The editors introduce the basic ideas of multiple model approaches in Chapter 1, where the existing paradigms for the application of multiple model and operating regime approaches to nonlinear modelling, identification and control are explored. The chapter also provides a survey and overview of the state of the art in terms of procedures, algorithms and tools for visualisation and interpretation.
Part II of the book deals predominantly with modelling methods and applications, including methods for estimation and experiment design.
In Chapter 2, Babuska and Verbruggen describe the Takagi-Sugeno fuzzy model, used as an interpolating scheduler for a set of multiple linear models which are valid locally around certain operating conditions. The antecedent of the fuzzy rule provides the local region and the interpolating mechanism, while the consequent is the locally valid model. The difficult task of learning the antecedents and consequents from data is reviewed and a constructive approach incorporating fuzzy clustering is developed. The identification methods are demonstrated using experimental data from problems in biotechnology and medicine.
Gollee, Hunt, Donaldson and Jarvis show in Chapter 3 that nonlinear models based on multiple local ARX models are able to capture the nonlinear effects apparent in experiments with electrically stimulated muscles, and provide high accuracy over a wide range of input signals. This chapter is an interesting example of local methods being applied to a problem which has been studied intensively with a variety of complex mathematical modelling techniques, and comparing well. The identification methods used are those from Chapter 7.
The `functional state' approach described by Halme, Visala and Zhang in Chapter 4 uses a discrete representation to describe the current `functional state' of the system. This concept is close to the idea of an operating regime, and each functional state has an associated local model. A finite state automaton is used to describe the possible transitions between operating regions of dynamical processes, and multi-layer perceptron neural networks with Laguerre filters are trained to recognise transitions between states. The concept is illustrated by experiments with a two-tank problem, and a fermentation process.
In Chapter 5 Meila and Jordan review the Mixture of Experts model structure and extend it to a Markov Mixture of Experts, where a Markov graph is used to define the transitions between multiple models in the system. This is similar in many ways to the functional state approach described in Chapter 4, but this time placed in a probabilistic framework where the transitions are described by a Markov model. The method is applied to fine motion control in robotics.
In Chapter 6 by Cohn, Ghahramani and Jordan, experiment design with the mixture of Gaussians model representation is studied. Local representations make it easier to produce local confidence limits, which can be used as the basis for an active learning algorithm, where the optimal search for new data can be guided by the model structure. Robotics simulations are used to illustrate the ideas. This chapter also, as with Chapter 6, gives useful insight into the probabilistic interpretation of multiple model approaches. The Expectation Maximisation algorithm is used for learning - this is also studied in Chapter 7. A further interesting aspect of the representation used in this chapter is that multiple local models are used to represent the joint input-output density of the data, which does not distinguish between inputs and outputs, unlike the other approaches which explicitly represent input-output or input-state mappings.
In Chapter 7, Murray-Smith and Johansen show that the commonly used global least squares method for parameter identification in multiple local models can be very sensitive and lead to ill-conditioning. They propose a cheaper locally weighted least squares identification method as a solution. The interactions between model structure and parameter identification methods are discussed - this theme reappears in several other chapters. The smoothing analysis used to illustrate the effects discussed, is also a general technique, which can be applied to other frameworks.
Chapter 8, by Shorten and Murray-Smith examines some of the side-effects of basis function normalisation - a common technique used in the weighting functions in local model and control structures to ensure that the operating range is completely covered. Normalisation also appears naturally in fuzzy and probabilistic representation of the weighting functions. Normalisation has a number of side-effects which alter the global properties of the model or controller, with respect to robustness and interpretability. These become especially important when automatic learning algorithms are used to adapt the basis functions. As well as the graphical and intuitive explanations of the side-effects, the chapter also describes some mathematical tools which can help gain a deeper understanding of the trade-offs involved.
Part III of the book is dedicated to applications of multiple model methods for nonlinear control.
In Chapter 9, Kuipers and Åström present methods for developing a nonlinear controller by combining multiple heterogeneous local control laws appropriate to different operating regions. Operating regions are described using fuzzy set membership, as in Chapter 2, but the local controllers can be classical control laws with their own internal states. Qualitative simulation is suggested as a method for validation of the global behaviour of the heterogeneous controls. Some aspects of the control law can, even in the case of incomplete knowledge, be represented as a qualitative differential equation, and qualitative simulation can be used to predict the possible behaviours of the system. The methods are demonstrated on a water level controller and a highly nonlinear chemical reactor.
In Chapter 10, Sbarbaro uses operating regime based models with multiple local Laguerre models for identification and control. Local Laguerre models potentially have advantages over the more common ARX local model as they can cope more easily with uncertainty in time delays and model orders. The method is compared in a simulation of a chemical reactor using a model predictive control algorithm.
Chapter 11 by Schott and Bequette describes multiple model adaptive control (MMAC), which is a classical model-based control strategy. Multiple models are used and a probabilistic weighting chooses which model or combination of models best represents the current plant input/output behaviour. The authors review MMAC theory, including model bank estimation and control, and describe applications to biomedical control problems.
In the work presented by Banerjee, Arkun, Pearson and Ogunnaike in Chapter 12, the composition of multiple linear state-space models is described as a parameter-varying model. The parameters of the global model are the local model weights which are estimated on-line using a Bayesian approach similar to Chapter 11. A globally stable controller composed of multiple local linear controllers is then designed for the linear parameter varying model using H-infinity design based on Linear Matrix Inequalities. The theory is applied to a simulated chemical reactor.
Zhao, Gorez and Wertz present in Chapter 13 a method for identification and structured analysis and design of Takagi-Sugeno fuzzy models and controllers. The Takagi-Sugeno fuzzy model is based on multiple local linear state-space models that are weighted using fuzzy membership functions. An identification method based on fuzzy clustering (see also Chapter 2) is presented and experimental results from application on a glass furnace are included. The control design method guarantees stability and robustness properties. The methods are based on modern tools such as Linear Matrix Inequalities, being closely related to Chapter 12. Simulation examples are used to illustrate the methods.
We hope that this book will bring to a wider audience the progress being made in both practical and theoretical use of the multiple models philosophy, and that the workers in the field will be able to gain a deeper understanding of the relations between the different existing approaches, and tools.
Roderick Murray-Smith,
Berlin.
Tor Arne Johansen
Trondheim,
March 1996.