Issues of numerical accuracy and stability in inverse simulation
Introduction
Inverse simulation aims to determine the system inputs required to produce a given response, defined in terms of the system output variables. It is commonly carried out either by a direct approach based on differentiation or iteratively using integration methods. The first published accounts of the problems of inverse simulation were concerned with aircraft applications and included papers by Kato and Sugiura [1] and Thomson [2]. Their methods involved numerical differentiation of the vehicle state variables, with respect to time. Although this approach gives rapid convergence it is not generic and requires restructuring when the model is modified due to coupling between the mathematical description of the system and the inverse solution algorithm. Sentoh and Bryson [3] defined the inverse process as a Linear Quadratic (LQ) optimal problem that involves minimisation of the integral of a weighted sum of the squares of deviations from a straight flight path and the squared values of control surface deflections. However, this method suffered from significant practical limitations and was relatively cumbersome to apply. Moreover, both these classes of methods are not suitable for handling redundancy problems where the number of control inputs is greater than the number of path constraints [4].
In the early 1990s members of a research group at the University of California, Davis [5], [6] proposed what is now the most commonly used approach to inverse simulation. In this methodology the inverse problem is solved by an iterative numerical scheme together with the Newton–Raphson (NR) algorithm. Unlike earlier methods based on the differentiation approach, the structure of this algorithm means that this integration-based method is less model-specific. It can therefore accommodate different models without any restructuring of the algorithm itself. One of the drawbacks of this approach is that it is an order of magnitude slower than the method involving differentiation with respect to time. Other approaches to inverse simulation include the Broyden, Fletcher, Goldfarb and Shannon (BFGS) quasi-Newton method [7], the Levenberg–Marquardt (LM) least-squares optimisation approach [8], a global optimisation method proposed by Celi [9], the two timescale simulation method of Avanzini, de Matteis et al. [4], [10], and a sensitivity-analysis (SA) based approach introduced recently by Lu et al. [11]. All of these approaches have shown advantages, to a greater or lesser extent, in terms of the solutions of specific inverse simulation problems.
It is known that the most widely used integration-based approach to inverse simulation [6] may give rise to high-frequency oscillations superimposed on the desired variables. Lin [12] suggested that these oscillations could excite uncontrolled state variables and showed, through an illustrative example, that this could stimulate further high-frequency oscillatory effects. In addition, the redundancy problem which arises in some cases introduces a further difficulty for this approach due to the non-square nature of the Jacobian matrix. In addition to the high-frequency oscillations and the redundancy problem, a phenomenon termed “constraint oscillations” has been found which can lead to oscillatory phenomena involving relatively low frequency oscillations in the results.
Furthermore, inverse simulation methods can also be used within output-tracking control systems and inversion-based feedforward controllers [3], [4], [13]. A more accurate model, which includes more physical information about the system such as input saturation, used to design a feedforward controller can lead to better tracking performance [14], [15]. However, in comparison with the effort applied in the past to exploring these issues concerned with oscillatory phenomena in inverse simulation solutions, previous investigations have given less consideration to situations involving saturation constraints or to discontinuities within the model or in the manoeuvres [4]. In fact, these effects present a challenge to traditional approaches to the inverse simulation problem, involving not only the integration-based approaches [6] but also the differentiation-based methods [1], and many of the other approaches outlined above [4], [7], [8], [11]. All of these techniques of inverse simulation require partial derivative calculations which depend on properties of continuity and smoothness of the model and of the manoeuvre. Therefore if a saturation situation is reached, the partial derivatives of all the tracked outputs with respect to the saturated variables become zero. The Jacobian matrix consequently will be singular and NR iterations will not converge. The same problem occurs in evaluation of the Hessian matrix used for the two-timescale method [4], [7].
To avoid the above problems and to achieve better convergence properties, a new algorithm for inverse simulation based on the constrained Nelder–Mead (NM) method has been developed. It is well known that the NM algorithm, which is a search-based approach, can handle discontinuities satisfactorily, particularly if they do not occur near the optimum solution [16], [17], [18]. Furthermore, the derivative-free property can facilitate investigation of numerical issues that exist in the traditional inverse simulation methods.
The paper begins, in Section 2, by reviewing the traditional integration method based on the NR algorithm. Section 3 discusses issues of numerical accuracy and stability for this algorithm. Section 4 investigates the constraint oscillation phenomenon on the basis of information gained from the application of analytical methods for the generation of inverse models. Section 5 presents the mathematical development of the new constrained NM method and describes applications involving two ship models.
Section snippets
The integration-based approach
This section provides a summary of the integration-based method of Hess et al. [6] which forms a useful reference against which other methods of inverse simulation can be compared.
A nonlinear system may be described by equations of the form:where is the set of nonlinear ordinary differential equations describing the original system, is the set of algebraic equations that construct the expected outputs, and is the input vector. The vector is the
Numerical issues and stability of inverse simulation with the integration-based NR iterative scheme
It is not a trivial task to obtain the inverse response from the equations of motion of a vehicle or other system and many problems can be encountered. Because of the widespread adoption of the integration-based approach to inverse simulation [5], [6] and the history of relatively successful applications with this type of method, this section focuses on numerical issues and the stability of this type of algorithm.
The investigation of constraint-oscillation phenomena
Lu et al. [13] have shown that inverse simulation and model inversion share the same objective in terms of calculation of the input from a defined output trajectory. This introduces the possibility of analysing inverse simulation techniques using the methods of model inversion and, thus, of explaining the constraint-oscillation phenomenon which has been found to exist in the inverse simulation process. The illustration presented here can be considered as an extension of the work of Thomson and
Development of the constrained NM method
The determination of the elements of the Jacobian matrix or Hessian matrix has already been discussed in Sections 1 Introduction, 2 The integration-based approach. Optimisation methods based on direct search algorithms, being derivative-free, eliminate the need to determine elements of these matrices and thus alleviate the difficulties associated with discontinuities and input saturation. This introduces the possibility of an alternative approach to inverse simulation that does not require
Conclusions
The paper has described systematically the issues of numerical accuracy and stability in inverse simulation. Firstly, the constraint oscillation phenomenon has been explained with the aid of model inversion methods. Analysis has shown that constraint oscillations relate to internal system properties. Secondly, instabilities and failure to converge for methods of inverse simulation based on the NR algorithm has been discussed in terms of discontinuous manoeuvres, discontinuities within the model
Acknowledgements
The authors wish to acknowledge helpful communications from Professor Lin Kuo-Chi of Department of Mechanical Materials and Aerospace Engineering (MMAE) at the University of Central Florida (UCF) and to the anonymous reviewers for all their helpful suggestions. Linghai Lu gratefully acknowledges the award of a University of Glasgow Postgraduate Scholarship and an Overseas Research Studentship from the British Government. D.G. Thomson and D.J. Murray-Smith acknowledge support from the UK
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