentially Stable (UGES) protocols are considered. This class includes several protocols, such Round Robin (RR), Try-Once-Discard (TOD) and sampled data systems. For further details on NCSs, protocols and their properties, see the overview papers [1]- [4].
The present note is focused on the implementation of a class of nonlinear observers on NCSs. Although the control of these systems has received considerable attention in the recent years, only few papers which deal with the observer design problem exist in the literature. For example, in deterministic case, the authors in [5] derived some conditions in order to maintain the observability of discrete-time linear systems under network communications. In [6], the authors proposed a method for the mutual design of protocols and observers for linear systems. The considered protocol gives the priority to the node where the error, between the measure and the estimate is the biggest, according to some weighted norms. Sufficient conditions for quadratic stability properties are then given using matrix inequalities. In [7], the authors consider the design of a linear observer for NCSs in the presence of time delay and sampling phenomenons without considering the constraints introduced by the protocols. They give sufficient conditions guaranteing asymptotic stability, by using a Lyapunov-Krasovskii approach. Recently, a small gain approach was used in [8] and [9] for the observers design via an emulation-like approach for some classes of networked control systems. The results contained in [8] and [9] state that if the observer has some robustness properties with respect to measurements errors then, for sufficiently small MATI, the stability in an appropriate sense of the observation error is guaranteed for several classes of network protocols. On the other hand, even without disturbances, the framework developed in [9] does not allow asymptotic stability of the observation error when the network is in zero order hold (ZOH) fashion. In the present study, we focus on a class of nonlinear triangular systems without considering disturbances. More precisely, we provide a framework and use a Lyapunov approach to guarantee an exponential stability of the observation error and improve the bounds of MATI compared to small gain approaches. We specially show that by using the framework presented below and a suitable Lyapunov-Krasovskii functional, we can derive an exponential stability of the observation error when the network is in ZOH fashion. Note that the results presented in the present letter can be easily extended to several classes of observers such, for example, linear observers and the class of observers described in [10]. The present note is organized as follows: In Section II, we present the notations which will be used throughout the letter. The framework that we propose is formulated in Section III. In Section IV, we present our results on the emulation of high gain observers on a class of networks.

II. NOTATIONS AND PRELIMINARIES
First some mathematical notations are introduced. Let R = (01; 1), R+ =[0; 1) and N is the set of natural numbers. The notation N 3 denotes the set of strictly positive integers. The usual euclidian norm of any vector v will be noted by jvj and the identity matrix of size p is denoted by Ip . The matrix 0p is a square matrix of size p with all values equal to zero. The notations min (S) and max (S) denote respectively the minimum and maximum eigenvalues of the square matrix S . A continuous function 0:R+ ! R+ is said to be of class K, if it is zero at zero, strictly increasing. It is of class K 1 , if it is of class K and unbounded. A continuous function ' : R + 2 R + ! R + is said to be of class 0KL,if '(:; t) is of class K for each t 0, and '(s; :) is non increasing and satisfies lim t!1 '(s; t)=0 . The notation e y (a + ) represents lim t!a;t>a e y (t) when it exists. The term W [t ;t) represents ess: sup s2[t ;t) jW(s)j. Throughout this letter the vectors x 2 R n and y 2 R p represent respectively the state vector and the outputs of the considered systems. We consider the following class of nonlinear systems: Throughout this letter, we assume that the following hypotheses are satisfied: Using these hypotheses, the following observer has been proposed in [11] for the class of systems (1) without considering network: where is a positive constant satisfying 1. S is a symmetric positive definite matrix, solution of the algebraic Lyapunov equation and 1 is a diagonal matrix which has the following form: 1=Diag I p ;; 1 I p ; ...; 1 q01 I p : The exponential convergence of this observer has been derived by using the Lyapunov function V = x T S x, where x =1 x. The authors prove in [11] that its time derivative satisfies

III. OBSERVERS FOR NETWORKED CONTROL SYSTEMS
In this section, we present a framework dedicated to implementation of a class of nonlinear observers on a class of networks. Let us consider the nonlinear systems and suppose that there exists an observer described by the dynamics is globally asymptotically stable. Now, let us consider the case where all outputs of system (10) are connected to observer (11) via a serial communication channel. We suppose that the signals of these sensors cannot be transmitted to observer (11) at the same time, then this problem can be modeled by the following framework: The vectorx is the continuous-time estimate of the system state x. The monotonically increasing sequence tj;j 2 N represents the transmission instants. The vectors w andx (which are re-initialized at each instant tj) represent respectively most recently transmitted output values via network and the prediction of the observer output. The vector y s (which is also re-initialized at t j ) allows us the possibility to describe the sampling phenomena. More precisely, ys can be used to represent the output of the zero order hold (ZOH) device. This variable is specially useful when the network operates in ZOH fashion. In this case we replace conjointly the outputs of the system and the observer by their sampled signals before considering the constraints introduced by the network. As, we will see in the sequel, this framework allows us to derive an asymptotic stability of the observation error when the network operates in ZOH fashion. The functions g y , g w andĝx represent prediction functions between two transmission instants. The error induced by the network is represented by the vector ey. The protocol hy is the algorithm by which the access to network of each node is determined. At each t j , the protocol h y selects which nodes k 2f1; ...;l g can transmit its data throughout the network. This algorithm is represented by the discrete time system: e y (t + j )=h y (j;e y (t j )) j 2 N.
Throughout this note, the variable =m a x ( t 0 t j ) represents the maximum allowable transfer interval (MATI) and MATI represents an upper bound of . To prevent zeno solution we suppose that the MATI satisfies >t j+1 0 tj >, where is an arbitrary positive constant.
Note that the sampled-data systems are a special case of NCSs since all sensor and control signals are transmitted at each transmission instant.
In this case, the protocol hy is equal to zero, and the maximum allowable time between any two successive transmissions is called maximum allowable sampling period MASP. Compared to [12] where a framework based on discrete-time approximate models is developed for the observer design for sampled-data systems, we can say that the framework (13) is an alternative approach to [12].

Hypothesis 3:
We suppose that the protocol hy is UGES. This means that there exists a positive function, W : N 2 R n ! R + and some positive constants 2 [0; 1), a1, a2 such that, for all j 2 N and for all e y 2 R n a 1 je y jW (j; e y ) a 2 je y j (14) W (j +1 ;h y (j; e y )) W (j; e y ): Hypothesis 4: We also assume that there exists a positive constant M 0 so that for all j 2 N and for all e y 2 R n @W(j; e y ) @e y M 0 : This condition is satisfied by many protocols such Round Robin (RR) and Try-Once-Discard (TOD).

IV. HIGH GAIN OBSERVER FOR NCSS
In this section we focus on the implementation of high gain observer (6) over a network with UGES protocols. We present two examples: The first one is an implementation in ZOH fashion, whereas the second one, is based on the introduction of an output predictor between two transmission instants. In both cases we prove exponential convergence of the high gain observer for sufficiently small MATI.

A. High Gain Observer in ZOH Fashion
When the network operates in ZOH fashion, we will consider that all prediction functions in (13) are equal to zero. This means that the corrector termŵ 0 w is held constant between two transmission instants.
Following framework (13), we propose this observer: As we can see, system (18) is disturbed by the term Cx(t + i01 ) 0Cx(t).
The Leibniz-Newton formula Provides  [4], [13] and [14] to the problem considered in the present note.

B. High Gain Observer With an Output Predictor
The idea of using a predictor of the output was introduced in [15] for the design of sampled-data observers. More precisely, this predictor is used by the observer between two transmission instants and its model is a copy of the output system model. This gives this differential equation _ w =(@hM(x)=@x)fM(x) where the initial condition w(t0) is arbitrary and at each transmission instant ti the value of w is reseated like this: w(t + i )=y(t i ). This idea has been extended to networked systems in [8] and [9] and some bounds of MATI have been derived using small gain approach. The aim of this section is to improve the bound of MATI for high gain observer when an output predictor is used. Following the framework (13), we can write the following observer: w t + i = ys(ti)+hy (i; ey(ti)) y s = ŷ w = Cx(t) Let us consider the observation errorx =x 0 x, then we have _ x =(A 0 1 01 S 01 C T C)x + f (x + x) 0 f (x) +1 01 S 01 C T ey t 2 [ti01;ti] _ ey =x 2 + f1(x 1 + x 1 ) 0 f1(x 1 ) t 2 [ti01;ti] x t + i =x(t i ) e y t + i = h y (i; e y (t i )) Theorem 2: Let us consider system (1) and suppose that hypotheses 1, 2, 3, 4 hold. Then for all >maxf; 1g, system (30) is a global exponential observer for system (1) Proof: As in the above section, using the same change of coordinates then (31) will be _ x = (A 0 S 01 C T C) x +1(f(x + x) 0 f (x)) +S 01 C T ey t 2 [ti01;ti] _ ey = x 2 + f1(x 1 + x 1 ) 0 f1(x 1 ) t 2 [ti01;ti] e y t + i = h y (i; e y (t i )) Let us consider the following candidate Lyapunov function as in [16]: where j 2 N, and are two positive constants which will be computed below. The positive function (t) is bounded and decreasing on the transmission interval (t i01 ;t i ] i 2 N 3 . This function satisfies the following conditions: t + i01 = 01 and t + i01 + MATI = 8i 2 N 3 where 2 (0; 1), to guarantee that U (t + i ) U (ti). Now, let us compute the time derivative of U between (ti01;ti], then we have _ U =2 x T S _ x + _ W 2 (i 0 1;ey)+2W(i 0 1;ey) _ W (i 0 1;ey) (34) after some computations, we deduce that _ U 0 ( 0 )min(S)j xj 2 + jeyj 2 + _ W 2 (i 0 1;e y )+2W(i 0 1;e y ) @W(i 0 1;ey) @ey j _ e y j: Let us remark that j _ e y jj x 2 j + f 1 (x 1 ) 0 f 1 (x 1 ) ( + 0 )j xj (35) using the property (16), we derive the following inequalities j@W (i 0 1;ey)=@eyjj_ eyjM0( + 0)j xj and _ U 0 ( 0 ) min (S)j xj 2 + je y j 2 + _ W 2 (i 0 1;e y )+2W(i 0 1;e y )M 0 ( + 0 )j xj: Using the fact that the protocol h y is UGES, and by adding and subtracting the term 0 W 2 (i 0 1;e y ), where 0 > 0, thus we have _ U 0 ( 0 )min(S)j xj Using the fact that j xj(1= min(S)) U (t) and from (41), then we deduce that there exist 1 > 0 and 2 > 0 such that j xj 1exp 0 (t0t ) 8t t0. This means that x converges exponentially towards zero, and the value of MATI is then derived by integrating  (42) and (47), then we have R =((=2) 0 2atan())=(1 0 ). Since 10 (=2)02atan() for all 2 [0; 1), then we deduce that R 1. This means that the bound derived from Lyapunov approach is larger than the one derived from small gain approach. This result means that we have also enlarged the interval of admissible values of the gain for any fixed MATI and therefore we have also improved the speediness of convergence. Note that in all formulas of MATI , it is clear that MATI depends inversely on . This obviously leads to small values of MATI for high values of the gain .
Constrained by the size of the present note, we focused on the simulations of observer (30) with Round-Robin (RR) protocols, that consists in granting access to each compenent of the vector ouputs after p transmission instants where p is the number of ouputs [2]. In the simulations presented below, we also suppose that unknown uncertainties (t) are represented by random function of Matlab with variance = 0:8. The following simulations are performed with u =0:1sin(t), =40and MATI =0 :01 (see Figs. 1 and 2). As we can see, even with some uncertainties, the results remain good with =4 0and MATI =0 :01. This is not a general conclusion. Indeed the uncertainties can cause robustness problems especially for high values of .

VI. CONCLUSION
In this letter, some results on the design of high gain observer for networked systems with UGES protocols were presented. These results can be easily extended to other observers. In future work, we will present some results concerning other classes of protocols.