不确定系统分析与综合及其在飞行控制中的应用
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摘要
自二十世纪50年代末现代控制理论诞生以来,控制理论得到了飞速的发展。现代控制理论中的许多结果都是基于一个精确的数学模型。然而,实际系统不可避免地要遇到各种不确定性,既包括系统本身的不确定性,诸如未建模动态、结构性的参数不确定、工作环境的变化、降阶及线性化近似等；又包括外部干扰的不确定性,如一般统计特性未知但能量有界等情形。另一方面,在实际的工业过程中,大惯性环节,传输过程以及复杂的在线分析通常会导致滞后现象,而这些滞后特性会严重影响系统的稳定性以及性能指标。因此,具有或不具有时滞的不确定系统的稳定性问题已经成为控制界的一个研究热点。
本文利用Lyapunov-Krasovskii泛函方法,以线性矩阵不等式(LMI)技术为工具,研究具有或不具有时滞的不确定系统分析与综合问题,并针对飞行控制的具体问题加以应用。论文主要包括以下内容：
利用LMI方法研究多面体系统指数非脆弱控制及其在飞行控制中的应用问题。首先,利用参数相关Lyapunov函数和描述模型变换给出了系统鲁棒非脆弱控制器存在的充分条件。其次,在LMI框架下分别给出高阶、低阶多工作点飞行器的俯仰轴操纵品质准则。最后,基于上述结果,分别在切换策略和多目标策略下设计了由前馈控制器和状态反馈控制器组成的非脆弱复合控制器。该控制器能够同时满足操纵品质指标和鲁棒指数稳定性要求。引入凸优化算法,求取影响飞行器响应灵敏度和LMI可行性的参数调整范围。仿真结果表明,得到的控制器能够满足设计要求。
基于Lyapunov-Krasovskii泛函研究不确定时变时滞系统的鲁棒稳定性及其镇定问题。首先,利用LMI技术得到了系统时滞—时滞导数相关鲁棒稳定性条件,该稳定性条件的保守性较小,能够允许更大的时滞上界。其次,针对含有飞行时滞的垂直起飞着陆(VTOL)直升机,利用所得结果设计了考虑控制器不确定性和外部扰动的时滞相关鲁棒非脆弱H∞控制器。仿真结果表明,所设计的控制器具有良好的鲁棒性和非脆弱性,能够适应实际更加恶劣的飞行环境。
利用LMI方法研究多面体时滞系统的鲁棒稳定性分析与综合问题。首先,在参数相关Lyapunov-Krasovskii泛函和描述模型变换的基础上,分别引入自由加权矩阵方法和时滞分割技术建立LMI形式的时滞相关鲁棒稳定性条件。得到的稳定性条件保守性较小,具有较小的计算负担。其次,针对具有飞行时滞的多工作点飞行器,在考虑执行器故障的情况下,利用上述结果和容错技术设计了时滞相关鲁棒可靠控制器。仿真结果表明,所设计的控制器在执行器发生故障情况下仍能确保系统的鲁棒稳定性。
基于考虑时滞概率分布特征的随机变量模型,研究了具有随机时滞的不确定系统鲁棒稳定性问题。通过适当的Lyapunov-Krasovskii泛函和积分不等式给出系统时滞及其分布相关的鲁棒稳定性条件。在不影响结果保守性的前提下,条件的推导无需引入自由权值矩阵,简化了条件并相应地减小了计算负担。此外,引入描述模型变换方法使条件的保守性进一步降低。数值算例验证了提出方法的有效性。
针对不确定区间时滞系统,基于随机变量模型研究了考虑时滞概率分布特征的鲁棒H∞控制问题。首先,通过具有时滞界分割特点的Lyapunov-Krasovskii泛函和积分不等式建立了保守性较小的时滞分布相关鲁棒稳定性条件。对于下界为零的一般时滞,该条件仍然具有较小的保守性。在此基础之上,给出了时滞分布相关的鲁棒H∞性能准则和相应的H∞控制器设计方法。数值算例验证了提出方法的有效性。
最后对全文进行总结,并提出今后需要进一步研究的方向。
Since the naissance of the modern control theory at the end of 19'50, the control theory has got rapid development. Many results of the modern control theory are based on an accurate math model. However, inevitably practical systems suffer different uncertainties which include either model imperfections, such as unmodeled dynamics, structured parametric uncertainties, changes of the operating environment, model reduction and linearization approximations, etc., or external disturbances, for examples, various disturbances with unknown statistical characteristic and bounded energy. On the other hand, large-lag links, transmission process and complicated on-line analysis usually lead to delay phenomenon in practical industrial processes, which influences stability and performance index of the system seriously. Therefore, the stability problem of uncertain systems with or without delay has been a hot topic in the control field.
Based on Lyapunov-Krasovskii functional method and linear matrix inequality(LMI) techniques, this thesis investigates the problem of analysis and synthesis for uncertain systems with or without delay and applies it to concrete problems of flight control. The main works of this thesis are as follows:
By virtue of LMI approach, the problems of non-fragile and exponential control for polytopic systems and its application to flight control are studied. Firstly, a sufficient condition for the existence of a robust non-fragile controller is given based on parameter-dependent Lyapunov functions combined with a descriptor system approach. Secondly, in the framework of LMI, handling quality criteria for pitch axis are derived for high and low order aircraft with multiple operating points, respectively. At last, on the basis of the results above, a compound and non-fragile controller, which consists of a feed-forward controller and a state feedback controller, is designed under the switch strategy and the multi-objective strategy, respectively. This controller can satisfy both handling quality index and robust exponential stability requirement simultaneously. The adjusting range of a parameter that influences the sensitivity of aircraft responses and feasibility of LMIs is obtained by introducing convex optimization algorithms. Simulation results show that the obtained controller can meet design requirements.
The problems of robust stability analysis and stabilization for uncertain systems with time-varying delay are investigated based on Lyapunov-Krasovskii functional. Firstly, a delay-dependent and rate-dependent robust stability criterion of the system is obtained in terms of LMI techniques. This stability criterion is less conservative and can ensure a larger upper bound of time delay. Secondly, in consideration of controller uncertainties and external disturbances, a delay-dependent robust non-fragile H∞controller is designed for a vertical take-off and landing (VTOL) helicopter with flight delay by taking advatages of obtained results. Simulation results show that the designed controller has good robust and non-fragile performance and is suitable for practical abominable flight environment.
The problem of robust stability analysis and synthesis for a polytopic system with time-delay is investigated by LMI approach. Firstly, on the basis of parameter-dependent Lyapunov-Krasovskii functional and a descriptor model transformation, free-weighting matrix method and delay partitioning technique are introduced to establish a delay-dependent robust stability criterion in terms of LMI, repectively. This stability criterion is less conservative and has less computational burden. Secondly, in consideration of actuator faults, a delay-dependent robust reliable controller is designed for an aircraft with multiple operating points and flight delay by means of the results above and fault tolerent techniques. Simulation results demonstrte that in the case of actuator faults, the designed controller can still secure robust stability of the system.
The problem of robust stability for an uncertain system with random time-delay is investigated based on the stochastic variable model with probability ditribution characteristics of dealy. A delay-dependent and delay-distribution dependent robust stability criterion of the system is derived by using an appropriate Lyapunov-Krasovskii functional and integral inequality. Without exercising any influence on conservatism of the result, no free-weighting matrix is needed in the derivation of the condition, which simplifies the result and reduces computational burden accordingly. In addtion, conservatism of the condition is reduced further by introducing a descriptor model transformation approach. Numerical examples illustrate the effectiveness of the proposed method.
In consideration of probability ditribution feature of dealy, the problem of robust H∞control for an uncertain system with interval delay is investegated based on the stochastic variable model. Firstly, a less conservative delay-distribution dependent robust stability criterion is established by integral inequality and Lyapunov-Krasovskii functional characterized by delay bound partitioning. For the routine delay whose low bound is zero, this criterion is still less conservative. On the basis of this, a delay-distribution dependent H∞performance criterion and the design method of the corresponding H∞controller are given. Numerical examples illustrate the effectiveness of the proposed method.
Finally, the conclusions of the dissertation are drawn and further research directions are put forward.
本文利用Lyapunov-Krasovskii泛函方法,以线性矩阵不等式(LMI)技术为工具,研究具有或不具有时滞的不确定系统分析与综合问题,并针对飞行控制的具体问题加以应用。论文主要包括以下内容：
利用LMI方法研究多面体系统指数非脆弱控制及其在飞行控制中的应用问题。首先,利用参数相关Lyapunov函数和描述模型变换给出了系统鲁棒非脆弱控制器存在的充分条件。其次,在LMI框架下分别给出高阶、低阶多工作点飞行器的俯仰轴操纵品质准则。最后,基于上述结果,分别在切换策略和多目标策略下设计了由前馈控制器和状态反馈控制器组成的非脆弱复合控制器。该控制器能够同时满足操纵品质指标和鲁棒指数稳定性要求。引入凸优化算法,求取影响飞行器响应灵敏度和LMI可行性的参数调整范围。仿真结果表明,得到的控制器能够满足设计要求。
基于Lyapunov-Krasovskii泛函研究不确定时变时滞系统的鲁棒稳定性及其镇定问题。首先,利用LMI技术得到了系统时滞—时滞导数相关鲁棒稳定性条件,该稳定性条件的保守性较小,能够允许更大的时滞上界。其次,针对含有飞行时滞的垂直起飞着陆(VTOL)直升机,利用所得结果设计了考虑控制器不确定性和外部扰动的时滞相关鲁棒非脆弱H∞控制器。仿真结果表明,所设计的控制器具有良好的鲁棒性和非脆弱性,能够适应实际更加恶劣的飞行环境。
利用LMI方法研究多面体时滞系统的鲁棒稳定性分析与综合问题。首先,在参数相关Lyapunov-Krasovskii泛函和描述模型变换的基础上,分别引入自由加权矩阵方法和时滞分割技术建立LMI形式的时滞相关鲁棒稳定性条件。得到的稳定性条件保守性较小,具有较小的计算负担。其次,针对具有飞行时滞的多工作点飞行器,在考虑执行器故障的情况下,利用上述结果和容错技术设计了时滞相关鲁棒可靠控制器。仿真结果表明,所设计的控制器在执行器发生故障情况下仍能确保系统的鲁棒稳定性。
基于考虑时滞概率分布特征的随机变量模型,研究了具有随机时滞的不确定系统鲁棒稳定性问题。通过适当的Lyapunov-Krasovskii泛函和积分不等式给出系统时滞及其分布相关的鲁棒稳定性条件。在不影响结果保守性的前提下,条件的推导无需引入自由权值矩阵,简化了条件并相应地减小了计算负担。此外,引入描述模型变换方法使条件的保守性进一步降低。数值算例验证了提出方法的有效性。
针对不确定区间时滞系统,基于随机变量模型研究了考虑时滞概率分布特征的鲁棒H∞控制问题。首先,通过具有时滞界分割特点的Lyapunov-Krasovskii泛函和积分不等式建立了保守性较小的时滞分布相关鲁棒稳定性条件。对于下界为零的一般时滞,该条件仍然具有较小的保守性。在此基础之上,给出了时滞分布相关的鲁棒H∞性能准则和相应的H∞控制器设计方法。数值算例验证了提出方法的有效性。
最后对全文进行总结,并提出今后需要进一步研究的方向。
Since the naissance of the modern control theory at the end of 19'50, the control theory has got rapid development. Many results of the modern control theory are based on an accurate math model. However, inevitably practical systems suffer different uncertainties which include either model imperfections, such as unmodeled dynamics, structured parametric uncertainties, changes of the operating environment, model reduction and linearization approximations, etc., or external disturbances, for examples, various disturbances with unknown statistical characteristic and bounded energy. On the other hand, large-lag links, transmission process and complicated on-line analysis usually lead to delay phenomenon in practical industrial processes, which influences stability and performance index of the system seriously. Therefore, the stability problem of uncertain systems with or without delay has been a hot topic in the control field.
Based on Lyapunov-Krasovskii functional method and linear matrix inequality(LMI) techniques, this thesis investigates the problem of analysis and synthesis for uncertain systems with or without delay and applies it to concrete problems of flight control. The main works of this thesis are as follows:
By virtue of LMI approach, the problems of non-fragile and exponential control for polytopic systems and its application to flight control are studied. Firstly, a sufficient condition for the existence of a robust non-fragile controller is given based on parameter-dependent Lyapunov functions combined with a descriptor system approach. Secondly, in the framework of LMI, handling quality criteria for pitch axis are derived for high and low order aircraft with multiple operating points, respectively. At last, on the basis of the results above, a compound and non-fragile controller, which consists of a feed-forward controller and a state feedback controller, is designed under the switch strategy and the multi-objective strategy, respectively. This controller can satisfy both handling quality index and robust exponential stability requirement simultaneously. The adjusting range of a parameter that influences the sensitivity of aircraft responses and feasibility of LMIs is obtained by introducing convex optimization algorithms. Simulation results show that the obtained controller can meet design requirements.
The problems of robust stability analysis and stabilization for uncertain systems with time-varying delay are investigated based on Lyapunov-Krasovskii functional. Firstly, a delay-dependent and rate-dependent robust stability criterion of the system is obtained in terms of LMI techniques. This stability criterion is less conservative and can ensure a larger upper bound of time delay. Secondly, in consideration of controller uncertainties and external disturbances, a delay-dependent robust non-fragile H∞controller is designed for a vertical take-off and landing (VTOL) helicopter with flight delay by taking advatages of obtained results. Simulation results show that the designed controller has good robust and non-fragile performance and is suitable for practical abominable flight environment.
The problem of robust stability analysis and synthesis for a polytopic system with time-delay is investigated by LMI approach. Firstly, on the basis of parameter-dependent Lyapunov-Krasovskii functional and a descriptor model transformation, free-weighting matrix method and delay partitioning technique are introduced to establish a delay-dependent robust stability criterion in terms of LMI, repectively. This stability criterion is less conservative and has less computational burden. Secondly, in consideration of actuator faults, a delay-dependent robust reliable controller is designed for an aircraft with multiple operating points and flight delay by means of the results above and fault tolerent techniques. Simulation results demonstrte that in the case of actuator faults, the designed controller can still secure robust stability of the system.
The problem of robust stability for an uncertain system with random time-delay is investigated based on the stochastic variable model with probability ditribution characteristics of dealy. A delay-dependent and delay-distribution dependent robust stability criterion of the system is derived by using an appropriate Lyapunov-Krasovskii functional and integral inequality. Without exercising any influence on conservatism of the result, no free-weighting matrix is needed in the derivation of the condition, which simplifies the result and reduces computational burden accordingly. In addtion, conservatism of the condition is reduced further by introducing a descriptor model transformation approach. Numerical examples illustrate the effectiveness of the proposed method.
In consideration of probability ditribution feature of dealy, the problem of robust H∞control for an uncertain system with interval delay is investegated based on the stochastic variable model. Firstly, a less conservative delay-distribution dependent robust stability criterion is established by integral inequality and Lyapunov-Krasovskii functional characterized by delay bound partitioning. For the routine delay whose low bound is zero, this criterion is still less conservative. On the basis of this, a delay-distribution dependent H∞performance criterion and the design method of the corresponding H∞controller are given. Numerical examples illustrate the effectiveness of the proposed method.
Finally, the conclusions of the dissertation are drawn and further research directions are put forward.
引文
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