An extension of LaSalle’s Invariance Principle for a class of switched linear systems

In this paper LaSalle’s Invariance Principle for switched linear systems is studied. Unlike most existing results in which each switching mode in the system needs to be asymptotically stable, in this paper the switching modes are allowed to be only Lyapunov stable. Under certain ergodicity assumptions, an extension of LaSalle’s Invariance Principle for global asymptotic stability of switched linear systems is proposed provided that the kernels of derivatives of a common quadratic Lyapunov function with respect to the switching modes are disjoint (except the origin).     

Optimal control of discrete-time bilinear systems with applications to switched linear stochastic systems

This paper aims at characterizing the most destabilizing switching law for discrete-time switched systems governed by a set of bounded linear operators. The switched system is embedded in a special class of discrete-time bilinear control systems. This allows us to apply the variational approach to the bilinear control system associated with a Mayer-type optimal control problem, and a second-order necessary optimality condition is derived. Optimal equivalence between the bilinear system and the switched system is analyzed, which shows that any optimal control law can be equivalently expressed as a switching law. This specific switching law is most unstable for the switched system, and thus can be used to determine stability under arbitrary switching. Based on the second-order moment of the state, the proposed approach is applied to analyze uniform mean-square stability of discrete-time switched linear stochastic systems. Numerical simulations are presented to verify the usefulness of the theoretic results.     

A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control

In this paper we consider the asymptotic stability of a class of discrete-time switching linear systems, where each of the constituent subsystems is Schur stable. We first present an example to motivate our study, which illustrates that the bilinear transform does not preserve the stability of a class of switched linear systems. Consequently, continuous time stability results cannot be transformed to discrete time analogs using this transformation. We then present a subclass of discrete-time switching systems that arise frequently in practical applications. We prove that global attractivity for this subclass can be obtained without requiring the existence of a common quadratic Lyapunov function (CQLF). Using this result, we present a synthesis procedure to construct switching stabilizing controllers for an automotive control problem, which is related to the stabilization of a vehicle’s roll dynamics subject to switches in the center of gravitys (CG) height.     

Speed regulation in steering-based source seeking

The simplest strategy for extremum seeking-based source localization, for sources with unknown spatial distributions and nonholonomic unicycle vehicles without position measurement, employs a constant positive forward speed. Steering of the vehicle in the plane is performed using only the variation of the angular velocity. While keeping the forward speed constant is a reasonable strategy motivated by implementation with aerial vehicles, it leads to complexities in the asymptotic behavior of the vehicle, since the vehicle cannot settle—at best it can converge to a small-size attractor around the source. In this paper we regulate the forward velocity, with the intent of bringing the vehicle to a stop, or as close to a stop as possible. The vehicle speed is controlled using simple derivative-like feedback of the sensor measurement (the derivative is approximated with a washout filter) to which a speed bias parameter V_c is added. The angular velocity is tuned using standard extremum seeking. We prove two results. For V_c in a certain range around zero, we show that the vehicle converges to a ring around the source and on average the limit of the vehicle’s heading is either directly away or towards the source. For other values of V_c>0, the vehicle converges to a ring around the source and it revolves around the source. Interestingly, the average heading of this revolution around the source is more outward than inward—this is possible because the vehicle’s speed is not constant, it is lower during the outward steering intervals and higher during the inward steering intervals. The theoretical results are illustrated with simulations.     

Stability of switched nonlinear systems via extensions of LaSalle’s invariance principle

This paper studies the extension of LaSalle’s invariance principle for switched nonlinear systems. Un-like most existing results in which each switching mode in the system needs to be asymptotically stable, this paper allows the switching modes to be only stable. Under certain ergodicity assumptions of the switching signals, two extensions of LaSalle’s invariance principle for global asymptotic stability of switched nonlinear systems are obtained using the method of common joint Lyapunov function.     

二阶切换线性系统的滑模控制

当前,切换系统的稳定控制问题作为控制领域的一个研究难点,引起了学术界和工业界的广泛关注.该文针对二阶切换线性系统,考虑切换时对象特性的变化,将其转化为一类特殊的非线性系统,并基于滑模控制理论,设计出适用于二阶切换线性系统的滑模控制器.最后通过具体的实例仿真验证、结果对比,表明滑模控制器比传统PID控制器在二阶切换线性系...     

Robust estimation algorithm for both switching signal and state of switched linear systems

We present a hybrid-type observer for detecting the switching time and estimating both the active mode and the states of continuous-time switched linear systems. The systems under consideration have external inputs and are affected by unknown disturbances. In addition, noise corrupts the output measurements. In this setting the switching cannot be detected immediately, and thus, this paper presents a condition that relates the amount of delay to the sizes of the unknown disturbances/noises, the external inputs, and the states, and the strength of the observability. Once the condition is satisfied, the proposed observer and algorithm return the exact active mode and approximate state information of the switched system. A numerical example is also presented to show the performance of our algorithm..     

Active modes and switching instants identification for linear switched systems based on Discrete Particle Swarm Optimization

In this paper, a methodology for identifying switching sequences and switching instants of switched linear systems (SLS) is derived. The identification problem of a SLS is a challenging and non-trivial problem. In fact, it involves interaction between binary, discrete and real-valued variables. A SLS switches many times over a finite time horizon and thus estimating the sequence of activated modes and the switches locations is a crucial problem for both control and Fault Detection and Isolation (FDI). The proposed methodology is based on the Discrete Particle Swarm Optimization (DPSO) technique. The identification problem is formulated as an optimization problem involving noisy data (system inputs and outputs). Both a set of binary variables corresponding to each sub-model before and after each switch, and the corresponding switching instants are iteratively adjusted by the DPSO algorithm. Thus, the DPSO algorithm has to classify which sub-system has generated which data. The efficiency of the proposed approach is illustrated through a numerical example and a physical one. The numerical example is a Switched Auto-Regressive eXogenous (SARX) system and the physical one is a buck–boost DC/DC converter.     

Graphic approach for the determination of the existence of sequences guaranteeing observability of switched linear systems

This paper deals with the design of sequences allowing the observability of the state of a switched linear system. The proposed method, based on a graph-theoretic approach, needs only the knowledge of the system’s structure. The necessary and sufficient conditions for the existence of at least one switching sequence guaranteeing the observability of a switched structured system are provided. Such conditions are particularly intuitive and they can be implemented with classic algorithms of the graph theory.     

夜晚车辆异常事件分析

夜晚车道模型是车辆跟踪和车辆行为分析的基础,但是当高速公路或者城市道路光线较暗时,很难通过车道检测的方法来建立车道模型,夜晚车辆快速行驶或相邻帧车辆之间重叠度较低时无法实现准确跟踪。针对此类问题提出了一种基于学习的车道模型建立方法和基于多帧的最佳匹配跟踪方法。首先利用自动多阈值分割方法提取场景中光亮的目标;其次,利用车灯的相关特征移除非车灯光亮区域;接着,利用空间信息把车灯聚类成一个车辆目标,利用多帧的最佳匹配跟踪方法进行跟踪;最后利用车辆跟踪参数与车道模型的融合对夜晚车辆异常事件进行分析。实验结果表明,该算法能够准确地检测出夜晚车辆换道、逆向行驶、交通拥挤、停车等异常事件,并且有很强的鲁棒性。     

Exponential stability of stochastic systems with hysteresis switching

For stochastic systems with state-dependent switching which are motivated by active regions of subsystems, the exponential stability is studied in this paper. Distinct from most of the existing references, the existence of the solution to stochastic switched systems is not given as a priori information but can be proved under some easily verified conditions. By the aid of Dynkin’s formula, Itô’s formula and exponential martingale inequality, the criteria on moment exponential stability and almost sure exponential stability of the stochastic switched system are established based on Lyapunov-like techniques. Simulation examples are presented to illustrate the validity of the results.     

Hybrid Model Predictive Power Management of A Fuel Cell‐Battery Vehicle

This paper considers optimal power management of a fuel cell‐battery hybrid vehicle (FCHV) powertrain having three distinct modal configurations (modes): electric motor propelling/battery discharging, propelling/charging, and generating/charging. Each mode has a distinct set of dynamics and constraints. Using component dynamical/algebraic models appropriate to power flow management, the paper develops a supervisory‐level switched system model as an interconnection of subsystems. Given the model, the paper sets forth a hybrid model predictive control strategy based on a minimization of a performance index (PI) that trades off tracking and fuel economy in each operational mode. Specifically, the PI trades off velocity tracking error, battery state of charge variance, and electric drive and hydrogen fuel usages while penalizing frictional braking to encourage regenerative braking. The optimization is performed using an embedded system model and collocation with matlab 's fmincon to compute mode switches and continuous time controls. The methodology avoids the computational complexity of alternate approaches based on, e.g., mixed integer programming. Projection methods for approximating the switched system solution from the embedded solution are empirically evaluated. To demonstrate the methodology, an example of a FCHV is simulated using three standard velocity driving profiles: a sawtooth profile with a hill climb, the EPA urban dynamic driving schedule, and the New European Driving Cycle. Also, drive cycle fuel usage is compared to that from the Equivalent Consumption Minimization Strategy.     

Observer-based stabilization of switching linear systems

In this paper, we present a “deep pole assignment method” to study the observer-based stabilization of switching linear systems where the dynamics of each mode are known a priori but the switching times of modes are arbitrary. The design can be used for both finite and infinite switched linear systems. We emphasize our paper on the case where the switchings of the observer and controller do not coincide with those of the system.     

Input‐to‐state stability of switched nonlinear systems with delay in the active mode detection

A switched nonlinear system subject to disturbances is considered in this paper. The switching signal admits an average dwell time and a state feedback control depending on the system operating modes, detected with a maximum time delay, is applied to the system. In this framework, the input‐to‐state stability problem of the closed‐loop system is addressed. Based on some established existence conditions of mode‐dependent Lyapunov‐like functions, the values of the maximum time delay and the average dwell time that allow to achieve the input‐to‐state stability of the closed‐loop system are determined. In order to obtain more tractable results, the existence conditions of the mode‐dependent Lyapunov‐like functions are given in terms of sum‐of‐squares programming in the case of polynomial nonlinearities. In the linear case, they are expressed in terms of linear matrix inequalities and a procedure for the synthesis of the mode‐dependent controller is provided in this situation. The established theoretical results are illustrated through a control problem of a building ventilation system and a switched control problem of a vehicle suspension system.     

A clustering-based approach for the identification of a class of temporally switched linear systems

The behaviours of hybrid dynamic systems (HDS) are determined by combining continuous variables with discrete switching logic. The identification of a HDS aims to find an accurate model of the system’s dynamics based on its past inputs and outputs. In pattern recognition (PR) methods, each mode is represented by a set of similar patterns that form restricted regions in the feature space. These sets of patterns are called classes. A pattern is a vector built from past inputs and outputs. HDS identification is a challenging problem since it involves the estimation of different sets of parameters without knowing in advance which sections of the measured data correspond to the different modes of the system. Therefore, HDS identification can be achieved by combining two steps: clustering and parameter estimation. In the clustering step, the number of discrete modes (i.e., the classes that input-output data points belong) is estimated. The parameter estimation step finds the parameters of the models that govern the continuous dynamics in each mode. In this paper, an unsupervised PR method is proposed to achieve the clustering step of the identification of temporally switched linear HDS. The determination of the number of modes does not require prior information about the modes or their number.     

Optimal control of continuous-time switched affine systems

This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the sequence of operating modes. We present two different approaches for solving such an optimal control problem. The first approach iterates between a procedure that finds an optimal switching sequence of modes, and a procedure that finds the optimal switching instants. The second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur, therefore providing a state feedback control law.     

Optimally switched linear systems

In this paper we address the problem of optimal switching for switched linear systems. The uniqueness of our approach lies in describing the switching action by multiple control inputs. This allows us to embed the switched system in a larger family of systems and apply Pontryagin’s Minimum Principle for solving the optimal control problem. This approach imposes no restriction on the switching sequence or the number of switchings. This is in contrast to search based algorithms where a fixed number of switchings is set a priori. In our approach, the optimal solution can be determined by solving the ensuing two-point boundary value problem. Results of numerical simulations are provided to support the proposed method.     

Robust stabilizing control laws for a class of second-order switched systems

For a class of second-order switched systems consisting of two linear time-invariant (LTI) subsystems, we show that the so-called conic switching law proposed previously by the present authors is robust, not only in the sense that the control law is flexible (to be explained further), but also in the sense that the Lyapunov stability (resp., Lagrange stability) properties of the switched system are preserved in the presence of certain kinds of vanishing perturbations (resp., nonvanishing perturbations). The analysis is possible since the conic switching laws always possess certain kinds of “quasi-periodic switching operations”. We also propose for a class of nonlinear second-order switched systems with time-invariant subsystems a switching control law which locally exponentially stabilizes the entire nonlinear switched system, provided that the conic switching law exponentially stabilizes the linearized switched systems (consisting of the linearization of each nonlinear subsystem). This switched control law is robust in the sense mentioned above.     

Stability and stabilization of switched stochastic systems under asynchronous switching

This paper studies the stability and stabilization problems for a class of switched stochastic systems under asynchronous switching. The asynchronous switching refers to that the switching of the candidate controllers does not coincide with the switching of system modes. Two situations are considered: (1) time-delayed switching situation, that is, the switching of the candidate controllers has a lag to the switching of the system modes; (2) mismatched switching situation, the switching of the candidate controllers does not match the switching of the system modes. Using average dwell time and Lyapunov-like function, sufficient conditions are established for stochastic input-to-state stability of the whole system. Also, the stabilizing controller design approach is proposed for switched stochastic linear systems. The minimal average dwell time and the controller gain are achieved. Finally, a numerical example is used to demonstrate the validity of the developed results.     

Optimal control of switched systems via non-linear optimization based on direct differentiations of value functions

This paper presents an approach for solving optimal control problems of switched systems. In general, in such problems one needs to find both optimal continuous inputs and optimal switching sequences, since the system dynamics vary before and after every switching instant. After formulating a general optimal control problem, we propose a two stage optimization methodology. Since many practical problems only concern optimization where the number of switchings and the sequence of active subsystems are given, we concentrate on such problems and propose a method which uses nonlinear optimization and is based on direct differentiations of value functions. The method is then applied to general switched linear quadratic (GSLQ) problems. Examples illustrate the results.