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Observability of the discrete state for dynamical piecewise hybrid systems
Nonlinear Analysis 63 (2005) 423 – 438 www.elsevier.com/locate/na
Observability of the discrete state for dynamical piecewise hybrid systems夡 Salim Chaiba,∗ , Driss Boutata , Abderraouf Benalia , Jean Pierre Barbotb a Laboratoire de Vision et Robotique, Ecole Nationale Supérieure d’Ingénieurs de Bourges, 10 Boulevard de
Lahitolle, 18020 Bourges, France b Equipe Commande des Systèmes, Ecole Nationale Supérieure de l’Electronique et de ses Applications, 6 Avenue
du Ponceau, 95014 Cergy-Pontoise, France Received 10 May 2005; accepted 10 May 2005
Abstract In this paper, we deal with the observability of piecewise-affine hybrid systems. Our aim is to give sufficient conditions to observe the discrete and continuous states, in terms of algebraic and geometrical conditions. Firstly, we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems. Secondly, we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems. Throughout this paper, we illustrate our propositions with examples and simulations. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Observability; Hybrid system; Piecewise-affine systems; Switching function; Transversality
1. Introduction Hybrid systems are systems containing mixtures of logic or discrete and continuous dynamics. The hybrid models have become very important in the last few years as tools for modelling systems. Hybrid behavior is generally described as intervals of piecewise 夡 This
work was supported by “la Région Centre de France”.
∗ Corresponding author.
E-mail addresses: [email protected] (S. Chaib), [email protected] (D. Boutat), [email protected] (A. Benali), [email protected] (J.P. Barbot). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.028
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continuous behaviors (modes) interspersed with discrete transitions that occur at points in time. Examples of hybrid systems include Networks, multi-agent systems, mechanical devices, air traffic management where different types of dynamics coexist and interact in nonobvious ways, interactive distributed simulation, robot design and path planning. The combination of continuous and the discrete dynamics makes these systems interesting and complex such that new mathematical tools are needed for their analysis, observation and control. Several works have been recently carried out on the algebraic properties of hybrid systems as well as their stability and hyperstability properties [9–12,15,16,20,21]. In the last decade many researchers were interested in the study of the observability of such systems, but the definitions and the testing criteria proposed in the literature varied depending on the class of hybrid systems considered and the type of knowledge on the output. In [2,3] we found sufficient conditions for final state observability and a methodology to design a dynamical observer of a class of hybrid systems that reconstructs the discrete state and the continuous state from the knowledge of the continuous and discrete outputs. It was introduced in [4,5] for the class of piecewise (PWA) systems. It implies that different initial states always give different outputs independent of the applied input, which is a stronger version of observability requiring a minimum amount of distinguishability between the states. Observability in the case of multiple discrete states was discussed in [1,7]. The authors in [13,14,24] proposed a definition of observability based on the concept of indistinguishability of continuous initial states and discrete states from the outputs in free evolution and gave the necessary and sufficient conditions for the observability of a class of hybrid systems called jump linear systems. Mixed logical dynamical (MLD) formulation of hybrid systems is given in [4,5,15]. Since the MLD formulation naturally lends itself to being analyzed through mixed-integer linear programming (MILP), and since very efficient algorithms have recently been developed for MILP, several computational approaches have recently been proposed to analyze various observability concepts for hybrid systems. In [15], an algorithm based on multi-parametric programming was proposed for computing the maximal observability region of a discrete-time hybrid system, i.e., the set of initial states that can only be determined by the output. Based on these works and on the results presented in [6] on the observability analysis for piecewise-affine hybrid systems without input, in this paper we give the algebraic and geometrical conditions to observe the discrete state. Our attention will be restricted to piecewise hybrid systems. The outline of this paper is as follows: in the next section, we give some definitions, notations and problem statement. In Section 3, we give the first results on the observability of the discrete state for linear hybrid systems, using the switch function reconstruction. In Section 4, we analyze the observability for nonlinear hybrid systems by means of a transversality concept. We give a simulation in Section 5. 2. Notations and problem formulation The class of systems considered in this section have the following form: x(t) ˙ = Aq x(t) + Bq u(t), y(t) = Cq x(t),
(1)
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where x(t) ∈ Rn is the continuous state, q ∈ Q = {1, 2} is the discrete state, u(t) ∈ R is the input, and y(t) ∈ R is the output. Aq ∈ Rn×n , Bq ∈ Rn×1 , Cq ∈ R1×n . In this paper we deal with a subclass of systems (1) where the discrete state is given by q = 1 if H x(t) 0, q = 2 if H x(t) > 0
(2)
and H is a 1 × n matrix. The observability matrix Oq of each subsystem q is given by Cq C q Aq . Oq .. . Cq An−1 q
(3)
Assumption 1. We assume that the systems under consideration have no Zeno phenomenon. Therefore, a solution to the hybrid system does not exhibit infinitely many discontinuities in a finite time interval. For each integer r 1, we define the following matrix: C2 C1 C2 A2 C1 A1 . Gr .. .. . . C1 Ar−1 1
(4)
C2 Ar−1 2
As we work in the finite-dimensional vector space Rn , there exists 2n, such that: r ∀ r < , rank(Gr ) = ∀ r .
(5)
More precisely, we have
= max[rank(Gr )].
(6)
r 1
We will call the global joint observability index, and G the joint observability matrix associated with system (1). For > 0, we set h (H H ).
(7)
Finally, we denote Y(t)
y y˙ .. .
y (n−1)
,
U(t)
u u˙ .. .
u(n−1)
,
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0 Cq B q Cq Aq Bq q (k) .. . Cq An−1 q Bq
0 0 Cq Bq ··· Cq An−2 q Bq
... ... ... .. . ...
0 0 0 .. . Cq Bq
0 0 0 . .. . 0
It is easy to show the following observation equation:
Y(t) = Oq x(t) + q U(t).
(8)
In this paper, we can say that system (1) is observable if we can recover the whole state: the continuous and the discrete states. Our problem is to deal with the observability of the discrete state. The idea to solve this problem is to give the sufficient conditions to know which subsystem of (1) evolves.
3. Algebraic conditions for discrete-state observability In this section, we give the sufficient conditions for the observability of the discrete state using the switch function reconstruction. For this we give the following theorem. Theorem 1. Let us assume the following conditions: (i) ∃ > 0 and ∃k, 0 k such that: hT ∈ I m[GTk ] (ii) C1 Ai1 B1 = C2 Ai2 B2 , ∀i = {0, . . . , k − 1}. Then, we have H1 x(t) = H2 x(t) and then from (2) we know which subsystem evolves. Therefore, the discrete state is observable for all admissible control inputs. Proof. If hT ∈ I m[GTk ] then rank[GTk hT ] = k. As rank[Gk ] = k this means that vector h is a linear combination of the rows of Gk . Then, there exists an = ( 0 . . . k−1 )T solution to the following algebraic equation: 0 . GTk .. = hT . (9) k−1 Then h =
k−1
i ((C1 Ai1 )T , (C2 Ai2 )T )
(10)
i=0
which gives two values of H x(t) H1 x(t) = T O(C1 , A1 )x(t) = T [Y(t) − 1 U(t)],
(11)
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H2 x(t) = T O(C2 , A2 )x(t) = T [Y(t) − 2 U(t)].
(12)
Both expressions can be written in the following form:
k m−1 m−1−i (m) (i) H1 x(t) = 0 y + , m y − C1 A 1 B1 u m=1
427
(13)
i=0
m−1 k 1 m−1−i (m) (i) H2 x(t) = . C2 A 2 B2 u 0 y + m y − m=1
(14)
i=0
If > 0 and thanks to condition (ii), H1 x(t) and H2 x(t) have the same sign. Thus, we know which subsystem of (1) evolves. Remarks. (i) If each subsystem is observable, rank(Oq )=n for q =1 and 2 and if conditions (i) and (ii) of Theorem 1 are fulfilled, then we can recover the whole hybrid state: the continuous and the discrete one. (ii) If = 1; then H1 x(t) = H2 x(t); thus, we measure the same switch function. (iii) When = 2n, we always have = 1. (iv) For k > 1, condition (ii) becomes (B1 , −B2 ) ∈ ker(G(k−1) ),
k 1.
(15)
Let us assume that C1 = C2 = C; then for k 1, condition (15) of the theorem is equivalent for 0 i k − 1 to Ai1 B1 − Ai2 B2 ∈ ker(C)
(16)
(v) If B1 = B2 = 0, then we obtain the same result given in [6] relative to piecewise-affine hybrid systems without control input. Example 1. Let us consider the hybrid system formed by the following two subsystems:
1 0 1 A1 = , C1 = ( 1 0 ) . , B1 = 0 −1 0
0 1 1 A2 = , B2 = , C2 = ( 1 0 ) . (17) −1 −1 0 Let us consider the following switch function: H x(t) = x1 (t) + x2 (t). We have k = = 4 and 1 0 0 1 T T [G4 h ] = 1 0 0 1
−1 0 0 −1 −1 1 −1 0
(18) 1 1 .
(19)
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Here = 1 and T = ( 1 1 0 0 ). Condition (ii) of Theorem 1 is fulfilled, which gives us a single value of the switch function given by H1 x(t) = y(t) + y(t) ˙ + u(t).
(20)
Then, we can know which subsystem evolves. Therefore, the discrete state is observable. Now, let us consider the system given by the parameters A1 , A2 , B1 , B2 as in (17); C1 and C2 are given by C1 = ( 1 0 ) ,
C2 = ( 0
1) .
(21)
We consider the same switch function as in (18), k = = 4, rankG4 = 4 and for = 1 we have rank[GT4 hT ] = 4. It is easy to see that: C1 Ai1 B1 = C2 Ai2 B2 ,
∀i = 0, 1, 2.
(22)
This means that condition (ii) of Theorem 1 is not fulfilled. Thus, we cannot determine the sign of the switch function H x(t). Therefore, the discrete state is not observable with the algebraic equality of Theorem 1. For the linear dynamical systems which do not satisfy the algebraic conditions of Theorem 1 and for the nonlinear systems, in the next section we will propose another method to observe the discrete state, based on a geometrical condition.
4. Geometrical conditions In this section, we give a geometrically sufficient condition to observe the discrete state. We consider nonlinear hybrid systems consisting of the two nonlinear subsystems: x(t) ˙ = fq (x(t)) + gq (x(t))u(t), y(t) = cq (x(t)), q = 1 if h(x(t)) 0, q = 2 if h(x(t)) > 0,
(23)
where fq (x) for q = 1, 2 is the drift smooth vector field, gq (x) is the input smooth vector fields, cq (x) is the smooth output and h(x) is a smooth switch function on Rn . As mentioned in Section 3, if we know at each moment which subsystem evolves then the discrete state is observable. In the following we will use dynamical equations satisfied by the output to give another method to observe the discrete state. In this section, we assume that each subsystem of (23) is observable i.e., for q = 1, 2 we have: dcq dL¯ fq +gq u cq rank (24) = n, .. . (n−1) dL¯ fq +gq ucq
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where L¯ is the Lie–Bäcklund derivative given by d + Lf + Lg u L¯ f +gu = dt
(25)
and Lf is the Lie derivative of a given function in the direction of the vector field f . Remark. As the following Lie–Bäcklund isomorphisms [18]: q
1 = cq , q 2 = L¯ fq +gq u cq , q = L¯ 2f +g u cq 3
q
q
.. . qn = L¯ n−1 fq +gq u cq
(26)
are different for each subsystem (q = 1 or 2) we have a jump on the canonical states when the system switches even if the original coordinates do not jump. For each subsystem we have (n)
y (n) = L¯ fq +gq u cq .
(27)
Next, we set y (n) = q (y, y, ˙ . . . , y (n−1) , u, u, ˙ . . . , u(n−1) )
(28)
as defined in (27). For a given input u, if the output y satisfies one and only one of the two dynamical equations of (28), then, we know which subsystem evolves, and we can thus observe the discrete state. Let us now define the following submanifold: ˙ . . . , u(n−1) ) = 2 (v, u, u, ˙ . . . , u(n−1) )}. Mu = {v ∈ Rn /1 (v, u, u,
(29)
The following assumption is one of the keys to the observability of the discrete state. Assumption 2. We set Lu to be the submanifold of the common singularities of the two subsystems of (23). If Lu contains more than one element, then we assume that Lu is included in only one of the two spaces shared by the switching function H x(t) i.e., for all x in Lu we have H x 0 or else H x > 0.
(30)
It is clear that Lu ⊂ Mu . The following result gives a sufficient condition to observe the discrete state. Theorem 2. Under Assumptions 1 and 2 and if the two subsystems of (23) are transverse to Mu except on a discrete subset, then, there exists a finite time when y(t) satisfies one and only one equation of (28).
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Fig. 1. The behavior of the vector fields on Mu at v between two instants.
Proof. Thanks to the assumption about the observability of each subsystem, the following change of coordinates: q
q
2 = y, ˙ . . . , qn = y (n−1)
1 = y,
transform (23) into the following forms: for subsystem 1: 1 ˙ i = 1i+1 for i = 1 : n − 1, 1 ˙ = (, u, u, ˙ . . . , u(n−1) ). n
(32)
1
for subsystem 2: 2 ˙ i = 2i+1 for i = 1 : n − 1, 2 ˙ = (, u, u, ˙ . . . , u(n−1) ), n
(31)
(33)
2
where q is given by (28). Let v ∈ / Mu , as Mu is a closed submanifold; then, there exists a neighborhood V of v such that V ∩ Mu = ∅. In V, we have
1 (v, u, u, ˙ . . . , u(n−1) )|v∈V = 2 (v, u, u, ˙ . . . , u(n−1) )|v∈V .
(34)
This inequality implies that the output satisfies one and only one equation of (28) for q =1, 2. If v ∈ Mu ∩ Lu then, by Assumption 2 we can conclude on the observability of the discrete state. Now, if v ∈ Mu and v ∈ / Lu at t = t ∗ then, 1 (v, u, u, ˙ . . . , u(n−1) ) = (n−1) ∗ (n−1) 2 (v, u, u, ˙ ...,u ) at t , but 1 (v, u, u, ˙ ...,u ) and 2 (v, u, u, ˙ . . . , u(n−1) ) are transverse to Mu except on a discrete subset and thanks to Assumption 1 we have
1 (v, u, u, ˙ . . . , u(n−1) ) = 2 (v, u, u, ˙ . . . , u(n−1) )
(35)
for a certain moment t = t ∗ + ( > 0) (see Fig. 1), which leads us to the previous case.
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x2 Mu Vector fields of subsysems 1 and 2
P1
x1
p2
Fig. 2. The behavior of the vector fields on M0 for u = 0.
Example 2. Let us consider the system described by the following subsystems:
x˙1 = x2 , x˙2 = −x23 + u = 1 (x1 , x2 , u), y = x1 ,
(36)
x˙1 = x2 , x˙2 = x1 + u = 2 (x1 , x2 , u), y = x1 .
(37)
We have
Mu = {(x1 , x2 ) ∈ R2 /x1 = −x23 }
(38)
The two systems (36) and (37) are transverse to Mu except on the common singularity (0, 0). At the points solution to the following equations:
x1 = −x23 , x2 (1 − 3x24 + 3x2 u) = 0.
(39)
The dynamics of the two subsystems are tangent to Mu , the vector fields of the two subsystems at these points have two nonnull components along the axis x1 and x2 andtherefore, the 1 1 , 31/4 discrete state is observable. For example for u=0 these points are given by p1 = − 33/4 1 1 and p2 = 33/4 , − 31/4 ; the vector fields of subsystems 1 and 2 are tangent to M0 (see Fig. 2). Therefore, the discrete state is observable for any switch function.
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Fig. 3. A double cart with elastic coupling.
5. Simulation: double cart with elastic coupling In this section, we give a linear example in order to illustrate our algebraic and geometrical methods. Let us consider the mechanical system described in Fig. 3. The linear hybrid model of the plant under investigation is as follows: for z2 (t) 0: m1 z¨ 1 = − 1 z1 + 2 (z2 − z1 ) + 1 (˙z2 − z˙ 1 ) + u and m2 z¨ 2 = − 2 (z2 − z1 ) − 1 (˙z2 − z˙ 1 )
(40)
for z2 (t) > 0: m1 z¨ 1 = − 1 z1 + 2 (z2 − z1 ) + 1 (˙z2 − z˙ 1 ) + u and m2 z¨ 2 = − 2 (z2 − z1 ) − 1 (˙z2 − z˙ 1 ) − 2 z˙ 2 .
(41)
Let x T = (x1 , x2 , x3 , x4 ) = (z1 , z˙ 1 , z2 , z˙ 2 ); then, the state representation of (40) and (41) is
0 − 1 + 2 m1 x(t) ˙ = 0 2
m2 y(t) = ( 1
0
0
0 + 2 − 1 m1 x(t) ˙ = 0 2
m2 y(t) = ( 1
0
0
1
− 1 m1 0
1 m2
0 2 m1 0 − 2 m2
0 0
1 m1 x(t) + 1 u(t), 1 0
1 0 − m2
0 ) x(t) if x3 0,
(43)
1
− 1 m1
0 2 m1
0
1 m1
0
1 m2
0 − 2 m2
1
1 + 2 − m2
0 ) x(t) if x3 > 0.
(42)
0 1 x(t) + u(t), 0 0
(44)
(45)
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0.6 0.4
Hx(t) estimed
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
5
10
15
20
25
Time[s]
Fig. 4. Real switching function.
We will consider the following two configurations: (1) The system without damper 1, i.e. ( 1 = 0). (2) The system with damper 1, i.e. ( 1 = 0). We will study the first one by using Theorem 1. In the second one, if the conditions of Theorem 1 are not fulfilled, we will then apply the geometrical method. Let m1 = m2 = 1 Kg, 1 = 2 = 1 N m−1 , 2 = 1 N s m−1 . Case 1: ( 1 = 0): The matrices A1 and A2 are given by: 0 1 0 0 0 1 0 0 0 −2 0 1 0 −2 0 1 A2 = (46) A1 = , . 0 0 0 1 0 0 0 1 1 0 −1 0 1 0 −1 −1 It is easy to show that, = 8, but k = 3 and = (2 0 1)T . We have C1 B1 = C2 B2 and C1 A1 B1 = C2 A2 B2 ; then, we estimate a single expression of the switch function given by ˙ H x(t) = 2y(t) + y(t) ¨ − C1 A1 B1 u(t) − C1 B1 u(t).
(47)
Thus, we can observe the discrete state. As rank(O1 ) = rank(O2 ) = 4 we can observe the whole state. The control input used for the simulation is sinusoidal. Fig. 4 shows the time evolution of the real switching function H x(t). The estimated switching function is represented by Fig. 5. Fig. 6 shows the evolution of the discrete state of the hybrid system. We observe that we have a series of fast switching at x3 = 0, which is explained by the vibrations at the percussion time. Fig. 7 shows the evolution of trajectory in space (x3 , x4 ).
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0.6 0.4
Hx(t) estimed
0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
5
10
15
20
25
20
25
Time[s]
Fig. 5. Estimated switching function.
3 2.5
q
2 1.5 1 0.5 0 0
5
10
15 Time[s]
Fig. 6. Discrete state.
Case 2: ( 1 = 0): Let 1 = 1N s m−1 ; in this case the two matrices A1 and A2 are
0 −2 A1 = 0 1
1 −1 0 1
0 1 0 −1
0 1 , 1 −1
0 −2 A2 = 0 1
1 −1 0 1
0 1 0 −1
0 1 . 1 −2
(48)
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435
0.6 subsystem 1
0.5
subsystem 2
0.4 0.3 x4[m/s]
0.2 0.1 0 -0.1 -0.2 -0.3 -0.4
Hx(t)=0 -1
-0.8
-0.6
-0.4 -0.2 x3[m]
0
0.2
0.4
0.6
Fig. 7. State trajectory (x3 , x4 ).
Using Eq. (28) we have y (4) (t) = −y(t) − y(t) ˙ − 3y(t) ¨ − 2y (3) (t) + u(t) + u(t) ˙ + u(t). ¨
(49)
For the second subsystem y (4) (t) = −y(t) − 5y(t) ˙ − 5y(t) ¨ − 4y (3) (t) + u(t) + 3u(t) ˙ + u(t). ¨
(50)
Mu is a subspace of dimension 3, given by ˙ Mu = {(2 , 3 , 4 ) ∈ R3 /22 + 3 + 4 = u}.
(51)
For piecewise continuous inputs, the transversality of (40) and (41) to Mu is verified except on the subset of Mu defined by 22 + 3 + 4 = u, ˙ (52) 1 − 2 = 0. In this simulation, we used the same control input as in the first case. Figs. 8 and 9 show, respectively, the time evolution of the output of the first and of the second subsystem. The real output is represented by Fig. 10. Fig. 11 shows the time evolution of the discrete state associated with each subsystem. 6. Conclusion The main contribution of this paper is the extension of the result presented in [6] to piecewise hybrid systems with control input. We have proposed a method to study the observability of a class of hybrid systems; a linear and a nonlinear case are considered. We
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0.4 0.3 0.2
y[m]
0.1 0 -0.1 -0.2 -0.3 -0.4 0
5
10
15
20
25
20
25
Time[s] Fig. 8. Output of subsystem 1.
0.8 0.6 0.4
y[m]
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
5
10
15 Time[s]
Fig. 9. Output of subsystem 2.
gave some sufficient conditions to observe the discrete state of such hybrid systems. We studied two cases: the first was related to the possibility of the switch function reconstruction from the output, input and their derivatives. In the second case, we studied the observability based on a transversality concept. The latter approach is promising for the hybrid observer design, where the discretestate observability will be necessary. The hybrid observer design will then be based on
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437
0.8 0.6 0.4
y[m]
0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
5
10
15
20
25
20
25
Time[s]
Fig. 10. Output of the hybrid system.
3 2.5
q
2 1.5 1 0.5 0
0
5
10
15 Time[s]
Fig. 11. Discrete state.
the observation of both the continuous and the discrete state. The continuous state will be observed using a family of continuous linear or nonlinear observers; each continuous observer will be designed using the subsystem forming the hybrid system. The design of the discrete part of the hybrid observer will be based on a discrete-state observability method which can be inspired by the concepts given in this paper.
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Acknowledgements The authors are grateful to professor Vincent Maki for his reading, corrections and remarks. References [1] M. Babaali, M. Egerstedt, Observability for switched linear systems, Hybrid Systems: Computation and Control, Springer, Philadelphia, PA, 2004. [2] A. Balluchi, L. Benvenuti, M.D. Di Benedetto, A.L. Sangiovanni-Vincentelli, Design of observers for hybrid systems, in: C.J. Tomlin, M.R. Green-street, (Eds.), Proceedings of the Fifth International Workshop on Hybrid Systems: Computation and Control, vol. 2289, Lecture Notes in Computer Science, Springer, Berlin, 2002, pp. 76–89. [3] A. Balluchi, L. Benvenuti, M.D. Di Benedetto, A.L. Sangiovanni-Vincentelli, Observability for hybrid systems, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, Hawaii, USA, 2003. [4] A. Bemporad, G. Ferrari-Trecate, M. Morari, Observability and controllability of piecewise affine and hybrid systems, in: Proceedings of the 38th IEEE Conference on Decision and Control, 1999, pp. 3966–3971. [5] A. Bemporad, G. Ferrari-Trecate, M. Morari, Observability and controllability of piecewise affine and hybrid systems, IEEE Trans. Automat. Control 45 (10) (2000) 1864–1876. [6] D. Boutat, A. Benali, J.P. Barbot, About the observability of piecewise dynamical systems, Symposium on Nonlinear Control Systems, Nolcos, Stuttgart, 2004. [7] P. Collins, J.H. van Schuppen, Observability of piecewise-affine hybrid systems, Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 2623, Springer, Berlin, 2004. [9] M. De La Sen, The reachability and observability of hybrid multirate sampling linear systems, Comput. Math. Appl. 31 (1996) 109–122. [10] M. De La Sen, A note on the hyperstability of a class of hybrid dynamic systems, IEEE Trans. Automat. Control 42 (1997) 1335–1339. [11] M. De La Sen, Robust adaptive stabilization of time-invariant first-order hybrid systems with covariance resetting, Int. J. Non-Linear Mech. 33 (1998) 47–57. [12] M. De La Sen, Sufficient conditions for Lyapunovs global stability of time-varying hybrid linear systems, Int. J. Control 72 (1999) 107–114. [13] E. De Santis, M.D. Di Benedetto, G. Pola, On observability and detectability of continuous-time linear switching systems, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Maui, HW, December 2003. [14] E. De Santis, M. D. Di Benedetto, S. Di Gennaro, G. Pola, Hybrid observer design methodology, distributed control and stochastic analysis of hybrid systems supporting safety critical real-time systems design (HYBRIDGE), IST-2001-32460 of European Commission, 19 August 2003. [15] G. Ferrati-Trecate, M. Gati, Computation observability regions for discrete-time hybrid systems, in: Proceedings of the 42nd IEEE Conference on Decision and Control, Hawaii, December 2003, pp. 1153–1158. [16] G. Ferrati-Trecate, D. Mignone, M. Morari, Moving horizon estimation for hybrid systems, IEEE Trans. Automat. Control 47 (10) (2002) 1663–1676. [18] M. Fliess, J. Lévine, P. Martin, P. Rouchon,A Lie–Bäcklund approache to equivalence and flatness of nonlinear systems, IEEE Trans. Automat. Control 44 (5) (1999) 922–937. [20] A. Isidori, Nonlinear Control Systems, Springer, Berlin, 1995. [21] P.T. Kabamba, S. Hara, Worst-case analysis and design of sampled-data control systems, IEEE Trans.Automat. Control 38 (1993) 1337–1357. [24] R. Vidal, A. Chiuso, S. Soatto, S.S. Sastry, Observability of linear hybrid systems, in: Hybrid Systems: Computation and Control, Lecture Notes in Computer Science, vol. 2623, Springer, Berlin, 2003, pp.526–539.
Observability of the discrete state for dynamical piecewise hybrid systems夡 Salim Chaiba,∗ , Driss Boutata , Abderraouf Benalia , Jean Pierre Barbotb a Laboratoire de Vision et Robotique, Ecole Nationale Supérieure d’Ingénieurs de Bourges, 10 Boulevard de
Lahitolle, 18020 Bourges, France b Equipe Commande des Systèmes, Ecole Nationale Supérieure de l’Electronique et de ses Applications, 6 Avenue
du Ponceau, 95014 Cergy-Pontoise, France Received 10 May 2005; accepted 10 May 2005
Abstract In this paper, we deal with the observability of piecewise-affine hybrid systems. Our aim is to give sufficient conditions to observe the discrete and continuous states, in terms of algebraic and geometrical conditions. Firstly, we will give the algebraic conditions to observe the discrete state based on the switch function reconstruction for linear hybrid systems. Secondly, we will give a geometrical condition based on the transversality concept for nonlinear hybrid systems. Throughout this paper, we illustrate our propositions with examples and simulations. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Observability; Hybrid system; Piecewise-affine systems; Switching function; Transversality
1. Introduction Hybrid systems are systems containing mixtures of logic or discrete and continuous dynamics. The hybrid models have become very important in the last few years as tools for modelling systems. Hybrid behavior is generally described as intervals of piecewise 夡 This
work was supported by “la Région Centre de France”.
∗ Corresponding author.
E-mail addresses: [email protected] (S. Chaib), [email protected] (D. Boutat), [email protected] (A. Benali), [email protected] (J.P. Barbot). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.028
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continuous behaviors (modes) interspersed with discrete transitions that occur at points in time. Examples of hybrid systems include Networks, multi-agent systems, mechanical devices, air traffic management where different types of dynamics coexist and interact in nonobvious ways, interactive distributed simulation, robot design and path planning. The combination of continuous and the discrete dynamics makes these systems interesting and complex such that new mathematical tools are needed for their analysis, observation and control. Several works have been recently carried out on the algebraic properties of hybrid systems as well as their stability and hyperstability properties [9–12,15,16,20,21]. In the last decade many researchers were interested in the study of the observability of such systems, but the definitions and the testing criteria proposed in the literature varied depending on the class of hybrid systems considered and the type of knowledge on the output. In [2,3] we found sufficient conditions for final state observability and a methodology to design a dynamical observer of a class of hybrid systems that reconstructs the discrete state and the continuous state from the knowledge of the continuous and discrete outputs. It was introduced in [4,5] for the class of piecewise (PWA) systems. It implies that different initial states always give different outputs independent of the applied input, which is a stronger version of observability requiring a minimum amount of distinguishability between the states. Observability in the case of multiple discrete states was discussed in [1,7]. The authors in [13,14,24] proposed a definition of observability based on the concept of indistinguishability of continuous initial states and discrete states from the outputs in free evolution and gave the necessary and sufficient conditions for the observability of a class of hybrid systems called jump linear systems. Mixed logical dynamical (MLD) formulation of hybrid systems is given in [4,5,15]. Since the MLD formulation naturally lends itself to being analyzed through mixed-integer linear programming (MILP), and since very efficient algorithms have recently been developed for MILP, several computational approaches have recently been proposed to analyze various observability concepts for hybrid systems. In [15], an algorithm based on multi-parametric programming was proposed for computing the maximal observability region of a discrete-time hybrid system, i.e., the set of initial states that can only be determined by the output. Based on these works and on the results presented in [6] on the observability analysis for piecewise-affine hybrid systems without input, in this paper we give the algebraic and geometrical conditions to observe the discrete state. Our attention will be restricted to piecewise hybrid systems. The outline of this paper is as follows: in the next section, we give some definitions, notations and problem statement. In Section 3, we give the first results on the observability of the discrete state for linear hybrid systems, using the switch function reconstruction. In Section 4, we analyze the observability for nonlinear hybrid systems by means of a transversality concept. We give a simulation in Section 5. 2. Notations and problem formulation The class of systems considered in this section have the following form: x(t) ˙ = Aq x(t) + Bq u(t), y(t) = Cq x(t),
(1)
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where x(t) ∈ Rn is the continuous state, q ∈ Q = {1, 2} is the discrete state, u(t) ∈ R is the input, and y(t) ∈ R is the output. Aq ∈ Rn×n , Bq ∈ Rn×1 , Cq ∈ R1×n . In this paper we deal with a subclass of systems (1) where the discrete state is given by q = 1 if H x(t) 0, q = 2 if H x(t) > 0
(2)
and H is a 1 × n matrix. The observability matrix Oq of each subsystem q is given by Cq C q Aq . Oq .. . Cq An−1 q
(3)
Assumption 1. We assume that the systems under consideration have no Zeno phenomenon. Therefore, a solution to the hybrid system does not exhibit infinitely many discontinuities in a finite time interval. For each integer r 1, we define the following matrix: C2 C1 C2 A2 C1 A1 . Gr .. .. . . C1 Ar−1 1
(4)
C2 Ar−1 2
As we work in the finite-dimensional vector space Rn , there exists 2n, such that: r ∀ r < , rank(Gr ) = ∀ r .
(5)
More precisely, we have
= max[rank(Gr )].
(6)
r 1
We will call the global joint observability index, and G the joint observability matrix associated with system (1). For > 0, we set h (H H ).
(7)
Finally, we denote Y(t)
y y˙ .. .
y (n−1)
,
U(t)
u u˙ .. .
u(n−1)
,
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0 Cq B q Cq Aq Bq q (k) .. . Cq An−1 q Bq
0 0 Cq Bq ··· Cq An−2 q Bq
... ... ... .. . ...
0 0 0 .. . Cq Bq
0 0 0 . .. . 0
It is easy to show the following observation equation:
Y(t) = Oq x(t) + q U(t).
(8)
In this paper, we can say that system (1) is observable if we can recover the whole state: the continuous and the discrete states. Our problem is to deal with the observability of the discrete state. The idea to solve this problem is to give the sufficient conditions to know which subsystem of (1) evolves.
3. Algebraic conditions for discrete-state observability In this section, we give the sufficient conditions for the observability of the discrete state using the switch function reconstruction. For this we give the following theorem. Theorem 1. Let us assume the following conditions: (i) ∃ > 0 and ∃k, 0 k such that: hT ∈ I m[GTk ] (ii) C1 Ai1 B1 = C2 Ai2 B2 , ∀i = {0, . . . , k − 1}. Then, we have H1 x(t) = H2 x(t) and then from (2) we know which subsystem evolves. Therefore, the discrete state is observable for all admissible control inputs. Proof. If hT ∈ I m[GTk ] then rank[GTk hT ] = k. As rank[Gk ] = k this means that vector h is a linear combination of the rows of Gk . Then, there exists an = ( 0 . . . k−1 )T solution to the following algebraic equation: 0 . GTk .. = hT . (9) k−1 Then h =
k−1
i ((C1 Ai1 )T , (C2 Ai2 )T )
(10)
i=0
which gives two values of H x(t) H1 x(t) = T O(C1 , A1 )x(t) = T [Y(t) − 1 U(t)],
(11)
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H2 x(t) = T O(C2 , A2 )x(t) = T [Y(t) − 2 U(t)].
(12)
Both expressions can be written in the following form:
k m−1 m−1−i (m) (i) H1 x(t) = 0 y + , m y − C1 A 1 B1 u m=1
427
(13)
i=0
m−1 k 1 m−1−i (m) (i) H2 x(t) = . C2 A 2 B2 u 0 y + m y − m=1
(14)
i=0
If > 0 and thanks to condition (ii), H1 x(t) and H2 x(t) have the same sign. Thus, we know which subsystem of (1) evolves. Remarks. (i) If each subsystem is observable, rank(Oq )=n for q =1 and 2 and if conditions (i) and (ii) of Theorem 1 are fulfilled, then we can recover the whole hybrid state: the continuous and the discrete one. (ii) If = 1; then H1 x(t) = H2 x(t); thus, we measure the same switch function. (iii) When = 2n, we always have = 1. (iv) For k > 1, condition (ii) becomes (B1 , −B2 ) ∈ ker(G(k−1) ),
k 1.
(15)
Let us assume that C1 = C2 = C; then for k 1, condition (15) of the theorem is equivalent for 0 i k − 1 to Ai1 B1 − Ai2 B2 ∈ ker(C)
(16)
(v) If B1 = B2 = 0, then we obtain the same result given in [6] relative to piecewise-affine hybrid systems without control input. Example 1. Let us consider the hybrid system formed by the following two subsystems:
1 0 1 A1 = , C1 = ( 1 0 ) . , B1 = 0 −1 0
0 1 1 A2 = , B2 = , C2 = ( 1 0 ) . (17) −1 −1 0 Let us consider the following switch function: H x(t) = x1 (t) + x2 (t). We have k = = 4 and 1 0 0 1 T T [G4 h ] = 1 0 0 1
−1 0 0 −1 −1 1 −1 0
(18) 1 1 .
(19)
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Here = 1 and T = ( 1 1 0 0 ). Condition (ii) of Theorem 1 is fulfilled, which gives us a single value of the switch function given by H1 x(t) = y(t) + y(t) ˙ + u(t).
(20)
Then, we can know which subsystem evolves. Therefore, the discrete state is observable. Now, let us consider the system given by the parameters A1 , A2 , B1 , B2 as in (17); C1 and C2 are given by C1 = ( 1 0 ) ,
C2 = ( 0
1) .
(21)
We consider the same switch function as in (18), k = = 4, rankG4 = 4 and for = 1 we have rank[GT4 hT ] = 4. It is easy to see that: C1 Ai1 B1 = C2 Ai2 B2 ,
∀i = 0, 1, 2.
(22)
This means that condition (ii) of Theorem 1 is not fulfilled. Thus, we cannot determine the sign of the switch function H x(t). Therefore, the discrete state is not observable with the algebraic equality of Theorem 1. For the linear dynamical systems which do not satisfy the algebraic conditions of Theorem 1 and for the nonlinear systems, in the next section we will propose another method to observe the discrete state, based on a geometrical condition.
4. Geometrical conditions In this section, we give a geometrically sufficient condition to observe the discrete state. We consider nonlinear hybrid systems consisting of the two nonlinear subsystems: x(t) ˙ = fq (x(t)) + gq (x(t))u(t), y(t) = cq (x(t)), q = 1 if h(x(t)) 0, q = 2 if h(x(t)) > 0,
(23)
where fq (x) for q = 1, 2 is the drift smooth vector field, gq (x) is the input smooth vector fields, cq (x) is the smooth output and h(x) is a smooth switch function on Rn . As mentioned in Section 3, if we know at each moment which subsystem evolves then the discrete state is observable. In the following we will use dynamical equations satisfied by the output to give another method to observe the discrete state. In this section, we assume that each subsystem of (23) is observable i.e., for q = 1, 2 we have: dcq dL¯ fq +gq u cq rank (24) = n, .. . (n−1) dL¯ fq +gq ucq
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where L¯ is the Lie–Bäcklund derivative given by d + Lf + Lg u L¯ f +gu = dt
(25)
and Lf is the Lie derivative of a given function in the direction of the vector field f . Remark. As the following Lie–Bäcklund isomorphisms [18]: q
1 = cq , q 2 = L¯ fq +gq u cq , q = L¯ 2f +g u cq 3
q
q
.. . qn = L¯ n−1 fq +gq u cq
(26)
are different for each subsystem (q = 1 or 2) we have a jump on the canonical states when the system switches even if the original coordinates do not jump. For each subsystem we have (n)
y (n) = L¯ fq +gq u cq .
(27)
Next, we set y (n) = q (y, y, ˙ . . . , y (n−1) , u, u, ˙ . . . , u(n−1) )
(28)
as defined in (27). For a given input u, if the output y satisfies one and only one of the two dynamical equations of (28), then, we know which subsystem evolves, and we can thus observe the discrete state. Let us now define the following submanifold: ˙ . . . , u(n−1) ) = 2 (v, u, u, ˙ . . . , u(n−1) )}. Mu = {v ∈ Rn /1 (v, u, u,
(29)
The following assumption is one of the keys to the observability of the discrete state. Assumption 2. We set Lu to be the submanifold of the common singularities of the two subsystems of (23). If Lu contains more than one element, then we assume that Lu is included in only one of the two spaces shared by the switching function H x(t) i.e., for all x in Lu we have H x 0 or else H x > 0.
(30)
It is clear that Lu ⊂ Mu . The following result gives a sufficient condition to observe the discrete state. Theorem 2. Under Assumptions 1 and 2 and if the two subsystems of (23) are transverse to Mu except on a discrete subset, then, there exists a finite time when y(t) satisfies one and only one equation of (28).
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Fig. 1. The behavior of the vector fields on Mu at v between two instants.
Proof. Thanks to the assumption about the observability of each subsystem, the following change of coordinates: q
q
2 = y, ˙ . . . , qn = y (n−1)
1 = y,
transform (23) into the following forms: for subsystem 1: 1 ˙ i = 1i+1 for i = 1 : n − 1, 1 ˙ = (, u, u, ˙ . . . , u(n−1) ). n
(32)
1
for subsystem 2: 2 ˙ i = 2i+1 for i = 1 : n − 1, 2 ˙ = (, u, u, ˙ . . . , u(n−1) ), n
(31)
(33)
2
where q is given by (28). Let v ∈ / Mu , as Mu is a closed submanifold; then, there exists a neighborhood V of v such that V ∩ Mu = ∅. In V, we have
1 (v, u, u, ˙ . . . , u(n−1) )|v∈V = 2 (v, u, u, ˙ . . . , u(n−1) )|v∈V .
(34)
This inequality implies that the output satisfies one and only one equation of (28) for q =1, 2. If v ∈ Mu ∩ Lu then, by Assumption 2 we can conclude on the observability of the discrete state. Now, if v ∈ Mu and v ∈ / Lu at t = t ∗ then, 1 (v, u, u, ˙ . . . , u(n−1) ) = (n−1) ∗ (n−1) 2 (v, u, u, ˙ ...,u ) at t , but 1 (v, u, u, ˙ ...,u ) and 2 (v, u, u, ˙ . . . , u(n−1) ) are transverse to Mu except on a discrete subset and thanks to Assumption 1 we have
1 (v, u, u, ˙ . . . , u(n−1) ) = 2 (v, u, u, ˙ . . . , u(n−1) )
(35)
for a certain moment t = t ∗ + ( > 0) (see Fig. 1), which leads us to the previous case.
S. Chaib et al. / Nonlinear Analysis 63 (2005) 423 – 438
431
x2 Mu Vector fields of subsysems 1 and 2
P1
x1
p2
Fig. 2. The behavior of the vector fields on M0 for u = 0.
Example 2. Let us consider the system described by the following subsystems:
x˙1 = x2 , x˙2 = −x23 + u = 1 (x1 , x2 , u), y = x1 ,
(36)
x˙1 = x2 , x˙2 = x1 + u = 2 (x1 , x2 , u), y = x1 .
(37)
We have
Mu = {(x1 , x2 ) ∈ R2 /x1 = −x23 }
(38)
The two systems (36) and (37) are transverse to Mu except on the common singularity (0, 0). At the points solution to the following equations:
x1 = −x23 , x2 (1 − 3x24 + 3x2 u) = 0.
(39)
The dynamics of the two subsystems are tangent to Mu , the vector fields of the two subsystems at these points have two nonnull components along the axis x1 and x2 andtherefore, the 1 1 , 31/4 discrete state is observable. For example for u=0 these points are given by p1 = − 33/4 1 1 and p2 = 33/4 , − 31/4 ; the vector fields of subsystems 1 and 2 are tangent to M0 (see Fig. 2). Therefore, the discrete state is observable for any switch function.
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Fig. 3. A double cart with elastic coupling.
5. Simulation: double cart with elastic coupling In this section, we give a linear example in order to illustrate our algebraic and geometrical methods. Let us consider the mechanical system described in Fig. 3. The linear hybrid model of the plant under investigation is as follows: for z2 (t) 0: m1 z¨ 1 = − 1 z1 + 2 (z2 − z1 ) + 1 (˙z2 − z˙ 1 ) + u and m2 z¨ 2 = − 2 (z2 − z1 ) − 1 (˙z2 − z˙ 1 )
(40)
for z2 (t) > 0: m1 z¨ 1 = − 1 z1 + 2 (z2 − z1 ) + 1 (˙z2 − z˙ 1 ) + u and m2 z¨ 2 = − 2 (z2 − z1 ) − 1 (˙z2 − z˙ 1 ) − 2 z˙ 2 .
(41)
Let x T = (x1 , x2 , x3 , x4 ) = (z1 , z˙ 1 , z2 , z˙ 2 ); then, the state representation of (40) and (41) is
0 − 1 + 2 m1 x(t) ˙ = 0 2
m2 y(t) = ( 1
0
0
0 + 2 − 1 m1 x(t) ˙ = 0 2
m2 y(t) = ( 1
0
0
1
− 1 m1 0
1 m2
0 2 m1 0 − 2 m2
0 0
1 m1 x(t) + 1 u(t), 1 0
1 0 − m2
0 ) x(t) if x3 0,
(43)
1
− 1 m1
0 2 m1
0
1 m1
0
1 m2
0 − 2 m2
1
1 + 2 − m2
0 ) x(t) if x3 > 0.
(42)
0 1 x(t) + u(t), 0 0
(44)
(45)
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433
0.6 0.4
Hx(t) estimed
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
5
10
15
20
25
Time[s]
Fig. 4. Real switching function.
We will consider the following two configurations: (1) The system without damper 1, i.e. ( 1 = 0). (2) The system with damper 1, i.e. ( 1 = 0). We will study the first one by using Theorem 1. In the second one, if the conditions of Theorem 1 are not fulfilled, we will then apply the geometrical method. Let m1 = m2 = 1 Kg, 1 = 2 = 1 N m−1 , 2 = 1 N s m−1 . Case 1: ( 1 = 0): The matrices A1 and A2 are given by: 0 1 0 0 0 1 0 0 0 −2 0 1 0 −2 0 1 A2 = (46) A1 = , . 0 0 0 1 0 0 0 1 1 0 −1 0 1 0 −1 −1 It is easy to show that, = 8, but k = 3 and = (2 0 1)T . We have C1 B1 = C2 B2 and C1 A1 B1 = C2 A2 B2 ; then, we estimate a single expression of the switch function given by ˙ H x(t) = 2y(t) + y(t) ¨ − C1 A1 B1 u(t) − C1 B1 u(t).
(47)
Thus, we can observe the discrete state. As rank(O1 ) = rank(O2 ) = 4 we can observe the whole state. The control input used for the simulation is sinusoidal. Fig. 4 shows the time evolution of the real switching function H x(t). The estimated switching function is represented by Fig. 5. Fig. 6 shows the evolution of the discrete state of the hybrid system. We observe that we have a series of fast switching at x3 = 0, which is explained by the vibrations at the percussion time. Fig. 7 shows the evolution of trajectory in space (x3 , x4 ).
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0.6 0.4
Hx(t) estimed
0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
5
10
15
20
25
20
25
Time[s]
Fig. 5. Estimated switching function.
3 2.5
q
2 1.5 1 0.5 0 0
5
10
15 Time[s]
Fig. 6. Discrete state.
Case 2: ( 1 = 0): Let 1 = 1N s m−1 ; in this case the two matrices A1 and A2 are
0 −2 A1 = 0 1
1 −1 0 1
0 1 0 −1
0 1 , 1 −1
0 −2 A2 = 0 1
1 −1 0 1
0 1 0 −1
0 1 . 1 −2
(48)
S. Chaib et al. / Nonlinear Analysis 63 (2005) 423 – 438
435
0.6 subsystem 1
0.5
subsystem 2
0.4 0.3 x4[m/s]
0.2 0.1 0 -0.1 -0.2 -0.3 -0.4
Hx(t)=0 -1
-0.8
-0.6
-0.4 -0.2 x3[m]
0
0.2
0.4
0.6
Fig. 7. State trajectory (x3 , x4 ).
Using Eq. (28) we have y (4) (t) = −y(t) − y(t) ˙ − 3y(t) ¨ − 2y (3) (t) + u(t) + u(t) ˙ + u(t). ¨
(49)
For the second subsystem y (4) (t) = −y(t) − 5y(t) ˙ − 5y(t) ¨ − 4y (3) (t) + u(t) + 3u(t) ˙ + u(t). ¨
(50)
Mu is a subspace of dimension 3, given by ˙ Mu = {(2 , 3 , 4 ) ∈ R3 /22 + 3 + 4 = u}.
(51)
For piecewise continuous inputs, the transversality of (40) and (41) to Mu is verified except on the subset of Mu defined by 22 + 3 + 4 = u, ˙ (52) 1 − 2 = 0. In this simulation, we used the same control input as in the first case. Figs. 8 and 9 show, respectively, the time evolution of the output of the first and of the second subsystem. The real output is represented by Fig. 10. Fig. 11 shows the time evolution of the discrete state associated with each subsystem. 6. Conclusion The main contribution of this paper is the extension of the result presented in [6] to piecewise hybrid systems with control input. We have proposed a method to study the observability of a class of hybrid systems; a linear and a nonlinear case are considered. We
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0.4 0.3 0.2
y[m]
0.1 0 -0.1 -0.2 -0.3 -0.4 0
5
10
15
20
25
20
25
Time[s] Fig. 8. Output of subsystem 1.
0.8 0.6 0.4
y[m]
0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
5
10
15 Time[s]
Fig. 9. Output of subsystem 2.
gave some sufficient conditions to observe the discrete state of such hybrid systems. We studied two cases: the first was related to the possibility of the switch function reconstruction from the output, input and their derivatives. In the second case, we studied the observability based on a transversality concept. The latter approach is promising for the hybrid observer design, where the discretestate observability will be necessary. The hybrid observer design will then be based on
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437
0.8 0.6 0.4
y[m]
0.2 0 -0.2 -0.4 -0.6 -0.8 -1
0
5
10
15
20
25
20
25
Time[s]
Fig. 10. Output of the hybrid system.
3 2.5
q
2 1.5 1 0.5 0
0
5
10
15 Time[s]
Fig. 11. Discrete state.
the observation of both the continuous and the discrete state. The continuous state will be observed using a family of continuous linear or nonlinear observers; each continuous observer will be designed using the subsystem forming the hybrid system. The design of the discrete part of the hybrid observer will be based on a discrete-state observability method which can be inspired by the concepts given in this paper.
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