不确定时滞非线性系统的最优滑模控制
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摘要
本文主要研究了不确定性时滞非线性系统的最优控制问题、最优滑模设计问题以及全局鲁棒最优滑模控制问题。
滑模变结构控制(SMC)的突出优点是滑动模态对于匹配的参数不确定性以及外界扰动具有完全的鲁棒性,并且滑动模态的动态品质是可以预先设计的。同时设计方法简单,易于实现。这些特点使其在处理不确定非线性和时滞问题上显示出强大的优越性。滑动模态的优化设计具有重要意义。线性二次型性能指标能够综合地反映系统对性能的要求,而且线性二次型最优控制器设计方法是最优控制的核心内容之一。但是它的设计是基于精确数学模型的,因此在研究最优控制的同时,必须考虑系统的鲁棒性。另外,时滞非线性系统基于二次型性能指标的最优控制问题的求解也是一个难题。
鉴于以上问题,本论文的研究思路是,首先研究时滞非线性系统最优控制律的一种近似解法,然后将其与滑模变结构控制理论相结合,达到以下目的:(1)将最优控制理论用于滑动模态的设计,以优化滑动运动;(2)将滑模变结构控制理论用于最优调节器的设计,使最优控制系统具有滑动模态的鲁棒性。这是一个非常有意义的研究课题,对于不确定时滞非线性系统,这方面的研究成果非常少见。现将本文的主要研究工作概括如下:
1.综述了滑模变结构控制理论的发展和研究现状,对国内外最优滑模控制的研究方法及最新成果进行了总结和分析,提出了存在的问题及研究方向。
2.分别针对一类时滞线性以及时滞非线性系统,研究了其二次型最优控制问题。将由最优控制的必要条件导出的既含时滞项又含超前项的线性或非线性两点边值(TPBV)问题,采用灵敏度法将其转化成一族不含时滞项和超前项的线性两点边值问题;通过递推求解线性两点边值问题族,得到最优控制律的近似解。提出并证明了近似最优控制律作用下闭环系统渐近稳定的充分条件。该部分的研究为后续研究打下理论基础。
3.研究了一类不确定时滞线性系统的最优滑模设计问题。将时滞系统的最优控制原理用于滑模面的构造,优化了滑动运动,并分析了最优滑动模态的稳定性。仿真结果验证了其有效性。
4.研究了一类不确定非线性系统的最优滑模设计问题。针对非线性系统滑动模态难以构造问题,首先通过微分同胚变换将非线性系统转换成标准型,然后把某些变量看作虚拟控制,从而将非线性滑模设计问题转化成非线性系统的最优控制问题。进而运用非线性系统的最优控制原理设计滑模面,使得滑动模态满足给出的最优二次型性能指标的要求,同时滑动模态对于已知上界的不确定性具有完全地鲁棒性。理论分析和仿真结果验证了其有效性。
5.分别针对不确定时滞线性系统和时滞非线性系统,研究了全局鲁棒最优滑模控制器的设计问题。受积分滑模控制可以实现全程滑动模态的思想的启发,提出了一种使最优控制器鲁棒化的设计策略:首先针对具有精确模型的标称系统设计最优控制律,然后考虑系统的不确定性,采用积分滑模控制,设计不连续补偿控制律(鲁棒化补偿控制律),使得系统对于不确定性具有积分滑模控制的全局鲁棒性,同时保持标称系统的最优控制性能。仿真将所提出的方案与最优控制进行了比较,结果充分显示了该方法的有效性以及在保持鲁棒最优性能方面的优越性。
6.总结论文的主要工作,并指出今后的研究方向。
For uncertain time-delay nonlinear systems, the problems of optimal control, optimal sliding mode design and global robust optimal sliding mode control are considered in this dissertation.
One of the most distinguished features of the sliding mode control (SMC) is that after reaching the sliding mode, the system has robustness to matched parameter uncertainties and external disturbances. The system dynamics performance on sliding mode can be pre-designed. And the control algorithm is simple and easy to be realized. All these features of SMC show a great superiority on controlling uncertain time-delay and nonlinear systems. Obviously, the optimization of sliding mode is a significant problem for a SMC system. Linear quadratic (LQ) performance index can synthetically express the requirement of the system performance, and the optimal LQ Regular theory is one of the most important contents of optimal control theory. Nevertheless, the design of optimal controllers is based on exact known models. Therefore it is necessary to take the robustness into account in the study of optimal controller design. In addition, the solution of the optimal control law for time-delay nonlinear systems is difficult to obtain.
Considering the problems above, the basic idea of this study is to find an approximate solution method for time-delay nonlinear systems first, and then to combine it with SMC theory to achieve the following objectives. Firstly, the sliding motion is optimized by applying optimal control technique to designing sliding mode. Secondly, SMC theory is introduced to optimal controller design to put the robustness of sliding mode into conventional optimal systems. This is a very significant subject. About uncertain time-delay nonlinear systems, such research has hardly been found in the literature. The study addresses the following topics:
1. The history of the development in sliding mode control theory is reviewed. The main methods and the latest research tendency are also summarized. The existing problems and research objective is presented.
2. The optimal control problem for a class of time-delay linear systems and time-delay nonlinear systems with quadratic performance indexes is considered, respectively. According to the necessary optimality conditions, the linear and nonlinear two-point boundary value (TPBV) problems with both time-delay and time-advance terms are induced, respectively. A sensitivity approach is adopted to convert the original TPBV problems into that of solving a series of linear TPBV iterative formulas without time-delay and time-advance variable terms. By the finite-step iteration, approximate solutions for the optimal control are obtained. Sufficient asymptotic stability conditions of the closed-loop systems are derived both for linear and nonlinear systems with time-delay. This part provides the foundation for the later study.
3. The problem of designing optimal sliding manifolds with a quadratic performance index for a class of uncertain systems with state time-delay is considered. The optimal control technique is employed to construct sliding manifold, so the sliding motion is optimized. The stability of the optimal sliding motion is analyzed. Simulation results demonstrate the efficiency of the proposed method.
4. The problem of designing nonlinear sliding manifolds with a quadratic performance index for a class of nonlinear systems is considered. As we know, it is difficult to choose an appropriate nonlinear sliding manifold on which the sliding motion has desired performance. To solve this problem, firstly, a nonsingular diffeomorphism transformation is used to convert the nonlinear system into a regular form. Then by viewing the some state variables as virtual control, the problem of optimal sliding mode design is transformed into optimal control problem for nonlinear systems. Furthermore, the optimal control theory is adopted to construct nonlinear sliding manifold. The dynamics of sliding motion could minimize the given quadratic performance index and is robust to uncertainties with known upper bounds.
5. For a class of uncertain linear systems and nonlinear systems with time delays, the problem of designing global robust optimal sliding mode controllers (GROSMC) is discussed. Inspired by the concept of integral sliding mode(ISM), which can ensure the global sliding mode during the entire response of the system, a design strategy which can robustify the optimal controllers is presented. First, the optimal controller is obtained for the nominal system ignoring uncertainties using optimal theory. Then based on ISM, a discontinuous compensation control law (also called robustification control law) is designed to suppress the uncertainties from the initial time moment. As a result, not only the optimal performance can be obtained but the global robustness is guaranteed also. Simulations are given to compare the proposed GROSMC with the conventional optimal control. Results show the effectiveness and superiority in robustness of the proposed method.
6. The conclusions and the directions for the future research work are given in the end of the paper.
滑模变结构控制(SMC)的突出优点是滑动模态对于匹配的参数不确定性以及外界扰动具有完全的鲁棒性,并且滑动模态的动态品质是可以预先设计的。同时设计方法简单,易于实现。这些特点使其在处理不确定非线性和时滞问题上显示出强大的优越性。滑动模态的优化设计具有重要意义。线性二次型性能指标能够综合地反映系统对性能的要求,而且线性二次型最优控制器设计方法是最优控制的核心内容之一。但是它的设计是基于精确数学模型的,因此在研究最优控制的同时,必须考虑系统的鲁棒性。另外,时滞非线性系统基于二次型性能指标的最优控制问题的求解也是一个难题。
鉴于以上问题,本论文的研究思路是,首先研究时滞非线性系统最优控制律的一种近似解法,然后将其与滑模变结构控制理论相结合,达到以下目的:(1)将最优控制理论用于滑动模态的设计,以优化滑动运动;(2)将滑模变结构控制理论用于最优调节器的设计,使最优控制系统具有滑动模态的鲁棒性。这是一个非常有意义的研究课题,对于不确定时滞非线性系统,这方面的研究成果非常少见。现将本文的主要研究工作概括如下:
1.综述了滑模变结构控制理论的发展和研究现状,对国内外最优滑模控制的研究方法及最新成果进行了总结和分析,提出了存在的问题及研究方向。
2.分别针对一类时滞线性以及时滞非线性系统,研究了其二次型最优控制问题。将由最优控制的必要条件导出的既含时滞项又含超前项的线性或非线性两点边值(TPBV)问题,采用灵敏度法将其转化成一族不含时滞项和超前项的线性两点边值问题;通过递推求解线性两点边值问题族,得到最优控制律的近似解。提出并证明了近似最优控制律作用下闭环系统渐近稳定的充分条件。该部分的研究为后续研究打下理论基础。
3.研究了一类不确定时滞线性系统的最优滑模设计问题。将时滞系统的最优控制原理用于滑模面的构造,优化了滑动运动,并分析了最优滑动模态的稳定性。仿真结果验证了其有效性。
4.研究了一类不确定非线性系统的最优滑模设计问题。针对非线性系统滑动模态难以构造问题,首先通过微分同胚变换将非线性系统转换成标准型,然后把某些变量看作虚拟控制,从而将非线性滑模设计问题转化成非线性系统的最优控制问题。进而运用非线性系统的最优控制原理设计滑模面,使得滑动模态满足给出的最优二次型性能指标的要求,同时滑动模态对于已知上界的不确定性具有完全地鲁棒性。理论分析和仿真结果验证了其有效性。
5.分别针对不确定时滞线性系统和时滞非线性系统,研究了全局鲁棒最优滑模控制器的设计问题。受积分滑模控制可以实现全程滑动模态的思想的启发,提出了一种使最优控制器鲁棒化的设计策略:首先针对具有精确模型的标称系统设计最优控制律,然后考虑系统的不确定性,采用积分滑模控制,设计不连续补偿控制律(鲁棒化补偿控制律),使得系统对于不确定性具有积分滑模控制的全局鲁棒性,同时保持标称系统的最优控制性能。仿真将所提出的方案与最优控制进行了比较,结果充分显示了该方法的有效性以及在保持鲁棒最优性能方面的优越性。
6.总结论文的主要工作,并指出今后的研究方向。
For uncertain time-delay nonlinear systems, the problems of optimal control, optimal sliding mode design and global robust optimal sliding mode control are considered in this dissertation.
One of the most distinguished features of the sliding mode control (SMC) is that after reaching the sliding mode, the system has robustness to matched parameter uncertainties and external disturbances. The system dynamics performance on sliding mode can be pre-designed. And the control algorithm is simple and easy to be realized. All these features of SMC show a great superiority on controlling uncertain time-delay and nonlinear systems. Obviously, the optimization of sliding mode is a significant problem for a SMC system. Linear quadratic (LQ) performance index can synthetically express the requirement of the system performance, and the optimal LQ Regular theory is one of the most important contents of optimal control theory. Nevertheless, the design of optimal controllers is based on exact known models. Therefore it is necessary to take the robustness into account in the study of optimal controller design. In addition, the solution of the optimal control law for time-delay nonlinear systems is difficult to obtain.
Considering the problems above, the basic idea of this study is to find an approximate solution method for time-delay nonlinear systems first, and then to combine it with SMC theory to achieve the following objectives. Firstly, the sliding motion is optimized by applying optimal control technique to designing sliding mode. Secondly, SMC theory is introduced to optimal controller design to put the robustness of sliding mode into conventional optimal systems. This is a very significant subject. About uncertain time-delay nonlinear systems, such research has hardly been found in the literature. The study addresses the following topics:
1. The history of the development in sliding mode control theory is reviewed. The main methods and the latest research tendency are also summarized. The existing problems and research objective is presented.
2. The optimal control problem for a class of time-delay linear systems and time-delay nonlinear systems with quadratic performance indexes is considered, respectively. According to the necessary optimality conditions, the linear and nonlinear two-point boundary value (TPBV) problems with both time-delay and time-advance terms are induced, respectively. A sensitivity approach is adopted to convert the original TPBV problems into that of solving a series of linear TPBV iterative formulas without time-delay and time-advance variable terms. By the finite-step iteration, approximate solutions for the optimal control are obtained. Sufficient asymptotic stability conditions of the closed-loop systems are derived both for linear and nonlinear systems with time-delay. This part provides the foundation for the later study.
3. The problem of designing optimal sliding manifolds with a quadratic performance index for a class of uncertain systems with state time-delay is considered. The optimal control technique is employed to construct sliding manifold, so the sliding motion is optimized. The stability of the optimal sliding motion is analyzed. Simulation results demonstrate the efficiency of the proposed method.
4. The problem of designing nonlinear sliding manifolds with a quadratic performance index for a class of nonlinear systems is considered. As we know, it is difficult to choose an appropriate nonlinear sliding manifold on which the sliding motion has desired performance. To solve this problem, firstly, a nonsingular diffeomorphism transformation is used to convert the nonlinear system into a regular form. Then by viewing the some state variables as virtual control, the problem of optimal sliding mode design is transformed into optimal control problem for nonlinear systems. Furthermore, the optimal control theory is adopted to construct nonlinear sliding manifold. The dynamics of sliding motion could minimize the given quadratic performance index and is robust to uncertainties with known upper bounds.
5. For a class of uncertain linear systems and nonlinear systems with time delays, the problem of designing global robust optimal sliding mode controllers (GROSMC) is discussed. Inspired by the concept of integral sliding mode(ISM), which can ensure the global sliding mode during the entire response of the system, a design strategy which can robustify the optimal controllers is presented. First, the optimal controller is obtained for the nominal system ignoring uncertainties using optimal theory. Then based on ISM, a discontinuous compensation control law (also called robustification control law) is designed to suppress the uncertainties from the initial time moment. As a result, not only the optimal performance can be obtained but the global robustness is guaranteed also. Simulations are given to compare the proposed GROSMC with the conventional optimal control. Results show the effectiveness and superiority in robustness of the proposed method.
6. The conclusions and the directions for the future research work are given in the end of the paper.
引文
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