Finite Memory Observer for State Estimation of Hybrid ...

IFAC

Copyright

Publications www.elsevier.comllocatelifac

FINITE MEMORY OBSERVER FOR STATE ESTIMATION OF HYBRID SYSTEM

F. KRATZ, D. AUBRY Laboratoire de Vision et Robotique - UP RES EA 2078 Universite d'Orleans - IUT de Bourges 63, avenue De Lattre De Tassigny - 18020 Bourges Cedex - France TeI. : (33) 2 48238479 Fax : (33) 2 48 238471 email : [email protected]

Abstract: Both industry and the academic world are becoming more and more interested in techniques to model, analyze, control and supervise complex systems that contain both analog and logical components. Such systems are called hybrid systems. This paper presents extensions and improvements on the finite memory observer. The main contribution is to incorporate the parity relation design and observer into a diagnosis scheme. The application of the finite memory observer to the sensor fault detection problem is illustrated by a numerical example. Simulation results show that the method is very effective for robust fault detection. Copyright © 2003 IFAC Keywords : Fault detection, Uncertain dynamic systems, State observers, Hybrid systems.

I. INTRODUCTION

2. PROBLEM FORMULATION

Hybrid dynamical systems are systems that contain both analog (continuous) and logical (discrete) components. Hybrid systems got a lot of attention in the last years. Their importance comes, among other abilities for describing autonomous or controlled switching. A very important class of hybrid systems is the class of systems subject to abrupt changes. Such systems switch among a finite set of linear timeinvariant realizations (models). Each model corresponds to a mode of the system and the system can jumps from a mode to another one. This type of hybrid system can be met in the case of synchronously switched linear systems and networks with periodically varying switches. Several authors have studied this class of hybrid model (Bemporad et aI., 2000; Johansson & Rantzer, 1998; Sontag, 1981).

Consider a class of linear switched systems (Liberzon & Morse, 1999; Morse, 1996), denoted as

x{k + 1) =A{p{k )~(k)+ B{p{k )~(k)+ f(p{k)) y{k) = C{p{k )~(k) + g(p{k)) where the state variable

x{k) E IR n ,

(I)

input variable

u{k) E IR m and output variable y{k) E /Rp .

p{k) :IR+ H I is a piecewise constant switching function mapping from real time line IR+ to an integer set I . Each element in I represents a kind of specific continuous dynamics . Matrices A(p), B(p) and C(p) are piecewise constant matrices depending on values of p, i.e.,

A{e) :IHIR nxn ,

B{e):IHIR nxm

C{e) : I H IRPxn .f(p) and g(p)

are constant vectors

of suitable dimension depending on values of p.

639

and

At the definition (I), we add the following assumptions (Yang, 2002): (i) The integer set 1 is finite; (ii) p(k) is left-continuous, and any time interval within which p(k) is constant is no less than a proper positive scalar 8 > 0, which is called the dwelling time (Morse, 1996); (iii) the switching time set

{t i n=1

(2)

j=i

and the

{n j }~=o

corresponding switching mode set

I

x{k - i)= ilA-I (p{k - j)}x{k)-

C{p{k - i))

Multiplying (2) by rearranging gives:

both

can be determined by the control design, as well as the continuous-time control signal within each selected mode; (iv) there are no discontinuous state jumps during mode switches.

from the left and

I

y{k - i)= C{p{k - i))TI A-I (p{k - j)}x(k)j=i

C(P(k-i))~J£I/-J(p(k-f)))B(P(k-j)~(k- j)+

The left continuity of p(k) guarantees that there are no successive switches at any time point; through the usage of the dwelling time we can avoid the Zeno/chattering phenomena (Lemmon, He, & Markovsky, 1999; Morse, 1996).

f{p{k - j))] + g{p{k - i))

(3)

System (I) will be represented in a batch form on the most recent time interval [k-L, k] called the horizon. On the horizon, the finite number of measurement is expressed in terms of the state x(k) at the current time k as follows (Kratz, Bousghiri, & Mourot, 1994):

In this paper a static supervisor is considered (Lunze, Nixdorf, & Richter, 2001). After an event (Y j, tj) has occurred the supervisor selects an input event (Uj, tj) in dependence upon the given control aim Wand the current output Y j, only. The control loop shown in Fig. I has the typical structure of hybrid systems consisting of continuous variable and discrete-event subsystems.

r{k -1)= ML(k}x{k)- HL(kp{k -1)NL(k )F{k - 1) + G{k - 1)

(4)

where

r{k -1)~~{k -1f y{k - 2f .. ·y{k - Lf

Supervisor

r Lf r

U{k -1)~~{k -1f u{k - 2f ... u{k - L)T

as

F{k -1)~[r{k _1)T f{k - 2f .. .j{k -

G{k-1)~[g{k-1f g{k-2f ... g{k-L)T INTERFACE

dx

-

r

r

and HL(k), ML(k) and NL(k) are matrices of dimension [(pxL) x (rnxL)] , [(pxL) x n] and [(pxL) x (nxL)] respectively.

= fk(X,u,t)

It is assumed that A(p(k)) is a nonsingular matrix for all

dt

y = gk(X,U,t)

y

k. When continuous-time systems x{t) = Ax{t) + iiu{t) are discretized with the sampling time T, we obtain sampled-data systems, x{k + 1) = Ax(k) + Bu{k), where

z = hk(X,U,t)

Continuous-vartiable System

A =eAT. So the assumption of the nonsingularity of A is not too restrictive to apply in practical view (we point out that our system is continuous per pieces and that we can use this discretization on each system).

Fig. I. Supervisory control of continuous-variable systems.

3. STATE ESTIMATION ALGORITHM

From (4), the state vector could be found through solution of the linear equation:

Assume that at the current time k we have the switching time sequence until the moment k-I and a set of the last L measurements until the moment k-I too.

ML(k}x{k) = r{k -1)+ HL(k p{k -1)+ NL{k)F{k -1)- G{k -1)

The relation between the current and delayed state vector values is given by the state space equation:

640

(5)

It is possible to solve this equation in the least square sense, then the least squares solution i(k) is given by T i(k)= (ML(k)ML(k))-J ML(kXr(k-/)+ T

H L (kp{k

the difference between the reliable estimate the estimate

(6)

-1)+ NL (k)F(k -1)- G((k -I)]

(8)

Now, if all sensors are operating correctly the estimate xL, (k) and the estimate i L2 (k) will be

Then, one can rewrite this matrix as

M [ (k )M dk ) =

which includes also unreliable

data:

The existence condition for the solution (6) is the existence of the inverse of square real matrix

M[ (k)M L(k).

XL2 (k)

xLf (k) and

H[I [j6/'oD C(P('

r

nearly identical. If however, a sensor fault occurs at the moment inside the fault detection window, the Er value would be of sufficient scale when the output change caused by this sensor fault is noticeable.

r (p(, - j))]c (p(, - e))

_f(~/-J (P(' - j))]l

5. AN ILLUSTRATIVE EXAMPLE Consider the switched systems given by

The existence of the inverse of this matrix is presented in (Sun, Ge & Lee, 2002).

x{k + 1)= A(p(k))x(k) Taking the expectation on both sides of (6), the following relation is obtained:

y(k)= C(p(k))x(k) where p is a function which takes for value I, 2 or 3. The function p used in this example is plotted in Fig. 2.

(7)

x(k)

has the deadbeat property, i.e.

i(k) = x(k)

if there

Al

arc no noises (Kwon, Kim, & Han, 2002).

A2

4. OBSERVER RESIDUALS GENERATION

sample

output

and

xL2 (k)

the

-I

- 3 ] C3 - 0.7

- 1.4] 0.9]

=[- 0.2 0.4]

2.8

Fig. 2 Switching index sequence

as the estimate computed from L,

values

0.8

=[-1.9 0.23] C2 =[0.1 0.5

We assume that there are no sensor failures before the moment k + L I. Our purpose is to detect and to isolate a faulty sensor inside the fault detection window by means of algorithm (6).

xLJ (k)

o

A3 =[I

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 parameters. The main idea of timeredundancy principle utilization in the sensor fault detection procedure is comparison of two estimates: one based on process output measurements from k to k .,.. L I, and the other based on output measurements available from k to k + L2, with L2 - LI = r. The value r will be referred as fault detection window, because it contains data suspected to be provided by a faulty sensor.

Define

=[ - 0.6 24] . Cl;; [- 0.9

estimate

computed from L2 sample output values. The whole information about possible sensor fault is contained in

641

~r-------~-------r-------'--------r-------.

~

40

40

30

30

20

20

:

,

r\

10

10

.;;;:::.,:;' ·10 .

·10

·20

-20

·30

-30

·r /1\/·;

. ....... .

.

V

._ .....

i

·40 -~

o

10

20

15

25

Fig. 3. Simulation of system: output trajectory

Fig. 5. Faulty measurements

In the first simulation, the window length (i.e. the horizon) is taken as L = 2. The state estimation errors are plotted in Fig. 4.

It can be seen that the fault is very small and it is hardly noticeable from the output. Also, let us be used the fault detection window. The horizon length are chosen as L\ = 2 and L2 = 3, so that r = 1.

3\

i2

:

... . ....

i,

,

c···· · ······ ··· · ··········,!··· ···············

:· ······ ........•... .. .

;........................... ...

The residual based on the difference between two estimates (8) are plotted in Fig. 6. We can see in this figure that each component of the residual is sensitive to the fault and it is very easy to detect.

~

~l~_~~ l ~_~~~~~ l

_,L-______

~

______

o

~

10

~

: : .••

.........1i .... ·.............. .......... ,

______

~

______

15

~

______

~

20

.... ............. ...... ,' ........................••••.•..•..........................';:......................................

25

~

~

1"5 -10 -15 -20 0

.

10

25

l.50~-------:---------:',0-:--------::,':-5-------2O~-------=. 25

Fig. 4. State estimation error This figure shows the deadbeat property. In the second is applied to the added to sensor; to 21. The figure

simulation, the finite memory observer corrupted system. One fault has been a step of magnitude 2 from instant 16 5 shows the faulty measurements.

_15L-______

o

~

______-':-______-:':--____

~~

______--:'

~

6. CONCLUSION In this paper, we considered Finite Memory Observer for state estimation for hybrid systems in the PieceWise-Affine form. The proposed algorithm is easy to implement. In the same way, the extension to the noisy systems should not pose problems. Among the prospects for research to be developed, we can quote: the taking into account of jumps of the states and the problem of the estimate of the switching index sequence.

642

REFERENCES Bemporad, A., Ferrari-Trecate, G., & Morari M. (2000). Observability and controllability of piecewise affine and hybrid systems. IEEE Automatic Control, 45 (l0), 1864-1876. Johansson, M. & Rantzer, A. (1998). Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Automatic Control, 43 (4),555-559. Kratz, F., Bousghiri, S., & Mourot, G. (1994). A finite memory observer approach to the design of fault detection algorithms. American Control Conference, Baltimore, June 29-July I . Kwon, W.H., Kim, P.S., & Han, S.H. (2002). A receding horizon unbiased FIR filter for discretetime state space models, Automatica, 38, 545-551. Lemmon, M.D., He, K.X., & Markovsky, I. (1999). Supervisory hybrid systems, IEEE Control Systems, 19 (4),59-70. Liberzon, D., & Morse, A.S . (1999). Basic problems in stability and design of switched systems. IEEE Control Systems, 19 (5), 59-70. Lunze, 1., Nixdorf, 8., & Richter, H. (2001). Process supervision by means of hybrid systems. Journal of Process Control, 11, 89-104. Morse, A.S. (1996). Supervisory control of families of linear set-point controllers, part I: Exact matching. IEEE Automatic Control, 41,1413-1434. Sontag, E.D. (1981). Nonlinear regulation: the piecewise linear approach, IEEE Automatic Control, 26 (2), 346-357. Sun, Z., Ge, S.S., & Lee, T.H. (2002). Controllability and reachability criteria for switched linear systems. Automatica, 38, 775-786. Yang, Z. (2002). An algebraic approach towards the controllability of controlled switching linear hybrid systems. Automatica, 38,1221-1228.

643