Predictive pole-placement control with linear models☆
Introduction
Most work on stabilising model-based predictive control (MPC) can be seen as the approximation of infinite horizon linear–quadratic control using a finite horizon optimisation with constraints. Stability results—for example those of Clarke and Scattolini (1991), Demircioglu and Clarke (1992), Muske and Rawlings (1993), Rawlings and Muske (1993), Chen and Allgöwer (1998) and Mayne, Rawlings, Rao, and Scokaert (2000)—are based on the idea of showing that, with suitable terminal constraints, this approximation is equivalent to the solution of a related infinite horizon cost function.
This paper takes a different approach. Although optimisation via model-based prediction is used, it is not as an approximation to a linear–quadratic cost function but rather as a means of solving linear constrained problems by approximating the behaviour of the classical linear state feedback control of a linear system with chosen closed-loop pole locations. These pole locations can be determined by any linear design method (including linear–quadratic). For this reason, the algorithm is named predictive pole placement (PPP). In common with many other MPC papers such as those of Muske and Rawlings (1993), Rawlings and Muske (1993), Gawthrop, Demircioglu, and Siller-Alcala (1998) and Chen and Allgöwer (1998), a state (as opposed to output) feedback approach is used; thus the method can be categorised as manipulating input–output behaviour using state feedback. In the linear context of this paper, output feedback may be readily accomplished using standard observer techniques.
Within a continuous-time setting, the basic PPP algorithm for a linear unconstrained system has the following features:
Feature (1) is not new: it has been used by Richalet, Rault, Testud, and Papon (1978) and Richalet (1993); the usual discrete-time choice of the control (or control move) at each sample time can be viewed as one such choice (Rawlings & Muske, 1993); a polynomial (in time) set of basis functions has been used in the continuous context by Demircioglu and Gawthrop (1991) and Gawthrop et al. (1998) and Laguerre functions have been used by Wang (2000).
Feature (2) can be viewed as an output-orientated version of the terminal state constraint shown to be important for stability in a number of papers, including those of Clarke and Scattolini (1991), Demircioglu and Clarke (1992), Muske and Rawlings (1993), Rawlings and Muske (1993) and Chen and Allgöwer (1998); we believe that this output-orientated approach is appropriate given the input–output focus of the paper. Not surprisingly, it is likewise important in creating a stable moving-horizon controller.
Scokaert and Rawlings (1998) show that (discrete-time) “constrained linear quadratic regulation” has the property that nominal closed-loop performance is identical to the open-loop predictions and thus shares feature (3) with the (continuous-time) PPP algorithm. One contribution of our paper is to prove feature (3) for the PPP algorithm.
Of course, standard techniques are available to design controllers for the linear unconstrained case and, in this case, the PPP algorithm would be yet another way of achieving the same result. However, the strength of the method is in its extension to constrained systems using the quadratic programming (QP) approach to optimisation (Fletcher, 1987); the linear unconstrained case representing an ideal situation with the corresponding nice properties listed above. Although outside the scope of this paper, we note that the method extends in principle to the nonlinear case and some preliminary results are available elsewhere (Gawthrop & Ronco, 2000).
Although much of the MPC literature uses a discrete-time formulation, this paper uses a continuous-time formulation and thus builds on the previous work of Demircioglu and Gawthrop 1991, Demircioglu and Gawthrop 1992, Demircioglu and Clarke (1992) and Gawthrop et al. (1998).
The paper is organised as follows. Section 2 considers the unconstrained optimisation problem and provides an explicit solution of the open- and closed-loop controllers. Section 3 gives conditions under which open- and closed-loop control are the same and shows that the PPP algorithm approximates this situation. Section 4 gives a selection of illustrative examples and Section 5 concludes the paper.
Section snippets
Unconstrained PPP
This section introduces the class of systems considered in this paper, the corresponding unconstrained optimisation problem and gives an explicit formula for its solution. Some special cases of the input and setpoint basis functions are considered in 2.1 Special forms of, 2.2 Special form of.
The linear systems considered in this paper are described bywhere , and . x0 is the system's initial condition. Given the state x(t) at time t, we are
Properties of unconstrained PPP
This section looks at the basic properties of the PPP algorithm and gives the fundamental result on the relationship between the open-loop control u★(t,τ) and the closed-loop control u(t).
Lemma 1 presents a general algorithm parameterised by the choice of input functions U★(τ) and setpoint functions W★(τ). A key idea in this paper is to choose the input functions U★(τ) (rewritten as in (37) to be the solutions of the autonomous system of (37). Some appropriate functions appear in Table 1
Examples
This section provides a number of illustrative examples of the main features of the PPP algorithm. Each example is chosen to illustrate a particular aspect of PPP as follows: Example 4.1 This use a simple example to allow hand calculation of the PPP algorithm. Example 4.2 This shows that PPP successfully controls a third order, unstable system with unstable inverse. The fact that PPP gives approximate pole-placement for finite horizon prediction is emphasised. Example 4.3 This illustrates PPP applied to a system with more outputs
Conclusion
A new MPC algorithm, predictive pole placement, has been shown to have a number of useful features as listed in the introduction and these features have been theoretically proved and illustrated by the examples of Section 4.
A popular approach to nonlinear control design is the exact linearisation approach of Isidori (1995). This has the disadvantage of cancelling zero dynamics and, in this sense, may be regarded as the nonlinear equivalent of linear design methods such as the minimum variance
Acknowledgements
This work was accomplished whilst the first author was a visitor at the Centre for Integrated Dynamics and Control, University of Newcastle, New South Wales. He would like to thank Professor Graham Goodwin for providing an excellent work environment and other members of the centre, in particular Liuping Wang, Rick Middleton and Will Heath, for helpful suggestions. Both authors would like to thank Professor David Hill of the University of Sydney for facilitating the collaboration between the two
Peter Gawthrop was born in Seascale, Cumberland, in 1952. He obtained his B.A. (first class honours), D.Phil. and M.A. degrees in Engineering Science from Oxford University in 1973, 1977, and 1979, respectively. Following a period as a Research Assistant with the Department of Engineering Science at Oxford University, he became W. W. Spooner Research Fellow at New College, Oxford. He then moved to the University of Sussex as a Lecturer, and later a Reader in control engineering. Since 1987, he
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2006, AutomaticaCitation Excerpt :Predictive pole-placement (PPP) control (Gawthrop & Ronco, 2002; Gawthrop, 2000) is a continuous-time model-based predictive control scheme which represents the moving-horizon control signal as a weighted sum of a finite set of basis functions which are the state impulse response of a stable linear time-invariant system of the same order as the controlled system. Under mild conditions, Gawthrop and Ronco (2002) show that the unconstrained closed-loop system has the same poles as the system generating the control signal components. Hence the name “predictive pole-placement” is used to describe the method.
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2005, AutomaticaCitation Excerpt :This exponential weighting converts an unstable system into a stable system with the same unknown parameters to which the two-stage approach is applicable. The motivation for this work is to generate models suitable for model-based predictive control (Mayne, Rawlings, Rao, & Scokaert, 2000; Rawlings, 2000), in particular models suitable for continuous time methods, such as those of Wang (2001) and Gawthrop and Ronco (2000, 2002). The outline of the paper is as follows.
Peter Gawthrop was born in Seascale, Cumberland, in 1952. He obtained his B.A. (first class honours), D.Phil. and M.A. degrees in Engineering Science from Oxford University in 1973, 1977, and 1979, respectively. Following a period as a Research Assistant with the Department of Engineering Science at Oxford University, he became W. W. Spooner Research Fellow at New College, Oxford. He then moved to the University of Sussex as a Lecturer, and later a Reader in control engineering. Since 1987, he has held the Wylie Chair of Control Engineering in the Department of Mechanical Engineering at Glasgow University where he was involved in founding the Centre for Systems and Control—a cross-departmental research grouping at Glasgow.
His research interests include self-tuning control, model-based predictive control, continuous-time system identification and system modelling—particularly using bond graphs in the context of partially-known systems. He is interested in applying modelling and control techniques to a number of areas, including process control, mechatronic systems, aerospace systems and anaesthesia. He has coauthored and authored numerous conference and journal articles and three books in these areas. He is a Fellow of the Institution of Electrical Engineers, a Fellow of the Institution of Mechanical Engineers, a Senior Member of the IEEE, a Chartered Engineer in the UK and a Eur.Ing. in the EU. In 1994 he was awarded the Honeywell International Medal by the Institute of Measurement and Control. In 1999 he spent a year in Australia at the Universities of Newcastle and Sydney.
Eric Ronco is director of a control group within a private company named SIMULOG, specialised in scientific computing. He has been academic researcher for several years focusing on the application of predictive control to large scale nonlinear systems such as power plants, human motor system, chemical process and various mechatronics systems. As a result he has developed a strong knowledge of system modelling and control in general. He is an expert in optimisation techniques for control, identification and system analysis. He is also an expert in physical modelling of systems, particularly through the use of the universal physical language of representation called “Bond Graphs”.
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Pre-Olof Gutman under the direction of Editor Tamer Basar.