基于LMI的滚动时域H_∞控制在三容实验系统中的应用研究
摘要
模型预测控制又称后退时域控制，它是70年代后期产生的一类新型计算机控制算法。它的问世,一方面是受到了计算机技术发展的推动,另一方面也来自复杂工业控制实践向高层优化控制提出的挑战。与其它控制算法相比，模型预测控制对模型的精度要求不高,建模方便,过程描述可由简单实验获得；采用非最小化描述的模型；采用滚动优化策略,而非全局一次优化,能及时弥补由于模型失配、畸变、干扰等因素引起的不确定性,动态性能较好；易将算法推广到有约束、大迟延、非最小相位、非线性等实际过程.尤为重要的是,它能有效处理多变量、有约束问题。
预测控制算法的种类多、表现形式多种多样,但都具有相同的三大本质特征:预测模型、滚动优化和反馈校正。各种预测控制算法具有类似的计算步骤:在当前时刻,基于过程的动态模型预测未来一定时域内每个采样周期的过程输出,这些输出为当前时刻和未来一定时域内控制量的函数。按照基于反馈校正的某个优化目标函数计算当前及未来一定时域的控制量大小。为了防止控制量剧烈变化及超调,一般在优化目标函数中都考虑使未来输出以一参考轨迹最优地去跟踪期望设定值。计算出当前控制量后输出给过程实施控制。至下一时刻,根据新测量数据重新按上述步骤计算控制量。从预测控制的基本原理可以看出,预测控制是不断滚动的局部优化,而非全局最优。随着其理论研究的不断深入,预测控制在工业过程的应用越来越广泛,应用范围遍及石油、化工、建材、冶金、航空、机械制造、食品加工、军事等几乎所有行业,控制技术、手段等也不断提高。
模型预测控制的鲁棒性理论研究很多，其中基于理论的预测控制是一种使约束系统鲁棒稳定的捷径，这是因为控制为系统的鲁棒性分析与设计提供了较完美的理论基础，而预测控制将系统约束显式地表述在
开环优化问题中并通过在线滚动优化使之动态满足，是约束系统优化控制的一个非常有效的方法。遗憾的是大多数已有的讨论都没有涉及系统的性能要求。文献[28,44,]在线性矩阵不等式(LMI)优化和多目标控制的框架下讨论存在控制量和输出约束时的性能控制，并融合预测控制的滚动优化原理讨论了一种滚动时域性能控制方法。把线性矩阵不等式与MPC联系起来主要有两个原因，第一，LMI很适合描述多约束或多目标的问题，这使得基于LMI方法对于经典的“分析”方法来说是一个很有价值的选择。第二，可以以线性矩阵不等式框架重新理解现存的鲁棒控制理论，意思是我们可以在每一个采样时刻设计MPC方案，求解LMI优化问题。滚动时域性能控制方法的优点是通过滚动优化在线协调控制性能要求和系统约束，充分利用有限的控制量提高控制性能。
传统的控制方法一般没有考虑系统的约束，而在控制系统中，普遍存在输入、状态等约束。约束系统的控制方法是在无约束系统的控制方法基础上，仿照预测控制解决约束的方法，把系统约束转化为优化参数的约束，把控制性能当作优化目标，解带约束的LMI优化问题，实现了控制的系统约束问题。约束系统的控制方法的缺点是控制性能在控制过程中固定不变，这就不能及时的根据系统的状态调整系统的控制性能，例如为了适应大的扰动，我们需要选择控制性能差的控制器，因为控制性能好的控制器不能满足系统约束，但当大的扰动过去后，控制性能依然保持不变，这将严重影响无扰动时的控制效果。为了进一步提高控制性能，充分利用被控系统的当前状态，滚动时域的控制方法利用预测控制的在线优化思想，把当前时刻的状态当作初始状态，在每个采样时刻都求解带约束的LMI优化问题，这样使系统能在线解调控制性能和系统约束——当有大扰动时降低控
制性能使约束满足，当无大扰动时提高控制性能。虽然采用滚动时域控制方法的控制性能有了较大的提高，但是它在在线求解的优化问题中所带的约束条件较多，这将无疑增加算法的保守性。混合控制算法根据系统的当前状态，在线在前三种控制算法间切换，减小了控制的保守性，进一步提高了控制性能。
把复杂算法应用到实际的物理设备上，需要编写复杂算法程序和控制平台程序。MATLAB是美国MathWorks公司推出的数学软件，具有强大的工程计算功能和丰富的工具箱，非常适合上层控制算法的编写，但对于循环处理和图形界面的处理比不上其它语言；Visual C++是美国Microsoft推出的可视化C++编程环境，是目前功能最强大的软件开发工具之一，被广泛应用于Windows 32位平台的基础应用程序开发,但用C++编写复杂的算法太繁琐。因此用VC++编写控制平台，用MATLAB编写法杂控制算法，通过MATLAB引擎函数库中提供的API完成控制算法与VC数据交换，可以实现复杂控制算法的实验应用。
约束系统的滚动时域控制方法还需要进一步的改进，比如不变椭圆域如何更好确定;对一些系统来说，加强了的耗散不等式条件所带来的算法保守性将严重影响系统的控制性能。所有这些都还需进一步研究。
由于数值求解LMI的成功，使得基于LMI的控制方法在实现上具有明显的优点，但是这种优化方法的求解非常复杂，这就带来了难以满足实时控制的问题，所以大部分现有的基于LMI的在线控制方法只能用于较缓慢的过程控制，尤其是存在系统状态?
Model Predictive Control (MPC) also known as Moving Horizon Control (M HC) or Receding Horizon Control (RHC) is a popular new control method in the later of 70’. It appeared due to two factors. One factor is the development of computer and the other is the requirement of shift from complicated industrial control to advanced optimal control. In the contrast with other control methods, the advantage of MPC are as follows: MPC needn’t accurate model, creating model is convenient and the depiction of process may be got from simple experiment.; MPC does not use least model; using moving horizon optimal policy, not optimization for a time, can make up for uncertainty caused by model mismatch, disturbance and so on. So the dynamic process is good; MPC is easily expanded into the system with constrained control, not least model, long delay and non-linear process. To the most important, MPC can deal with constrained and multi-variables system.
MPC appears in variable forms, but its essence is made of three properties: predictive model, moving horizon optimization and feedback revise. All kinds of predictive arithmetic are nearly same: At every instant, MPC requires the on-line solution of an optimization problem to compute optimal control inputs over a fixed number of future time instants, known as the “time horizon”. Although more than one control move is generally calculated, only the first one is implemented. At the next sampling time, the optimization problem is reformulated and solved with new measurements obtained from the system. The on-line optimization can be reduced to either a linear program or a quadratic program.
With the development of MPC theory, it is used in wider and wider areas. It involves in chemical process control in the petrochemical, paper industries and gas pipeline control, aviation, metallurgy, military affairs and so on. At the same time, both the control technology and control measure are improved.
The theory on the robustness of MPC is mature, among which the MPC theory based is shortcut for robust stability of constrained system. There are two reasons for saying so. On the one hand, control provides perfect theoretic
foundation for the analyse and design of robustness. On the other hand, MPC is a efficient way to deal with the constraints of system, because MPC expressly describes constraints in the open-loop optimization and satisfies dynamic performance by moving horizon. Unfortunately, most literature on this didn’t discuss the performance of the system. Document [28,44] discussed control of the constrained system, in the form of LMI and multi-objection control. They also showed the moving horizon control arithmetic by combing MPC and control. There are two reasons why LMI optimization is relevant to MPC. Firstly, LMI-based optimization problems can be solved in polynomial-time, often in times comparable to that required for the evaluation of an analytical solution for a similar problem. Thus, LMI optimization can be implemented on-line. Secondly, it is possible to recast much of existing robust control theory in the framework of LMIs. The implication is that we can devise an MPC scheme where at each time instant; an LMI optimization problem (as opposed to a conventional linear or quadratic programs) is solved, which incorporates input and output constraints. The main advantage of moving horizon control arithmetic is its capability of automatically relaxing or tightening the performance specification in order to obey hard control constraints while achieving the best possible performance in a suitable class of LMI-generated control gains.
Traditional control does not consider the constraints of system, but the constraints of inputs and state exit in a common control system. On the base of control, control of constrained system is formed. It solves the problem of constraints just as MPC does, by transforming the constraints of the input or state into the constraints of optimized parameters, taking the performance of as optimized objection and resolving the
预测控制算法的种类多、表现形式多种多样,但都具有相同的三大本质特征:预测模型、滚动优化和反馈校正。各种预测控制算法具有类似的计算步骤:在当前时刻,基于过程的动态模型预测未来一定时域内每个采样周期的过程输出,这些输出为当前时刻和未来一定时域内控制量的函数。按照基于反馈校正的某个优化目标函数计算当前及未来一定时域的控制量大小。为了防止控制量剧烈变化及超调,一般在优化目标函数中都考虑使未来输出以一参考轨迹最优地去跟踪期望设定值。计算出当前控制量后输出给过程实施控制。至下一时刻,根据新测量数据重新按上述步骤计算控制量。从预测控制的基本原理可以看出,预测控制是不断滚动的局部优化,而非全局最优。随着其理论研究的不断深入,预测控制在工业过程的应用越来越广泛,应用范围遍及石油、化工、建材、冶金、航空、机械制造、食品加工、军事等几乎所有行业,控制技术、手段等也不断提高。
模型预测控制的鲁棒性理论研究很多，其中基于理论的预测控制是一种使约束系统鲁棒稳定的捷径，这是因为控制为系统的鲁棒性分析与设计提供了较完美的理论基础，而预测控制将系统约束显式地表述在
开环优化问题中并通过在线滚动优化使之动态满足，是约束系统优化控制的一个非常有效的方法。遗憾的是大多数已有的讨论都没有涉及系统的性能要求。文献[28,44,]在线性矩阵不等式(LMI)优化和多目标控制的框架下讨论存在控制量和输出约束时的性能控制，并融合预测控制的滚动优化原理讨论了一种滚动时域性能控制方法。把线性矩阵不等式与MPC联系起来主要有两个原因，第一，LMI很适合描述多约束或多目标的问题，这使得基于LMI方法对于经典的“分析”方法来说是一个很有价值的选择。第二，可以以线性矩阵不等式框架重新理解现存的鲁棒控制理论，意思是我们可以在每一个采样时刻设计MPC方案，求解LMI优化问题。滚动时域性能控制方法的优点是通过滚动优化在线协调控制性能要求和系统约束，充分利用有限的控制量提高控制性能。
传统的控制方法一般没有考虑系统的约束，而在控制系统中，普遍存在输入、状态等约束。约束系统的控制方法是在无约束系统的控制方法基础上，仿照预测控制解决约束的方法，把系统约束转化为优化参数的约束，把控制性能当作优化目标，解带约束的LMI优化问题，实现了控制的系统约束问题。约束系统的控制方法的缺点是控制性能在控制过程中固定不变，这就不能及时的根据系统的状态调整系统的控制性能，例如为了适应大的扰动，我们需要选择控制性能差的控制器，因为控制性能好的控制器不能满足系统约束，但当大的扰动过去后，控制性能依然保持不变，这将严重影响无扰动时的控制效果。为了进一步提高控制性能，充分利用被控系统的当前状态，滚动时域的控制方法利用预测控制的在线优化思想，把当前时刻的状态当作初始状态，在每个采样时刻都求解带约束的LMI优化问题，这样使系统能在线解调控制性能和系统约束——当有大扰动时降低控
制性能使约束满足，当无大扰动时提高控制性能。虽然采用滚动时域控制方法的控制性能有了较大的提高，但是它在在线求解的优化问题中所带的约束条件较多，这将无疑增加算法的保守性。混合控制算法根据系统的当前状态，在线在前三种控制算法间切换，减小了控制的保守性，进一步提高了控制性能。
把复杂算法应用到实际的物理设备上，需要编写复杂算法程序和控制平台程序。MATLAB是美国MathWorks公司推出的数学软件，具有强大的工程计算功能和丰富的工具箱，非常适合上层控制算法的编写，但对于循环处理和图形界面的处理比不上其它语言；Visual C++是美国Microsoft推出的可视化C++编程环境，是目前功能最强大的软件开发工具之一，被广泛应用于Windows 32位平台的基础应用程序开发,但用C++编写复杂的算法太繁琐。因此用VC++编写控制平台，用MATLAB编写法杂控制算法，通过MATLAB引擎函数库中提供的API完成控制算法与VC数据交换，可以实现复杂控制算法的实验应用。
约束系统的滚动时域控制方法还需要进一步的改进，比如不变椭圆域如何更好确定;对一些系统来说，加强了的耗散不等式条件所带来的算法保守性将严重影响系统的控制性能。所有这些都还需进一步研究。
由于数值求解LMI的成功，使得基于LMI的控制方法在实现上具有明显的优点，但是这种优化方法的求解非常复杂，这就带来了难以满足实时控制的问题，所以大部分现有的基于LMI的在线控制方法只能用于较缓慢的过程控制，尤其是存在系统状态?
Model Predictive Control (MPC) also known as Moving Horizon Control (M HC) or Receding Horizon Control (RHC) is a popular new control method in the later of 70’. It appeared due to two factors. One factor is the development of computer and the other is the requirement of shift from complicated industrial control to advanced optimal control. In the contrast with other control methods, the advantage of MPC are as follows: MPC needn’t accurate model, creating model is convenient and the depiction of process may be got from simple experiment.; MPC does not use least model; using moving horizon optimal policy, not optimization for a time, can make up for uncertainty caused by model mismatch, disturbance and so on. So the dynamic process is good; MPC is easily expanded into the system with constrained control, not least model, long delay and non-linear process. To the most important, MPC can deal with constrained and multi-variables system.
MPC appears in variable forms, but its essence is made of three properties: predictive model, moving horizon optimization and feedback revise. All kinds of predictive arithmetic are nearly same: At every instant, MPC requires the on-line solution of an optimization problem to compute optimal control inputs over a fixed number of future time instants, known as the “time horizon”. Although more than one control move is generally calculated, only the first one is implemented. At the next sampling time, the optimization problem is reformulated and solved with new measurements obtained from the system. The on-line optimization can be reduced to either a linear program or a quadratic program.
With the development of MPC theory, it is used in wider and wider areas. It involves in chemical process control in the petrochemical, paper industries and gas pipeline control, aviation, metallurgy, military affairs and so on. At the same time, both the control technology and control measure are improved.
The theory on the robustness of MPC is mature, among which the MPC theory based is shortcut for robust stability of constrained system. There are two reasons for saying so. On the one hand, control provides perfect theoretic
foundation for the analyse and design of robustness. On the other hand, MPC is a efficient way to deal with the constraints of system, because MPC expressly describes constraints in the open-loop optimization and satisfies dynamic performance by moving horizon. Unfortunately, most literature on this didn’t discuss the performance of the system. Document [28,44] discussed control of the constrained system, in the form of LMI and multi-objection control. They also showed the moving horizon control arithmetic by combing MPC and control. There are two reasons why LMI optimization is relevant to MPC. Firstly, LMI-based optimization problems can be solved in polynomial-time, often in times comparable to that required for the evaluation of an analytical solution for a similar problem. Thus, LMI optimization can be implemented on-line. Secondly, it is possible to recast much of existing robust control theory in the framework of LMIs. The implication is that we can devise an MPC scheme where at each time instant; an LMI optimization problem (as opposed to a conventional linear or quadratic programs) is solved, which incorporates input and output constraints. The main advantage of moving horizon control arithmetic is its capability of automatically relaxing or tightening the performance specification in order to obey hard control constraints while achieving the best possible performance in a suitable class of LMI-generated control gains.
Traditional control does not consider the constraints of system, but the constraints of inputs and state exit in a common control system. On the base of control, control of constrained system is formed. It solves the problem of constraints just as MPC does, by transforming the constraints of the input or state into the constraints of optimized parameters, taking the performance of as optimized objection and resolving the
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