Optimal Control of Hybrid SystemsMain research topics:
Even though discretetime hybrid systems (and in particular Piecewise Affine (PWA) systems) are a special class of nonlinear systems most of the nonlinear system and control theory does not apply because it requires certain smoothness assumptions. For the same reason we also cannot simply use linear control theory in some approximate manner to design controllers for PWA systems. In the past most tools for the analysis and control of hybrid systems were ad hoc supported by extensive simulation. Our aim is to further advance systematic procedures and develop efficient algorithmic implementations, see e.g. MPT, that give the exact solution to optimal control problems for the above mentioned constrained or hybrid systems. Model Predictive Control (MPC)Model predictive control, or MPC, is a control paradigm with a motley background. The underlying ideas for MPC originated already in the sixties as a natural application of optimal control theory. The true birth of MPC took place in industry in the mid 70's to mid 80's where the MPC strategy became popular in the petrochemical industry. During this period, there was a flood of new variants of MPC. Despite the vast number of variants, not much differed between the algorithms. Typically, they differed in the process model (impulse response, step response, statespace, etc.), disturbance (constant, decaying, altered white noise, etc.), and adaptation to time varying models. During the 90's, the theory of MPC has matured substantially, the main reason being the use of statespace models instead of inputoutput models. MPC is an optimization based control law, and the performance measure J is almost always based on quadratic forms or linear norms. By using a quadratic or linear performance measure efficient optimization problems arise and mathematical analysis simplifies. By defining the performance weight L(x,u) our underlying goal is to find the optimal control input that minimizes an infinite horizon performance measure subject to system dynamics (the prediction model) and constraints: In the general constrained case, there does not exist any simple closedform expression for the solution. Instead, the first step in MPC is to define a prediction horizon T and approximate the performance measure by using a finite horizon, The solution to the CFTOC problem is a finite input sequence U_{T}(x(t)). Due to the finite horizon the optimization problem is finite dimensional and easier to solve. The second idea is to apply only the first control move of the obtained sequence U_{T}(x(t)) to the plant and resolve a new finite horizon problem when we obtain new measurements of the current state x(t). Putting this conceptual idea of the socalled receding horizon policy into an algorithm yields the following basic MPC controller
The finite horizon approximation used in the MPC scheme introduces lack of guaranteed stability. To enforce closedloop stability in an MPC framework, the above mentioned finite dimensional optimization problem usually is modified by choosing an appropriate terminal cost and constraint set. Online MPC vs. Offline MPC (Explicit Solution)As outlined above, MPC is usually performed by solving an optimization problem online in between two sampling instances and applying the first optimal control move to the plant. Due to the computational complexity of the optimization problem, MPC is limited to systems with long sampling periods, systems with only few state variables and rather short prediction horizons, although significant advances in the development of online solvers have been made in the last decade. Nevertheless, it can be beneficial to move computational burden from the online implementation to an offline stage. However, the optimal control problem of a constrained PWA system can be formulated as a multiparametric program by treating the current state x(t) as a parameter. Solving such a multiparametric program yields an explicit solution, i.e. an optimal lookup table, for the MPC problem which inherits all the stability and performance properties of the original MPC problem. This optimal solution for the considered problem class is a piecewise affine statefeedback control law that is defined over a partition of the feasible statespace. Therefore, the timeconsuming and complex computation can be carried out offline, while the online implementation (control action computation) reduces to a simple setmembership test and evaluation of a linear function. Depending on the type of performance measure used, either multiparametric linear or quadratic programs have to be solved. A. Bemporad, M. Morari, V. Dua, and E.N. Pistikopoulos, "The Explicit Solution of Model Predictive Control via Multiparametric Quadratic Programming". In Proceedings of the American Control Conference, vol. 2, pp. 872876, Chicago, USA, June 2000. D. Frick, A. Domahidi, and M. Morari, "Embedded Optimization for Mixed Logical Dynamical Systems". To appear in Computers and Chemical Engineering, 2014. Constrained Finite Time Optimal Control (CFTOC)The research results of our group include:
Constrained Infinite Time Optimal Control (CITOC)In contrast to the aforementioned constrained finite time optimal control the Constrained Infinite Time Optimal Control (CITOC) problem focuses on the optimization problem defined over an infinite prediction/control horizon. The main advantages of the infinite time solution, compared to the corresponding finite time solution, are the inherent guaranteed stability and feasibility as well as optimality of the closedloop system. We have studied the explicit solution of the CITOC problem for
Nominal vs. Robust ControlAlthough a rich theory has been developed for the robust control of linear systems, relatively few results exist for the robust control of constrained linear systems and even less for hybrid systems. Therefore, this remains a hot topic of an ongoing research. A seminal early result, presented in [Kothare et.al. 1996], defines the problem of robust optimal control for constrained linear systems as an LMI problem. Later results of our research group address the problem of robust optimal control with an emphasis on the explicit solution. The most relevant results are listed below:
