A Multi–Parametric Optimization Strategy for Multilevel Hierarchical Control problems ⋆ Nuno P. Fa´ ısca, Konstantinos I. Kouramas, Efstratios N. Pistikopoulos Centre for Process Systems Engineering, Department of Chemical Engineering, Imperial College London, SW7 2AZ, UK (e-mail: e.pistikopoulos@imperial.ac.uk). Abstract: In this work the three–level hierarchical control problem and the decentralised control problem are investigated and a general optimisation strategy is developed for solving these problems based on recent developments on multi–parametric programming. The main idea is to recast each optimisation subproblem in the multilevel hierarchy as a multi–parametric programming problem and then transform the multilevel problem into a single-level optimisation problem. This allows for the control policies (decisions) at each level of the multilevel optimisation problem to be obtained as explicit functions of the state of the dynamic systems involved in each level and the control policies of the higher levels. A three person dynamic optimisation problem is presented to illustrate the mathematical developments. Keywords: Hierarchical control; decentralised control; multilevel and multi-follower optimization; multi–parametric programming. 1. INTRODUCTION In optimisation and control of large–scale dynamic systems, hierarchical and decentralised control allow for the decom- position of the original problem into smaller, interconnected problems which are typical arranged into a multilevel hier- archy (Mesarovic et al. [1970], Cohen [1977], Morari et al. [1980], N.R. Sandell et al. [1978], Stephanopoulos and Ng [2000], Venkat et al. [2005]). Various applications of hierarchi- cal and decentralised control arise in process systems engineer- ing (Morari et al. [1980], Stephanopoulos and Ng [2000]), me- chanical and power systems (Delaleau and Stankovi´ c [2004]), aeronautics (Tolani et al. [2004], Li et al. [2002]), traffic con- trol (Shimizu et al. [1995]) and large-scale systems control (N.R. Sandell et al. [1978], Roberts and Becerra [2001]). In most of these problems the general formulations of the decom- posed multilevel hierarchy is given in Figures 1-2. Examples of the hierarchical structure in Figure 1 can be found in Delaleau and Stankovi´ c [2004], Singh et al. [1975], Stankovi´ c and ˇ Siljak [1989]. In Delaleau and Stankovi´ c [2004] the control problem of a PM synchronous motor is decomposed into a bi–level hierarchical control problem (similar to Figure 1 without the third level). A high–level controller, corresponding to the slow dynamics of the motor’s mechanical system, is de- signed to obtain the right set points for the low–level controller which controls the fast dynamics of the electrical system. In [Singh et al., 1975, Section 5] and Stankovi´ c and ˇ Siljak [1989] the same hierarchical decomposition was used to deal with opti- mal LQG and optimal LQR control of sequentially (or serially) interconnected linear dynamical systems. Decentralised control of large-scale systems also yields a two–level structure where in the lower level more than one subproblems are considered (Figure 2) (N.R. Sandell et al. [1978], Venkat et al. [2005]). ⋆ This work is supported by EPSRC (GR/T02560/01) and Marie Curie Euro- pean Project PRISM (MRTN-CT-2004- 512233). Controller 1 Controller 2 Controller 3 Fig. 1. Three-level controller structure Central Controller Local Local Local Local Controller 1 Controller i − 1 Controller i Controller s Nash equilibrium ... ... Fig. 2. Hierarchical control configuration. Nash equilibrium is often a preferred strategy to coordinate such decentralised systems Venkat et al. [2005]. Similarly, this is also the hierarchical structure that is found in a typical leader – multi–follower problem Li et al. [2002]. Generally, it is widely recognised that the successful design of large and complex systems involves some type of decompo- sition of the original problem into smaller and intercommuni- cating subsystems, typically arranged in one of the multilevel hierarchies described in Figures 1 and 2. Multilevel and decen- tralised optimisation problems, which typically arise in many engineering Clark [1983], Morari et al. [1980], Stephanopoulos Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 6307 10.3182/20080706-5-KR-1001.2289