Structural Observability of Controlled Switching Linear Systems

In this paper, a new methodology for analysis of structural observability of controlled switching linear systems modelled by bond graphs is proposed. Causal manipulations on the bond graph model enable to determine graphically the observable subspace. A novel definit ion of observability is proposed. Finally, two sufficient conditions of observability are derived. The proposed method, based on a bond graph theoretic approach, assumes only the knowledge of the systems structure. These conditions can be implemented by classical bond graph theory algorithms based on finding particular paths and cycles in a bond graph.


Introduction
Hybrid dynamic systems constitute a particular class of dynamic systems in which some elements (called switching components) or phenomena evolve much faster than the time scale at wh ich the system behaviour needs to be analysed [16]. Such systems, also called switched systems, are widespread in physical and engineering fields (hydraulic systems with valves, electrical systems containing diodes, relays or transistors,…, mechanical systems with clutches or collisions...). In this present work, I investigate the observability issue for controlled switching linear systems modelled by bond graph. My obtained results present a graphical method based on an energy concept.
Observability is a fundamental concept in modern control theory of systems, has been extensively studied both in the continuous and in discrete domains. More recently various researchers have approached the study of observability for hybrid systems [2,14,15,18,19].
Ezzine and Hadd ad [12] first stud ied the on e-period controllab ility and observability for period ically switched systems, so me sufficient and necessary cond it ions were estab lish ed . Then , Xie an d Zh en g [6] in tro du ced the mu lt iple-period contro llability and observability concepts naturally extended fro m the one-period ones, necessary, and sufficient criteria were derived. It was also pointed out that controllability can be realized in n periods at most, where n is the state dimension. As to arbitrarily switched linear systems, Sun and Zheng [21] first gave a sufficient condition and a necessary condition for controllability and proved that the necessary condition is also sufficient fo r three-d imensional systems with on ly two subsystems. Following the work in [21] and [22] extended the result to three-dimensional systems with arbit rary number o f subsystems. Then, necessary and sufficient geometric type criteria for controllability and observability were derived in [22] and [7]. Vidal and al. [18] considered autonomous switching systems and proposed a definit ion of observability based on the concept of indistinguishability of continuous initial states and discrete state evolutions from the outputs in free evolution. Incremental observability was introduced in [2] for the Class of Piecewise Affine (PWA) systems. Incremental observability imp lies that different init ial states always give different outputs independently of the applied input. In [1], a methodology was presented for the design of dynamic observers of hybrid systems that reconstructs the discrete state and the continuous state from the knowledge of the continuous and discrete outputs. In [10], the defin itions of observability of [19] and the results of [1] on the design of an observer for deterministic hybrid systems are extended to discrete-time stochastic linear autonomous hybrid systems.
In order to obtain a more realistic model for the analysis of system properties, the concept of structural property has been introduced in [5], is true for almost all values of the parameters. This framework is consistent with physical reality in the sense that system parameter values are never exactly known, with an exception for zero values that express the absence of interactions or connections.
The elements of a structural matrix [A] are either fixed at zero o r indeterminate values that are assumed to be independent of one another. Hence it is desirable to investigate system properties that are only determined by the structure of the system and not by the parameter nu merical values. A useful tool for this purpose is the bond graph approach, which has received a g reat deal o f attention in the last decade [5,17]. This approach has been used for the analysis of structural properties of linear systems [3,4,9]. This paper is organized as follows: The second section, formulates the Controlled Switching Linear Systems (CSLS) observability. Section three recalls some background about bond graph modelling of hybrid systems with ideal switches. In section four the structural observability of these systems is discussed using bond graph model. Graphical conditions and procedures are also proposed. Finally, a simp le examp le which illustrates the previous results is discussed.

Observability of Controlled Switching Linear Systems
Consider a controlled switching linear systems [23], described by If we consider this system in a part icular mode i, the equation (1) can be written as :

An Algebraic Sufficient Condi tion
In order to investigate observability of (1), the following zero input system is considered.
It is obvious that the observability of (1) is equivalent to that of (3).
The observability comb ined matrix O [9] of system (1) is given by equation (4), Re mark 2 Fro m this theorem, we can deduce that: 1) The system (1) can be observable, if there is only one observable sub-system (mode).
2) However, it is possible that no sub-system is observable but that the system (1) is observable.

A necessary and Sufficient Algebraic Conditi on
References [9] and [22] define the subspace sequence . This theorem is a geometric criterion, thus, it is easy to transform it into algebraic form.
Definiti on 2The joint observability matrix of system (1) is defined as : Im G = G . Theorem 3 [9]System (1) is observable, if and only if rank( ) G n = . Proof.For 0 = j , we have : In a similar way one finds : Of another share, we have : ■. We exposed algebraic and geometric criteria of analysis of the properties of observability of CSLS. The next section is devoted to the graphic interpretation of these results by using the Bond graph approach.

Bond Graph Approach
The bond graph structure junction contains informat ions on the type of the elements constituting the system, and how they are interconnected, whatever the numerical values of parameters. The structure junction of a switching bond graph can be represented by Figure 1. Five fields model the components behaviour, four fields that belong to the standard bond graph formalism; -source field which produces energy, -detector field; -R field which d issipates it, -I and C field which can store it, -and the Sw field that is added for switching components. Figure 1 represents the block d iagram that is deduced fro m the causal bond graph.
The following key variables are used : -the state vector x(t)is composed of the energy variables on the bond connected to an element in integral causality (the

Assumptions 2
To take into account the absence of discontinuities (Assumption 1), we suppose that there are no elements in derivative causality in the bond graph model in integral causality, before and after co mmutation.
Using the structure junction, the follo wing equation is given [11] : 11 L is a positive matrix. Let assume that The third line of (7) g ives : Then substituting also in the first line of (7) gives : 11

S H S Fx t S S H S T t S S H S u t
The output vector is given by : [ ) where 11 13 13 ( )

Structural Observability
The bond graph concept is an alternate representation of physical systems. So me recent works permit to highlight structural properties of these systems [4]and [3]. In [4], the structural observability property is studied using simp le causal manipulat ions on the bond graph model. It is shown that the structural ran k concept is so mewhat d ifferent for bond graph models because it is more precise than for other representations. Our objective is to extend these properties to CSLS systems.
In the following we note that: -BG: acausal (without causality) bond graph model, -BGI: bond graph model when the preferential integral causality is affected, -BGD: bond graph model when the preferential derivative causality is affected, -i t : the number of elements in integral causality in BGD i . i indicate the mode i. To study structural observability of CSLS modelled by bond graph, graphical methods are proposed in the form of two sufficient conditions. In fact, formal representation of observability subspace is given for bond graph models. It is calculated through causal man ipulations. The base of this subspace is used to propose a procedure to study the system observability.

Graphical Sufficient Condition 1
A system (1) with q modes is observable if only one system (2) is observable. Th is condition can be interpreted by using the result of structural observability of LTI system. Indeed, this result is a simp le recovery of those giving the necessary and sufficient condition of structural observability of LTI system modelled by bond graph approach [4].
Theorem 4 The CSLS system is structurally state observable if: -On the BGI i , all dynamical elements in integral causality are causally connected with a continuous output De or Df associated to y(t) or a discrete output Sw associated to ( ) Example 1 We consider the following acausal bond graph model. Shown in Figure 2:  x P P P P q q = .
The application of the derivative causality, for examp le on mode 1 (Figure 3.a), give the following BGD 1 (Figure 4). This result can be verified using formal calculation on the bond graph model in integral causality [3].
To study the observability of system (12), it is necessary to apply this result to all modes; if one observable mode exists, the procedure is stopped. The case where no mode is observable, but when the system is observable, can be verified by formal calculat ion of co mbined matrix (4).
This calculation can be formally effected : -by using the bond graph model in integral causality [3], or -by calculating the observability subspace fro m bond graph model in derivative causality.
We chose to translate the latter in the form of a second sufficient condition. For that, formal representation of structural observability subspace, denoted as 0 Ω , is given for BG model. It is calculated using causal manipulat ions. The base of this subspace is used to propose a procedure to study the structural observability of system.

Graphical Sufficient Condition 2
On the BGD i (and dualization of continuous and discrete    The coefficients of the algebraic relation are mu ltiplied by the inductance or capacitance parameter, because of the form of the output matrix in the state equation. Thus we obtain In order to calculate a 0 i Ω basis,it is enough to find  (14).

Procedure 2 Calculat ion of 0
i Ω 1) On the BGD i , dualize the maximu m nu mber of output detectors in order to eliminate the elements in integral causality.
2) For each element remaining in derivative causality, write the algebraic relat ion with elements in integral causality, (Equation (14)).
3) Write a row vector ir w for each algebraic relat ion with the different gains of the causal paths, (Equation (14)).
Some calculation is carried out for mode 2. We obtain ( ) , with 21 11 w w = , 22 12 w w = , 23 The graphical calculation of structural observability subspaces and remark 2 lead to theorem 5. After co mmutation fro m i th mode to (i +1) th mode, implement a derivative causality on the bond graph model and dualization the maximu m nu mber of continuous and discrete outputs. We can write another algebraic relation (equation 15). (15) Its base is given by

Example
Let us consider the fo llo wing acausal BG model ( figure 6).   Step 1 : Verificat ion of sufficient condition 1 This step is applied to the 4 modes, but only the mode 1 is presented.
-On the BGI 1 , all state variables are in integral causality and are causally connected with the detectors, -On the BGD 1 , two elements 1 I and 3 I stays in integral causality. After dualizat ion of the discrete output 2 2 Sw f T o = associated to 2 Sw , only one element 1 I remain ing in integral causality (figure 8.A), So this mode is not observable.
In the same way, the other modes are not observable, therefore, step 1 is not verified.
Step 2 : Verificat ion of sufficient condition 2 ▪ Calculation of W  The system is structurally observable. Re mark 3 If these conditions are not checked, it is necessary to use a necessary and sufficient condition. This result will be done in a future work.

Conclusions
The structural observability property of controlled switching linear systems is studied using simp le causal man ipulations on the bond graph model. Thus, formal calculation enables us to know the reachable variables; its checking is immediate on the bond graph model in integral causality. On the other hand the bond graph model in derivative causality enables us to characterize graphically the structural observability subspaces relating to each mode. Two sufficient conditionswas given by explo iting these various bases. Finally procedures were p roposed.
In fact, the proposed method, based on a bond graph theoretic approach, assumes only the knowledge of the systems structure. The subspaces can be employed to propose structured state feedback matrices in the context of pole assignment by static state feedback. This result can be implemented by classical bond graph theory.