Structural analysis of switching systems modelled by bond graph

Abstract

This paper deals with some structural properties for switching devices modelled by bond graphs. To calculate the structural rank of the state matrices, to point out impulse modes and to study the structural controllability and observability, graphical methods are proposed. These methods are based on simple causal paths and causal manipulations on the bond graph model.

Keywords:

• Bond Graph
• Switching Systems
• Infinite Mode
• Structural Controllability
• Structural Observability

1 Introduction

Switching devices are very common in several fields (diodes, relays, clutches, valves, …). Their changes of state can be either controlled or autonomous. Various researchers have investigated this problem using the bond graph tool [1, 3 –  6, 8, 10, 12 –  15, 18, 20, 21]. Two main approaches, the ideal and the non-ideal ones, are recalled.

a.	In the non-ideal approach [6], switches are modeled as resistive elements what point out the loss of power in the ON mode eventually associated with a modulated transformer. The modulation is done using a Boolean variable. The main advantage of such an approach is that it leads to a unique causal bond graph model, no matters the state that switches can have.
b.	In the ideal approach (no power dissipation) [3, 20, 21], each switch is modelled by a null source: an effort source for a closed switch state, and a flow source for an open one. The state can be discontinuous at commutation times. This is the approach used in this work.

In order to obtain a more realistic model for the analysis of system properties, the concept of structural property has been introduced by [11]. This framework is consistent with physical reality because system parameter values are never exactly known, with an exception made for zero values expressing the absence of interactions or connections. The elements of a structural matrix are fixed either as zero or as indeterminate values that are assumed to be independent. Hence it is desirable to investigate system properties that are determined by the system structure and not by the parameter numerical values. These systems are called “structured systems”. The bond graph approach is used for the analysis of structural properties of linear multivariable time invariant models. In this approach, the structural property concept has been used, however, the hypothesis of independence of the structural matrix terms, commonly used in the structural property study for structured systems by the digraph approach [7, 11, 19] is not stated. The present work is an extension of the structural analysis of generalized systems modelled by bond graph [14, 15, 17]. Here we consider switching devices modelled by bond graph as a good specific case of generalised systems.

This paper is organized as follows: The second section recalls some useful background about bond graph modelling of switching devices. The modelling is done using the structure junction equation, leading to an implicit model E[xdot](t) = Ax(t) + Bu(t). In the third section, structural ranks of the E- and A-matrices are determined using causal manipulations on the bond graph model. This allows determination of the minimum number of input sources (respectively outputs) necessary for the control and optimisation of their positions to simplify state feedback control laws. In section 4, a procedure that allows an easy determination of impulse modes from a bond graph model is presented. This procedure is based on simple causal manipulations on the bond graph model. Graphical methods for structural R- controllability and structural impulse controllability are proposed in section 5. In the same way, in section 6, graphical methods for structural R-observability and structural impulse observability are proposed. Finally, section 7 gives an application example.

2 Modelling

In the ideal approach, all switched bond graph elements are classical bond graph elements with one exception, the switch. The switch is modelled as a null source; its type depends on its current state (flow source for an open switch, effort source for a closed one). The junction structure of a bond graph model contains information on the types of elements which compose it and on the way they are interconnected. The different vectors involved in a bond graph are linked through the junction structure equation (equation 1).

With,

. S ₁₁ and S ₃₃ are skew-symmetric matrices [3, 18]. Elementary laws are associated with these components.

X_i and X_d are the state vectors associated with the I and C elements respectively in integral causality (p for I-elements and q for C-elements) and derivative causality. Z_i and Z_d are called complementary state vectors, F_i and F_d are diagonal matrices composed of I and C parameters. These matrices are always invertible for bond graph models without multiport elements. D_in and D_out denote the vectors composed by variables flowing respectively into and out of the R-components; L is a diagonal matrix composed by resistance and conductance parameters. Vector T_in contains the variables imposed by switches in the initial configuration: flow in the closed position and effort in the open position. The vector T_out is the complementary vector of T_in . Vector U contains the variables imposed by the effort or flow control sources MSe, MSf (“M” for modulated). Vector Y contains the variables associated with the effort or flow detectors De, Df.

The switching law can be written as equation (3).

Where Λ is a square diagonal matrix

; its elements are equal to 1 if the switches commute, else they are equal to 0 [3].

Some hypotheses are taken into account.

a.	Two dependent elements (I,C) in derivative causality are not directly causally connected, S 22 = 0. Actually, if this occurs, it is possible to change simultaneously their two causalities in order to obtain two dynamical elements in integral causality.
b.	An element (I,C) in derivative causality is not causally connected to linear R- elements (otherwise it would be possible to exchange the causalities), S 23 = S 32 = 0.
c.	The output is not directly connected to the input S 33 = 0.
d.	No signal bond in the bond graph model is made. For sake of simplicity, we will suppose from now on that there are no algebraic loops between R-elements in the bond graph model studied.

Using these assumptions, equation (1) can be written under the singular form E{\dot X} = AX + BU (equation 4).

(I = identity matrix of appropriate dimension).

2.1 State equation in the initial configuration

The initial configuration (or reference configuration [3]) is chosen in order to maximize the number of elements in integral causality in the bond graph in preferred integral causality, denoted BGI. In this configuration (before commutation), Λ = 0 in equation (3) implies T_in  = 0, so equation (4) can be rewritten as equations (5) and (6).

With dim X = dim X_i  +  dimX_d  = n = n_i  + n_d , dim U = m and r = rank[E].

The output relation is given by equation 7.

Where C_i and C_d are the output sub-matrices related respectively to X_i and X_d , and dim y = p.

2.2 State equation in the configuration after the commutation

After the commutation Λ = I implies T_out  = 0. From (4) we obtain eqution (8).

With dim X = dim X_i  +  dim X_d  + dim T = n = n_i  + n_d  + n_w ; n_w denotes the number of switches.

The output is given by equation (9).

Where C_w is the output sub-matrix related to T_in .

The expression for the sub-matrices composing E, A, and B proposed in equation (8), is deduced from the junction structure matrix built from the bond graph model and elementary physical laws. For a 1-port bond graph model, the formal procedure to obtain these matrices directly from the bond graph is given in [15].

Remark 1. It is important to note that a new model is obtained after each commutation. The dimensions of the vectors and the matrices are different. In this work, for sake of simplicity, we keep the same notations of dimensions and matrices before and after a commutation.

3 Structural ranks

In order to show how a bond graph model of a switching system is a good tool for structural analysis, we begin by studying the structural rank of the state matrix A denoted bg_rank [A] [16].

3.1 Rank of A-matrix

The bond graph rank of the state matrix of switching systems is determined in the same way as for generalized systems [14]. The difference here is that we have to consider two cases by switch, before and after its commutation.

Proposition 1

The bond graph rank of the state matrix A obtained in the initial configuration (respectively after the commutation) denoted by bg_rank [A] is equal to (n_i – q) + n_d, (respectively (n_i – q) + n_d  + n_w ) with q : the number of elements initially in integral causality staying in integral causality in the bond graph in preferred derivative causality, denoted BGD.

Proof

bg_rank[A] = N – q, with N = n_i  + n_d (equation (5)) or N = n_i  + n_d  + n_w (equation (7)). This proposition is equivalent to the one proposed in the case of model with neither elements in derivative causality nor switches (N = n_i ) in the BGI [22].

3.2 Rank of E-matrix

Proposition 2. The bond graph rank of the singular matrix E (noted r) obtained in the initial configuration (equation (5)) is equal to n_i , where n_i is the number of dynamical elements in integral causality in the BGI.

Proof

Proposition 3. The bond graph rank of the singular matrix E (noted r) obtained after the commutation (equation (7)) is equal to n_i  + n_wd , where n_i is the number of dynamical elements in integral causality in the BGI and n_wd the number of disjoint causal paths between switches Sw and (I,C) elements in derivative causality.

Proof

After commutation, the bond graph rank of the E matrix is given by equation (11).

In a bond graph model (using equation (1)),

represents the causal paths between switches Sw and (I,C) elements in derivative causality. So

is given by the number of disjoint causal paths connecting switches Sw and (I,C) elements in derivative causality.

4 Infinite mode

Let us consider the state model, described by the equation (1), if the matrix [sE-A] is regular, output and input variables under null initial conditions (E x(0) = 0) are related by the non strictly proper transfer function, as equation (12).

The response of the system can exhibit d exponential modes and (r- d) infinite-frequency modes : impulsive (or distributional) modes corresponding essentially to (sE-A) losing rank at s = ∞ hence to poles at infinity. d is equal to the degree of det[sE – A] and r = rank[E] is the actual order of the model; hence d ⩽ r, the equality occurs in the regular case (i.e. E is not singular).

4.1 Algebraic characterisation

Using rank matrix conditions, the existence of infinite modes can be pointed out.

Proposition 4

In the initial configuration, the number of impulsive modes is null.

Proof

In this case d = n _i and bg_rank[E] = n_i , so r - d = 0.

Proposition 5

After the commutation, the number of impulsive modes is equal to

.

Proof

After the commutation d = n_i and some elements change from integral causality to derivative causality.

1.	⇒r = bg_rank[E] = ni  +
2.

4.2 Graphical interpretation

A graphical interpretation of the number of impulse mode can be done using proposition 3 to calculate bg_rank[E] and [14] to determine the degree of the characteristic polynomial using the causal cycle families concept. The presence of infinite modes can be verified in the BGI in terms of causal paths. After the commutation, it can happen that some elements in integral causality lose their integral causality. Their number corresponds to the reduction of the order of the characteristic polynomial.

Proposition 6

The number of impulse modes is given by the number of elements that appear in derivative causality after the commutation in the BGI.

Proof

In the initial configuration d=n _i. After the commutation, if p elements change to derivative causality, the number of elements in integral causality is then equal to n _i −p. d=n _i −p and bg_rank[E] = r is unchanged. Another equivalent proposition can be given as a graphical interpretation of

.

Proposition 7

The number of impulse modes is given by the number of disjoint causal paths which appear after commutation between elements in derivative causality and switch elements in the BGI.

Proof

In the same way bg_rank[E] = r is unchanged and thus it is equal to n_i in the initial configuration and to n_i  + 

after commutation. So the number of impulse modes is equal to

.

is given by the number of disjoint causal paths between elements in derivative causality and switch elements.

Remark 2

Our purpose concerns only structural analysis, it is clear that for particular symbolic values of parameters the model can exhibit non structural impulse modes.

5 Structural controllability

The objective of this part is to present graphical methods using the bond graph methodology to derive information on structural controllability. For the switching system, the controllability property is decomposed into R-controllability, complete controllability and impulse controllability. Currently, we limit the study to the R-controllability and impulse controllability.

5.1 R-controllability

R-controllability is related to the ability to control the finite dynamic modes (classical controllability of exponential modes for regular system). It is associated with the differential part composing the state space.

Proposition 8

A switching system is structurally state R-controllable if and only if the two following conditions are verified:

1.	All (I,C) elements are causally connected with a source in the BGI.
2.

Proof:

This result is derived from digraph theory [19].

The bond graph rank of the matrix [sE – A,B] is given using causal manipulations in the bond graph model.

Proposition 9

bg_rank[sE – A,B] = n − t_c where t_c is the number of dynamical elements remaining in integral causality when a dualisation of the maximal number of input sources is performed in the BGD in order to cancel these integral causalities.

Remark 3

The elements initially in derivative causality in the BGI keep their causalities unchanged in the BGD.

Proof

This result is a simple extension of the previous one, associated with regular model [22].

5.2 Impulse controllability

Definition 1

A switched system is impulse controllable if all the impulsive modes can be excited by suitably chosen non-impulsive inputs.

Proposition 10

A switched system is impulse controllable if and only if:

Proof

Using the classical impulse controllability condition [23].

A graphical interpretation of equation (13) leads to the graphical interpretation of the impulse controllability.

Proposition 11

A switched system is impulse controllable if and only if the number of impulse modes is equal to the number of disjoint causal paths between input sources and switches passing through (I,C) elements in derivative causality in the BGI.

Proof

Using equation (7), and the substitution of the E, A and B matrices,

Bg_rank

.

From equation (13),

⇒ n +  n_i  + 

 = n +  rank [E],

⇒ n +  n_i  + 

 = n_i  + 

,

⇒

 = 

.

S ₂₄ represents causal paths between (I,C) elements in derivative causality and switches elements, B_d is the input sub-matrix connecting input sources and (I,C) elements in derivative causality in the BGI,

is composed by causal paths between input sources and switches passing through elements in derivative causality in the BGI. So

corresponds to the number of disjoint causal paths between input sources and switches passing through elements in derivative causality.

On the other hand, we showed in section 4 that

is equal to the number of impulse modes.

6 Structural observability

Using duality with controllability, observability can also be decomposed into: R-observability, complete observability and impulse observability. The complete observability is not considered in this work.

6.1 R-observability

Definition 2

A switching system is R-observable if the initial condition x(0) can be determined by a given input control u(t) and a given output y(t).

Proposition 12

A switching system is structurally state R-observable if and only if the two following conditions are verified:

1.	All (I,C) elements are causally connected with a detector.
2.

Proof

This result is derived from digraph theory [19].

The bond graph rank of the matrix

is given using causal manipulations in the bond graph model.

Proposition 13

, where, t_o is the number of dynamical elements remaining in integral causality when a dualisation of the maximal number of output detectors is performed in the BGD in order to eliminate these integral causalities.

Proof:

This result can be considered as an extension of the classical observability of regular systems modelled by bond graphs [22].

Remark 4

The elements initially in derivative causality in the BGI keep their causalities unchanged in the BGD.

6.2 Impulse-observability

Definition 3

A switching system is impulse observable if at each time τ the knowledge of the output impulse and the input discontinuities at τ are sufficient to determine the discontinuities of the state vector x at the same time.

Proposition 14

A switching system is impulse observable if and only if:

Proof:

This result is the classical impulse observability condition [23].

The graphical interpretation of this result is given using causal paths in the bond graph model.

Proposition 15

A switching system is structurally impulse observable if and only if, the number of impulse modes is equal to the number of disjoint causal paths between the switches and the output detectors passing through (I,C) elements in derivative causality in the BGI.

Proof

Using equation (7) and the substitution of the E, A and C matrices,

From equation (15)

⇒ n +  n_i  + 

 = n +  rank [E],

⇒ n +  n_i  + 

= n_i  + 

,

⇒

 = 

.

S ₂₄ represents causal paths between (I,C) elements in derivative causality and switches in the BGI, C_w is the output sub-matrix connecting switches and output detectors in the BGI,

is composed by causal paths between switches and output detectors passing through elements in derivative causality in the BGI. So

corresponds to the number of disjoint causal paths between switches and output detectors passing through elements in derivative causality in the BGI.

7 Example

In order to illustrate the methods proposed for controllability and observability analysis, let us consider the simple example of the electrical circuit shown in Figure 1. Sw1 and Sw2 represent diodes. The associated bond graph models represent the commutations.

Figure 1. Electrical circuit with two switching elements Sw1 and Sw2.

Remark 5

The impulse controllability and impulse observability analysis is necessary only in the case that exhibits infinite modes.

When Sw1 is closed and Sw2 open, the elements (I,C) are in integral causality. This configuration is chosen as the initial one (Figure 2). If Sw1 is open and Sw2 closed, the second configuration (Figure 4) is obtained.

Figure 2. BGI in the configuration : Sw1 closed and Sw2 open.

In the initial configuration : Sw1 closed and Sw2 open (Figure 2)

a.	All the elements are in integral causality, thus ni  = 4 and nd  = 0 = so n = ni  + nd  = 4.
b.	The number of inputs sources is m = 1 and the number of outputs detectors is p = 1.
c.	Bg_rank [E] = ni  = 4.
d.	Bg_rank [A] : in the BGD all the (I,C) elements become in derivative causality (Figure 3), so bg_rank [A] = n - q = 4.
e.	R-controllability : in the BGI (Figure 2) all the (I,C) elements are causally connected to MSe and in the BGD (Figure 3) they become in derivative causality, so the model is R-controllable by the input E which has then to be a controlled input MSe.
f.	R-observability : in the same way the model is R-observable by De.

Figure 3. BGD in the configuration : Sw1 closed and Sw2 open.

Figure 4. BGI in the configuration Sw1 open and Sw2 closed.

In the configuration Sw1 open and Sw2 closed (Figure 4)

a.	In the BGI (figure 4), the number of elements in integral causality is ni  = 2 and the number of elements in derivative causality is nd  = 2.
b.	The number of switching elements is nw  = 2, so n = ni  + nd  + nw  = 6.
c.	The number of inputs sources is m = 1 and the number of outputs detectors is p = 1.
d.	Bg_rank [E] : in the BGI (Figure 4), the number of elements in integral causality is equal to 2, and there exits two disjoint causal paths between switches Sw and (I,C) elements in derivative causality ((Sw1, I1) and (Sw2, C2)), so r = 4.
e.	Bg_rank [A] : in the BGD (figure 5) the element I2 is in integral causality, q = 1, so bg_rank [A] = n − q = 5.
f.	R-controllability and R-observability can be verified as in the initiale configuration.
g.	Infinite mode: After the commutation two elements in derivative causality (I1 and C2) appear in the BGI (figure 4), so there exists 2 infinite modes. This result can be obtained by computing the number of disjoint causal paths between the switches Sw and (I,C) elements in derivative causality in the BGI. Here two disjoint causal paths appear ((Sw1, I1) and (Sw2, C2)).
h.	Impulse controllability: the existence of two infinite modes implies that two input controls are necessary to control these impulse modes. In the BGI (Figure 4) m = 1, so this model is not impulse controllable.
i.	Impulse observability: in the same way two output detectors are necessary to observe the two impulse modes. But in the BGI (Figure 4) p = 1, so this model is not impulse observable.

Figure 5. BGD in the configuration Sw1 open and Sw2 closed.

8 Conclusions

In this paper we have shown how the bond graph approach can be used as a good tool for structural analysis of switching devices modelled by bond graph. We have studied some graphical methods essentially for pointing out infinite modes and for structural analysis of controllability and observability. These methods are based exclusively on causality handling of the bond graph model. Simulation of the switching systems [2,8] is not considerd in this paper.

Our next work will concern the compensation of infinite modes. Actually, in the case where the model is impulse controllable, a static state feedback control can solve the problem, otherwise it is necessary to introduce a dynamic compensator. On the other hand switching systems are considerd as hybrid systems (see [13] for example). Our objective is to propose some structural properties of hybrid systems modelled by bond graph.