离散时间切换系统控制方法研究
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摘要
在控制工程中，存在着具有模式切换特性的系统，因此，切换系统的控制问题应运而生，其研究具有重要的理论意义和工程应用价值。
本文针对一类离散模式切换由外部离散扰动输入事件触发，且连续动态系统用离散时间差分方程描述的切换系统展开讨论，分别对其状态反馈和输出反馈控制问题进行了深入研究。在控制器的设计过程当中，考虑了影响系统动态的参数不确定性、外部干扰、非线性以及多时域约束和性能要求等因素，并将所提出的理论方法应用到汽车发动机怠速控制问题中，探讨了本文提出的理论方法在实际问题中的有效性。
在系统受到状态和控制量约束的情况下，研究了线性切换系统的状态反馈镇定问题。基于正定不变性的概念，提出了一种状态反馈控制器设计方法。主要思想是将状态约束集设计成为闭环系统的一个受控不变集合，并将其设计在控制量的线性未饱和区域中。在上述给出的镇定控制器当中，选取出使得闭环系统具有较快收敛速度的控制器具有重要意义。通过构造与闭环系统收敛速度对应的一个等价条件，将最优镇定控制器的选取问题转化为受约束的优化问题，易于实现。本章设计的控制器在满足约束的同时，能够保证闭环系统的稳定性，且使得闭环系统具有较快的收敛速度。
实际系统往往受到不确定性的影响，如不考虑这些因素，将难以实现期望的控制目标。针对具有参数不确定性或外部连续干扰情况下的线性切换系统，分别研究了状态反馈镇定问题。针对具有参数不确定性的情况，建立了系统鲁棒稳定性与存在鲁棒受控不变集合的关系，并将鲁棒受控不变集合的判定转化为可通过有限步计算实现的检测问题。进而，基于扩展的正定不变方法，给出了判定某一能够实现镇定名义系统的状态反馈控制器是否具有鲁棒性的检测方法，获得鲁棒镇定控制器。针对具有外部连续干扰的情况，为了保证受扰系统的ISS (Input-to-State Stability)稳定性，引入了受控D不变性的概念。采用类似的研究思路，给出了一种满足系统约束且使得闭环系统ISS稳定的状态反馈控制器设计方法。本章提出的控制器形式简单，可以通过在有限个点处的验证或对有限个不等式的求解而获得。
系统的状态并不是都可测量的，根据系统的测量输出，设计输出反馈控制器保证闭环系统的稳定性是一个重要的控制问题。针对线性切换系统，考虑到最小驻留时间特性，本章提出了保证稳定性的一种新的条件，即存在δm-受控不变集。在考虑椭球体描述集合情况下，给出了这一集合易于求解的判定条件。以此为基础，当线性切换系统的连续状态不可测量时，分别提出了基于连续状态观测器的输出反馈和具有一般形式的输出反馈控制器设计方法。本章提出的控制器设计方法均可以通过LMI (Linear Matrix Inequality)的可行性问题而给出，易于计算获得。
在实际系统中，非线性动态广泛存在，考虑系统的非线性动态更具有一般性。针对一类具有Lipschitz-like非线性的切换系统，分别研究了全状态反馈控制器、连续状态观测器及基于连续状态观测器的输出反馈控制器设计问题。主要思想是构造闭环系统的不同形式的混杂Lyapunov函数，使得其沿着闭环系统的执行是不增的以保证稳定性。在LMI理论框架下，将控制器和观测器的设计转化为LMI的可行性问题。本章给出的控制器及观测器设计方法不依赖于具体的非线性动态形式，只与其Lipschitz-like常值矩阵有关，易于求解。
针对具有自环模式跃迁及受控和仿射项重置函数的约束切换系统，在受到参数不确定性及外部连续扰动影响下，研究了系统输出的l∞性能问题，提出了保证系统具有最小l∞输出性能的控制器设计方法。主要思想是将求取系统的输出性能问题转化为系统状态的安全性控制问题，通过设计保证安全性的最大安全集合及其控制律，并结合二分法，获得最小的l∞输出性能。同时，将本章的理论方法应用于汽车发动机怠速控制器设计中。针对四缸四冲程火花塞点火直列式发动机，在考虑外部负载力矩及大气温度变化的影响下，建立了发动机怠速工况下的不确定切换模型，并将发动机怠速要求转化为本章切换系统的l∞输出性能问题，给出了发动机怠速控制器的设计方法。通过在不同情况下的仿真表明，设计出的怠速控制器使得发动机怠速具有良好的控制性能。
In control engineering, there exist control systems which often have the character-istic of mode transitions, therefore, the control problems of switching systems are takeninto consideration, its research has both theoretical significance and engineering value.
In this thesis, a class of switching systems is discussed, whose discrete mode tran-sitions are caused by discrete disturbance events and the continuous dynamical systemis described by discrete-time difference equation, the state-feedback control and output-feedback control problems are investigated deeply. During the controller designs, thefactors which affect the dynamics of switching systems are considered, such as the pa-rameter uncertainty、disturbances、nonlinear dynamics、time-domain satisfaction andthe required performances. Accordingly, the feasibility of applying the proposed methodsto idle speed control problem (ISCP) is discussed.
In the presence of state and control input bounds, the state-feedback stabilizing prob-lem of linear switching systems is researched. Based on the concept of positive invariance,a state-feedback controller design methodology is proposed. The basic idea of this methodis to design the state constraint set as a controlled invariant one for closed-loop systems,and guarantee the state constraint set contained in a polyhedral domain of linear behaviorwhere the control input does not saturate. Among the stabilizing controllers above, it isof great importance to select the controller which can guarantee the closed-loop switch-ing systems have the optimal convergence rate. An equivalent condition which impliesthe closed-loop systems have the optimal convergence rate is proposed, then the selectionof optimal stabilizing controller is converted to an optimization problem, which is easyto implement. The proposed controller design method in this chapter can guarantee thestability of systems, and the closed-loop switching systems have the optimal convergencerate, also the constraints on state and control input are satisfied.
In practice, the control systems always suffer uncertainties, if these influences are nottaken into account, the expected control objectives cannot be achieved. For linear switch-ing systems with parameter uncertainty or continuous disturbances, the state-feedbackstabilizing problems are discussed. In the case of parameter uncertainty, the relation be-tween robust stability of systems and the existence of robust controlled invariant set is given, and the condition which can test the robust controlled invariance is converted toa problem that can be judged in finite-step calculations. Then, based on the extendedpositive invariant approach, a condition to test the robustness of a state feedback con-troller which can stabilize the nominal linear switching systems is proposed, and obtainthe robust stabilizing controller. In the presence of continuous disturbances, in order toguarantee the ISS (Input-to-State Stability) of systems, the concept of controlled D invari-ance is introduced. Taking the similar idea, a procedure which can stabilize the disturbedlinear switching systems and also satisfy the state and control input constraints, is given.The form of controller in this chapter is simple, and can be obtained from the judgment ata finite number of points or solving of a finite number of inequalities.
Usually, the state of systems cannot be directly measured, then it is an importantcontrol problem to design output feedback controller by using the measured output ofsystems to guarantee the stability of closed-loop systems. For linear switching systems,taking into account the concept of the minimum dwell time, a new condition of stabilityis presented, which is the existence of δm-controlled invariant set. In the case of invariantellipsoids, existence conditions to obtain such a set are given. Then, under the assumptionthat the continuous state is unavailable for feedback, this result is used to find the observer-based output feedback controller and ordinary output feedback controller design methods.The proposed approaches in this chapter can be obtained from the feasible problem ofLMI, and easy to calculate.
In practice, the nonlinear dynamics is widespread existence, it is practically impor-tant to consider the effects of nonlinear terms in the dynamical system. For switching sys-tems with Lipschitz-like nonlinear term, full-state feedback controller design、continuousstate observer design and observer-based output feedback controller design problems areconsidered. The basic idea of the proposed approaches is to construct the different types ofHybrid Lyapunov function which are not increasing to guarantee the stability of closed-loop systems. In the framework of LMI, all of those methods are given in the form offeasible problem of LMI. The results obtained in this chapter are only dependent on theLipschitz-like constant matrices without regard to the specific nonlinear forms in switch-ing systems, easy to implement.
For the constrained switching systems with in-loop transitions and controlled affine-resets, in the presence of parameter variations and additive disturbances, the control prob- lem of determining the non-conservative bounds on the l∞performance of the controlledoutput is considered, and the controller design methodology which can obtain the minimall∞performance is proposed. The basic idea is to translate the required level of perfor-mance into the safety problem of systems, then the algorithm for maximal safe set andcontroller design methodology are given, together with the bisection method, the minimall∞performance can be obtained. Meanwhile, this method is applied to controller designof ISCP. For the4-cylinder4-stroke spark ignition in-line engine in idle mode, taking intoaccount the effect of disturbance torque and air temperature, a reduced switching model ofISCP is given, also, the control problem of ISCP is considered as the minimal l∞perfor-mance problem of switching systems in this chapter, then a controller design technologyof ISCP is presented. Different simulation results show that the controller of ISCP has agood control performances.
本文针对一类离散模式切换由外部离散扰动输入事件触发，且连续动态系统用离散时间差分方程描述的切换系统展开讨论，分别对其状态反馈和输出反馈控制问题进行了深入研究。在控制器的设计过程当中，考虑了影响系统动态的参数不确定性、外部干扰、非线性以及多时域约束和性能要求等因素，并将所提出的理论方法应用到汽车发动机怠速控制问题中，探讨了本文提出的理论方法在实际问题中的有效性。
在系统受到状态和控制量约束的情况下，研究了线性切换系统的状态反馈镇定问题。基于正定不变性的概念，提出了一种状态反馈控制器设计方法。主要思想是将状态约束集设计成为闭环系统的一个受控不变集合，并将其设计在控制量的线性未饱和区域中。在上述给出的镇定控制器当中，选取出使得闭环系统具有较快收敛速度的控制器具有重要意义。通过构造与闭环系统收敛速度对应的一个等价条件，将最优镇定控制器的选取问题转化为受约束的优化问题，易于实现。本章设计的控制器在满足约束的同时，能够保证闭环系统的稳定性，且使得闭环系统具有较快的收敛速度。
实际系统往往受到不确定性的影响，如不考虑这些因素，将难以实现期望的控制目标。针对具有参数不确定性或外部连续干扰情况下的线性切换系统，分别研究了状态反馈镇定问题。针对具有参数不确定性的情况，建立了系统鲁棒稳定性与存在鲁棒受控不变集合的关系，并将鲁棒受控不变集合的判定转化为可通过有限步计算实现的检测问题。进而，基于扩展的正定不变方法，给出了判定某一能够实现镇定名义系统的状态反馈控制器是否具有鲁棒性的检测方法，获得鲁棒镇定控制器。针对具有外部连续干扰的情况，为了保证受扰系统的ISS (Input-to-State Stability)稳定性，引入了受控D不变性的概念。采用类似的研究思路，给出了一种满足系统约束且使得闭环系统ISS稳定的状态反馈控制器设计方法。本章提出的控制器形式简单，可以通过在有限个点处的验证或对有限个不等式的求解而获得。
系统的状态并不是都可测量的，根据系统的测量输出，设计输出反馈控制器保证闭环系统的稳定性是一个重要的控制问题。针对线性切换系统，考虑到最小驻留时间特性，本章提出了保证稳定性的一种新的条件，即存在δm-受控不变集。在考虑椭球体描述集合情况下，给出了这一集合易于求解的判定条件。以此为基础，当线性切换系统的连续状态不可测量时，分别提出了基于连续状态观测器的输出反馈和具有一般形式的输出反馈控制器设计方法。本章提出的控制器设计方法均可以通过LMI (Linear Matrix Inequality)的可行性问题而给出，易于计算获得。
在实际系统中，非线性动态广泛存在，考虑系统的非线性动态更具有一般性。针对一类具有Lipschitz-like非线性的切换系统，分别研究了全状态反馈控制器、连续状态观测器及基于连续状态观测器的输出反馈控制器设计问题。主要思想是构造闭环系统的不同形式的混杂Lyapunov函数，使得其沿着闭环系统的执行是不增的以保证稳定性。在LMI理论框架下，将控制器和观测器的设计转化为LMI的可行性问题。本章给出的控制器及观测器设计方法不依赖于具体的非线性动态形式，只与其Lipschitz-like常值矩阵有关，易于求解。
针对具有自环模式跃迁及受控和仿射项重置函数的约束切换系统，在受到参数不确定性及外部连续扰动影响下，研究了系统输出的l∞性能问题，提出了保证系统具有最小l∞输出性能的控制器设计方法。主要思想是将求取系统的输出性能问题转化为系统状态的安全性控制问题，通过设计保证安全性的最大安全集合及其控制律，并结合二分法，获得最小的l∞输出性能。同时，将本章的理论方法应用于汽车发动机怠速控制器设计中。针对四缸四冲程火花塞点火直列式发动机，在考虑外部负载力矩及大气温度变化的影响下，建立了发动机怠速工况下的不确定切换模型，并将发动机怠速要求转化为本章切换系统的l∞输出性能问题，给出了发动机怠速控制器的设计方法。通过在不同情况下的仿真表明，设计出的怠速控制器使得发动机怠速具有良好的控制性能。
In control engineering, there exist control systems which often have the character-istic of mode transitions, therefore, the control problems of switching systems are takeninto consideration, its research has both theoretical significance and engineering value.
In this thesis, a class of switching systems is discussed, whose discrete mode tran-sitions are caused by discrete disturbance events and the continuous dynamical systemis described by discrete-time difference equation, the state-feedback control and output-feedback control problems are investigated deeply. During the controller designs, thefactors which affect the dynamics of switching systems are considered, such as the pa-rameter uncertainty、disturbances、nonlinear dynamics、time-domain satisfaction andthe required performances. Accordingly, the feasibility of applying the proposed methodsto idle speed control problem (ISCP) is discussed.
In the presence of state and control input bounds, the state-feedback stabilizing prob-lem of linear switching systems is researched. Based on the concept of positive invariance,a state-feedback controller design methodology is proposed. The basic idea of this methodis to design the state constraint set as a controlled invariant one for closed-loop systems,and guarantee the state constraint set contained in a polyhedral domain of linear behaviorwhere the control input does not saturate. Among the stabilizing controllers above, it isof great importance to select the controller which can guarantee the closed-loop switch-ing systems have the optimal convergence rate. An equivalent condition which impliesthe closed-loop systems have the optimal convergence rate is proposed, then the selectionof optimal stabilizing controller is converted to an optimization problem, which is easyto implement. The proposed controller design method in this chapter can guarantee thestability of systems, and the closed-loop switching systems have the optimal convergencerate, also the constraints on state and control input are satisfied.
In practice, the control systems always suffer uncertainties, if these influences are nottaken into account, the expected control objectives cannot be achieved. For linear switch-ing systems with parameter uncertainty or continuous disturbances, the state-feedbackstabilizing problems are discussed. In the case of parameter uncertainty, the relation be-tween robust stability of systems and the existence of robust controlled invariant set is given, and the condition which can test the robust controlled invariance is converted toa problem that can be judged in finite-step calculations. Then, based on the extendedpositive invariant approach, a condition to test the robustness of a state feedback con-troller which can stabilize the nominal linear switching systems is proposed, and obtainthe robust stabilizing controller. In the presence of continuous disturbances, in order toguarantee the ISS (Input-to-State Stability) of systems, the concept of controlled D invari-ance is introduced. Taking the similar idea, a procedure which can stabilize the disturbedlinear switching systems and also satisfy the state and control input constraints, is given.The form of controller in this chapter is simple, and can be obtained from the judgment ata finite number of points or solving of a finite number of inequalities.
Usually, the state of systems cannot be directly measured, then it is an importantcontrol problem to design output feedback controller by using the measured output ofsystems to guarantee the stability of closed-loop systems. For linear switching systems,taking into account the concept of the minimum dwell time, a new condition of stabilityis presented, which is the existence of δm-controlled invariant set. In the case of invariantellipsoids, existence conditions to obtain such a set are given. Then, under the assumptionthat the continuous state is unavailable for feedback, this result is used to find the observer-based output feedback controller and ordinary output feedback controller design methods.The proposed approaches in this chapter can be obtained from the feasible problem ofLMI, and easy to calculate.
In practice, the nonlinear dynamics is widespread existence, it is practically impor-tant to consider the effects of nonlinear terms in the dynamical system. For switching sys-tems with Lipschitz-like nonlinear term, full-state feedback controller design、continuousstate observer design and observer-based output feedback controller design problems areconsidered. The basic idea of the proposed approaches is to construct the different types ofHybrid Lyapunov function which are not increasing to guarantee the stability of closed-loop systems. In the framework of LMI, all of those methods are given in the form offeasible problem of LMI. The results obtained in this chapter are only dependent on theLipschitz-like constant matrices without regard to the specific nonlinear forms in switch-ing systems, easy to implement.
For the constrained switching systems with in-loop transitions and controlled affine-resets, in the presence of parameter variations and additive disturbances, the control prob- lem of determining the non-conservative bounds on the l∞performance of the controlledoutput is considered, and the controller design methodology which can obtain the minimall∞performance is proposed. The basic idea is to translate the required level of perfor-mance into the safety problem of systems, then the algorithm for maximal safe set andcontroller design methodology are given, together with the bisection method, the minimall∞performance can be obtained. Meanwhile, this method is applied to controller designof ISCP. For the4-cylinder4-stroke spark ignition in-line engine in idle mode, taking intoaccount the effect of disturbance torque and air temperature, a reduced switching model ofISCP is given, also, the control problem of ISCP is considered as the minimal l∞perfor-mance problem of switching systems in this chapter, then a controller design technologyof ISCP is presented. Different simulation results show that the controller of ISCP has agood control performances.
引文
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