A finite memory observer structure of continuous descriptor systems

F. KRATZ, S. BOUSGKIRI, W. NUNINGER

Centre de Recherche en Automatique de Nancy - CNRS UA 821 Institut National Polytechnique de Lorraine 2, avenue de Feet de Haye- 54 515 Vandoeuvre Cedex - France

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2. State estimation algorithm In this paper, we consider the continuous descriptor system:

In this paper, we show how the finite memory o b server structure can be extended to a continuous descriptor system. The decomposition of a singular system into two subsystems, a slow and a fast subsystems, allows us to generalize the used technique for standard systems. Using the concept of standardized least square residuals, residuals can be generated in a number of ways and the design of fault detection and fault isolation scheme can be presented. The performances of the proposed method are illustrated on a simulated process: a simple circuit network W C ) .

Ei(t) = Ax(t) + Bu(t) y(t) = Cx(t) + v(t) where x E Rn, U E A E Rnxn, B E ]Rnx resent a zero-mean white noise process (measurement noise) with constant covariance matrix V. For any solvable singular system (i.e., det(E+hA)#O for all except a finite number of h E Q,[17]), there exists nonsingular matrices Q and P ([Sj) such that (1) is r e stricted system equivalent to ([3]):

failure detection, singular system, state estimation, finite memory observer. 1. Introduction

=Alxl + Blu Ni2 = x2 + B2u y = (21x1 + c2x2 + v

In the last two decades, a large variety of methods have been proposed for solving the failure detection and isolation @I) problem ([l], [4], [6] and [15]). In one way or another, the procedure of FDI essentially consists of two stages. The first stage is the generation of signals, called residuals, whose values are nominally zero or close to zero when no failure in system is presented and must be distinguishably different from zero under failure conditions. The second stage involves using the residuals to make the appropriate decisions. Different methods to generate residuals have been proposed ([16], [13] and [7]). These methods use the direct r e lationships between the inputs and the outputs of the system (transfer function approach), the error of reconshuction or the redundancy generated as a linear combination of the system input/output data over a finite horizon (parity space technique) ([141,121 and [SI). In ([9], [lo], [ l l ] and [12]), a finite memory observer approach has been proposed to generate FDI residuals. We a~ interested in this method developed for standard linear systems. We show how extend this method to singular systems.

(2.a) (2.4 (24

with the coordinate transformation:

[:I QB =

= PIX with x1

E

CP=[C1 c21

where nl + n2 = 11, N E Zxn2 is nilpotent whose nilpotent index is denoted by h. Note that the slow subsystem (2.a) is an ordinary differential equation. Assume that we have a set of time-de lay values q,i = 0, ..., k and Q > zk-1 > ...> .ro (in the general case, one takes Q = 0). And suppose that the system (2) is observable ([17]). 3900

The relation between the current and delayed state vector values is given by the state-space equation ([9]): It follows from the assumption of the measurement noise that S is the matrix of the form:

[1; :: :"1

Suppose that the function u(t) is h times piecewise continuously differentiable and that we know an analytical formulation of u(t). By continuously taking derivatives with respect to ton both sides of (2.b), and left multiplying both sides by matrix N, we obtain the following equations ([3]) :

....* ......* ... 0 0 .......v

Where

t

+ IC1 exp(A1(t-q-v))Bl u(v)dv t-Ti From the addition of these equations and the fact Nh = 0, we know readily that the solution x2(t) is given by:

It is possible to solve equation (5) in the least-squares sense so that Q> is minimized.

h-1

x2(t) = - CNi&U(i)

(4)

i=O

The solution

(t) is given by:

&multiplying (3) by C1 and rearranging gives: The equation (4) immediately yields the state estimation for the substate x2:

If we introduce the measurement equation (2.e) in this last expression, we can write: To sum up, the state of singular system (1) is given

by:

3. Sequential observer form

If the output of system (1) is measured k+l times, then the state-vector XI could be found through solution of the linear equation:

To clarify the influence of the number of measure ments k on the observer performance, we propose in this section a sequential form for the observer with respect to k. From (8), we can see that the estimate ?2(t) is not influenced by the parameter k. We study the influence only on the substate (t). Define the finite memory observer that performing state vector estimation from the k+l outputs values:

Let the weighting matrix S be block-diagonal positive definite and of the form as follows:

with

3901

T 0 rlrl

... 0 . . . . . . . . . . .. T 0 0 ...rk'k Moreover, let:

Now, if all sensors are operating correctly the estimate &(t) and the estimate Gr(t) will be nearly identical. If

however, a sensor fault occurs at the moment inside the fault detection window, the E(t) value would be of sufficient scale when the output change caused by this sensor fault is noticeable. 5.

The sequential observer form is given by an equation similar to the conventional Kalman-filter:

Application

In this section, we describe application which permits to exhibit the main abilities of the detection algorithm d e scribed in this paper. This example is a circuit network ([3], p. 10-11)as shown in Figure 1.

where

4. Observer residuals generation

Figure 1

In this section we outline the general procedure of residual generation using observers. Since the estimate includes the information on plant inputs and outputs history, it could be implemented as the peculiarity of a plant performance over the determined time interval. A number of state estimates referred to the same time instant but taken from different data gives the potentiality to form residuals that describe possible changes in the plant p a rameters. The main idea of time-redundancyprinciple utilisation in the sensor fault detection procedure is comparison of two estimates: one based on process output measurements from t - % to t - TO, and the other based on output measurements available from t - z, to t - W. The value r will be referred as fault detection window, because it contains data suspected to be provided by a faulty sensor. We assume that there are no sensor failures before the moment t - q. Our purpose is to detect and to isolate a faulty sensor inside the fault detection window by means of algorithm (9). Define & as the estimate computed from k+l sample output values and & the estimate computed from r+l sample output values. The whole information about possible sensor fault is contained in the difference between the reliable estimate ?r and the estimate Gk which includes also unreliable data: E(t) = $(t)

- ?r