CONTINUOUS AND DISCRETE STATE ESTIMATION FOR A CLASS OF HYBRID NONLINEAR SYSTEMS

1st IFAC Workshop on Dependable Control of Discrete Systems (DCDS'07) ENS Cachan, France - June 13-15, 2007

CONTINUOUS AND DISCRETE STATE ESTIMATION FOR A CLASS OF HYBRID NONLINEAR SYSTEMS 1 Hao Yang ∗,∗∗ , Vincent Cocquempot ∗∗ , Bin Jiang ∗ ∗

College of Automation Engineering Nanjing University of Aeronautics and Astronautics Nanjing, 210016, P.R. China Email: [email protected] ∗∗

LAGIS-CNRS, UMR 8146 Universit´e des Sciences et Technologies de Lille, 59655 Villeneuve d’Ascq cedex, France Email: [email protected]

Abstract: State estimation and fault diagnosis problems are discussed for a class of hybrid nonlinear systems modelled by hybrid automata, which have uncontrollable discrete mode transitions and parametric uncertainties. Two kinds of faults are considered: Continuous faults that affect each mode; Discrete faults that affect the mode transition. To estimate both the continuous and discrete states, a novel observer is designed for each mode whose estimation error is not affected by continuous faults and sensitive to discrete mode transitions. Sufficient conditions are given to achieve the state estimation goal for the overall hybrid c 2007 IFAC. system.Copyright ° Keywords: Hybrid system, Observers, State estimation, Fault diagnosis.

1. INTRODUCTION Hybrid Systems (HS) arise naturally in a number of engineering applications, which have fuelled the development of theories for modeling and simulation, verification, stability and controller synthesis for HS (Lygeros et al., 2003), (Gu´eguen and Zaytoon, 2004). However, most of techniques about safety verification only consider HS with full state measurements and do not involve the on-line fault diagnosis (FD) scheme. As for the case that only partial state measurements are available, an observer has to be designed. In (Balluchi et al., 2002) and (Saadaoui et al., 2006), a design method which combines continuous observer and location observer has been proposed. However, the location observer can not often provide enough information to identify 1

This work is partially supported by National Natural Science Foundation of China (NSFC) and Key Laboratory of Process Industry, Ministry of Education of China.

the mode transition. The observability of mode transitions has also been investigated under the framework of discrete event systems (Lin and Wonham, 1988) and Petri net (Giua et al., 2004), where the continuous dynamics are not considered. On the other hand, faults may lead to unacceptable system behaviors. FD is aimed at detecting, isolating and estimating the faults (Blanke et al., 2003). Two kinds of faults have been defined for HS in (Cocquempot et al., 2004): Continuous faults corrupt the behavior of the continuous mode; Discrete faults affect the discrete mode transitions. Only a few literature is devoted to FD of hybrid systems: finite memory observer (Kajdan et al., 2006), parity space method (Cocquempot et al., 2004). In our previous work (Yang et al., 2007), observer-based FD schemes were designed for HS with time-dependent switching (i.e., switching instants are fixed or can be designed arbitrarily).

In this paper, we consider hybrid nonlinear systems with uncontrollable state dependent switching and parametric uncertainties. One challenge of observer design for such systems is to distinguish the effects of the continuous faults and mode transitions (may include discrete faults) on the system. From the abnormal change of continuous state estimates provided by the observer, we should first identify whether continuous faults in the current mode occur or another mode is switched into (discrete state is changed). Compared with (Balluchi et al., 2002)(Saadaoui et al., 2006)(Lin and Wonham, 1988) and (Giua et al., 2004), a new state estimation scheme of hybrid systems is proposed from continuous modes point of view, which can provide rapid and accurate estimates of both continuous and discrete states. The main contributions of this paper are as follows: 1. Under geometric conditions, each mode of HS is transformed into a new form, on which, a novel observer is designed whose estimation error is not affect by continuous faults and sensitive to mode transitions. 2. Sufficient conditions are given to achieve the state estimation goal for the overall hybrid system in spite of both continuous and discrete faults, and uncertainties.

Definition 1 : A Hybrid Automaton is a 10-tuple H = (Q, X, U, F, Y, Init, Inv, E, G, R), where • Q = {1, 2, . . . , N } is the finite set of discrete states; • X is the set of continuous states; • U is the set of continuous inputs; • F: Q × X × U → X is the set of vector fields for each mode; • Y is the set of continuous outputs; • Init ∈ Q × X is the set of initial states; • Inv: Q → 2X is the invariant set for each mode. • E ∈ Q × Q is the set of discrete transitions between modes; • G : E → 2X is a guard condition related to each (j, j 0 ) ∈ E, where the system can be switched from mode j to j 0 . • R : E × X → X is the set of reset maps. 2 In this paper, the plant is modelled as a class of H with following properties: • H is always sequence deterministic. i.e., the set of executions of H contains one and only one switching sequence (Lygeros et al., 2003). • The vector field F for mode j is of the form x˙ = g0j (x) + g j (x)uj + φj (x, u)θj + ej (x)f j 1≤j≤N

We first discuss the case where the system lies in mode j (∀j ∈ Q) of H in sections 3-4, then consider the overall HS in Section 5. The superscript j in (1) is omitted in sections 3, 4.1 and 4.2. The arguments of functions will be added if necessary.

3. SYSTEM DECOMPOSITION Consider the vector field F for mode j x˙ = g0 (x) + g(x)u + φ(x, u)θ + e(x)f y = h(x)

2. PROBLEM FORMULATION

y j = hj (x),

where x ∈ 0, where | · | is Euclidean norm The continuous fault is described by the distribution matrix ej (x) and a “fault signal” j f j ∈
(1)

(2)

where x ∈
(3)

z˙2 = ψ0 (z) + γ3 (z2 , y)u + e¯(z)f + ψ2 (z, u)θ y¯2 = z2

(4)

> where z = [z1> , z2> ]> , z1 = [ξ1> , ξ2> , . . . , ξm−q ]> ∈ n−q > > > q < , z2 = [ξm−q+1 , . . . , ξm ] ∈ < are the states > > ] ∈ of system (3)-(4), with ξ = [ξ1> , ξ2> , . . . , ξm n %i > > > > < , ξi ∈ < = [ξi1 , . . . , ξi%i ] . y = [¯ y1 , y¯2 ] with y¯1 ∈
> γ1 (z1 , y) = [¯ g1> , g¯2> , . . . , g¯m−q ]> with   g¯i1 (ξ1 , . . . , ξi−1 , ξi1 , yi+1 , . . . , ym )  g¯i2 (ξ1 , . . . , ξi−1 , ξi1 , ξi2 , yi+1 , . . . , ym )    g¯i =   (5) ..   .

g¯i%i (ξ1 , . . . , ξi−1 , ξi , yi+1 , . . . , ym ) ¯> ¯> > and γ2 (z1 , y) = [g¯> 1 , g 2 , . . . , g m−q ] with g¯i = [0, . . . , 0, L%g0i hi (ξ1 . . . ξi , yi+1 . . . ym )]> (6) Assumption 1 : There exist a set of integer Pm numbers {%1 , %2 , . . . , %m } such that i=1 %i = n and ξ = T (x) ∈

where Lg h denotes Lie derivative of h along a ∂g2 1 vector field g, [g1 , g2 ] = ∂g ∂x · g2 − ∂x · g1 , denotes the Lie bracket. Moreover, the relative degree of yr , denoted as ρr , is such that ρr = %r = 1, m − q + 1 ≤ r ≤ m. ♦ Under Assumption 1, dT (x) is invertible ∀x ∈
∂ ∂ ∂ , ,..., } ∂ξhv¯ii ∂ξhv¯i −1i ∂ξhv¯i −%¯i +2i

Condition III can be represented in ξ−coordinate as

[

m X

Lg0 hi

i=m−q+1

∂ ∂ ∂ , ] ∈ span{ , ∂ξhvi i ∂ξhki ∂ξhvm−q i

∂ ∂ ∂ ... ... } (7) ∂ξhvm−q −1i ∂ξhv¯i −%¯i +2i ∂ξhv1 −%1 +2i for %¯i−1 + 2 ≤ k ≤ %¯i . Pm ∂ Note that i=m−q+1 Lg0 hi ∂ξhv , ∂ξ∂hki ] ii Pm ∂ ∂ = i=m−q+1 Lg0 hi ∂ξhki ∂ξhvi i , which, together with (7) implies that Lg0 hi

∂ = 0, m − q + 1 ≤ i ≤ m ∂ξhki

followed by the property that γ3 (z2 , y) is independent on z1 \¯ y1 . Finally, condition VI decouples the subsystems (3) from f . This completes the proof. 2 After transformation, the subsystem (3) is not affected by continuous faults, an observer can be designed for this subsystem to estimate z1 , θ, f , and meanwhile, to detect the mode transitions. 4. STATE ESTIMATION 4.1 Continuous state estimation In this section, an observer based on (Lee and Park, 2003) is proposed, which can overcome the large transient oscillation due to high gain observer as in (Shim et al., 2001). The observer will be constructed firstly through the following several steps. ¯ i for 1 ≤ i ≤ m − q as Step 1 : Define M  Ci  Ci Fi (z1 , u, y) ¯ i (z1 , u, y) =  M  ..  .

    

Ci Fi%i −1 (z1 , u, y) where Fi (z1 , u, y) = Ai + Gij (z1 , y)u, Gij (z1 , y) = ∂¯ gi /∂ξj for 1 ≤ j ≤ m − q. ¯ i Fi M ¯ −1 − Ai )> Ri , where Step 2 : Let Ni = Ri (M i %i −1 βi ] with βi = [0 . . . 0 1]> . Ri = [βi Ai βi Ai From the construction, one can show that   Ni%i 0 . . . 0  Ni(%i −1) 0 . . . 0    Ni =  .. .. . . ..   . . . . Ni1 0 . . . 0 Moreover, Ni can be decomposed into Ni = Li Ci , where Li ∈ <%i × 1. Step 3 : Define Wi as



 Ci  Ci A¯i (z1 , u, y)    Wi (z1 , u, y) =   ..   . %i −1 ¯ Ci A (z1 , u, y)

where A¯i = Ai + Ni , and also define Mi = ¯ i. W −1 M i

V˙ i ≤ −²i λmax (Qi )|˜ e2 |2 +

(8)

Assumption 2 : 2.1 The partial derivatives of g¯i w.r.t. z1 and their respective time derivatives are bounded. 2.2 There exists a function B(z1 , u, y) such that ψ1 = B ψ¯1 (z1 , u, y), where B ∈ <(n−q)×(m−q) is Lipschitz w.r.t. z1 , |B| ≤ b0 , and |ψ¯1 | ≤ q¯(z1 , u, y) ≤ q0 for a function q and numbers b0 , q0 > 0. 2.3 There exist matrices P = P > ∈ ∈ <%i ×%i and a function R(z1 , u, y) ∈ < such that

ˆ i (ψ1i θ +2˜ eTi Pi ∆²i M

i−1 X

µji |˜ ej |2

j=1

− ψˆ1i θˆ − Υi )

Now, consider the Lyapunov candidate function as W (ez , eθ ) = Vz + Vθ for the overall system, Pm−q > −1 where Vz = eθ . Denote i=1 Vi , Vθ = eθ Γ > > > e˜ = [˜ e1 , . . . , e˜m−q ] . The time derivative of W along (3), (9) and (11) is X

m−q

˙ ≤ W

Ã

!

X

m−q

(−²i λmax (Qi ) + i=1

µij )|˜ ei |

2

j=1

ˆ − 2˜ ˆ (ψ1 θ − ψˆ1 θ) ˆ Υ − 2e> ψ¯> R ˆ > Cez + 2˜ e> P ∆² M e> P ∆² M θ 1

{z

|

}

Ψ

Based on Assumption 2.3 and (8), we have −1 ˆ¯ e − 2e> ψ ˆ¯> ˆ > ˆB ˆψ Ψ = 2˜ e> P ∆² M ˜ 1 θ θ 1 R C∆² e

∆² P ∆² M B = C > R (Ai − Ki Ci )> Pi + Pi (Ai − Ki Ci ) = −Qi where ∆² = diag[∆²1 , . . . , ∆²m−q ], ∆²i = diag[1/²i , . . . , 1/²%i i ], with ² a design parameter. M = diag[M1 , . . . , Mm−q ], Ki ∈ <%i ×1 are such that (Ai − Ki Ci ) is stable. ♦

ˆ¯ )θˆ + 2˜ ˆ B( ˆ ψ¯1 − ψ ˆ (B − B) ˆ ψ¯1 θˆ +2˜ e> P ∆² M e> P ∆² M 1 ˆ Bsgn( ˆ ˆ > )[θ0 (q0 + q(ˆ ˆ¯1 )] −2˜ e> P ∆² M R z1 , y))sgn(¯ y1 − y ˆ (B − B) ˆ ψ¯1 θˆ ≤ 2˜ e> P ∆² M

zˆ˙ 1 = Aˆ z1 + ψ1 (ˆ z1 , u, y)θˆ + γ1 (ˆ z1 , y)u + γ2 (ˆ z1 , y) −1 −1 +M (ˆ z1 , u, y)[L(ˆ z1 , y) + ∆² K](¯ y1 − yˆ¯1 ) > ˆ ˆ +Bsgn(R )[θ0 (q0 + q(ˆ z1 , y))sgn(¯ y1 − yˆ¯1 )] (9) (10) (11)

where L = diag[L1 , . . . , Lm−q ], and K = diag[K1 , ˆ . . . , Km−q ]. The weighting matrix Γ = Γ> > 0. Ξ > > > denotes Ξ(ˆ z1 , u, y). Denote ez = [e1 , . . . , em−q ] ˆ with ei = ξi − ξˆi , 1 ≤ i ≤ m − q, eθ = θ − θ. Theorem 1 : Under Assumption 2, the observer described by (9)-(10) together with the adaptive algorithm (11) can realize limt→∞ ez = 0 and limt→∞ eθ = 0 under a persistent excitation. Proof (sketch): The proof of the theorem follows the recursive way. Consider the ith subsystem of (3) and (9), we have ˆ i + ∆−1 e˙ i = Ai ei + (¯ gi − gˆ¯i )u − (Mˆi−1 )(L ²i Ki )Ci ei ˆ ˆ +ψ1i θ − ψ1i θ − Υi (12) > > > where Υ = [Υ> 1 , . . . , Υm−q ] , Bsgn(R )[θ0 (q0 + > > q(ˆ z1 , y))sgn(¯ y1 − yˆ¯1 )]. ψ1 = [ψ11 , . . . , ψ1(m−q) ]> . ˆ i ei and Consider the transformation e˜i , ∆²i M choose a Lyapunov candidate function Vi = e˜Ti Pi e˜i . Based on (Lee and Park, 2003), it can be shown that the time derivative of Vi along (12) satisfies

(13)

Based on Assumption 2.2, we further obtain ˙ ≤ −η|˜ W e|2

The observer is constructed as

yˆ¯1 = C zˆ1 ˙ θˆ = Γψ¯1> (ˆ z1 , u, y)R> (ˆ z1 , u, y)(¯ y1 − yˆ¯1 )

µij |˜ ei |2 +

j=1

From the above procedure, we can obtain the following equations (Lee and Park, 2003) Ci Mi−1 = Ci

i X

(14)

where ²i , 1 ≤ i ≤ m−p, is chosen such that η > 0. Since M and ∆² are all bounded and nonsingular, inequality (14) implies the stability of the origin ez = 0, eθ = 0. one can get limt→∞ ez = 0. On the other hand, the persistent excitation condition means that there exist two positive constants σ and t0 such that for R t+t all t t 0 ψ1> (z1 (s), y(s))ψ1 (z1 (s), y(s))ds ≥ σI, which, together with (11) and the uniform boundedness of eθ , leads to limt→∞ eθ = 0. 2 4.2 Diagnosis of continuous faults Considering that the effect of continuous faults on outputs y¯2 is independent leads to suppose that e¯(z) is invertible. The fault estimates can be obtained from z2 subsystem as −1 fˆ = ˆe¯ [y¯˙ 2 − ψˆ0 − ψˆ2 θˆ − γ¯2 u]

(15)

It can be obtained that ˆe¯fˆ−¯ ef = (ψ0 −ψˆ0 )+(ψ2 θ− ˆ ˆ ψ2 θ). Since limt→∞ ez = 0, limt→∞ eθ = 0, due to the continuity of e¯, ψ0 , and ψ2 , there always exist two numbers kz , kθ > 0 such that for all bounded z, zˆ, if |ez | and |eθ | are sufficient small (which can be achieved by observer), then the following inequality holds |ˆe¯fˆ − e¯f | ≤ kz |ez | + kθ |eθ | Moreover, we have limt→∞ |ˆe¯fˆ − e¯f | = 0.

(16)

4.3 Discrete state estimation The discrete state is determined by the guard set G or Gf (discrete faults). Based on the observer analysis in sections 4.1, we provide a discrete SE approach in this section. Assumption 3 : All modes of H are discernable, i.e., for mode j, the estimation error |ejz | is convergent as in (14) only under the observer (9)-(11) which is associated with mode j. ♦ j It follows from Theorem 1 that limt→∞ y¯j − yˆ¯1 = 0 before the mode transition occurs. Similar to fault detection problem (Jiang et al., 2006), based on j Assumption 3, y¯j −yˆ¯1 can be regarded as a residual for mode j to detect the switching instants, and estimate the discrete state.

For mode j, note that inequality (14) can be rewritten as V˙ zj ≤ −¯ η1j Vzj − V˙ θj , for η¯1j > 0. Using the differential inequality theory, we have Zt Vzj (t)

j

−¯ η1 (t−tj (k))

≤e

Vzj (tj (k))

Theorem 2 : Under Assumptions 3, consider the HS (1) under a family of observers (9)-(11) and diagnostic scheme (15), where all the modes satisfy the conditions in Theorem 1. If, at switching instants tj (k) , the following inequality holds : ejbound (tj (k + 1)) < ejbound (tj (k))

(20)

then the estimation error of the overall HS (2) converges to zero. Proof (sketch): From (20), it can be obtained that limt→∞ ebound (t) = 0, which, together with (17), leads to the global convergence of ez to zero. 2

6. AN EXAMPLE In this section, a three tank system is employed to illustrate our approach. The system consists of three cylindrical tanks linked to each other through connecting cylindrical pipes as shown in Fig. 1. Two pumps control two incoming flows.

j

η1 (t−τ ) ˙ j e−¯ Vθ (τ )dτ

− tj (k)

j −¯ η1 t

≤e

Vzj (tj (k)) j

Zt

h

η1 t +e−¯ Vθj (tj (k)) + η¯1j

i

j

eη¯1 τ Vθj (τ )dτ (17)

tj (k)

ηj

where η¯1j , λmax1(P j ) . tj (k) denotes the kth time when the mode j is switched into. We further have p |Cejz (t)| ≤ mj − q j ejbound (tj (k)) (18)

The following two modes are considered

with ³

Fig. 1. The three tank system

´2

ejbound (tj (k))

−¯ η1j t

,e

h

η¯2j |ejz (tj (k))|2

³ ´2 +¯ η3j θ0j + |θˆj (tj (k))| Zt +¯ η4j

´2 i ³ j eη¯1 τ θ0j + |θˆj (τ )| dτ (19)

tj (k)

where η¯2j , η¯1j η¯3j .

λmax (P j ) λmin (P j ) ,

η¯3j ,

λmax ((Γj )−1 ) λmin (P j ) ,

and η¯4j ,

Given an initial z1 (0) (or a norm bound of z1 (0)), inequality (18) can be considered as a time varying threshold to detect the mode transition for mode j, and provide accurate estimates of discrete states under G or Gf .

5. OVERALL HYBRID SYSTEM The following theorem guarantees that the SE goal can be achieved for the overall HS.

Mode 1 (save water): Valves V1 , V2 are opened, V3 , V4 are closed. Levels h2 and h3 rise. Mode 2 (lose water): Valves V1 , V2 , V3 , V4 are opened. Levels h2 and h3 drop. h1 , h2 , h3 are continuous states of the system with h2 and h3 as outputs for both modes 1, 2. G(12) = {h1 ≥ 0.545} which is achieved by admissible incoming flows. Suppose that the system is initialized at mode 1. Only f and θ in mode 1 and Gf (12) are considered. The continous fault corresponds to sediment deposit in P1 and P4 , i.e., sections of P1 and P4 progressively change. The uncertainty denotes the modelling error and input disturbance related to Q1 . The system models can be transformed as in Definition 2. The physical parameters are omitted due to space. In the simulation, the initial levels are setting as [0.48 0.37 0.25]> . As for mode 1, f = 0.3 − 0.3e−0.05(t−10) which is assumed to occur at t =

10s, θ = 0.09m/s is unknown. Fig. 2 shows the continuous SE performance, from which, we can see that the rapid and accurate estimates of the unknown parameter θ and the continuous fault f can be obtained. 1 0.8 0.6 0.4 0.2 0

parameter estimates parameter 0

1

2

3

4

5 t/s

6

7

8

0.3

9

10

fault estimates continuous fault

0.25 0.2 0.15 0.1 0.05 0 −0.05

0

2

4

6

8

10 t/s

12

14

16

18

20

Fig. 2. Continuous SE performance −6

11

x 10

threshold residual

10 9 8 7 6 5 98.25

98.26

98.27

98.28

98.29

98.3 t/s

98.31

98.32

98.33

98.34

98.35

86.52

86.53

86.54

86.55 t/s

86.56

86.57

86.58

86.59

86.6

−5

2.6

x 10

2.4

threshold residual

2.2 2 1.8 1.6 86.5

86.51

Fig. 3. Discrete SE performance Under G(12), the switching detection threshold is exceeded and switching is detected at t = 98.298s when h1 reaches the guard set as in Fig.3. Consider Gf (12) = {h1 ≥ 0.543}, the switching is detected at t = 86.578s as in Fig.3. It can be seen that, the estimations of continuous and discrete states are effective under G(12) and Gf (12). 7. CONCLUSION In this paper, we discussed SE and FD problems for a class of hybrid nonlinear systems. We provide a clue that the estimation error of the observer for HS is required to be not affected by (or say robust to ) continuous faults and sensitive to mode transitions. REFERENCES Balluchi, A., L. Benvenuti, M. D. Di. Benedetto and A. L. Sangiovanni-Vincentelli (2002). De-

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