长时滞采样系统非脆弱控制与滤波问题研究
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摘要
随着数字技术的迅速发展,采样控制技术广泛应用于焊接工业、航空航天、信号处理、地震预测等领域,将采样控制技术和非脆弱控制理论相结合产生了采样系统的非脆弱控制与滤波研究方向，在采样系统中，当采样的对象是具有时滞项的，并且时滞大于一个采样周期，通常被称为长时滞采样系统。对此系统的研究内容可分为稳定性分析、非脆弱控制器设计、观测器设计以及滤波等问题。针对长时滞采样系统的时滞处理方法有分布直接处理方法和集中增广处理方法。本文利用这两种方法，对长时滞采样系统进行了深入研究和讨论，主要工作包括如下几个方面。
首先，在离散域框架下，建立长时滞采样系统模型，在模型建立过程中充分考虑了系统时滞与采样周期的关系，从而将采样系统被控对象的状态长时滞，转化成离散系统的两个连续状态时滞，并对模型中的两个连续状态时滞分别采取集中增广方法和分布直接方法进行处理，并将Lyapunov稳定性理论和线性矩阵不等式技术相结合，提出长时滞采样系统稳定的充分条件，并在分布直接处理方法中，引入自由权矩阵和合同变换矩阵，使得求解稳定充分条件的计算负担减小，并基于线性矩阵不等式的求解，提出了非脆弱控制器的设计方法。
其次，对于已建立的长时滞采样系统模型，当系统的极点位于在单位圆内的圆域时，通过线性变换就可将单位圆内的圆域变换为单位圆，从而，将其系统的D稳定性问题转化为与其等价的渐近稳定问题，通过构建二次Lyapunov函数和求解线性矩阵不等式，提出了长时滞采样系统的D稳定分析方法，和非脆弱控制器设计方法。
再次，对已建立的长时滞采样系统，当系统的状态未知或部分已知时，利用状态反馈已不能解决其系统的分析与综合问题。因此，需要用观测器对系统进行状态重构，基于Lyapunov函数，研究了观测误差系统的稳定性分析和综合问题。在LMI的求解过程中，解决了非线性项与线性项的转化问题，进而，提出了非脆弱观测器和控制器的设计方法，并使其系统满足一定的H∞扰动衰减性能指标。
最后，对已建立的长时滞采样系统，当系统状态未知，并且系统存在非高斯有界噪声时，对系统进行状态估计。通过建立滤波系统的误差模型，并将Lyapunov稳定性理论与线性矩阵不等式技术相结合，提出滤波误差系统稳定的充分条件，并保证系统具有一定的H∞扰动衰减性能和无源性能，并通过求解LMI，分别提出了非脆弱H∞滤波器和非脆弱无源滤波器的设计方法。
通过对长时滞采样系统的各个问题的数值仿真，验证了所提出方法的有效性和可行性。
With rapid development of digital technology, sampled-data control technology has been extensively applied to welding industry, aeronautics and astronautics, signal processing, earthquake prediction and so on. Combing sampled-data control technology and non-fragile control theory yields non-fragile control theory of sampled-data system. In sampled-data system, sampling plant is provided with time-delay and time-delay is longer than a sampling period, which is called by sampled-data with long time-delay. Research content of such system may be divided into stability analysis, design of non-fragile controller, observer and filtering. Aimed to time-delay's processing approach, there are usually two approaches, one is distributed direct approach, the other is concentrated augmented approach. In-depth research and discussion of sampled-data system with long time-delay are presented by making use of two afore-mentioned approaches. This work mainly includes:
Firstly, in the framework of discrete domain, sampled-data system with long time-delay model is established. In the process, relation of system's time-delay and sampling period is taken fully into account, Consequently, State's long time-delay is transformed into two successive state time-delay of discrete-time system. Two successive state time-delay is processed by concentrated augmented approach and distributed direct approach, and then Lyapunov stability theory and LMI technology is combined with LMI technology, and stability condition of sampled-data system with long time-delay are presented. Specially, applying distributed direct approach, free weighting matrix and congruent transformation matrix are introduced, so that computational burden of stability's sufficient condition is less, and based on solving linear matrix inequality, designed approach of non-fragile controller is provided.
Secondly, according to sampled-data system with long time-delay model which has been established, whose pole is located in a round domain inside unit circle. By linear transformation, a round domain inside unit circle is in conversion to unit circle. Thereby,D-stability of system is transformed into equivalent asymptotically stable problem, and then, by constructing quadratic Lyapunov functions and solving linear matrix inequality, analysis approach of D-stability and designed approach of non-fragile controller is presented.
Thirdly, towards sampled-data system with long time-delay model which has been established, when system's state is unknown and partly known, the problem of analysis and synthesis can't be solved by state feedback. Thus, observer is designed to reconstruct state. Based Lyapunov function, stable analysis and synthesis of observe error system is researched. Solving linear matrix inequality, nonlinear term of LMI is in conversion of linear term, thereby, designed approach of non-fragile observer and controller are presented, which ensure H∞performance index of system is satisfied.
Finally, according to sampled-data system with long time-delay model which has been established, whose state is unknown and whose noise is non-gaussian bounded, state estimation is used. By constructing filtering error system, and combining Lyapunov stability theory and linear matrix inequality technique, stable condition of filtering error system is presented and H∞and passive performance index are satisfied. By solving linear matrix inequality, designed approach of non-fragile H∞filter and non-fragile passive filter are proposed respectively.
The simulation examples in every issues of sampled-data system with long timedelay verify the effectiveness and feasibility of the proposed results and methods.
首先，在离散域框架下，建立长时滞采样系统模型，在模型建立过程中充分考虑了系统时滞与采样周期的关系，从而将采样系统被控对象的状态长时滞，转化成离散系统的两个连续状态时滞，并对模型中的两个连续状态时滞分别采取集中增广方法和分布直接方法进行处理，并将Lyapunov稳定性理论和线性矩阵不等式技术相结合，提出长时滞采样系统稳定的充分条件，并在分布直接处理方法中，引入自由权矩阵和合同变换矩阵，使得求解稳定充分条件的计算负担减小，并基于线性矩阵不等式的求解，提出了非脆弱控制器的设计方法。
其次，对于已建立的长时滞采样系统模型，当系统的极点位于在单位圆内的圆域时，通过线性变换就可将单位圆内的圆域变换为单位圆，从而，将其系统的D稳定性问题转化为与其等价的渐近稳定问题，通过构建二次Lyapunov函数和求解线性矩阵不等式，提出了长时滞采样系统的D稳定分析方法，和非脆弱控制器设计方法。
再次，对已建立的长时滞采样系统，当系统的状态未知或部分已知时，利用状态反馈已不能解决其系统的分析与综合问题。因此，需要用观测器对系统进行状态重构，基于Lyapunov函数，研究了观测误差系统的稳定性分析和综合问题。在LMI的求解过程中，解决了非线性项与线性项的转化问题，进而，提出了非脆弱观测器和控制器的设计方法，并使其系统满足一定的H∞扰动衰减性能指标。
最后，对已建立的长时滞采样系统，当系统状态未知，并且系统存在非高斯有界噪声时，对系统进行状态估计。通过建立滤波系统的误差模型，并将Lyapunov稳定性理论与线性矩阵不等式技术相结合，提出滤波误差系统稳定的充分条件，并保证系统具有一定的H∞扰动衰减性能和无源性能，并通过求解LMI，分别提出了非脆弱H∞滤波器和非脆弱无源滤波器的设计方法。
通过对长时滞采样系统的各个问题的数值仿真，验证了所提出方法的有效性和可行性。
With rapid development of digital technology, sampled-data control technology has been extensively applied to welding industry, aeronautics and astronautics, signal processing, earthquake prediction and so on. Combing sampled-data control technology and non-fragile control theory yields non-fragile control theory of sampled-data system. In sampled-data system, sampling plant is provided with time-delay and time-delay is longer than a sampling period, which is called by sampled-data with long time-delay. Research content of such system may be divided into stability analysis, design of non-fragile controller, observer and filtering. Aimed to time-delay's processing approach, there are usually two approaches, one is distributed direct approach, the other is concentrated augmented approach. In-depth research and discussion of sampled-data system with long time-delay are presented by making use of two afore-mentioned approaches. This work mainly includes:
Firstly, in the framework of discrete domain, sampled-data system with long time-delay model is established. In the process, relation of system's time-delay and sampling period is taken fully into account, Consequently, State's long time-delay is transformed into two successive state time-delay of discrete-time system. Two successive state time-delay is processed by concentrated augmented approach and distributed direct approach, and then Lyapunov stability theory and LMI technology is combined with LMI technology, and stability condition of sampled-data system with long time-delay are presented. Specially, applying distributed direct approach, free weighting matrix and congruent transformation matrix are introduced, so that computational burden of stability's sufficient condition is less, and based on solving linear matrix inequality, designed approach of non-fragile controller is provided.
Secondly, according to sampled-data system with long time-delay model which has been established, whose pole is located in a round domain inside unit circle. By linear transformation, a round domain inside unit circle is in conversion to unit circle. Thereby,D-stability of system is transformed into equivalent asymptotically stable problem, and then, by constructing quadratic Lyapunov functions and solving linear matrix inequality, analysis approach of D-stability and designed approach of non-fragile controller is presented.
Thirdly, towards sampled-data system with long time-delay model which has been established, when system's state is unknown and partly known, the problem of analysis and synthesis can't be solved by state feedback. Thus, observer is designed to reconstruct state. Based Lyapunov function, stable analysis and synthesis of observe error system is researched. Solving linear matrix inequality, nonlinear term of LMI is in conversion of linear term, thereby, designed approach of non-fragile observer and controller are presented, which ensure H∞performance index of system is satisfied.
Finally, according to sampled-data system with long time-delay model which has been established, whose state is unknown and whose noise is non-gaussian bounded, state estimation is used. By constructing filtering error system, and combining Lyapunov stability theory and linear matrix inequality technique, stable condition of filtering error system is presented and H∞and passive performance index are satisfied. By solving linear matrix inequality, designed approach of non-fragile H∞filter and non-fragile passive filter are proposed respectively.
The simulation examples in every issues of sampled-data system with long timedelay verify the effectiveness and feasibility of the proposed results and methods.
引文
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