Brief PaperOptimal control of nonlinear systems: a predictive control approach☆
Introduction
Optimal control of nonlinear systems is one of the most active subjects in control theory. One of the main difficulties with classic optimal control theory is that, to determine optimal control for a nonlinear system, the Hamilton–Jacobi–Bellman (HJB) partial differential equations (PDEs) have to be solved Bryson & Ho, 1975. There is rarely an analytical solution although several numerical computation approaches have been proposed; for example, see Polak, 1997. The same difficulty also occurs in recently developed “nonlinear” H∞ control theory where Hamilton–Jacobi–Issacs PDEs have to be solved (for example see Ball, Helton, & Walker, 1993).
As a practical alternative approach, model-based predictive control (MPC) has received a great deal of attention and is considered by many to be one of the most promising methods in control engineering Garcia, Prett, & Morari, 1989. The core of all model-based predictive algorithms is to use “open-loop optimal control” instead of “closed-loop optimal control” within a moving horizon. Among them, long-range generalised predictive control (GPC) is one of the most promising algorithms Clarke, 1994. Following the successes with linear systems, much effort has been taken to extend GPC to nonlinear systems (for state of the art of nonlinear predictive control see Allgöwer & Zheng, 1998). Various nonlinear GPC methods for discrete-time systems have been developed; for example, see Bequette (1991), Biegler & Rawlings (1991), Mayne (1996), and Allgöwer & Zheng (1998). The main shortcoming of these methods is that on-line dynamic optimisation is required, which, in general, is non-convex. As pointed by Chen, Ballance, & O'Reilly (2000), heavy on-line computational burden is the main obstacle in the application of GPC in nonlinear engineering systems. This causes two main problems. One problem is a large computational delay and the other problem is that global minimum may not be achieved, or even worse a local minimum cannot be achieved due to time limitation in each optimisation cycle.
To avoid the online computational issue, one way is to develop a closed-form optimal GPC. To this end, Lu (1995), Soroush and Soroush (1997) and Siller-Alcala (1998) limit the control order to be zero, that is, to limit the control effort to be a constant in the predictive interval. Then the closed-form optimal GPC laws are given. However, it is difficult to predict the system output over a long horizon since the output order is limited to be the relative degree of a nonlinear system in this approach. Moreover, as shown in this paper, however small the predictive horizon is chosen, the closed-loop system is unstable for plants with large relative degree, i.e., ρ>4. To obtain adequate performance, the control order should be chosen to be reasonably large. When this approach is used to deal with the control order larger than zero by augmenting the derivatives of the control as additional state, the control law derived depends on the derivatives of the control that are unknown and thus is impossible to implement. Alternatively, it is shown by Gawthrop, Demircioglu and Siller-Alcala (1998) that the special case of zero prediction horizon also leads to an analytic solution related to those obtained by the geometric approach (Isidori, 1995).
In a similar vein, this paper looks at another special case of the nonlinear GPC of Gawthrop et al. (1998) where the degree of the output prediction is constrained in terms of both relative degree and control order. The approach gives an analytic solution for a class of multivariable nonlinear systems in terms of a generalised predictive control performance index. The result is based on four concepts: prediction via Taylor series expansion, receding horizon control, control constraints (within the moving horizon time frame) and optimisation. In order to avoid the numerical computation difficulties in optimisation, an analytical solution to a set of nonlinear equations arising in optimisation is derived. As a result, an optimal generalised predictive control law is presented in a closed form, which turns out to be a time invariant nonlinear state feedback control law. By showing that the closed-loop system is linear, the stability of the closed-loop system is established. Moreover, the design parameters in this nonlinear control design method can be directly chosen according to desired time-domain transient and thus a trade-off between performance specifications and control effort is possible.
Section snippets
Nonlinear generalised predictive control (NGPC)
Consider the nonlinear systemwhere x∈Rn, u∈Rm and y=[y1,y2,…,ym]T∈Rm are the state, control and output vectors, respectively. In general, an optimal tracking problem can be stated as follows: design a controller such that the closed-loop system is asymptotically stable and the output, y(t), of the nonlinear system (1) optimally tracks a prescribed reference, w(t), in terms of a given performance index.
To avoid the difficulties in solving PDEs in
Design procedure based on time-domain specifications
In the nonlinear predictive control design method developed above there are two design parameters: the control order, r, and the predictive time, T. How to choose these parameters according to time-domain specifications is discussed in this section.
In many cases the time-domain specification of a system is given in terms of its step response. There are two important indices, overshoot and settling time. Since the error equation for a nonlinear system under the nonlinear GPC (18) is given by (27)
Example: predictive control of a missile
The method of this paper is demonstrated by the design of an autopilot for a high angle of attack missile. The model of the longitudinal dynamics of a missile is taken from Reichert (1990), given bywhere α is the angle of attack (deg), q the pitch rate (deg/s), and δ the tail fin deflection (deg). The nonlinear functions f1(α), f1(α), b1(α) and b2 are determined by the aerodynamic coefficients (Reichert, 1990).
The tail fin actuator dynamics are approximated by a
Conclusion
This paper presents a systematic method for designing an optimal controller to achieve prescribed time-domain specifications for a nonlinear system that satisfies Assumptions (A1)–(A4). By defining an optimal tracking problem in terms of a GPC performance index, an optimal control law for a continuous-time nonlinear plant has been derived.
The significant features of this new optimal predictive control law are:
- •
the optimal control is given in a closed form, which only depends on the states of a
Wen-Hua Chen holds a Lectureship in Flight Control Systems in Department of Aeronautical and Automotive Engineering at Loughborough University, UK. He received his MSc and Ph.D degrees from Department of Automatic Control at Northeast University, China, in 1989 and 1991, respectively. From 1991 to 1997, he was a Lecturer in Department of Automatic Control at Nanjing University of Aeronautics and Astronautics. He held a research position and then a Lectureship in Control Engineering in Center
References (25)
- et al.
Continuous-time generalised predictive control
Automatica
(1991) - et al.
Multivariable continuous-time generalised predictive control
Automatica
(1992) - et al.
Model predictive controlTheory and practice — a survey
Automatica
(1989) - Allgöwer, F., & Zheng, A. (Eds). (1998). Proceedings of international symposium on nonlinear model predictive control:...
- et al.
H∞ control for nonlinear systems with output feedback
IEEE Transactions on Automatic Control
(1993) Nonlinear control of chemical processesa review
Industrial Engineering Chemical Research
(1991)- Biegler, L. T., & Rawlings, J. B. (1991). Optimisation approaches to nonlinear model predictive control. In Proceedings...
- et al.
Applied optimal control
(1975) - et al.
Model predictive control of nonlinear systemscomputational burden and stability
IEE Proceedings Part D: Control Theory and Applications
(2000) - Chen, W.-H., Ballance, D. J., & Gawthrop, P. J. (1999a). Nonlinear generalised predictive control and optimal dynamic...
Nonlinear PID predictive controller
IEE Proceedings Part D: Control Theory and Applications
Cited by (0)
Wen-Hua Chen holds a Lectureship in Flight Control Systems in Department of Aeronautical and Automotive Engineering at Loughborough University, UK. He received his MSc and Ph.D degrees from Department of Automatic Control at Northeast University, China, in 1989 and 1991, respectively. From 1991 to 1997, he was a Lecturer in Department of Automatic Control at Nanjing University of Aeronautics and Astronautics. He held a research position and then a Lectureship in Control Engineering in Center for Systems and Control at University of Glasgow, UK, from 1997 to 2000. He has published one book and more than 40 papers on journals and conferences. His research interests include robust control, nonlinear control and their applications in automotive and aeronautical engineering.
Donald Ballance is a Senior Lecturer in Control Engineering. He has undertaken research in control theory, the application of this theory and system modelling. Areas of interest in control systems include Quantitative Feedback Theory (QFT), Model Predictive Control (MPC), and the development of these techniques for copntinuous-time nonlinear systems. He has worked extensively with Prof. Peter Gawthrop on the development of bond graph techniques for modelling, simulation, analysis and control. He is also interested in real-time systems.
Peter J. Gawthrop was born in Seascale, Cumberland, in 1952. He obtained his BA, D.Phil. and MA 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. In 1987, he took up the Wylie Chair of Control Engineering in the Department of Mechanical Engineering at Glasgow University. He was involved in founding the Centre for Systems and Control — a cross-departmental research grouping at Glasgow with about 12 full time academic staff including four professors. His research interests include self-tuning control, continuous-time system identification and system modelling — particularly using bond graphs in the context of partially-known systems. He is interested in applying control techniques to a number of areas, including process control, robotics aerospace systems and anaesthesia. He has coauthored and authored some 130 conference and journal articles and three books in these areas. He was an associate editor of Automatica and an honorary editor of IEE Proceedings Pt. D, and serves on the editorial boards of journals including the IMechE Journal of Systems and Control Engineering. He is a Fellow of the IEE and I Mech E, 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. He has help visiting appointments at the Universities of Newcastle (Australia), Syney and New South Wales.
- ☆
This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Per-Olof Gutman under the direction of Editor Tamer Basar.