约束离散时间分段仿射系统的鲁棒预测控制
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摘要
控制理论中的一个重要和具有挑战性的问题就是得到一般的约束非线性系统或混杂系统控制器计算的系统性方法,并确保闭环系统的稳定性、可行性以及最优性。而无论在理论上还是工业实践上,对这类系统应用最成功的现代控制策略无疑是模型预测控制,或称为滚动时域控制。由于种种原因,用来描述被控系统动态特性的模型往往具有某种不确定性。要保证系统的鲁棒稳定性,在控制器的设计上就必须考虑不确定性的影响,因此,对这类系统的鲁棒预测控制研究正得到越来越多学者的关注。
分段仿射系统是通过将扩展的状态-输入空间分割成多面体区域,并在每个区域上定义一个状态空间表达式得到,它可以描述一大类的非线性系统,并且在一定的条件下,可以和很多类混杂系统进行等价转换,如混合逻辑动态模型、线性互补模型、混杂自动机等。
本论文着重讨论在附加有界扰动下,一类约束离散时间分段仿射系统的鲁棒预测控制问题。在鲁棒预测理论已有研究成果的基础上,利用鲁棒不变集、鲁棒收缩序列集、多参数规划、多面体的几何运算和线性矩阵不等式等相关理论和方法,分别研究具有鲁棒可行性和稳定性保证的预测控制器的在线和离线设计方法。具体而言,本文的贡献主要在以下几个方面:
1.研究了基于开环优化的鲁棒预测控制问题,提出了一种鲁棒双模的控制方法。该方法基于不确定演变集,即在任意可能的扰动下,系统的预测状态演变集。把它作为预测优化问题的状态约束,并选择一个鲁棒正不变集作为终端约束集,使得优化问题的可行性即保证了系统的鲁棒稳定性,从而可大大减小优化问题的在线计算量。
2.利用多参数规划计算系统的鲁棒一步集,同时得到系统的鲁棒一步可达集和相关的控制器。进而通过迭代计算,得到系统的最大鲁棒正不变集、最大鲁棒受控不变集和最大鲁棒可稳定集。
3.研究了降低闭环优化计算复杂性的鲁棒预测控制问题,提出了一种具有稳定性保证的模型预测控制方法。基于鲁棒正不变集,计算系统的鲁棒收缩序列集,把它作为优化问题的稳定约束,使得在次优解的情况下,可保证系统的鲁棒可行性和稳定性。并在确保鲁棒稳定性的前提下,进一步简化了预测控制器的约束条件,减小了控制器的在线计算。
4.提出了一种扩大鲁棒预测控制吸引域的新方法,将基于鲁棒正不变集的鲁棒收缩序列集作为优化问题的终端约束集,扩大了终端约束域,从而扩大了优化问题的可行域。
5.研究了约束分段仿射系统鲁棒预测控制的离线计算问题,利用多参数规划和多面体的几何运算,讨论了鲁棒时间最优控制问题和鲁棒滚动时域控制问题,并给出了最优显式解的一般几何特征。
6.为减小鲁棒预测控制的离线计算复杂性,提出了一种低复杂性的控制策略:鲁棒一步控制。把系统的最大鲁棒可稳定集作为第一步预测状态的约束集,保证了优化问题的鲁棒可行性,并使得优化问题的可行域覆盖最大鲁棒可稳定集。在鲁棒稳定性分析中,给出了用线性矩阵不等式求解二次李雅普诺夫函数的一般方法。
One of the most important and challenging problem in control is the derivation of systematic tools for the computation of controllers for general constrained non-linear or hybrid system that can guarantee closed-loop stability, feasibility, and optimality. The most successful modern control strategy both in theory and in practice for this class of systems is undoubtedly Model Predictive Control (MPC), also interchangeably called Receding Horizon Control (RHC). On the other hand, the model, which is used to describe the dynamics of controlled system, always has some uncertainty. In order to guarantee the robust stability when uncertainties are present, they must be taken into account in the computation of the control law and hence, the robust perdictive control of this class of systems has garnered increasing interest in the research community.
Piecewise Affine (PWA) systems are obtained by partitioning the extended state-input space into polyhedral regions and associating with each region a different affine state update equation. PWA systems represent a powerful tool for approximating non-linear systems and are (under very mild assumptions) equivalent to many other hybrid systems, such as Mixed Logical Dynamical systems, Linear Complementary systems, Hybrid Automation and so on.
In this thesis, the focus lies on robust predictive contorl for a class of constrained discrete-time PWA systems with bounded disturbances. Based on the existing theoretical results on model predictive control, the thesis is devoted to the study on the on- and off-line robust predictive control with robust feasibility and stability guaranteed. To achieve this, the relevant theory and approaches, such as robust invariant set, robust contractive sequence of sets , multi-parametric programming, geometry operations on polytopes, and linear matrix inequalities (LMI), are employed in the study. Specifically, the main contributions of this thesis are as follows:
1. A robust MPC based on open-loop formulation is studied and a robust dual-model control method is presented. The method is based on so-called uncertain evolution sets, which are the sets containing the predicted evolution of the uncertain system under any admissible uncertainty. By considering these sets as the sate constrain of optimization problem of MPC and choosing as terminal constrain a robust positively invariant set, the robust stability is guaranteed by the feasibility of optimization problem . This property allow us to greatly reduce the on-line computational burden.
2. It is demonstrated that how multi-parametric programming can be used to simultaneously obtain robust one step reachable set and the associated PWA feedback controller. Based on robust one step set, the maximal robust positively invariant set, maximal robust control invariant set and maximal robust stabilizable set are computed by iteration.
3. A robust MPC focused on the reduction of the complexity of closed-loop optimization problem is studied and a MPC scheme with stability guaranteed is proposed. Based on the robust positively invariant set of the PWA system, the robust contractive sequence of sets are computed and is incorporated as a stabilizing constraint in the optimization problem. As a result, robust feasibility and stability is guaranteed in the case of suboptimal solutions. Finally, the simplification of stability conditions is made to reduce the computational complexity of associated optimization problem.
4. A new method for enlarging the domain of attraction of robust MPC for constrained PWA systems with bounded disturbances is presented. Considering a contractive sequence of robust stabilizable sets, which is computed off-line based on robust positively invariant set, as the terminal constraint of predictive state in the optimization problem, robust stability and the enlargement of domain of attraction of robust MPC are guaranteed.
5. The off-line computation of robust MPC controller for constrained PWA systems is studied. Using multi-parametric programming, dynamic programming and geometry operations on polytopes, the robust time-optimal and robust receding horizon control prolems are addressed and the resulting solutions are characterised.
6. In order to reduce the off-line computation complexity of robust MPC, a low complexity control scheme, referred to as robust one-step control,is proposed. The maximal robust stabilizable set is chosen as the constraint set of the first predicted state in MPC formulation, such that the resulting feasible region cover the maximal robust stabilizable set and the robust feasibility is guaranteed for all time. In the sequent stability analysis, a general formulation for searching the common quadratic Lyapunov function with LMIs is presented.
分段仿射系统是通过将扩展的状态-输入空间分割成多面体区域,并在每个区域上定义一个状态空间表达式得到,它可以描述一大类的非线性系统,并且在一定的条件下,可以和很多类混杂系统进行等价转换,如混合逻辑动态模型、线性互补模型、混杂自动机等。
本论文着重讨论在附加有界扰动下,一类约束离散时间分段仿射系统的鲁棒预测控制问题。在鲁棒预测理论已有研究成果的基础上,利用鲁棒不变集、鲁棒收缩序列集、多参数规划、多面体的几何运算和线性矩阵不等式等相关理论和方法,分别研究具有鲁棒可行性和稳定性保证的预测控制器的在线和离线设计方法。具体而言,本文的贡献主要在以下几个方面:
1.研究了基于开环优化的鲁棒预测控制问题,提出了一种鲁棒双模的控制方法。该方法基于不确定演变集,即在任意可能的扰动下,系统的预测状态演变集。把它作为预测优化问题的状态约束,并选择一个鲁棒正不变集作为终端约束集,使得优化问题的可行性即保证了系统的鲁棒稳定性,从而可大大减小优化问题的在线计算量。
2.利用多参数规划计算系统的鲁棒一步集,同时得到系统的鲁棒一步可达集和相关的控制器。进而通过迭代计算,得到系统的最大鲁棒正不变集、最大鲁棒受控不变集和最大鲁棒可稳定集。
3.研究了降低闭环优化计算复杂性的鲁棒预测控制问题,提出了一种具有稳定性保证的模型预测控制方法。基于鲁棒正不变集,计算系统的鲁棒收缩序列集,把它作为优化问题的稳定约束,使得在次优解的情况下,可保证系统的鲁棒可行性和稳定性。并在确保鲁棒稳定性的前提下,进一步简化了预测控制器的约束条件,减小了控制器的在线计算。
4.提出了一种扩大鲁棒预测控制吸引域的新方法,将基于鲁棒正不变集的鲁棒收缩序列集作为优化问题的终端约束集,扩大了终端约束域,从而扩大了优化问题的可行域。
5.研究了约束分段仿射系统鲁棒预测控制的离线计算问题,利用多参数规划和多面体的几何运算,讨论了鲁棒时间最优控制问题和鲁棒滚动时域控制问题,并给出了最优显式解的一般几何特征。
6.为减小鲁棒预测控制的离线计算复杂性,提出了一种低复杂性的控制策略:鲁棒一步控制。把系统的最大鲁棒可稳定集作为第一步预测状态的约束集,保证了优化问题的鲁棒可行性,并使得优化问题的可行域覆盖最大鲁棒可稳定集。在鲁棒稳定性分析中,给出了用线性矩阵不等式求解二次李雅普诺夫函数的一般方法。
One of the most important and challenging problem in control is the derivation of systematic tools for the computation of controllers for general constrained non-linear or hybrid system that can guarantee closed-loop stability, feasibility, and optimality. The most successful modern control strategy both in theory and in practice for this class of systems is undoubtedly Model Predictive Control (MPC), also interchangeably called Receding Horizon Control (RHC). On the other hand, the model, which is used to describe the dynamics of controlled system, always has some uncertainty. In order to guarantee the robust stability when uncertainties are present, they must be taken into account in the computation of the control law and hence, the robust perdictive control of this class of systems has garnered increasing interest in the research community.
Piecewise Affine (PWA) systems are obtained by partitioning the extended state-input space into polyhedral regions and associating with each region a different affine state update equation. PWA systems represent a powerful tool for approximating non-linear systems and are (under very mild assumptions) equivalent to many other hybrid systems, such as Mixed Logical Dynamical systems, Linear Complementary systems, Hybrid Automation and so on.
In this thesis, the focus lies on robust predictive contorl for a class of constrained discrete-time PWA systems with bounded disturbances. Based on the existing theoretical results on model predictive control, the thesis is devoted to the study on the on- and off-line robust predictive control with robust feasibility and stability guaranteed. To achieve this, the relevant theory and approaches, such as robust invariant set, robust contractive sequence of sets , multi-parametric programming, geometry operations on polytopes, and linear matrix inequalities (LMI), are employed in the study. Specifically, the main contributions of this thesis are as follows:
1. A robust MPC based on open-loop formulation is studied and a robust dual-model control method is presented. The method is based on so-called uncertain evolution sets, which are the sets containing the predicted evolution of the uncertain system under any admissible uncertainty. By considering these sets as the sate constrain of optimization problem of MPC and choosing as terminal constrain a robust positively invariant set, the robust stability is guaranteed by the feasibility of optimization problem . This property allow us to greatly reduce the on-line computational burden.
2. It is demonstrated that how multi-parametric programming can be used to simultaneously obtain robust one step reachable set and the associated PWA feedback controller. Based on robust one step set, the maximal robust positively invariant set, maximal robust control invariant set and maximal robust stabilizable set are computed by iteration.
3. A robust MPC focused on the reduction of the complexity of closed-loop optimization problem is studied and a MPC scheme with stability guaranteed is proposed. Based on the robust positively invariant set of the PWA system, the robust contractive sequence of sets are computed and is incorporated as a stabilizing constraint in the optimization problem. As a result, robust feasibility and stability is guaranteed in the case of suboptimal solutions. Finally, the simplification of stability conditions is made to reduce the computational complexity of associated optimization problem.
4. A new method for enlarging the domain of attraction of robust MPC for constrained PWA systems with bounded disturbances is presented. Considering a contractive sequence of robust stabilizable sets, which is computed off-line based on robust positively invariant set, as the terminal constraint of predictive state in the optimization problem, robust stability and the enlargement of domain of attraction of robust MPC are guaranteed.
5. The off-line computation of robust MPC controller for constrained PWA systems is studied. Using multi-parametric programming, dynamic programming and geometry operations on polytopes, the robust time-optimal and robust receding horizon control prolems are addressed and the resulting solutions are characterised.
6. In order to reduce the off-line computation complexity of robust MPC, a low complexity control scheme, referred to as robust one-step control,is proposed. The maximal robust stabilizable set is chosen as the constraint set of the first predicted state in MPC formulation, such that the resulting feasible region cover the maximal robust stabilizable set and the robust feasibility is guaranteed for all time. In the sequent stability analysis, a general formulation for searching the common quadratic Lyapunov function with LMIs is presented.
引文
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