Optimal Control of Internally Forced Switching Systems With Guaranteed Feasibility

Abstract

An efficient dynamic optimization approach for internally forced switching systems is provided in this work. The distinguishing characteristic of the internally forced switching systems is that once the trajectory of the state of a mode hits a switching surface in the state space, the current mode stops operating immediately and then the next mode is activated automatically. To effectively address the optimal control of such systems, first, the continuous control input is approximated by piecewise constant functions utilizing the control vector parametrization (CVP) technique. Sensitivity analysis is subsequently used to derive the gradient of the cost function with respect to (w.r.t.) parameterized control input, while the state transition matrix is introduced for determining the gradients of the objective function w.r.t. switching instants. Second, a dynamic optimization method with the aid of gradient information is presented to locate the optimal solution for internally forced switching systems with a guarantee of rigorously satisfying the path constraints. Third, it is demonstrated that the designed algorithm terminates finitely to generate a feasible solution satisfying the Karush-Kuhn-Tucker (KKT) conditions of the dynamic optimization of internally forced switching systems to a specified tolerance. Finally, the proposed optimization approach is applied to the fed-batch fermentation process to obtain a high concentration of 1,3-propanediol production, while ensuring that the glycerol concentration rigorously satisfies the required boundary during the whole feed process. Note to Practitioners—The motivation of this work is to develop an efficient optimization method for path-constrained internally forced switching systems, which have a wide range of applications in practice, e.g., obstacle avoidance robots, chemical batch feed fermentation, etc. Although research methods for dynamic optimization of switched systems have been well-established, to the best of the author’s knowledge, there is almost no research directly dealing with internally forced switching systems. Since the switching instant of such systems is strongly dependent on the inputs and the state of the system, which makes it difficult to analyze the gradient of the objective function w.r.t. the switching instant. Moreover, it is necessary to consider the inequality path constraints that guarantee the requirements such as system safety and product quality. Therefore, this work aims at the dynamic optimization of the path-constrained internal forced switching system, considering the two important requirements of rigorous satisfaction of path constraints and a finite number of iterations of the algorithm. An efficient dynamic optimization approach is proposed based on the variational method, semi-infinite program technique, and right-handed constraint method, which can simultaneously obtain the optimal switching instants and optimal control inputs while guaranteeing that the path constraints are rigorously satisfied.