基于有限精度求解的非线性预测控制算法研究
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摘要
非线性预测控制算法具有很高的实时性要求,需要反复进行最优控制命题的在线求解,并且要在一个采样周期内得到合适的解,反馈于被控对象。然而该类命题往往是带有复杂约束条件的大规模非线性规划问题,快速求解非常困难；而求解耗时长、计算代价大产生的计算延时,使控制作用不能及时反馈,最终导致控制器性能下降,甚至破坏闭环稳定性。为缩短计算时间,降低计算延时,本文从优化算法角度对非线性预测控制算法进行了研究。
本文的主要研究工作有以下几方面：
1)提出了能够显著提高计算效率的有限精度求解(reduced precision solution,RPS)准则。动态优化问题无法快速求解是当前基于非线性机理模型的预测控制算法发展的瓶颈问题之一。现有的优化算法均是基于传统的Himmelblau收敛准则。这类刚性准则,仅能给出“收敛”和“不收敛”两种结论,不能有效地反映求解的进展程度,常常使得迭代过程进入到计算代价远大于解值改善程度的阶段,从而导致求解缓慢甚至最终收敛失败。本文提出RPS准则,对求解过程进行指标量化,通过衡量求解的收敛程度及改进空间来判断是否停止迭代。数值实验表明将RPS准则嵌入SQP算法,可以快速求解带有非线性微分方程约束的最优控制问题。
2)提出了基于RPS准则的非线性预测控制算法——rps-NMPC。当求解过程被提前终止时,得到的解是否满足约束,是否可以作为有效的控制作用,反馈到被控对象,是值得考虑的。本文提出了基于RPS准则的可行摄动序列二次规划(feasibility perturbed sequential quadratic programming, FP-SQP)算法,并将其应用于非线性预测控制。该方法保证在求解终止时得到次优的可行解,且可以作为控制作用反馈到被控系统。仿真实验表明该方法能够有效缩短计算时间,改善因计算延时造成的控制性能下降。
3)基于输入状态稳定性(input-to-state stability,ISS)理论,本文对采用RPS准则的NMPC (rps-NMPC)作用下的系统ISS稳定性进行了详细论证,就次优解对稳定性的影响进行了理论分析。通过数值实验,说明了当存在模型失配及外部扰动时,rps-NMPC体现了良好的控制性能。
4)提出了基于小波去噪的滚动时域估计(]moving horizon estimation, MHE)算法。考虑到实际中存在模型失配、外部扰动以及测量噪声,MHE是有效的状态估计方法,但是估计效果受到测量数据品质好坏的影响。通过小波分析方法对带噪测量数据进行去噪预处理,并将处理后的数据传递给MHE估计器进行计算,可以有效改善估计效果。仿真实验验证了本方法的有效性。
5)提出了基于RPS准则的滚动时域估计(moving horizon estimation, MHE)算法。针对计算延时对状态估计问题求解产生的负面影响,本文将RPS准则应用到MHE的求解中。实验表明该方法可以缩短求解时间,估计结果却并没有因为求解精度的降低而遭到严重破坏。实验结果表明基于RPS准则的MHE算法在降低计算消耗的同时,能够及时得到较好的状态估计值。
Nonlinear model predictive control (NMPC) with high real-time requirements requires repeatedly solving the optimal control problems (OCPs) in one sampling interval, so that the input can be injected into the controlled system. It is difficult to solve the nonlinear OCPs with differential equations as its constraints timely. Computational delay arising in the solution procedure may destroy the controller performance even the stability of the closed loop. To shorten the computation time and reduce the computation delay, the paper studies NMPC algorithm from the perspective of optimization algorithm. The main research work are the following:
1) A kind of termination criteria named Reduced Precision Solution (RPS) Criteria is proposed to reduce the computation time for solving the nonlinear OCPs. Solving the dynamic optimization problem is one of the bottleneck problem for development of NMPC algorithm based on nonlinear first-principle model equations. Most of existing optimization algorithms are based on the traditional convergence criteria, that is, Himmelblau rules. This is a kind of rigid criteria, which can only give "convergence" and "not convergence" as conclusions and can not effectively reflect the extent of convergence and margin of improvement. By defining a series of indices to measure the degree of convergence and the margin of improvement, RPS criteria can decide whether to iterate deeply. The numerical experiments illustrate that the SQP algorithm with RPS criteria can quickly solve the OCPs with good approximate solution.
2) The NMPC framework based on feasibility-perturbed sequential quadratic programming (FP-SQP) algorithm with RPS criteria is proposed. When the optimization procedure of OCPs is required to stop prematurely, the obtained iterate should be feasible so that can be used as input acting on system. The FP-SQP with RPS criteria is suitable to deal with this problem. Simulation results show that this method can save computation time, reduce the computational delay, and improve the control performance eventually;
3) Based on input-to-state stability (ISS) theory, the stability of NMPC with RPS criteria (rps-NMPC) is proved theoretically. Moreover, the effect of the reduced precision suboptimal solution on the stability is also analyzed. The simulations illustrate that the rps-NMPC owns robustness and stability when model mismatch and external disturbances exist.
4) Moving horizon estimation (MHE) algorithm based on wavelet denoising is proposed. Taking the fact that there are external disturbances, model mismatch and the impact of noise in the measurement process into account, MHE is an effective method for state estimation. However, the estimation results are affected by the quality of the measurement data. In this paper, measurement data de-noising based on wavelet analysis is proposed to eliminate the noise and get more accurate output infomation, then the de-noised data is passed to the MHE estimator to get the state estimation. Experiments show that this method can effectively reduce the adverse effect caused by noise on the measured signals.
5) MHE algorithm based on RPS criteria is proposed to get good estimation by reducing computational delay. MHE problem needs to be solved in one sampling interval, long computation time makes the state estimation can not be passed to the controller timely, thereby degrading the controller performance. Therefore, the RPS criteria are used to solve the nonlinear MHE problem, the experiments show that this method can reduce the solution time, and the state estimation doesn't get worse. Finally, simulation results illustrate that the computation time is reduced and proper state estimation is obtained simultaneously.
本文的主要研究工作有以下几方面：
1)提出了能够显著提高计算效率的有限精度求解(reduced precision solution,RPS)准则。动态优化问题无法快速求解是当前基于非线性机理模型的预测控制算法发展的瓶颈问题之一。现有的优化算法均是基于传统的Himmelblau收敛准则。这类刚性准则,仅能给出“收敛”和“不收敛”两种结论,不能有效地反映求解的进展程度,常常使得迭代过程进入到计算代价远大于解值改善程度的阶段,从而导致求解缓慢甚至最终收敛失败。本文提出RPS准则,对求解过程进行指标量化,通过衡量求解的收敛程度及改进空间来判断是否停止迭代。数值实验表明将RPS准则嵌入SQP算法,可以快速求解带有非线性微分方程约束的最优控制问题。
2)提出了基于RPS准则的非线性预测控制算法——rps-NMPC。当求解过程被提前终止时,得到的解是否满足约束,是否可以作为有效的控制作用,反馈到被控对象,是值得考虑的。本文提出了基于RPS准则的可行摄动序列二次规划(feasibility perturbed sequential quadratic programming, FP-SQP)算法,并将其应用于非线性预测控制。该方法保证在求解终止时得到次优的可行解,且可以作为控制作用反馈到被控系统。仿真实验表明该方法能够有效缩短计算时间,改善因计算延时造成的控制性能下降。
3)基于输入状态稳定性(input-to-state stability,ISS)理论,本文对采用RPS准则的NMPC (rps-NMPC)作用下的系统ISS稳定性进行了详细论证,就次优解对稳定性的影响进行了理论分析。通过数值实验,说明了当存在模型失配及外部扰动时,rps-NMPC体现了良好的控制性能。
4)提出了基于小波去噪的滚动时域估计(]moving horizon estimation, MHE)算法。考虑到实际中存在模型失配、外部扰动以及测量噪声,MHE是有效的状态估计方法,但是估计效果受到测量数据品质好坏的影响。通过小波分析方法对带噪测量数据进行去噪预处理,并将处理后的数据传递给MHE估计器进行计算,可以有效改善估计效果。仿真实验验证了本方法的有效性。
5)提出了基于RPS准则的滚动时域估计(moving horizon estimation, MHE)算法。针对计算延时对状态估计问题求解产生的负面影响,本文将RPS准则应用到MHE的求解中。实验表明该方法可以缩短求解时间,估计结果却并没有因为求解精度的降低而遭到严重破坏。实验结果表明基于RPS准则的MHE算法在降低计算消耗的同时,能够及时得到较好的状态估计值。
Nonlinear model predictive control (NMPC) with high real-time requirements requires repeatedly solving the optimal control problems (OCPs) in one sampling interval, so that the input can be injected into the controlled system. It is difficult to solve the nonlinear OCPs with differential equations as its constraints timely. Computational delay arising in the solution procedure may destroy the controller performance even the stability of the closed loop. To shorten the computation time and reduce the computation delay, the paper studies NMPC algorithm from the perspective of optimization algorithm. The main research work are the following:
1) A kind of termination criteria named Reduced Precision Solution (RPS) Criteria is proposed to reduce the computation time for solving the nonlinear OCPs. Solving the dynamic optimization problem is one of the bottleneck problem for development of NMPC algorithm based on nonlinear first-principle model equations. Most of existing optimization algorithms are based on the traditional convergence criteria, that is, Himmelblau rules. This is a kind of rigid criteria, which can only give "convergence" and "not convergence" as conclusions and can not effectively reflect the extent of convergence and margin of improvement. By defining a series of indices to measure the degree of convergence and the margin of improvement, RPS criteria can decide whether to iterate deeply. The numerical experiments illustrate that the SQP algorithm with RPS criteria can quickly solve the OCPs with good approximate solution.
2) The NMPC framework based on feasibility-perturbed sequential quadratic programming (FP-SQP) algorithm with RPS criteria is proposed. When the optimization procedure of OCPs is required to stop prematurely, the obtained iterate should be feasible so that can be used as input acting on system. The FP-SQP with RPS criteria is suitable to deal with this problem. Simulation results show that this method can save computation time, reduce the computational delay, and improve the control performance eventually;
3) Based on input-to-state stability (ISS) theory, the stability of NMPC with RPS criteria (rps-NMPC) is proved theoretically. Moreover, the effect of the reduced precision suboptimal solution on the stability is also analyzed. The simulations illustrate that the rps-NMPC owns robustness and stability when model mismatch and external disturbances exist.
4) Moving horizon estimation (MHE) algorithm based on wavelet denoising is proposed. Taking the fact that there are external disturbances, model mismatch and the impact of noise in the measurement process into account, MHE is an effective method for state estimation. However, the estimation results are affected by the quality of the measurement data. In this paper, measurement data de-noising based on wavelet analysis is proposed to eliminate the noise and get more accurate output infomation, then the de-noised data is passed to the MHE estimator to get the state estimation. Experiments show that this method can effectively reduce the adverse effect caused by noise on the measured signals.
5) MHE algorithm based on RPS criteria is proposed to get good estimation by reducing computational delay. MHE problem needs to be solved in one sampling interval, long computation time makes the state estimation can not be passed to the controller timely, thereby degrading the controller performance. Therefore, the RPS criteria are used to solve the nonlinear MHE problem, the experiments show that this method can reduce the solution time, and the state estimation doesn't get worse. Finally, simulation results illustrate that the computation time is reduced and proper state estimation is obtained simultaneously.
引文
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