Optimal control under uncertainty by use of invariant sets

Research project P 13706 Optimal control under uncertainty by use of invariant sets Franz KAPPEL 28.06.1999 Many problems for controller design are characterized by the fact that parameters of the system and perturbation acting on the system are not precisely known. In principle there are two ways of looking `at such a problem. The first one is to assume that the perturbations and the parameters are within known bounds. This leads to a socalled worst-case analysis. The second one is to assume that one has -information on the probability distribution of the parameters resp. the perturbations can be considered as stochastic processes. This point of view leads to a socalled average-case analysis. Clearly, both approaches have their advantages and disadvantages. The choice between worst-case analysis and average-case analysis must be based on the characteristics of the measured data and on the control requirements, i.e., if it is required to obtain a solution for the control problem such that the objectives of the problem are guaranteed in any case or a solution where it is highly probable. A characteristic feature for problems which require a robust control design, is that the model parameters and perturbations are only known to satisfy relatively wide deterministic bounds, whereas the controls, because of quality criteria or of safety considerations, have to meet very strict deterministic specifications. The theory of worst case parameter identification, in the presence of bounded noise, has seen a recent burst of activity. Deterministic formulations of the identification problem are motivated by the need to develop a theory of system identification that is compatible with modern robust control. As a result, system identification techniques should be required to provide a set of systems as the model in general. In cases where the objective of the identification is to provide the basis for a robust controller design, the most important issue is the development of control oriented identification methods. The basic idea of the latter is that the identified model should have low bounds for uncertainties where closed loop control specifications require this, but can have large bounds for uncertainties elsewhere. One of the major objectives of the proposed research is to develop practically applicable control oriented identification algorithms and computer programs. In contrast to identification and adaptive control design, considerably less efforts have been paid to the worst-case approach to automatic control synthesis. The reason for this goes back to the necessity of calculating minimal invariant sets, which has been considered to be too difficult. Due to recent achievements, this becomes reasonable and practically applicable. It is suggested to use the radius of a minimal invariant set as a measure for the stability degree of a system and apply such a criterion for optimal control synthesis by means of minimizing this radius. Within the proposed project, it is also planned to find a solution of an approaching (pursuit) problem in presence of bounded noise in the measurements. This problem is one of those target problems attracting the attention of the mathematicians and engineers for a long time. The final goal of the project is to design a software toolbox for optimal "hardbound" control problems.

1. Control-oriented set-membership identification The difference between the two existing approaches, stochastic and hard-bound, lays only in resulting estimates, a vector and a convex compact set respectively. A set estimate could be evaluated by a scalar measure such as volume, diameter, etc. Commonly, a scalar criterion has a purely geometrical value. In contrary, control-oriented identification dictates using another criterion adequate to a given control objective. As a result, control oriented identification provides a set-valued estimate which is optimal by means of a given control goal. In the case of linear systems, it has been proven that optimal estimates have to be sets of a minimum diameter as soon as a quadratic control objective function (with respect to control and uncertain system parameters) is given. On the other hand, the use of another control objective may result in obtaining estimates of various shapes; therewith a model (parameter vector) may have a large uncertainty in any direction but those fixed by the given objective functional. The designed algorithm of control-oriented set-membership identification calculates a control at a discrete-time instance, which is optimal in view of the given quadratic function. 2. Development of the invariant sets theory and its application to optimal control This concerns further development of the invariant sets theory with its application to analysis and synthesis of control systems. Particularly, the results obtained for linear control systems have been extended to the class of models including bounded nonlinear components, i.e. those items allowing description with bounded nonlinear functions. These models could be roughly classified as the catena of a scalar nonlinear "saturation" item and a linear one. The considered class of plants is a natural generalization of the Lurè class except the nonlinearities are bounded. It is also assumed that the exact parameter values for the linear item of the model description are unknown and the parameters are given by their set-valued estimates. 3. Solution of the pursuit-evasion problem for hard-bound parameter estimates The control synthesis problem for a pursuit-evasion process, which is complicated in itself, becomes even more complicated as measurements of phase states of the players are affected by uncertain errors. In the known publications, this problem is analyzed under the common assumption that observation errors are random values independent on systems dynamics and having a priori known probabilistic properties. However, the hypothesis on statistic nature of observation errors is not applicable generally, particularly it is not applicable when errors (disturbances) are known to be dependent on phase coordinates (states) of the moving plants. This is particularly true for the cases of 3-D measurements in the air with the radio aids and under water with the acoustic instrumentation since measurement errors are the larger in these cases the longer is the distance between the players. This research continues the investigations fulfilled by V.M.Kuntsevich and B.N.Pschenitchnyi and utilizes their definitions (particularly of a minimal invariant set) and theorems. It resulted in obtaining an elegant analytic solution to the pursuit-evasion problem for two controllable moving plants under uncertainty conditions which are (i) uncertain evader`s controls and (ii) uncertain observation errors dependent on pursuer`s and evader`s states. In other words, the pursuit-evasion problem is solved for the worst-case scenario, meaning the evader takes optimal (and unknown for the pursuer) controls from the given bounded set and the observation errors are least favorable for the pursuer. 4. Developing of the Matlab toolbox on control-oriented set-membership identification and optimal adaptive control The toolbox is a successor of the Matlab toolbox on adaptive control synthesis (1996). It includes many new features due to implementation of the new control criteria and it is considerably improved by means of handling polyhedral set estimates. This toolbox can be used for both educational and engineering purposes.