基于凸优化的分层控制系统性能评估
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摘要
随着过程工业生产规模的不断扩大化和被控对象的复杂化,直接控制层加约束控制层的分层控制方案以其灵活的组态结构和更强的鲁棒性等优势在过程工业中广泛应用,保障了过程运行中安全、平稳、高效等不同层面的需求。分层控制策略中把扰动抑制、设定值跟踪、经济性能优化等各个方面的控制任务分配给各层次的子系统来完成。由于控制目标相互关联,所以该系统各方面控制性能的实现程度相互影响且难以被直观评判。如何实时、准确、全面地评估分层控制系统的控制性能具有十分重要的意义。为了解决分层控制系统的性能评价问题,本文基于凸优化理论提出了一整套关于控制性能基准建模、评估与优化的研究框架。位于直接控制层的PID控制器受到结构限制,给控制性能优化问题引入了非凸约束,给问题求解带来较大困难。在充分考虑控制器结构约束的基础上,本文将关于控制器参数的最优可达性能基准非凸优化问题重新描述为更清晰的数学结构形式,设法将带有非凸约束的优化问题近似为易于求解的凸优化问题,降低了求解算法的复杂度。本文的主要工作包括以下方面：
1.针对直接控制层中由PID(Proportional-Integral-Derivative)控制的单变量线性时不变系统,在设定值恒定时,提出一种以最小可达输出方差为调节性能基准的性能评价框架。单变量线性时不变PID控制系统最小可达输出方差的估计问题被归结为一个带有非凸约束的凸优化问题。为求解该问题,将非凸约束线性化,采用罚函数法处理,并设计出一种全新的低计算复杂度的迭代凸规划求解算法。从而建立起PID控制回路的调节性能基准。该方法适用于受PID控制的连续过程调节性能评估。
2.针对直接控制层中由PID控制的单变量线性时不变系统,在设定值受约束控制层影响出现跳变时,提出一种以最小可达调节误差方差为调节性能评估基准,以最小绝对跟踪误差积分为跟踪性能评估基准,并以调节／跟踪性能折中曲线作为均衡性能基准的性能评估框架。将线性时不变系统PID控制器的最小可达调节误差方差估计问题重新归结为一个带有双线性矩阵不等式约束的二次型规划问题,将最小绝对跟踪误差积分估计问题归结为一个带有双线性矩阵不等式约束的线性凸规划问题,再通过目标函数加权和的形式,把调节／跟踪性能均衡最优化问题构造为一个关于PID参数的带有双线性矩阵不等式约束的二次型优化问题。采用通用的序列二次规划算法求解上述问题,建立起受单一PID控制的多目标性能基准。该方法适用于直接控制层的调节／跟踪综合性能评估。
3.针对由PID控制的单变量线性时变间歇过程,考虑一个完整批次包含多个运行阶段,提出一种基于最小可达结构化残差的性能基准建模与求解方法。为了简化线性时变系统的分析与综合,使基于线性时不变系统PID控制性能评估的方法可移植到线性时变系统中,建立了线性时变系统的时变脉冲响应矩阵模型。根据设定值的动态特性将一个批次划分为调节与跟踪阶段,通过阶段的权重矩阵辅助建立关于各运行阶段和完整批次的结构化残差,线性时变系统PID控制的最小可达结构化残差方差的估计问题被归结为一个非凸优化问题。分析了引入的结构化矩阵对该优化问题求解的影响,分别针对相应的带有秩一约束的凸优化问题和带有双线性矩阵不等式约束的二次型规划问题设计了相应的求解算法。仿真实验表明该优化问题变型大幅提高了求解效率。该方法适用于受PID控制的间歇过程综合性能评估。
4.针对分层控制系统中约束控制层的经济性能评估问题,提出了一种基于最优-最差经济性能区间的评估方法。通过分析分层控制的直接控制、约束控制和实时优化三层垂直结构的信息交互方式,阐述了基本控制性能与总体经济性能的关联关系。通过计算该控制系统的最优-最差经济性能区间,给出了一种面向经济性能的评估基准。提出一种根据过程开环模型和直接层控制器参数计算约束控制层中模型预测控制(Model Predictive Control, MPC)广义对象模型的方法,有效减少模型与真实对象失配引起的控制性能下降。给出了实时监测和评估分层控制系统经济效益的步骤方法,对Shell重油分馏塔模型的仿真结果表明了该方法的有效性。
With the emerging growth of the scale of industrial processes and the complexity of the plant in recent years, traditional centralized control strategies are gradually replaced by the newly devel-oped hierarchical control systems due to its advantages of flexible configuration and robustness. A Hierarchical control system including direct control layer and constraint control layer is a prospec-tive scheme in processes control practice, which could guarantee control demands in different aspects, including the safety, stability, reliability and efficiency of the plant operation. In hierarchi-cal control system, control tasks are composed of disturbances depression, set-point tracking, and economic performance optimization. And they are assigned to three individual subsystem layers to fulfill. Because there exists mutual influences between the realization degree of these tasks, and also interactions of control layers, it is not convenient for control engineers to determine how well the performance of local controllers and the plantwide system are. It is important and of practical significance to comprehensively evaluate the overall performance of a hierarchical con-trol system. To assess performance of direct control layer and constraint control layer, the most important components in a hierarchical control system, we proposed in this thesis a framework of benchmarking, assessing and optimizing the direct control and economic performance based on convex optimization technology. Especially, the non-convexity brought in by PID structure of the minimum achievable error variance problem in assessing direct control performance is refor-mulated as a new non-convex constraint with more explicit mathematical structure with respect to PID parameters.The original nonconvex optimization problem is then approximated by a convex optimization which can be easily solved with low-complexity algorithms. The main contributions of this thesis are as following:
1. Minimum achievable output variance (MAOV) is a common benchmark for control per-formance assessment. Finding the MAOV of proportional-integral-derivative (PID) control systems is computationally expensive due to the inherent non-convexity of the associated optimization problem. We present a new method for computing the MAOV of PID control systems. The problem of estimating the MAOV of a PID control system is novelly for-mulated as a convex program with an additional non-convex constraint. The non-convex constraint is linearized and handled by the penalty approach. Based on this, a customized low-complexity algorithm, which relies on the iterative convex programming technique, is developed to solve the MAOV problem. The new algorithm is proved to be convergent. We show via numerical examples that the new approach yields close-to-optimal solutions that are better than (or as good as) the solutions generated by the existing methods.
2. Considering the set-point of direct control layer is often affected by upper constraint control layer, a novel comprehensive performance assessment framework is proposed. Minimum achievable regulatory error variance (MAREV) is used as regulatory performance bench-mark, and minimum integrated absolute tracking error (MIATE) is established as benchmark of tracking performance. A regulatory/tracking performance trade-off curve is obtained to evaluate how balance the total control effect is. The problem of estimating the MAREV of a PID control system is formulated as a quadratic programming (QP) with a bilinear ma-trix inequality (BMI) constraint, and the problem of estimating the MIATE is formulated as a linear programming (LP) with a BMI constraint. Accordingly the problem of searching the balanced tracking/regulatory performance benchmark is formulated as a QP with BMI constraints with respect to PID parameters. These optimization problems are dealt with a standard sequential quadratic programming (SQP) algorithm. Thus facing multiple tasks in direct control layer, a flexible control performance benchmark suited for PID controllers is obtained. We show via numerical examples that compared to commercial software PENBMI which is specially designed for solving BMI problems, the approach we presented yields the same optimal solutions with less time consumption.
3. Batch processes usually have strong nonlinearity or time-varying property, and are often operated in multiple phases. For those batch processes which can be described as linear time varying (LTV) systems, we suggest a new control performance benchmark which is based on the minimum achievable structured residual variance (MASRV). Firstly, for the ease of elaboration, a model called time-varying impulse response matrix is established to describe the LTV systems, which makes the LTV systems own similar formation with LTI systems. Secondly, a duration time of a batch is divided into regulatory and tracking stages according to set-point dynamics, then a structured residual is defined based on the weight matrix of the stage. The problem of estimating the MASRV of LTV system with PID controller is formulated as a non-convex optimization. The fact that the introduced structured matrix is whether invertible or not affects the choice of optimization algorithm. Iterative semidefinit programming (SDP) is selected to handle the convex optimization with a rank one constraint if structured matrix is invertible, otherwise, sequential quadratic programming (SQP) method is used to solve the QP problem with BMI constraints. Numerical simulations study prove that the reformulation of the problem improves the efficiency of optimization.
4. In order to ensure multivariable industrial processes operating in a safe and economic mode, a method for control performance assessment of hierarchical control systems was proposed. The three-layer structure of a hierarchical control system:direct control layer, constraint con-trol layer and real-time optimization layer, was analyzed to formulate the control objective functions of three aspects:suppressing disturbances, keeping constraints and maximizing process profits, respectively. A control performance assessment benchmark called "best to worst performance range" was established to monitor the economic performance of industrial processes, and to evaluate how much potential would be improved. To avoid the degrada-tion of control performance due to model-plant mismatch, a method to compute generalized object model through open loop model and regulatory parameters was presented. The relia-bility and efficacy of the proposed performance assessment technique is demonstrated on a case study on Shell heavy oil fractionator control problem.
1.针对直接控制层中由PID(Proportional-Integral-Derivative)控制的单变量线性时不变系统,在设定值恒定时,提出一种以最小可达输出方差为调节性能基准的性能评价框架。单变量线性时不变PID控制系统最小可达输出方差的估计问题被归结为一个带有非凸约束的凸优化问题。为求解该问题,将非凸约束线性化,采用罚函数法处理,并设计出一种全新的低计算复杂度的迭代凸规划求解算法。从而建立起PID控制回路的调节性能基准。该方法适用于受PID控制的连续过程调节性能评估。
2.针对直接控制层中由PID控制的单变量线性时不变系统,在设定值受约束控制层影响出现跳变时,提出一种以最小可达调节误差方差为调节性能评估基准,以最小绝对跟踪误差积分为跟踪性能评估基准,并以调节／跟踪性能折中曲线作为均衡性能基准的性能评估框架。将线性时不变系统PID控制器的最小可达调节误差方差估计问题重新归结为一个带有双线性矩阵不等式约束的二次型规划问题,将最小绝对跟踪误差积分估计问题归结为一个带有双线性矩阵不等式约束的线性凸规划问题,再通过目标函数加权和的形式,把调节／跟踪性能均衡最优化问题构造为一个关于PID参数的带有双线性矩阵不等式约束的二次型优化问题。采用通用的序列二次规划算法求解上述问题,建立起受单一PID控制的多目标性能基准。该方法适用于直接控制层的调节／跟踪综合性能评估。
3.针对由PID控制的单变量线性时变间歇过程,考虑一个完整批次包含多个运行阶段,提出一种基于最小可达结构化残差的性能基准建模与求解方法。为了简化线性时变系统的分析与综合,使基于线性时不变系统PID控制性能评估的方法可移植到线性时变系统中,建立了线性时变系统的时变脉冲响应矩阵模型。根据设定值的动态特性将一个批次划分为调节与跟踪阶段,通过阶段的权重矩阵辅助建立关于各运行阶段和完整批次的结构化残差,线性时变系统PID控制的最小可达结构化残差方差的估计问题被归结为一个非凸优化问题。分析了引入的结构化矩阵对该优化问题求解的影响,分别针对相应的带有秩一约束的凸优化问题和带有双线性矩阵不等式约束的二次型规划问题设计了相应的求解算法。仿真实验表明该优化问题变型大幅提高了求解效率。该方法适用于受PID控制的间歇过程综合性能评估。
4.针对分层控制系统中约束控制层的经济性能评估问题,提出了一种基于最优-最差经济性能区间的评估方法。通过分析分层控制的直接控制、约束控制和实时优化三层垂直结构的信息交互方式,阐述了基本控制性能与总体经济性能的关联关系。通过计算该控制系统的最优-最差经济性能区间,给出了一种面向经济性能的评估基准。提出一种根据过程开环模型和直接层控制器参数计算约束控制层中模型预测控制(Model Predictive Control, MPC)广义对象模型的方法,有效减少模型与真实对象失配引起的控制性能下降。给出了实时监测和评估分层控制系统经济效益的步骤方法,对Shell重油分馏塔模型的仿真结果表明了该方法的有效性。
With the emerging growth of the scale of industrial processes and the complexity of the plant in recent years, traditional centralized control strategies are gradually replaced by the newly devel-oped hierarchical control systems due to its advantages of flexible configuration and robustness. A Hierarchical control system including direct control layer and constraint control layer is a prospec-tive scheme in processes control practice, which could guarantee control demands in different aspects, including the safety, stability, reliability and efficiency of the plant operation. In hierarchi-cal control system, control tasks are composed of disturbances depression, set-point tracking, and economic performance optimization. And they are assigned to three individual subsystem layers to fulfill. Because there exists mutual influences between the realization degree of these tasks, and also interactions of control layers, it is not convenient for control engineers to determine how well the performance of local controllers and the plantwide system are. It is important and of practical significance to comprehensively evaluate the overall performance of a hierarchical con-trol system. To assess performance of direct control layer and constraint control layer, the most important components in a hierarchical control system, we proposed in this thesis a framework of benchmarking, assessing and optimizing the direct control and economic performance based on convex optimization technology. Especially, the non-convexity brought in by PID structure of the minimum achievable error variance problem in assessing direct control performance is refor-mulated as a new non-convex constraint with more explicit mathematical structure with respect to PID parameters.The original nonconvex optimization problem is then approximated by a convex optimization which can be easily solved with low-complexity algorithms. The main contributions of this thesis are as following:
1. Minimum achievable output variance (MAOV) is a common benchmark for control per-formance assessment. Finding the MAOV of proportional-integral-derivative (PID) control systems is computationally expensive due to the inherent non-convexity of the associated optimization problem. We present a new method for computing the MAOV of PID control systems. The problem of estimating the MAOV of a PID control system is novelly for-mulated as a convex program with an additional non-convex constraint. The non-convex constraint is linearized and handled by the penalty approach. Based on this, a customized low-complexity algorithm, which relies on the iterative convex programming technique, is developed to solve the MAOV problem. The new algorithm is proved to be convergent. We show via numerical examples that the new approach yields close-to-optimal solutions that are better than (or as good as) the solutions generated by the existing methods.
2. Considering the set-point of direct control layer is often affected by upper constraint control layer, a novel comprehensive performance assessment framework is proposed. Minimum achievable regulatory error variance (MAREV) is used as regulatory performance bench-mark, and minimum integrated absolute tracking error (MIATE) is established as benchmark of tracking performance. A regulatory/tracking performance trade-off curve is obtained to evaluate how balance the total control effect is. The problem of estimating the MAREV of a PID control system is formulated as a quadratic programming (QP) with a bilinear ma-trix inequality (BMI) constraint, and the problem of estimating the MIATE is formulated as a linear programming (LP) with a BMI constraint. Accordingly the problem of searching the balanced tracking/regulatory performance benchmark is formulated as a QP with BMI constraints with respect to PID parameters. These optimization problems are dealt with a standard sequential quadratic programming (SQP) algorithm. Thus facing multiple tasks in direct control layer, a flexible control performance benchmark suited for PID controllers is obtained. We show via numerical examples that compared to commercial software PENBMI which is specially designed for solving BMI problems, the approach we presented yields the same optimal solutions with less time consumption.
3. Batch processes usually have strong nonlinearity or time-varying property, and are often operated in multiple phases. For those batch processes which can be described as linear time varying (LTV) systems, we suggest a new control performance benchmark which is based on the minimum achievable structured residual variance (MASRV). Firstly, for the ease of elaboration, a model called time-varying impulse response matrix is established to describe the LTV systems, which makes the LTV systems own similar formation with LTI systems. Secondly, a duration time of a batch is divided into regulatory and tracking stages according to set-point dynamics, then a structured residual is defined based on the weight matrix of the stage. The problem of estimating the MASRV of LTV system with PID controller is formulated as a non-convex optimization. The fact that the introduced structured matrix is whether invertible or not affects the choice of optimization algorithm. Iterative semidefinit programming (SDP) is selected to handle the convex optimization with a rank one constraint if structured matrix is invertible, otherwise, sequential quadratic programming (SQP) method is used to solve the QP problem with BMI constraints. Numerical simulations study prove that the reformulation of the problem improves the efficiency of optimization.
4. In order to ensure multivariable industrial processes operating in a safe and economic mode, a method for control performance assessment of hierarchical control systems was proposed. The three-layer structure of a hierarchical control system:direct control layer, constraint con-trol layer and real-time optimization layer, was analyzed to formulate the control objective functions of three aspects:suppressing disturbances, keeping constraints and maximizing process profits, respectively. A control performance assessment benchmark called "best to worst performance range" was established to monitor the economic performance of industrial processes, and to evaluate how much potential would be improved. To avoid the degrada-tion of control performance due to model-plant mismatch, a method to compute generalized object model through open loop model and regulatory parameters was presented. The relia-bility and efficacy of the proposed performance assessment technique is demonstrated on a case study on Shell heavy oil fractionator control problem.
引文
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