Optimal control problems involving constrained, switched, and delay systems

Abstract

In this thesis, we develop numerical methods for solving five nonstandard optimal control problems. The main idea of each method is to reformulate the optimal control problem as, or approximate it by, a nonlinear programming problem. The decision variables in this nonlinear programming problem influence its cost function (and constraints, if it has any) implicitly through the dynamic system. Hence, deriving the gradient of the cost and the constraint functions is a difficult task. A major focus of this thesis is on developing methods for computing these gradients. These methods can then be used in conjunction with a gradient-based optimization technique to solve the optimal control problem efficiently.The first optimal control problem that we consider has nonlinear inequality constraints that depend on the state at two or more discrete time points. These time points are decision variables that, together with a control function, should be chosen in an optimal manner. To tackle this problem, we first approximate the control by a piecewise constant function whose values and switching times (the times at which it changes value) are decision variables. We then apply a novel time-scaling transformation that maps the switching times to fixed points in a new time horizon. This yields an approximate dynamic optimization problem with a finite number of decision variables. We develop a new algorithm, which involves integrating an auxiliary dynamic system forward in time, for computing the gradient of the cost and constraints in this approximate problem.The second optimal control problem that we consider has nonlinear continuous inequality constraints. These constraints restrict both the state and the control at every point in the time horizon. As with the first problem, we approximate the control by a piecewise constant function and then transform the time variable. This yields an approximate semi-infinite programming problem, which can be solved using a penalty function algorithm. A solution of this problem immediately furnishes a suboptimal control for the original optimal control problem. By repeatedly increasing the number of parameters used in the approximation, we can generate a sequence of suboptimal controls. Our main result shows that the cost of these suboptimal controls converges to the minimum cost.The third optimal control problem that we consider is an applied problem from electrical engineering. Its aim is to determine an optimal operating scheme for a switchedcapacitor DC-DC power converter—an electronic device that transforms one DC voltage into another by periodically switching between several circuit topologies. Specifically, the optimal control problem is to choose the times at which the topology switches occur so that the output voltage ripple is minimized and the load regulation is maximized. This problem is governed by a switched system with linear subsystems (each subsystem models one of the power converter’s topologies). Moreover, its cost function is non-smooth. By introducing an auxiliary dynamic system and transforming the time variable (so that the topology switching times become fixed), we derive an equivalent semi-infinite programming problem. This semi-infinite programming problem, like the one that approximates the continuously-constrained optimal control problem, can be solved using a penalty function algorithm.The fourth optimal control problem that we consider involves a general switched system, which includes the model of a switched-capacitor DC-DC power converter as a special case. This switched system evolves by switching between several subsystems of nonlinear ordinary differential equations. Furthermore, each subsystem switch is accompanied by an instantaneous change in the state. These instantaneous changes—so-called state jumps—are influenced by control variables that, together with the subsystem switching times, should be selected in an optimal manner. As with the previous optimal control problems, we tackle this problem by transforming the time variable to obtain an equivalent problem in which the switching times are fixed. However, the functions governing the state jumps in this new problem are discontinuous. To overcome this difficulty, we introduce an approximate problem whose state jumps are governed by smooth functions. This approximate problem can be solved using a nonlinear programming algorithm. We prove an important convergence result that links the approximate problem’s solution with the original problem’s solution.The final optimal control problem that we consider is a parameter identification problem. The aim of this problem is to use given experimental data to identify unknown state-delays in a nonlinear delay-differential system. More precisely, the optimal control problem involves choosing the state-delays to minimize a cost function measuring the discrepancy between predicted and observed system output. We show that the gradient of this cost function can be computed by solving an auxiliary delay-differential system. On the basis of this result, the optimal control problem can be formulated—and hence solved—as a standard nonlinear programming problem.