Physical interpretation of inverse dynamics using bicausal bond graphs

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Abstract

A physical interpretation of the inverse dynamics of linear and non-linear systems is given in terms of the bond graph of the inverse system. It is argued that this interpretation yields physical insight to guide the control engineer. Examples are drawn from both robotic and process systems.

Introduction

It has been argued by Sharon et al. [1] that control system design could, and should, be based on physical system insight. Bond graphs [2], [3], [4], [5], [6], [7] provide a high-level modelling language for describing dynamic systems in a graphical form which retains such physical insight. For these reasons, physical-model-based control using bond graphs has been suggested by Karnopp [8], Roberts et al. [9], Karnopp [10] and Gawthrop [11] as an approach to control system design. This paper focuses on one aspect of this, the physical interpretation of zero dynamics [12] (system zeros in the case of linear systems) in the context of control system design.1

This concept is, in turn, related to a classical problem in control (particularly in the field of process engineering): the determination of the configuration of the control system in the sense of determining inputs, outputs and feed-forward terms for the purposes of effective control. This is called the control loop pairing problem. This is not primarily to do with system parameters, but rather with system structure; thus this is largely a qualitative, as opposed to a quantitative, problem. Bond graphs make a clear distinction between system structure (as represented by the bond graph) and numerical parameters; therefore bond graphs are an appropriate tool in this context. There is a particular interest at the moment in including control considerations at the process design stage (see, for example [13], [14], [15] and the references therein) to complement the relatively well-established conceptual design on process plant alone as, for example, discussed by Douglas [16].

There are many transfer-function-based approaches to the control configuration problem including the relative gain array of Bristol [17] and a number of singular-value decomposition-based methods such as that of Havre and Skogestad [18]. Following the ‘design in the physical domain’ approach of Sharon et al. (1991), it is argued here that it is better to stick to the physical model itself than to immediately use an abstraction, such as a transfer function; bond graphs provide an appropriate physical domain representation. A bond graph interpretation of the relative gain array, and related indicators, has already been given by Gawthrop et al. [19].

The approach taken here is to investigate the possibility of ideal control; that is, is it possible in principle to set system outputs to desired values and deduce the corresponding inputs: and this leads to the notion of system inversion (or partial inversion) with respect to input–output pairs [7], [20], [21]. As discussed by Gawthrop [22] bicausal bond graphs have an important role in investigating the inverses of a system described by bond graphs; in particular, the bond graph of the system inverse is often best represented by a bicausal bond graph. A contribution of this paper is to give a physical interpretation of zeros (zero dynamics in the non-linear case) in the context of bond graphs.

The approach differs from the work of Huang and Youcef-Toumi [23] and the earlier work of Gawthrop and Smith [17] in that the concept of bicausality is used to represent the causal implications of inversion [20], [21]. Without bicausality, the bond graph does not completely represent the inverse system and the resulting zero dynamics due to the need to solve an additional set of algebraic equations which are not explicitly represented in the bond graph.

Building on the basic ideas of bond-graph-based control presented by Gawthrop [11], this paper provides an alternative approach to control system configuration which is based on bond graph inversion and bicausal bond graphs. To support this, extensions of the bicausal bond graph theory to systems with modulation are given.

Because of the multi-disciplinary nature of bond graphs, the method is applicable to a range of engineering systems. Hence the ideas are illustrated using both a mechanical and process engineering examples. The figures are generated using a bicausal version of a set of Model transformation tools Gawthrop [24] which is currently under development.

The outline of the paper is as follows. Section 2 discusses the role of zero dynamics in determining the controllability (in a loose sense) of a system. Section 3 gives the background theory relating to bicausal bond graphs and Section 4 shows how the bicausal bond graphs can be used to give the bond graph of the inverse of a dynamic system. Section 5 gives a mechanical (robotic) example due to Sharon [1] illustrating lightly damped zeros. Section 6 gives a physical interpretation of unstable linear zero dynamics. Section 7 takes a detailed look at control loop pairing for a two-input two-output non-linear model of a non-isothermal chemical reactor and gives a physical interpretation of the (non-linear) zero dynamics.

Some of this material has already appeared in conference papers [22], [25].

Section snippets

System controllability and zero dynamics

The controllability of a system has at least two meanings:

  • a strict state-space concept which in the linear case corresponds to the non-singularity of a certain matrix [26], [27] and

  • a somewhat broader (and practically more useful) concept developed over the years in the process engineering community to do with how difficult a system is to control.

It is the latter meaning that is used here.

There are at least two reasons why a particular input/output pair of a linear system described by a rational

Bicausal bond graphs

Bicausal bond graphs were introduced by Gawthrop [22] to give a foundation for deriving system properties relating to

  • system inversion,

  • state estimation and,

  • parameter estimation,

directly from the system bond graph. This paper expands on the first of these: system inversion.

This section provides a brief review of the material and extends the concepts to include modulation and active bonds. Some of the techniques are related to those introduced by Cornet and Lorenz [33]. Further developments have

System inversion

System inversion is discussed by Gawthrop and Smith [7] and shown to correspond to the reversal of causality on SS components. However, the method presented there implies the use of constraint equations not directly represented on the bond graph; the resultant bond graph cannot be easily used to extract qualitative information about the system inverse.

The introduction of bicausal bond graphs by Gawthrop [22] allows all the equations describing the system inverse to be directly represented on

Example: Macro–micro manipulator

In his Ph.D. Thesis Sharon [37] uses the ‘design in the physical domain’ approach (discussed by Sharon et al. [1]) to design the force control system for a ‘Macro/Micro Manipulator’. This is a prototype of one-dimensional, two degrees of freedom translational manipulator with two main components:

  • a macromanipulator responsible for gross motion at the tip of which is mounted

  • a micromanipulator responsible for fine motion.

The macromanipulator has a proportional + derivative (PD) position control;

Example: a system with unstable inverse

A standard example of a system with an unstable inverse arises from the parallel combination of two first-order systems: one slow, with large steady-state gain and the other faster, with a smaller steady-state gain. The output of the overall system is the difference in the outputs of the two subsystems. In transfer function terms, an example of such a systems isG(s)=11+2s12+s=1−s(1+2s)(2+s).Although G(s) is stable (with poles at s=−2 and −0.5), the inverse of this system has a pole at s=1 and

Non-linear example: chemical reactor

An example of a non-linear, unstable chemical reactor with unstable zero dynamics (due to Trickett and Bogle [39]) is used in this section. The schematic diagram appears in Fig. 15. The reactor has two reaction mechanisms: A→B→C and 2A→D. The reactor mass inflow and outflow f are identical. q represents the heat inflow to the reactor. Following Trickett and Bogle [39], the consequences of two possible choices of the two system outputs are examined:

(1)either ca (the concentration of substance A),

Conclusion

It has been shown, and illustrated by three examples, that bond graphs (when augmented by the bicausality concept) provide a powerful approach to examining the controllability properties of a dynamic system, expressed in terms of the system zeros, directly from the system bond graph.

In particular, the bicausality concept allows the bond graph of the inverse to be deduced directly from that of the system itself; examination of this bond graph gives information about the system zeros (in the

Acknowledgements

I would like to acknowledge the discussions at the Department of Chemical Engineering of Edinburgh University with Murray Laing on controllability and Jack Ponton on systems with inverse response. The reactor problem arose partly from discussions with Des Costello (same department) and partly from a seminar given by David Bogle of University College, London, when visiting Edinburgh.

I am grateful to the two anonymous referees for their detailed comments and suggestions which have improved the

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