Optimal control of switched systems based on parameterization of the switching instants

This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a prespecified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. In order to search for the optimal switching instants, the derivatives of the optimal cost with respect to the switching instants need to be known. The most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarity equations, the boundary and continuity conditions, along with their differentiations. This method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed. An extension of the method is also applied to problems with internally forced switching. Examples are shown to illustrate the results in the paper.     

A moving switching sequence approach for nonlinear model predictive control of switched systems with state‐dependent switches and state jumps

In this paper, we propose an approach for real‐time implementation of nonlinear model predictive control (NMPC) for switched systems with state‐dependent switches called the moving switching sequence approach. In this approach, the switching sequence on the horizon moves to the present time at each time as well as the optimal state trajectory and the optimal control input on the horizon. We assume that the switching sequence is basically invariant until the first predicted switching time reaches the current time or a new switch enters the horizon. This assumption is reasonable in NMPC for systems with state‐dependent switches and reduces computational cost significantly compared with the direct optimization of the switching sequence all over the horizon. We update the switching sequence by checking whether an additional switch occurs or not at the last interval of the present switching sequence and whether the actual switch occurs or not between the current time and the next sampling time. We propose an algorithm consisting of two parts: (1) the local optimization of the control input and switching instants by solving the two‐point boundary‐value problem for the whole horizon under a given switching sequence and (2) the detection of an additional switch and the reconstruction of the solution taking into account the additional switch. We demonstrate the effectiveness of the proposed method through numerical simulations of a compass‐like biped walking robot, which contains state‐dependent switches and state jumps.     

Dynamic buffer management using optimal control of hybrid systems

This paper studies a dynamic buffer management problem with one buffer inserted between two interacting components. The component to be controlled is assumed to have multiple power modes corresponding to different data processing rates. The overall system is modeled as a hybrid system and the buffer management problem is formulated as an optimal control problem. Different from many previous studies, the objective function of the proposed problem depends on the switching cost and the size of the continuous state space, making its solution much more challenging. By exploiting some particular features of the problem, the best mode sequence and the optimal switching instants are characterized analytically using a variational approach. Simulation results based on real data shows that the proposed method can significantly reduce the energy consumption compared with another heuristic scheme in several typical situations.     

Optimal switching instants for a switched-capacitor DC/DC power converter

We consider a switched-capacitor DC/DC power converter with variable switching instants. The determination of optimal switching instants giving low output ripple and strong load regulation is posed as a non-smooth dynamic optimization problem. By introducing a set of auxiliary differential equations and applying a time-scaling transformation, we formulate an equivalent optimization problem with semi-infinite constraints. Existing algorithms can be applied to solve this smooth semi-infinite optimization problem. The existence of an optimal solution is also established. For illustration, the optimal switching instants for a practical switched-capacitor DC/DC power converter are determined using this approach.     

Inversion-based optimal output tracking–transition switching with preview for nonminimum-phase linear systems

In this article, the problem of nonperiodic tracking–transition switching with preview is considered. Such a control problem exists in applications including nanoscale material property mapping, robot manipulation, and probe-based nanofabrication, where the output needs to track the desired trajectory during the tracking sections, and rapidly transit to another point during the transition sections with no post-transition oscillations. Due to the coupling between the control of the tracking sections and that of the transition ones, and the potential mismatch of the boundary system state at the tracking–transition switching instants, these control objectives become challenging for nonminimum-phase systems. In the proposed approach, the optimal desired output trajectory for the transition sections is designed through a direct minimization of the output energy, and the needed control input that maintains the smoothness of both the output and the system state across all tracking–transition switching is obtained through a preview-based stable-inversion approach. The needed preview time is quantified by the characteristics of the system dynamics, and can be minimized via the recently developed optimal preview-based inversion technique. The proposed approach is illustrated through a nanomanipulation example in simulation.     

On stabilization of switched nonlinear systems with unstable modes

This paper addresses stabilization issue of switched nonlinear systems where some modes are stable and others may be unstable. A new stabilizing switching law that determines the initial states and the switching instants for any given switching sequence is proposed. The developed technique relies on the tradeoff among the functions’ gains of continuous modes, and does not depend on the constant ratio condition required in “dwell-time scheme”.     

Optimal control of switching time in switched stochastic systems with multi-switching times and different costs

In this paper, we solve an optimal control problem for a class of time-invariant switched stochastic systems with multi-switching times, where the objective is to minimise a cost functional with different costs defined on the states. In particular, we focus on problems in which a pre-specified sequence of active subsystems is given and the switching times are the only control variables. Based on the calculus of variation, we derive the gradient of the cost functional with respect to the switching times on an especially simple form, which can be directly used in gradient descent algorithms to locate the optimal switching instants. Finally, a numerical example is given, highlighting the validity of the proposed methodology.     

Control discrete-time switched singular systems with state delays under asynchronous switching

This article is concerned with the problem of state feedback control for a class of discrete-time switched singular systems with time-varying state delays under asynchronous switching. The asynchronous switching considered here means that the switching instants of the candidate controllers lag behind those of the system modes. The concept of mismatched control rate is introduced. By using the multiple Lyapunov function approach and the average dwell time technique, a sufficient condition for the existence a stabilising switching law is first derived to guarantee the regularity, causality and exponential stability of the closed-loop system in the presence of asynchronous switching. The stabilising switching law is characterised by a upper bound on the mismatched control rate and a lower bound on the average dwell time. Then, the corresponding solvability condition for a set of mode-dependent state feedback controllers is established by using the linear matrix inequality (LMI) technique. Finally, a numerical example is provided to illustrate the effectiveness of the proposed method.     

Stabilization and second‐order optimization for multimodule impulsive switched linear systems

In this work, we investigate stabilization and optimization issues for a class of multimodule impulsive switched linear systems. First, we establish a necessary and sufficient criterion on asymptotic stabilizability via a pathwise state‐feedback scheme, which achieves the merits of both time‐driven and state‐feedback mechanisms. Then, we present an impulse/switching instant optimization problem that usually arises in finite‐horizon optimal control. To reduce the computational burden, we propose a novel method via developing efficiently computable expressions for the cost function, the gradient vector, and the Hessian matrix. Next, we design a second‐order optimization algorithm searching for the optimal impulse/switching instants. Finally, a numerical example is provided to illustrate the effectiveness of the proposed approach.     

Synthesis of optimal controllers for piecewise affine systems with sampled-data switching

This paper discusses the optimal control problem of the continuous-time piecewise affine (PWA) systems with sampled-data switching, where the switching action is executed based upon a condition on the state at each sampling time. First, an algebraic characterization for the problem to be feasible is derived. Next, an optimal continuous-time controller is derived for a general class of PWA systems with sampled-data switching, for which the optimal control problem is feasible but whose subsystems in some modes may be uncontrollable in the usual sense. Finally, as an application of the proposed approach, the high-speed and energy-saving control problem of the CPU processing is formulated, and the validity of the proposed methods is shown by numerical simulations.     

Optimal Control of Switching Times in Switched Stochastic Systems

In this paper, we solve an optimal control problem for switched stochastic systems using calculus of variations, where the objective is to minimize a cost functional defined on the state and the switching times are the sole control variables. In particular, we focus on the problem in which a pre‐specified sequence of active subsystems is given. For one switching time case, the derivative of the cost functional with respect to the switching time is derived, which has an especially simple form and can be directly used in gradient descent algorithms to locate the optimal switching instant. Then, we propose an approach to deal with the problem with multi‐switching times case. Finally, two numerical examples are given, highlighting the viability and advantages of the proposed methodology.     

Stabilization of switched continuous-time systems with all modes unstable via dwell time switching

Stabilization of switched systems composed fully of unstable subsystems is one of the most challenging problems in the field of switched systems. In this brief paper, a sufficient condition ensuring the asymptotic stability of switched continuous-time systems with all modes unstable is proposed. The main idea is to exploit the stabilization property of switching behaviors to compensate the state divergence made by unstable modes. Then, by using a discretized Lyapunov function approach, a computable sufficient condition for switched linear systems is proposed in the framework of dwell time; it is shown that the time intervals between two successive switching instants are required to be confined by a pair of upper and lower bounds to guarantee the asymptotic stability. Based on derived results, an algorithm is proposed to compute the stability region of admissible dwell time. A numerical example is proposed to illustrate our approach.     

切换系统的无扰切换控制及其在航空发动机中的应用

利用多Lyapunov函数方法, 本文研究了一类切换线性系统的状态跟踪无扰切换控制问题.首先, 刻画了控制信号在切换时刻处的抖振抑制水平.其次, 通过控制器与切换律的同时设计, 实现了系统的状态跟踪和控制信号抖振抑制.最后, 将所提出的状态跟踪无扰切换控制策略应用于一个涡扇航空发动机模型的控制设计上, 说明了所提出方法的有效性.     

Computational Method for a Class of Switched System Optimal Control Problems

We consider an optimal control problem with dynamics that switch between several subsystems of nonlinear differential equations. Each subsystem is assumed to satisfy a linear growth condition. Furthermore, each subsystem switch is accompanied by an instantaneous change in the state. These instantaneous changes-called ldquostate jumpsrdquo-are influenced by a set of control parameters that, together with the subsystem switching times, are decision variables to be selected optimally. We show that an approximate solution for this optimal control problem can be computed by solving a sequence of conventional dynamic optimization problems. Existing optimization techniques can be used to solve each problem in this sequence. A convergence result is also given to justify this approach.     

Economically optimal batch diafiltration via analytical multi-objective optimal control

This paper studies the problem of economically oriented optimal operation of batch membrane diafiltration processes that are designed to concentrate the valuable components of the solution and to purge the impurities from it. We consider a complex economical objective that accounts for the total operational costs comprising a cost of consumed diluant, costs related to duration of processing, and a cost of product loss. The optimization problem is formulated as a multi-objective optimal control problem in order to investigate the impact of operational cost factors on optimal operation policy. This is achieved thanks to the use of the analytical approach that exploits Pontryagin's minimum principle. We show that the economically optimal control strategy is to carry out an operation involving saturated (bang-bang or constraint-tracking) control modes and a singular arc. For the most common cases of diafiltration problems, it turns out that the switching of the consecutive control modes can be realized in the state feedback fashion, i.e. the entire optimal operation is defined analytically in the space of process states. We demonstrate the applicability of the presented approach and we illustrate achievable benefits, over traditional control methods for the batch diafiltration processes, on two case studies taken from the literature.     

一类切换系统的输入-状态稳定性分析与优化控制

利用驻留时间法和 Gronwall-Bellman不等式研究了一类切换系统的输入一状态稳定性分析与优化控制问题.在保证切换系统输入-状态稳定的前提下,将切换时刻和切换次数约束条件转化为线性约束,提出了一种新的切换系统优化问题目标函数的形式.与已有的方法相比,该方法无需引入新的状态变量,无需同时满足构造输入-状态稳定控制李亚普诺夫函数和所有子系统都是输入-状态稳定的条件,为控制器的优化设计提供了便利.最后,通过算例仿真证实了文中所提方法的可行性.     

State estimation in stochastic hybrid Systems with sparse observations

In this note we study the problem of state estimation for a class of sampled-measurement stochastic hybrid systems, where the continuous state x satisfies a linear stochastic differential equation, and noisy measurements y are taken at assigned discrete-time instants. The parameters of both the state and measurement equation depend on the discrete state q of a continuous-time finite Markov chain. Even in the fault detection setting we consider-at most one transition for q is admissible-the switch may occur between two observations, whence it turns out that the optimal estimates cannot be expressed in parametric form and time integrations are unavoidable, so that the known estimation techniques cannot be applied. We derive and implement an algorithm for the estimation of the states x, q and of the discrete-state switching time that is convenient for both recursive update and the eventual numerical quadrature. Numerical simulations are illustrated.     

HIERARCHICAL DYNAMIC OPTIMIZATION FOR LINEAR DISCRETE SYSTEMS

In this paper an on-line multilevel suboptimal control procedure is developed for large-scale, time-invariant, and discrete linear systems with quadratic performance criterion, and measurement disturbances having arbitrary statistical distributions. The on-line coordination procedure as reported here employs an upper-level intervention through interaction estimates and goal modifications in order to modify lower-level subsystem optimization problems. Their individual controls are applied to the system and the resulting output is estimated by Kalman Type filter and performance information is fedback to the upper level unit. A reference control approach is used to up-date intervention variables over a sequence of time instants to improve on the control being applied to the process. In the computer simulation of the process, it was found that the actual overall cost for systems using this on-line multilevel control procedure is within ±10% of the cost resulting from a conventional overall optimal control policy.