Abstract
Gradient‐based algorithms are efficient to compute numerical solutions of optimal control problems for hybrid systems (OCPHS), and the key point is how to get the sensitivity analysis of the optimal control problems. In this paper, optimality condition‐based sensitivity analysis of optimal control for hybrid systems with mode invariants and control constraints is addressed under a priori fixed mode transition order. The decision variables are the mode transition instant sequence and admissible continuous control functions. After equivalent transformation of the original problem, the derivatives of the objective functional with respect to control variables are established based on optimal necessary conditions. By using the obtained derivatives, a control vector parametrization method is implemented to obtain the numerical solution to the OCPHS. Examples are given to illustrate the results.
Highlights
In many fields of applications, such as powertrain systems of automobiles and multistage chemical processes, dynamics of the systems involve a sequence of distinct modes with fixed mode transition order, forming a hybrid system characterized by the coexistence and interaction of discrete and continuous dynamics the mode is commonly denoted by a discrete state of the systems in hybrid systems literature
To achieve some overall optimal performance for the systems, the duration and the admissible continuous control function of each mode must be determined as a whole 1–3 ; it necessitates the use of theories and techniques for the analysis and synthesis of hybrid dynamical systems
When control vector parametrization methods are implemented to obtain numerical solution to the OCPHS, updating the parameters of control profiles should be at the same time point when iterative procedure is running
Summary
In many fields of applications, such as powertrain systems of automobiles and multistage chemical processes, dynamics of the systems involve a sequence of distinct modes with fixed mode transition order, forming a hybrid system characterized by the coexistence and interaction of discrete and continuous dynamics the mode is commonly denoted by a discrete state of the systems in hybrid systems literature. As far as switching hybrid systems without external continuous control function are concerned, Egerstedt et al 6 and Johnson and Murphey 18 derived the gradients and second-order derivatives of the cost functional, respectively, and used them to design an associated algorithm to get the mode transition instants. From the view of dynamic programming, Seatzu et al 16 provided an optimal state feedback control law to switched piecewise affine autonomous systems These algorithms pose the hierarchy 17, 21, 22 , and the basic module of the hierarchical algorithms is how to get optimal continuous control and optimal mode transition instants, though the main challenge of OCPHS is how to get the optimal mode transition order. A Optimality conditions-based sensitivity analysis of optimal control for hybrid systems with mode invariants are given explicitly, and b following the given derivatives, a control vector parameterization method is designed to obtain the numerical solution. B , Rn denotes the family of continuous functions f from a, b to Rn with up to l order derivatives. · denotes the Euclidean norm
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