Dynamic optimization of state-dependent switched systems with free switching sequences

Abstract

Without necessarily assuming that switching sequences are fixed, a dynamic optimization method is proposed for optimal control of state-dependent switched systems. First, a parameterization method is developed to parameterize the switching instants and control vectors to facilitate the calculation of the gradient information, and then the original problem becomes a finite-dimensional mixed discrete–continuous nonlinear program as the switching sequence is discrete and the other variables are continuous. Secondly, the mixed discrete–continuous nonlinear program is transformed into an equivalent problem that contains only continuous variables by relaxing 0 and 1 discrete variables into continuous variables between 0 and 1 and adding proper linear and quadratic constraints. Thirdly, the formulas to compute the gradients of the objective function with respect to all the arguments are derived by solving the variational systems and a two-point boundary value differential algebraic equations (DAEs). Fourthly, an algorithm is proposed to locate a feasible point satisfying the Karush–Kuhn–Tucker (KKT) conditions to a specified tolerance of dynamic optimization of switched systems (DOSS) while guaranteeing feasibility of inequality path constraints, and the finite convergence of the algorithm is proved. Finally, the performance of the algorithm is analyzed via a numerical example.

Introduction

State-dependent switched systems are switched once a modal switching occurs whenever the state of the system reaches a switching surface in the state space. Application domains of state-dependent systems include biomedical engineering (Haenny et al., 1988, Young, 2004), robotics (Egerstedt, 2000, Luca et al., 2012), manufacturing systems (David and Buzacott, 1985, Youssef et al., 2006), and electrical circuit systems (Flieller et al., 1998, Wu et al., 2006). Among all problems studied for state-dependent switched systems, optimal control of such systems is a very important one (Axelsson et al., 2008, Boccadoro et al., 2005, Boccadoro and Wardi, 2005, Feng and Antsaklis, 2015, Liu et al., 2018, Xu and Antsaklis, 2003), and all of these works note that state-dependent switched systems are difficult to handle because the switching times are not the free parameters but depend on the input parameters in a complicated way. In Liu et al. (2018), dynamic optimization of time-delay switched systems with state-dependent switching has been studied, and the gradient-based optimization algorithm to determine the optimal parameter values is developed. The authors of Axelsson et al., 2008, Boccadoro and Wardi, 2005 and Boccadoro et al. (2005) investigated the problem on optimal switching surface design for hybrid dynamical systems. The above-mentioned results are only applicable to autonomous switched systems (i.e. no control input signals) and also require that the switching sequences are pre-fixed. In many practical switched systems, the switching sequences should also be determined optimally. However, besides obtaining optimal switching instants and optimal control, determining the switching sequences of switched systems is a very difficult task (Hante and Falk, 2015, Shaikh and Caines, 2003, Song, 2012, Xu and Antsaklis, 2004, Zhai et al., 2017) due to the fact that the variables determining the switching sequence are discrete, which results that the optimal control of switched systems is a mixed discrete–continuous nonlinear dynamic program. To our best knowledge, there are only few works to optimize the switching sequences (Bengea and DeCarlo, 2005, Feng et al., 2010, Wei et al., 2007, Yu et al., 2013, Zhai et al., 2017). In Bengea and DeCarlo, 2005, Wei et al., 2007, Yu et al., 2013 and Zhai et al. (2017), the mixed discrete–continuous nonlinear program can be embedded into a large family of systems where the switching function takes values in the interval $[0,1]$ as opposed to the discrete set $\{0,1\}$, and this essentially transforms the discrete optimal control problem into a smooth problem that can be solved using traditional numerical methods (e.g. sequential quadratic programming). The authors of Feng et al. (2010) proposed an algorithm that combines discrete filled function with descent method to solve the dynamic optimization of switched systems by introducing a metric in the space of switching sequences and then constructing an appropriate discrete filled function. The common features of the above works, however, are that they cannot obtain the optimal switching sequences within a finite number of iterations and only focus on time-dependent switched systems. In addition, the authors in Xu and Antsaklis (2002) noted that the switching sequences should be optimized, but they do not give a specific method to deal with this issue.

Moreover, rigorous satisfaction of inequality path constraints in the optimal control of switched systems is another important issue, since they often reflect the safety and quality limits of a process. Recently, an effective algorithm is proposed in Fu and Zhang (0000) and Zhang and Fu (2021) to solve the optimal control of switched systems with guaranteed satisfaction of inequality path constraints within finite iterations. However, the system under consideration is still limited to a pre-defined switching sequence. Thus, one may wonder whether it is possible to design an algorithm simultaneously optimizes the switching times, control inputs and switching sequences for state-dependent switched systems while guaranteeing the rigorous satisfaction of path constraints within finite iterations? This is the main motivation of this paper.

A dynamic optimization method is proposed for state- dependent switched systems with free switching sequences to locate a feasible point, which satisfies the KKT conditions to a specified tolerance within a finite number of iterations. First, the control vector parameterization (CVP) technique and the switching times parameterization (STP) method are used to reduce the infinite-dimensional optimal control problem to finite- dimensional mixed discrete–continuous nonlinear program. Then, the mixed discrete–continuous nonlinear program is transformed into an equivalent a continuous nonlinear dynamic program by relaxing $0$ and $1$ discrete variables into continuous variables between $0$ and $1$ and adding proper linear and quadratic constraints (Wang, Teo, & Lee, 1996). By solving the variational systems and the boundary value DAEs, the gradients of objective function w.r.t. all decision variables are derived. With these gradient information, an algorithm is proposed based on iteratively approximating the dynamic optimization of state-dependent path-constrained switched systems by restricting the right-hand side of the path constraints and enforcing the path constraints at finitely many time points. Finally, it is proven that the algorithm terminates within a finite number of iterations.

The main contributions can be summarized as follows. (1) A dynamic optimization method is proposed to solve the optimal control for state-dependent path-constrained switched systems with free switching sequences. (2) This paper provides the accurate gradient values of the cost function with respect to switching instants for state-dependent switched systems. (3) The switching instants, the control input, and the switching sequences are optimized at the same iteration, which greatly reduces the number of nonlinear programs compared to the two-level optimization algorithm (Fu and Zhang, 0000, Xu and Antsaklis, 2000, Xu and Antsaklis, 2004) and thus saves computational time. (4) It is mathematically proven that the designed algorithm for the optimal control of state-dependent path-constrained switched systems converges within finite iterations.

The rest of the article is structured as follows. In Section 2, the switched system and its optimal control problem are stated. In Section 3, the mixed discrete–continuous nonlinear program is transformed into a continuous dynamic optimization problem. In Section 4, the gradients for the objective function w.r.t. all arguments are derived. With the gradient formulas, in Section 5, the algorithm to locate a feasible approximate KKT point of dynamic optimization of switched systems with guaranteed satisfaction of path constraints is designed, and moreover, a proof of finite convergence of the algorithm is presented. In Section 6, the property of guaranteed satisfaction of path constraints for dynamic optimization of switched systems is illustrated and the effect of tuning parameters in the algorithm using a numerical simulation example is analyzed. Conclusions and an outlook on future work are given in Section 7.