摘要
计算机控制是最典型的离散控制系统。近几十年来,随着科学技术的高速发展,尤其是计算机和信息学科的飞速发展,极大地推动了微处理机和微型计算机在控制系统中的应用。一方面,高效微处理器的出现使得时滞非线性控制器的应用变得简单;另一方面,现代技术要求控制系统有更严格的设计规定。因此离散系统的分析与设计已经成为控制理论的一个重要组成部分。
时滞非线性系统的最优控制问题普遍存在于生命科学、工程科学、计算机科学及经济学等学科,它是以时滞非线性动力系统为约束的泛函优化问题,对这类问题的数值优化理论与算法的研究,不仅是非线性领域的前沿课题,也是控制论与其他学科交叉发展的前沿课题。目前该领域的主要成果集中在定性理论研究,极需实用、有效的数值优化理论与算法。
本文首先综述了国内外非线性、时滞系统最优控制问题的研究现状。然后分别研究了离散非线性系统、离散状态时滞非线性系统以及离散控制时滞非线性系统的最优控制问题。在此基础上,针对一类参考输入信号由外系统给定的离散多时滞线性系统和离散控制时滞非线性系统,深入地研究了其最优输出跟踪问题。给出了系统控制律的存在唯一性条件及其近似设计方法,并讨论了其物理可实现问题。全文主要研究内容如下:
1.首先研究了具有一般形式的离散非线性系统的基于二次型性能指标最优控制问题。对于非线性系统的一般形式,可以通过Maclaurin展开,将系统转化为具有线性项和非线性项的形式。其次,将系统的非线性项当作系统的附加扰动,通过引入一个伴随向量对非线性项加以补偿,可以将由极大值原理的必要条件导出的非线性两点边值问题变换为由伴随向量方程和状态方程组成的新的两点边值问题。通过逐次逼近法将此问题转化为一族解耦的线性非齐次两点边值问题序列,从而通过求解该问题序列得到系统的前馈-反馈次优控制律。证明了该线性两点边值问题的解序列一致收敛于原最优控制问题的解。通过有限次迭代,可以得到系统的近似的最优控制律。得到的组合控制律由反馈项和以非线性补偿向量的极限形式表示的前馈项组成。最后,给出并证明了无限时域情形下系统控制律的存在唯一性条件,并给出了多个仿真实例来验证效果。
2.研究了状态变量含有时滞的离散非线性系统,提出一种次优控制律的无时滞转换设计方案。首先构造一个其解收敛于原时滞系统的无时滞离散系统序列。然后将离散时滞非系统的最优控制问题化为求解无时滞系统最优控制序列问题。采用所提出的逐次逼近法对系统进行解耦,通过截取最优控制序列解的有限项,从而得到系统的前馈-反馈次优控制律。实例仿真表明,即使对于时滞较大的离散时间系统也能取得了良好效果。
3.研究了控制变量含有时滞及系统带有非线性项的离散非线性系统的最优控制问题,引入连续系统的Artstein模型化简方法对控制时滞进行处理,将原系统状态方程转化为无时滞的非线性系统,从而采用前述的逐次逼近法设计了该系统的近似最优控制器并通过仿真验证了该方法的有效性。
4.解决了对于系统的参考输入信号的动态特性有外系统描述的具有多状态时滞和控制时滞的离散线性系统最优输出跟踪控制问题。利用参考输入外系统的状态来构造前馈控制作用,前馈增益可以通过求解Stein矩阵方程而精确得到,避免了利用构造增广系统的方法带来的维数增高,计算复杂的困难。同时,对前馈的参考输入外系统的状态物理不可实现问题,通过构造降维观测器加以实现。通过逐次逼近法解决了既含有时间超前项又有时间滞后项的两点边值问题的求解,给出了无限时域下系统组合最优控制律的存在唯一性条件及其证明。仿真实例证明,得到的近似最优控制律能有效地补偿时滞的影响并跟踪参考输入。
5.研究了含有控制时滞和非线性项的离散时间系统的最优输出跟踪问题。由于系统的输出方程经过转化后具有多控制时滞项,因此通过引入时滞及非线性补偿项,来补偿系统的输出方程中存在的控制时滞以及转化的系统的中的非线性项,通过逐次逼近法解决了具有时滞项和非线性项的两点边值问题,得到了可将线性部分项与非线性部分及时滞部分相分离的非线性系统的最优输出跟踪控制律。最后通过仿真验证了该方法的有效性。
最后部分总结了论文的主要工作,并对今后进一步的研究工作进行了展望。
It is well known that computer control is the most typical discrete-time control system. In recent years, the rapid development of computer technology has greatly facilitated the applications of microprocessors and microcomputers in control systems. On the one hand, the emergences of the high efficiency microprocessor make the application of the nonlinear time-delay controller become simple. On the other hand, the modern techniques request stricter controllers. Thus the analysis and synthesis of the discrete-time systems have become an important component in the control theory.
The optimal control of nonlinear time-delay systems exists in life science, engineering science, computer science and economics etc. It is an optimal functional problems which subject to nonlinear time-delay dynamic systems constraints. The research of numerical optimization methods and algorithms in this field is not only the front subjects of nonlinear control, but also the leading edges of cross subjects. At present the results are mostly focus on the qualitative analysis, and it is imperious to study the numerical optimization methods and algorithms.
The dissertation first reviews the relative studies on the optimal control and optimal output tracking control (OOTC) problem for nonlinear time-delay systems up to now in detail. The latest research tendency and the main methods are also introduced. The major results of this dissertation are summarized as follows.
1. The optimal control problem for discrete-time nonlinear systems with a quadratic performance index is studied. A successive approximation approach (SAA) is proposed to find a solution sequence of the nonlinear two-point boundary value (TPBV) problem, which is obtained from the necessary optimality conditions. We take the nonlinear term as an additional disturbance of the system and turn the system model into an iterative form. By introducing a nonlinear compensation vector and using the SAA, we transform the nonlinear TPBV problem into a sequence of nonhomogeneous linear TPBV problems. By taking the finite iterative value, we obtain a suboptimal control law. The conditions of existence and uniqueness of the optimal control law are presented for infinite-time horizon problems. Simulation examples are employed to test the validity of the optimal control algorithm.
2. The optimal control problem of finite-time and infinite-time nonlinear discrete systems with state delay is developed. A sequence of non-delay discrete systems is constructed, which uniformly converges to the original discrete system with time-delay. Then the optimal control for the original discrete nonlinear system is transformed into an optimal control sequence for non-delay linear systems. By truncating a finite term of the optimal sequence, a suboptimal control law is obtained. The suboptimal laws consist of linear analytic terms and a time-delay compensation term, and the time-delay compensation term is described by a limit of the solution sequence of the adjoint state vector equations. Simulations show the algorithm has lower computation complexity and can be easily implemented. Moreover, even there is large delay in the systems, the accuracy of the optimal solution and the computation speed is satisfied.
3. Systems subject to an input delay or measurement delay are more common. The so-called“Artstein model reduction”is often involved when one considers systems with input delay. By introducing the new variable the original system is reduced to a system free-of-delay. The optimal control problem of a class of discrete-time nonliear system with input delay is considered based on the thoughts of Artstein model reduction of the continuous systems. Then the optimal controller is designed by SAA.
4. The OOTC problem of the discrete-time system with multiple state and input delays whose reference input is generally produced by an exosystem is addressed. The state of the exosystem is introduced into the feedforward control, instead of constructing an augment system as the classical optimal control theory. Thus the tracking error can be reduced and the feedback control effort can be decreased. The SAA is then applied to the OOTC problem. The existence and uniqueness of the optimal control law in infinite-time horizon is proved and the detail design process is proposed. The obtained optimal control laws consist of linear analytic terms and time-delay compensation terms. The linear analytic terms can be found by solving a Riccati equation and a Stein equation respectively, and the time-delay compensation term is described by the limit of the solution sequence of the adjoint state vector. In this case, the OOTC law obtained contains the state variable of the Exoystem which is unrealizable in physical. In order to solve this problem, a reduced observer is introduced.
5. The OOTC problem of the discrete-time system with input delays is addressed. Firstly the system is transformed into a nonlinear system without delay in state equations by using the model reduction in discrete-time systems. Then an adjoint vector is introduced to compensate the input delays in output equation and the nonlinear part in state equation. Using SAA, the original nonlinear TPBV problem is transformed into a sequence of nonhomogeneous linear TPBV problems without unknown time-delay terms and nonlinear terms and then an approximate OOTC law is obtained. Simulation results show that the proposed algorithm is effective and has better convergence properties at different time-delays.
Finally, the conclusions are given, and a proposition is indicated on the research work in the future.
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