约束非线性预测控制算法及其鲁棒稳定性研究
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摘要
约束非线性系统的优化控制正逐渐成为非线性控制领域内的研究热点。模型预测控制可以显式地处理系统约束和目标优化,是对约束系统进行优化控制的最有效的先进控制技术之一。然而,对于具有强非线性、强扰动或大范围操作的系统,线性预测控制或基于工作点附近线性化模型的预测控制常常无法满足系统控制品质的要求。因此,研究约束非线性模型预测控制具有重要的理论意义和应用价值。
由于非线性系统的复杂多样,非线性预测控制研究无论在理论上还是应用上都比线性预测控制落后,特别是在稳定性、鲁棒性和在线优化计算量方面一直是预测控制和优化算法研究者们所共同关注的热点问题。本文在非线性预测控制理论已有的研究成果基础上,利用微分对策理论、(鲁棒)不变集理论、输入状态稳定性、控制Lyapunov函数、反步设计等方法和概念,对有约束的非线性系统,研究非线性预测控制的优化可行性、稳定性、鲁棒性、计算量和稳定域估计等5个基本问题,以期得到更具实际应用意义的理论性成果和控制算法设计。本文的主要贡献由以下几个部分组成:
(1)考虑约束不确定离散时间非线性系统,对非线性预测控制进行minimax鲁棒次优算法设计,以期从策略上降低鲁棒非线性预测控制的在线计算量。引入微分对策理论和输入状态稳定性概念,证明该次优控制算法在向量范数意义下的闭环鲁棒稳定性。
(2)以约束不确定离散时间非线性仿射系统为对象,系统地研究H_∞非线性预测控制的扰动抑制问题和实现问题。利用(1)的结论分别研究H_∞预测控制在最优解和次优解存在下的鲁棒稳定性问题。通过引入非线性系统L_(2-)增益概念,定量地研究H_∞预测控制在信号范数意义下的扰动抑制问题。为了减轻非线性预测控制在线优化的计算量,在算法的实现中引入了有限维参数化设计。
(3)以连续时间约束非线性系统为对象,提出一种新的具有闭环稳定的非线性预测控制——构造性非线性模型预测控制。算法的主要思想是通过构造性设计离线得到预测控制的一个稳定控制类,再根据目标函数在线优化其中的可调参数,以期降低非线性预测控制的在线计算量,同时实现闭环系统稳定性与性能指标优化的分离。
(4)基于(3)中提出的非线性预测控制策略,利用反步设计法构造约束严格反馈非线性系统的稳定控制类,实现约束非线性模型预测控制设计。最后将该非线性预测控制算法用于非完整轮式移动机器人的优化控制。
Optimization control of constrained nonlinear systems has increasingly been a hot topic of research in the field of nonlinear control. Model predictive control (MPC) has an ability to deal with system constraints and the optimization of costs explicitly, and then is one of the most effective, advanced control techniques to be selected for optimization control of constrained systems. However, for highly nonlinear processes with disturbances or large operating regions, linear predictive control (LMPC) or linearization models-based predictive control, in general, does not meet the requirement of control performances. Hence, the research on constrained nonlinear model predictive control (NMPC) is of great significance theoretically and practically.
Due to the complexities of nonlinear systems, the research of NMPC has lagged behind that of LMPC both in theoretical and practical ways. Especially, the issues of stability, robustness and the computational burden of online optimization on NMPC have been still the focuses of the researchers for both predictive control and optimization algorithms. Based on the existing theoretical results on NMPC, this dissertation investigates the basic 5 topics of NMPC for constrained nonlinear systems, that is, feasibility of optimization, stability, robustness, computational burden and estimate of stability region. The goal of the work is to obtain some theoretical results and algorithms with more practical value. To achieve the goal, the relevant theory and approaches are exploited, such as differential game, (robust) invariant set theory, input-to-state stability, control Lyapunov function, backstepping method, etc. The main contribution of this dissertation includes:
(1) A suboptimal, minimax robust NMPC algorithm is proposed for discrete-time nonlinear systems subject to constraints and uncertainties, the goal of which is to reduce the on-line computational load of robust NMPC schemes. By introducing the differential game theory and the concept of input-to-state stability, the closed-loop robust stability for this suboptimal NMPC algorithm is achieved in the vector norm way.
(2) The problems of disturbance rejection and implementation of H_∞NMPC is systematically investigated for discrete-time nonlinear affine systems subject to constraints and uncertainties. Based on the results in (1), the robust stability of H_∞NMPC is studied in both cases of optimal and suboptimal solutions, respectively. By introducing L_2-gain of nonlinear systems, the problem of disturbance rejection of H_∞NMPC is investigated quantitatively in the signal norm way. In order to lessen the computational burden of online optimization in NMPC, finite dimension parameterizations are incorporated into the design of the NMPC algorithm.
(3) A novel NMPC guaranteed closed-loop stability——constructive NMPC is proposed for continuous-time, constrained nonlinear systems. The main idea of this algorithm is to derive a stable control class off-line via the design of construction, and then to calculate its adjustable parameters on-line via the optimization of costs. The desire goal is to lessen the on-line computational load of NMPC and to decouple the stability of closed-loop systems from the Optimality of performance indexes.
(4) Based on the NMPC scheme addressed in (3), a NMPC algorithm is presented for constrained, strict-feedback nonlinear systems, where the stable control class is constructed by using backstepping method. Then the NMPC algorithm is employed to optimize the control of nonholonomic wheeled mobile robots.
由于非线性系统的复杂多样,非线性预测控制研究无论在理论上还是应用上都比线性预测控制落后,特别是在稳定性、鲁棒性和在线优化计算量方面一直是预测控制和优化算法研究者们所共同关注的热点问题。本文在非线性预测控制理论已有的研究成果基础上,利用微分对策理论、(鲁棒)不变集理论、输入状态稳定性、控制Lyapunov函数、反步设计等方法和概念,对有约束的非线性系统,研究非线性预测控制的优化可行性、稳定性、鲁棒性、计算量和稳定域估计等5个基本问题,以期得到更具实际应用意义的理论性成果和控制算法设计。本文的主要贡献由以下几个部分组成:
(1)考虑约束不确定离散时间非线性系统,对非线性预测控制进行minimax鲁棒次优算法设计,以期从策略上降低鲁棒非线性预测控制的在线计算量。引入微分对策理论和输入状态稳定性概念,证明该次优控制算法在向量范数意义下的闭环鲁棒稳定性。
(2)以约束不确定离散时间非线性仿射系统为对象,系统地研究H_∞非线性预测控制的扰动抑制问题和实现问题。利用(1)的结论分别研究H_∞预测控制在最优解和次优解存在下的鲁棒稳定性问题。通过引入非线性系统L_(2-)增益概念,定量地研究H_∞预测控制在信号范数意义下的扰动抑制问题。为了减轻非线性预测控制在线优化的计算量,在算法的实现中引入了有限维参数化设计。
(3)以连续时间约束非线性系统为对象,提出一种新的具有闭环稳定的非线性预测控制——构造性非线性模型预测控制。算法的主要思想是通过构造性设计离线得到预测控制的一个稳定控制类,再根据目标函数在线优化其中的可调参数,以期降低非线性预测控制的在线计算量,同时实现闭环系统稳定性与性能指标优化的分离。
(4)基于(3)中提出的非线性预测控制策略,利用反步设计法构造约束严格反馈非线性系统的稳定控制类,实现约束非线性模型预测控制设计。最后将该非线性预测控制算法用于非完整轮式移动机器人的优化控制。
Optimization control of constrained nonlinear systems has increasingly been a hot topic of research in the field of nonlinear control. Model predictive control (MPC) has an ability to deal with system constraints and the optimization of costs explicitly, and then is one of the most effective, advanced control techniques to be selected for optimization control of constrained systems. However, for highly nonlinear processes with disturbances or large operating regions, linear predictive control (LMPC) or linearization models-based predictive control, in general, does not meet the requirement of control performances. Hence, the research on constrained nonlinear model predictive control (NMPC) is of great significance theoretically and practically.
Due to the complexities of nonlinear systems, the research of NMPC has lagged behind that of LMPC both in theoretical and practical ways. Especially, the issues of stability, robustness and the computational burden of online optimization on NMPC have been still the focuses of the researchers for both predictive control and optimization algorithms. Based on the existing theoretical results on NMPC, this dissertation investigates the basic 5 topics of NMPC for constrained nonlinear systems, that is, feasibility of optimization, stability, robustness, computational burden and estimate of stability region. The goal of the work is to obtain some theoretical results and algorithms with more practical value. To achieve the goal, the relevant theory and approaches are exploited, such as differential game, (robust) invariant set theory, input-to-state stability, control Lyapunov function, backstepping method, etc. The main contribution of this dissertation includes:
(1) A suboptimal, minimax robust NMPC algorithm is proposed for discrete-time nonlinear systems subject to constraints and uncertainties, the goal of which is to reduce the on-line computational load of robust NMPC schemes. By introducing the differential game theory and the concept of input-to-state stability, the closed-loop robust stability for this suboptimal NMPC algorithm is achieved in the vector norm way.
(2) The problems of disturbance rejection and implementation of H_∞NMPC is systematically investigated for discrete-time nonlinear affine systems subject to constraints and uncertainties. Based on the results in (1), the robust stability of H_∞NMPC is studied in both cases of optimal and suboptimal solutions, respectively. By introducing L_2-gain of nonlinear systems, the problem of disturbance rejection of H_∞NMPC is investigated quantitatively in the signal norm way. In order to lessen the computational burden of online optimization in NMPC, finite dimension parameterizations are incorporated into the design of the NMPC algorithm.
(3) A novel NMPC guaranteed closed-loop stability——constructive NMPC is proposed for continuous-time, constrained nonlinear systems. The main idea of this algorithm is to derive a stable control class off-line via the design of construction, and then to calculate its adjustable parameters on-line via the optimization of costs. The desire goal is to lessen the on-line computational load of NMPC and to decouple the stability of closed-loop systems from the Optimality of performance indexes.
(4) Based on the NMPC scheme addressed in (3), a NMPC algorithm is presented for constrained, strict-feedback nonlinear systems, where the stable control class is constructed by using backstepping method. Then the NMPC algorithm is employed to optimize the control of nonholonomic wheeled mobile robots.
引文
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