Constrained optimal control via multiparametric quadraticprogramming

The standard way of solving a mathematical program, is to feed the program into some numerical solver, which usually is custom-made for a class of problems. When several optimization problems, which only differ by the value of some parameter vector, are to be solved, an alternative may be to consider the problem as a parametric program. Some classes of parametric programs can be solved explicitly, giving the optimal solution as an explicit function of the parameter vector. When the mathematical program originates from a control problem, such an explicit solution may give real-time implementational advantages compared to using a numerical solver.This thesis treats topics within parametric programming, and control problems which can be addressed by this method. The main motivation for our work on parametric programming has been the possibility of formulating a constrained receding horizon optimal control problem (RHC) as a parametric program. RHC has been a successful control method in the process industries, where the relatively low sampling rates have allowed realtime numerical solvers to be applied. However, with the explicit solution obtained by solving a parametric program, the range of applications where RHC can be applied is extended.Chapters 2 and 3 treat topics within multiparametric quadratic programming (mpQP), which can be used to obtain an explicit piecewise linear (PWL) solution to the RHC problem when a linear model is used to characterize the underlying dynamics subject to linear inequality constraints. An algorithm to solve such problems is developed, giving an off-line execution speed which is an order of magnitude faster than existing solvers. Moreover, some theoretical aspects of mpQP are explored, in particular related to degeneracy of the problem.The explicit solutions to some of the main classes of multiparametric programs are PWL functions of the parameter vector. When the problem size increases, these functions tend to be complex. When such a complex PWL function is to be evaluated in a real-time control application, efficient and reliable implementation is needed. In Chapter 4 we suggest to represent general PWL functions as binary search trees. This method gives a evaluation time which is logarithmic in the number of regions in the PWL function. The method has given good results on practical problems.Some of the advantages of having an explicit PWL solution to a control problem are low hardware and software complexity, verifiability of the solution and possible high sampling rates. These are qualities which make these methods attractive for embedded solutions, such as in the automotive industry. In Chapter 5 we address the problem of lateral stabilization of automotive vehicles. The idea is that in situations where the driver is about to lose control of the vehicle the controller can retain steerability by applying appropriate braking forces to the individual wheels. The control law is obtained by formulating an RHC problem with a nonlinear dynamical model and constraints that capture physical limitation and represent safe operation of the vehicle. This can be considered a multiparametric nonlinear program (mpNLP), to which an approximate solution is computed by giving several local mpQP approximations. Satisfactory performance is obtained on simulations.Constrained control allocation for over-actuated marine vessels, aircraft or other mechanical systems is an application area which is well suited for solution via parametric programming. This is the topic of Chapter 6. MpQPs are formulated for control allocation in a high-performance aircraft, and compared to the methods found in the existing literature, good results were obtained, both in terms of optimality and real-time properties. Experimental results were obtained from constrained control allocation on a dynamically positioned model supply ship.