Observer Design for a Class of Discrete Time Piecewise-Linear Systems

Copyright © Analysis and Design of Hybrid Systems Alghero, Italy 2006

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IFAC PUBLICATIONS

OBSERVER DESIGN FOR A CLASS OF DISCRETE TIME PIECEWISE-LINEAR SYSTEMS Abderazik Birouche, Jamal Daafouz, Claude Iung

Institut National Polytechnique de Lorraine, URAN UMR 7039 CNRS INPL UHP E N S E M 2, Avenue de la For~t de Haye 5~516 Vandoeuvre-l~s-Nancy Cedex, France. Ernail:{ A bdearzik.Birouche, Jamal.Daaf ouz, Claude.Iung} @ensern. inplnancy.fr Abstract" In this paper we consider a class of discrete time piecewise-linear systems composed by linear discrete time LTI subsystems with autonomous switching, T h e aim is to propose a m e t h o d for the synthesis of a hybrid observer. The approach proposed here consists on the combination of a discrete observer and a continuous observer. It is shown that under conditions related to minimum dwell time of each mode, one can express the switching law as a linear combination of the system i n p u t / o u t p u t samples. The continuous observer is a piecewise-linear observer whose dynamics depend on the current active mode. Two estimation schemes are analyzed ' o n - l i n e estimation and off-line estimation. Copyright © 2006 IFA C. Keywords: Hybrid systems, Observer design, Piecewise-linear affine systems.

1. I N T R O D U C T I O N

framework, and a continuous observer, based on the classical state observation theory. The former identifies the current mode, while the latter produces an estimate of the evolution of the continuous state of the hybrid system. In this paper, we study the observation problem for a class of piecewise-atTine hybrid systems where the switching law depends on the continuous state vector. The observation scheme proposed here consists on the combination of an active mode detection and continuous observer. This paper is organized as follows. First, we present a solution for detecting the switching time instant and the corresponding active mode. This method is the transposition in discrete time of the idea introduced in (Benali et al., 2004) for the continuous time case. The adaptation to the discrete time case is not immediate and introduces some specificities. The continuous observer is based on the switching observer developed in (Daafouz et al., 2002), whose dynamics depend on the active mode provided by the discrete observer. Two configurations are considered

Observer design for hybrid systems is an important and challenging problem. Applications for control or fault detection purposes is of first importance. Recently, many papers have considered such a problem. A discussion on observability conditions for switched linear systems is proposed in (Vidal et al., 2004). Moving horizon estimation strategy is discussed in (Ferrari-Trecate et al., 2002) for piecewise-linear systems. Piecewiselinear systems are an important class of hybrid dynamic systems which has attracted a growing interest (Bemporad et al., 2000),(Johansson, 2003),(Liberzon, 2003). In (Balluchi et al., 2002), a hybrid observer is proposed for systems with a hybrid automaton description. The scheme of the proposed observer consists of two blocs: a discrete observer, based on the discrete event dynamic 1 Work partially done in the framework of the HYCON Network of Excellence, contract number FP6-IST-511368

12

3. D I S C R E T E O B S E R V E R : A C T I V E M O D E DETECTION

: o n - l i n e gain attribution, where the two parts of the observer work simultaneously, off-line gain attribution, where the hybrid observer works with a delay compared to the real system. We finish by an illustrative example and a conclusion.

In this section we present a m e t h o d for detecting the switching time instant and the corresponding active mode using the i n p u t / o u t p u t d a t a on a time horizon. First, the autonomous case is considereal. A generalization is proposed in the next section for the n o n - a u t o n o m o u s case. We consider the following piecewise-linear system:

2. P R O B L E M S T A T E M E N T We consider the class of discrete time piecewiselinear systems given by :

Xk+

-

-

Aixk if H x k

y~ = Cix#. Xkq-1

A~xk + Biuk if H x k C [ai, ai+l] Yk=Cix~ for i = l , 2 , . . . , s -

-

(1)

characterizes the dynamics of the system in a region of the state space, s is the number of subsystems (also the number of regions). The switching strategy is specified by the linear form Hxk -- E ~ I h~x~, H - [hi, ..., h~] which indicates the active subsystem and defines a partition of the state-space where each region is delimited by H x k = a~, with a~ c IR and al < as < ... < as.

As in the continuous cas, we try to express H x k as a linear combination of the o u t p u t s Yk taken on a time horizon of length N. The relation between the o u t p u t s and the switching function will be then written as : N-1

The hybrid observer has to provide an evaluation of the active mode (discrete state)~/'~ c Q, and an estimate of the state vector 2k. The hybrid observer proposed here consists of two parts: discrete observer and continuous observer (see figure 1). Knowing the system input and o u t p u t (u~,y~,k = 1,...,N) on a time horizon N, an evaluation ~ of the active mode index 0k c {1, 2..., s} is calculated. The result is used by the piecewise-linear observer to determine 2 an estimation of the continuous state vector x.

[

j=0 where c~ = [c~0 as "'" C~N-S ] are o u t p u t weighting coefficients to be computed. Where in the continuous time case the successive o u t p u t derivatives y ( n ) n = 1 , . . . , rt involve always the same active mode, in the discrete time case the samples Yk+j,j = 1 , . . . , N do not necessarily correspond to a same active mode. In a consequence it is not possible to find c~ such t h a t equation (3) is valid when a c o m m u t a t i o n occurs

Y~"'~

o~ [0, ..., N -

of Hx k

i].

The idea is, first to find c~ independent of the discrete state such t h a t the equation (3) is valid whenever the discrete state keeps constant on the interval [0, ..., N - 1], second to detect the commutation when it occurs, at last find the new discrete

Discrete observer Evaluation

(2)

The switching time instants are determined by the value of the quantity Hxk. As we have only the collected o u t p u t d a t a Yk on a fixed horizon, the simplest idea is to express H x k function of these available outputs. This idea has been investigated in (Benali et al., 2004) in the continuous time case and appears quite easy to exploit. It is possible to express H x ( t ) as a linear combination of the o u t p u t y(t) and its successive derivatives w i t h o u t any assumptions. We show t h a t the transposition in the discrete time case cannot be used without additional assumptions.

where xk c IRn is the state vector, Yk is the o u t p u t vector, k refers to the sample index, qk is the active mode index, and the discrete state of the hybrid system, it takes its values in the finite set Q = { 1 , 2 , . . . ,s}. Each triplet (A~,Bi, Ci)

Z'lk~[ {Xk+l=4Xkq-j =C,xk~i~kyk

C [ai, ai+l] for i = 1, 2, ' . . , s

qk )

state.

C°ntinu°us °bserver I Xk+l = 4X'k

-'[-Bi~k -']-Li ( yk -- .~2k)

Before starting the t h e o r e m let us recall a few definitions associated with the problem of joint observability (Vidal et al., 2004). We define Gk

)

Cl

Fig. 1. Hybrid observer structure

Gk--

The two parts of the observer are described in the following sections.

c2 ... c~ /

CIA1 C2A2 . ............

." .

13

.

UsA8

/2--1

D e f i n i t i o n 1. (The joint observability index) T h e joint observability index is defined as

,-

H- E 4
m~x(~a~(c))

Theorem 1. S u p p o s e t h a t t h e following assumptions are verified :

D e f i n i t i o n 2. (The joint observability matrix) G/2 is called joint observability m a t r i x of t h e s y s t e m Si = (Ai, Ci), i = 1, ..., s L e m m a 1. T h e r e exist ct c -- [ a 8 (~ "'" s . t " Hx~ = [Hx~]a~, where

~/2--1 c

(1) Si, i = 1,..., s observable.

(2)

]

(~) ~ /2-1

(~)

Then" w h e n e v e r the discrete s t a t e q e Q keeps c o n s t a n t on [k, ..., k ÷ p - 1] if and only if

Proof

(~)

T hT ... H ]

s times

Proof of lemma.1 : I f : if rank ( [ G ~ h 2P ]) - # t h e r e h -

.... G-. 1 ~ch th~

/2--1

(~) j=O

--

of theorem.1

(ai s V l d i d ~ 2

/2--1

[Hxk]c~

Else the switching instant is te - k + # -

/2--1

E ctjCiAixk c j

_

j=0

i 1Cid~-l)xs o~/2

[Hxs]~ E ~ - o ajyy+xc _ E J _ o ajCiA~ ~ Jxs + (a~ sC1dld~ -2 a~/2--1 Cid~ S)xs /2 l i ~ A j /2 1 c J we have ~ 0 - 0 c~jwi i x/~ -- }-23_ 0 ayCiA i x/~ t h e n

~ ctcjYk+j j=0

where y~+j is the o u t p u t of s y s t e m Si. O n l y if : if t h e r e exist [(~0c ogC~ . . . ozC/2-s ] s . t

[ H x s ] ~ - IHXl]~ - (c~; s c~; 1)(ClAz c ozi CiAi)A~ exs 0 only if c~, 1 , 1 or (CIAz CiAi) - 0 which c o n t r a d i c t the a s s u m p t i o n s (4,5).

/2--1

H.~ = ~

:

C a s e 1: [HXl]c~ ~ ¢ [HXl]c~q0 gives the information of a c o m m u t a t i o n at >. C a s e 2: [ H x s ] ~ [Hxs]~q0 • A s s u m e t h a t a c o m m u t a t i o n (from q0 i to qs l -¢ i) occurs at # t h e n [Hxs]~ - }--~-~=0/2 s a ij Y j + I -- ~-2j=O /2-s c~jL~i ~ i f, A JiX1 ÷

is a linear c o m b i n a t i o n of the rows of the m a t r i x

then

¢ o~/2-s"

[ H H ... H ] s times

t h ~ ~xi~t [~; <

,.

First it is i m p o r t a n t to notice t h a t we only prove the result on [0, N], N > p. In fact w h e n a comm u t a t i o n will be d e t e c t e d at t i m e k e [0, N] we will reinitiMize t h e d e t e c t i o n procedure. [ H x 0 ] ~ gives the discrete s t a t e q0 - i on [0, p 1] t h g n k s to l e m m a 1 and dwell-time hypothesis. Let us e x a m i n e [Hxx](~c.

If all Si - (Ai, C i ) i s observable # > n.

~..

h ]) =

If [H~k]~ - [H~k]~, t h e . q~ - i

j=O

H

(IG

(3) the hybrid s y s t e m exhibits t r a n s i t i o n s w i t h t i m e s e p a r a t i o n g r e a t e r t h a n or equal to some p > 0 (dwell t i m e hypothesis). (4) CiAi ¢ CSAj, Vi, j e Q.

/2--1

withh-[H

(7)

j=O

~?v~+~. W~

j=0

Remark : T h e previous t h e o r e m gives conditions u n d e r which the switching i n s t a n t can be detected. T h e only s i t u a t i o n where no conclusion can be m a d e is A~ 2xs e Nul(CiA1 CiAi) which is far from being realistic.

while y~+s is the o u t p u t of any s y s t e m Si (q is c o n s t a n t on [k, ..., k + # - 1]), t h e n /2--1 j=O /2--1

and H --

c J Vie ctjCiAi,

3.1 Generalization to the non-autonomous case

Q and h is linear

j=0

According to the same p r o c e d u r e described for the a u t o n o m o u s piecewise-linear systems, we define Qk expressed as a linear c o m b i n a t i o n of t h e o u t p u t samples t a k e n on a time horizon #:

c o m b i n a t i o n of the rows G/2. [] If & = (Ai, Ci) is observable t h e n x~ is linear c o m b i n a i s o n of t h e o u t p u t s Yk, Yk+s, ..., Yk+/2-s and the Hx~ is also a linear combinMson of Y~,Y~+s, ...,Yk+/2-s, so t h e r e exist i i " " . ~i/2--1 ] , for all i e Q s.t o~i - [ O~00~1 [Hxk]~-

/2--1 E

j=o

>--1

Q~ - Z

/2--1

c~jyk+j ~

-

~jyk+j - ~Y~

(s)

j=O

with Y/~ [Yk Y/~+I " " Yk+/2-1 ]T mode, we h a v e :

~ c~jCi ~ A ji x~ t h e n j=0

14

For each

1

Y~ - O~z~ + P~U~

The equation (12a) describes the situation where the active mode is correctly identified, and the second equation (12b) describes a situation where the estimated mode is different from the actual active mode. In this paper, we will study two possible estimation schemes :

(9)

where Oi the observability matrix and Fi is the Toeplitz matrix" I

G C~A~ O~ -

.

,F~

0

......

0

C~B~

".

".

•

•

,

(1) O f f - l i n e e s t i m a t i o n : The previously presented analyses a set [Ys, ..., Yt~] and gives the discrete state 0k - qk gh c [1, ....,tf]. The continuous observer uses the exact knowledge of qk to construct 2k. (2) O n - l i n e e s t i m a t i o n : The hybrid observer and the hybrid system works simultaneously• In this case the i n p u t / o u t p u t data provided by the hybrid system are analyzed by the hybrid observer in real time.

C~A~ -~

U k - [uk uk+s "" u k + . - s IT By substitution of equation ( 9 ) i n (8), we obtain Qk - ctOizk + >--1

aF~Uk with c~O~ = ~ c t j G d ! . If a a c, then j=0 Qk - H x k + acF~Uk. Using (8) with a - a c, the switching function H x k is then given by

where/3[--c~Fi• The term/3[ in (10) states that s different evaluations of H x k are computed. As in the autonomous case, different coefficients a i are used. For each subsystem, the quantity H x ~ is evaluated in two different ways, [ H x k ] ~ , ~ and [Hxk]~,,~,, given by

~.1 Off-line state estimation To analyze the off line estimation case, let us consider a time instant tf > p. The i n p u t / o u t p u t data information is available for k < tf. Using a moving window of size p, the discrete observer anMyzes the data and provides q'k. The method proposed in the previous sections guarantees, under section 3, t h a t qk qk, this information is used by the continuous observer. The error dynamic is governed by the following equation"

[H~k]~ ~ - a~Y~ + Z~U~ where/3 i -

-cdF~

The active subsystem qk and the switching instant tc are given by

(Ai

ek+l

If [ H x k ] ~

-- [Hxk]~,~,~ then q'k - i Else the switching instant is tc = k + p - 1

(14)

LiG)ek

The observer gains L~ are computed so that the estimated state xk converges towards the state of the system xk for all the initial conditions i•e •

4. HYBRID OBSERVER Ve0 c ]R~

lira

ek

0

(15)

k---+ oo

In this section the continuous observer is analyzed considering the complete hybrid system obtained by composing the hybrid system and the hybrid observer. The continuous observer is a piecewiselinear observer of the form:

To obtain these observer gains, we use the LMI (Linear Matrix Inequalities (Boyd et al., 1994) approach developed in (Daafouz et al., 2002). An indicator vector ~k [~, ~ . . . . . ~8k] T is defined as

~ A i x k -Jr-B i u k -~ Li(yk -- Yk) y% = G 2 k if 0 k - i c Q

Xk+l

{ 0~ if ~k~ i

(16)

(~)

=

The estimation error is • The hybrid observer has (s × s) situations of type (qi, ~ ) C Q x Q• To each (qk, ~/'k), the error dynamics is given by :

ek+l - ~

ek+s = (Ai - LiCi)ek if q k = 0 k = i (12a) ek+s = (Ai - L i G ) e k + vk if qk = j, 0k i (12b) -

(~(Ai

LiCi)ek

(17)

i=1

Global convergence of (17) is ensured by selecting the gains Li;i - 1 , . . . ,s such that the error stability condition established in (Daafouz et al., 2002) are satisfied. It consists in finding positive definite matrices & and matrices Fi and Gi for i - 1 , . . . , s solution of the following inequalities :

-

with

(13)

15

+ aT AT. Gt - C T Ft

cTA

- F?c

Sy

]> 0

1 Pi G i A t - FtCt G t A t - FtCt -

(is)

with j - 1 , . . . , s. In this case, the matrices Lt are given by

' AiGi-

s

<0

(21)

Li -- o ~ - l F ?

e r ~ ~ P i e with Pi = /=1

S-1 t

]

0 2Gt - q l

if the minimization problem has a solution P [ c IRn×n, C~ C R n×n, F? C IRn×m et q* c]l, ~ [ , the gains Li are given by :

The asymptotic stability is g u a r a n t e e d by the Lyapunov function V -

-

c ' ir : i

Pj - 2Gt 0

The I n p u t - t o - S t a t e stability is guaranteed by the Lyapunov function V(ek, ~ ) = eTPkekk with Pk --

"

The estimation error verifies:

~P[.

~.2 O n - l i n e state estimation i=1

We recall that to detect the active mode, we use the input/output samples available on a time horizon of size #. Let us consider a switch occurring at time tc. The discrete observer detects at tc H- 1 this commutation. However the new mode is only identified at the instant tcH-#. In the on-line state estimation case the continuous observer work simultaneously with the discrete observer, therefore on the interval [tc, tc + # - 1], we must provide qk for the continuous observer. Different schemes can be investigated during the interval time Its, tc + # - 1] as the discrete state is unknown and the continuous state cannot be correctly calculated. In this paper we propose to keep the discrete state with its last known value t c - 1 and estimate the continuous state with the observer designed for value qt~-l, with this option it can prove t h a t the estimation error remains bounded.

(22) --

~7"

As the LMI resolution implies t h a t q* is necessarily greater t h a n 1, when k > ~ the estimation error is b o u n d e d Ilekll < ~*11~11~ if vk is bounded. exist X > 0 and g > 0, such that Ilxkll~ < x,

I1~11~ < g ~nd vk ~ < V - m a x

i#j

5. I L L U S T R A T I V E E X A M P L E To illustrate the observation scheme proposed here, we consider a piecewise-linear system given by (1) with:

[

09,9,

- 0 . 2 2 0.80

~ ~{k(Ai-

LiCi)c k @

g2=

'

[1

1],

- 0 . 2 9 0.50

1

[0-0.500]

H=

I0

1]

The switching function H x k is defined by :

(19)

if

the estimation error is given by: Ck+l --

[

'

r020]

Bz-/0.20 C1=C2=

~-

X+

(B 5 - B~) U

The corresponding error dynamics is given by the equation (12). A new indicator vector ~k = ( ~ , ~ , " " ,~)T is associated in order to establish a correspondence between discrete 0t and the triplet (At, Bt, Ct) •

1 if 0k = i 0A. if ,,, 0 k ~ i ~k -- ~*k - 1 if 0k is not available.

((Aj-Ai)-Li(Cj-Ci))

(0.2
For this system we find # = 4 and vk

~c

(20)

i= 1

: [-

11.87

39.65

-45.18

18.65

]

In off-line state estimation, t h e gains of the observer are obtained solving the LMIs (18):

This error cannot be asymptotically stable because of the t e r m vk. However, we can calculate t h e gain Lt, such t h a t the error estimation is I n p u t - t o - S t a t e stable (Jiang et al., 1999). For that, we use the result proposed in (Daafouz et al., 2005). The gain Lt is calculated by solving the optimization p r o b l e m :

L~ =

-0.7660 2.5341

L2 = '

0.5441 0.1030

Figure (2) shows t h e actual active mode (full line), the reconstructed active mode (stars ,) and estimation error norm. Notice t h a t the modes are perfectly identified, and the error estimation norm decreases towards zero. A zoom shows t h a t after the 15 th sample the error norm is almost null.

Min q P,~ = P/ a~ = al Fi, a

under

16

2.:t...

6. C O N C L U S I O N In this paper, we propose a hybrid observer for a class of piecewise-linear systems. The association of a discrete state detection m e t h o d and a piecewise-linear switched observer leads to a hybrid observer witch may operates off-line or online. In the off-line case, the observation error is guaranteed to converge toward zero whereas in the on-line case, it is guaranteed to be bounded.

~'1.4 '"

l~e~ ~o81

5

10

15

k

20

25

! ..........i:°:! ~ 1_::~m]- ~

............. [[]! ............... ioi! '~ ............... !i .................. !i

REFERENCES Balluchi, A., L. Benvenuti and A.L. SangiovanniVincentelli {2002}. Observers for hybrid systems with continuous state resets. Proceedings of the l Oth Mediterranean Conference on Control and Automation (MED2002), Lisbon, Portugal. Bemporad, A., G. Ferrari-Trecate and M. Morari (2000). Observability and controllability of piecewise affine and hybrid systems. I E E E Trans. Autorn. Contr. 45, 1864-1876. Benali, A., D. B o u t a t and J.P. B a r b o t (2004). Une condition alg6brique pour l'observabilit6 d'une classe de syst~mes hybrides. Proceedings of I E E E CIFA, Tunisie. Boyd, S., E. Feron L. E1 Ghaoui and V. Balakrishnan (1994). Linear Matrix Inequalities in System and Control Theory. Studies in Applied Mathematics. Daafouz, J., G. Millerioux and C. Iung (2002). A poly-quadratique stability based approch for linear switched systems. Special issue on switched, piecewise and polytopic linear systerns. Int.& Control. Daafouz, J., G. Millerioux and L. Rosier (2005). Observer design with guaranteed b o u n d for lpv systems, in Proceedings of IFAC World Congress on Automatic Control. Ferrari-Trecate, G., D. Mignone and M. Morari (2002). Moving horizon estimation for hybrid systems. I E E E Trans. Autorn Contr 47, 1663 1676. Jiang, Z-P., E. Sontag and Y. Wang (1999). Inputto-state stability for discrete-time nonlinear systems. Proc. l~th triennal World Congress. Johansson, M. (2003). Piecewise linear control systems: Lecture notes in control and information sciences. Springer. Germany. Liberzon, D. (2003). Switching in systems and control. Birkh~user Boston, Systems & control: foundations & applications,. Vidal, R., R. Chiuso and S. Soatto (2004). Observability and identifiability of j u m p linear systerns. International Conference on Decision and Control.

Fig. 2. off-line estimation case : mode evaluation (in the top) and the error estimation norm (in the b o t t o m ) . In o n - l i n e state estimation, the gains of the piecewise-linear observer are obtained by solving t h e optimization problem (21) :

11.3122,, L 1 -

q * - -

[

- 0 . 0 6 2 1

, L2 =

1.3035

0.0524 0.8879

The actual mode (full line) and the evaluated mode (dotted line) and estimation error norm are given on the figure (3). 21 Z2 t

~1.8~_~1.6

,

~

m

t

|... i ........

~ ............

: ...........

.I. .......

:.... ! ................

: ........

~ ..........

|

'

;

I

i

'

i

,

4 I

F

'o18/[ ..........

|...:

;

........

!

|

......~-w...........

I

...... i ~ ..........

, ............

: ...........

t .......

:. . . . ' . . . . . . . . . . . . . .

m

!

/

:

=

:. . . . . . . .

:

, ..........

~

4

/

' .............

! ...........

"1" .......

!....

' ..............

! ........

' ...........

"7

~ ~4 I- .......... 1 : ....... ~............ : ........... 1 ....... :.-..m.............. !........ ~........... -t ~1.2

...........

,1""'! i ........

,

5

i

/

10

i

,

15

i

,

20

l

25

k

15.~-] lO L!

I

•E l o

5[-

|I

........ :.................... : .................... '..................... : :.................... ........ ::................. o,2:1-.................... ;..................... ;.................

I

::

I

::

~ " l i :

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

o.1 ~ : : : : : : : : : : ~ 1.0

~I~--, I

oi- ............... i.... ~

5

1.~

: ....................

:.....

:

i

i 10

k

A

2.0

:

i ..................................

Zoomi i i i 15

.......... 20

25

Fig. 3. o n - l i n e estimation case" mode evaluation (in the top) and the error estimation norm (in the b o t t o m ) . At the beginning of the estimate k = 1, we initialized the discrete state estimated at ~'~ 2, k = {1, 2, 3}, whereas the true discrete state is q~ = 1. Since t h e acquisition of the 4 th sample, the discrete observer provides ~'~ = 1, at t h e instant time k = 5 a switch occurs: qk = 2. The discrete observer keeps its old value ~ = 1 for k = {5, 6, 7}. For the continuous component, we notes t h a t starting from the sample k = 10 the norm of the estimation error is stabilized around zero. A zoom on the interval [i0, 25], shows t h a t the error n o r m is not null but it is bounded.

17