Markov跳变时滞It（？）随机微分系统的稳定性与鲁棒控制
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摘要
当系统结构和参数遭遇突变时,如元器件的损坏或修复、子系统关联结构改变和突然的环境变换,通常用连续时间Markov链驱动的复杂系统来进行描述和分析。在实际的系统中,时滞和不确定性现象是难以避免的,而且常常会导致系统不稳定和系统性能变坏。此外,由于实际系统的工作环境,外界环境噪声是客观存在的,因此建模时就必须考虑这类噪声因素。因此,关于Markov跳变时滞随机微分系统的鲁棒分析与控制问题的研究,不仅具有重要的理论意义,同时也具有重要的实际价值。另一方面,由于许多实际系统中的时滞一般都是有界的,无穷时滞很少出现,因此,不考虑时滞大小的结论,当时滞比较小时,是相对保守的。本文系统地研究了Markov跳变时滞It(o|^)随机微分系统的时滞无关/相关的稳定性、鲁棒性和鲁棒控制等问题。主要创新性研究成果如下:
1、研究了Markov跳变时滞It(o|^)随机微分系统的均方意义下的随机指数稳定性和几乎必然指数稳定性。对于时滞无关型判据,利用“小标量”方法,引入了更多的参量矩阵,从而降低此类判据的保守性;同时,应用不同的Lyapunov函数构造方法,提出了此类系统的时滞相关均方意义下随机指数稳定性判据。
2、研究了Markov跳变中立型It(o|^)随机微分系统的随机均方指数稳定性和几乎必然指数稳定性。根据此类系统的特殊结构,选取相应的Lyapunov函数,应用推广的广义It(o|^)公式,给出了此类系统的时滞无关与时滞相关的均方意义下随机指数稳定性和几乎必然指数稳定性的判据。
3、研究了非线性Markov跳变It(o|^)随机微分系统的均方意义下随机指数稳定性问题。分别针对非时滞、时滞和中立型随机系统,利用Grown-Wall不等式和广义It(o|^)公式等工具,给出了非线性跳变It(o|^)随机微分系统的条件更为宽松的均方意义下随机指数稳的性判据。
4、研究了Markov跳变时滞It(o|^)随机微分系统的鲁棒控制问题。基于稳定性分析,给出了此类系统的时滞无关与时滞相关的镇定控制器设计方法;同时,于随机二次最优(SLQ)理论,研究了基于SLQ最优控制器的镇定问题。对于范数有界不确定性,基于不同的Lyapunov函数,给出了系统的具有鲁棒H_∞性能时滞相关判据,及相应的控制器设计方法。最后,提出并解决了此类随机系统的鲁棒保性能控制问题,给出了对应控制器设计方法。
The hybrid systems driven by continuous-time Markov chains have been usedto model many practical systems, where they may experience abrupt changes in theirstructure and parameters, such as component failures or repairs, changing subsysteminterconnections, and abrupt environmental disturbances. Moreover, it has been wellrecognized that time-delay and uncertainties cannot be avoided in practice and oftenresults in instability and poor performance. Besides, because the environmental noiseactually exists, it is necessary to take this element into account. Therefore, robustanalysis and control for delay It(o|^) stochastic differential systems with Markov jumpingare very important in control theory and its applications. On the other hand, sincethe delay of many practical systems is bounded and the infinite delay is very less,for the case of the small delay, the results which does not consider the length of thedelay are more conservative. This dissertation deals systematically with the delay-dependent stability analysis, robust control and design filtering of concern systems.The contributions can be concluded as follows:
1. The stochastic exponential stability in the mean square and the almost-surestability of delay It(o|^) stochastic differential systems with Markov jumping are investi-gated. For the delay-independent criterion, based on”small scalar method”, some suf-ficient criteria are obtained with less conservativeness by introducing more parametri-cal matrices; And some sufficient delay-dependent criteria for stochastic exponentialstability are obtained by constructing different stochastic Lyapunov functionals.
2. The stochastic exponential stability in the mean square and the almost-surestability of It(o|^) stochastic differential systems of neutral type with Markov jumping areinvestigated. Based on the special structure of such systems, applying the generalizedIt(o|^) formula, the criterion for the stochastic exponential stability in the mean squareand the almost-sure stability of It(o|^) stochastic differential systems of neutral type withMarkov jumping are obtained by constructing Lyapunov functional in term of LMIs.
3. The stochastic exponential stability in the mean-square sense of nonlinear It(o|^)stochastic differential systems with Markov jumping are investigated. For the case ofnon-delay, delay and neutral type, some new criteria are given based on Grown-Wall inequality and generalized It(o|^) formula, which improve the existed results.
4. The robust control problem of delay It(o|^) stochastic differential systems withMarkov jumping is investigated. Based on the stability analysis, the approach of sta-bilization controller in the delay-independent and delay-dependent case is obtained.And based on the SLQ theory, the stabilization problem based on optimal SLQ con-troller is discussed. For norm-bounded uncertainty, the problem of robust H_∞analysisand the design approach of robust H_∞controller are investigated. And the problemof robust guaranteed cost control of such systems is proposed and solved.
1、研究了Markov跳变时滞It(o|^)随机微分系统的均方意义下的随机指数稳定性和几乎必然指数稳定性。对于时滞无关型判据,利用“小标量”方法,引入了更多的参量矩阵,从而降低此类判据的保守性;同时,应用不同的Lyapunov函数构造方法,提出了此类系统的时滞相关均方意义下随机指数稳定性判据。
2、研究了Markov跳变中立型It(o|^)随机微分系统的随机均方指数稳定性和几乎必然指数稳定性。根据此类系统的特殊结构,选取相应的Lyapunov函数,应用推广的广义It(o|^)公式,给出了此类系统的时滞无关与时滞相关的均方意义下随机指数稳定性和几乎必然指数稳定性的判据。
3、研究了非线性Markov跳变It(o|^)随机微分系统的均方意义下随机指数稳定性问题。分别针对非时滞、时滞和中立型随机系统,利用Grown-Wall不等式和广义It(o|^)公式等工具,给出了非线性跳变It(o|^)随机微分系统的条件更为宽松的均方意义下随机指数稳的性判据。
4、研究了Markov跳变时滞It(o|^)随机微分系统的鲁棒控制问题。基于稳定性分析,给出了此类系统的时滞无关与时滞相关的镇定控制器设计方法;同时,于随机二次最优(SLQ)理论,研究了基于SLQ最优控制器的镇定问题。对于范数有界不确定性,基于不同的Lyapunov函数,给出了系统的具有鲁棒H_∞性能时滞相关判据,及相应的控制器设计方法。最后,提出并解决了此类随机系统的鲁棒保性能控制问题,给出了对应控制器设计方法。
The hybrid systems driven by continuous-time Markov chains have been usedto model many practical systems, where they may experience abrupt changes in theirstructure and parameters, such as component failures or repairs, changing subsysteminterconnections, and abrupt environmental disturbances. Moreover, it has been wellrecognized that time-delay and uncertainties cannot be avoided in practice and oftenresults in instability and poor performance. Besides, because the environmental noiseactually exists, it is necessary to take this element into account. Therefore, robustanalysis and control for delay It(o|^) stochastic differential systems with Markov jumpingare very important in control theory and its applications. On the other hand, sincethe delay of many practical systems is bounded and the infinite delay is very less,for the case of the small delay, the results which does not consider the length of thedelay are more conservative. This dissertation deals systematically with the delay-dependent stability analysis, robust control and design filtering of concern systems.The contributions can be concluded as follows:
1. The stochastic exponential stability in the mean square and the almost-surestability of delay It(o|^) stochastic differential systems with Markov jumping are investi-gated. For the delay-independent criterion, based on”small scalar method”, some suf-ficient criteria are obtained with less conservativeness by introducing more parametri-cal matrices; And some sufficient delay-dependent criteria for stochastic exponentialstability are obtained by constructing different stochastic Lyapunov functionals.
2. The stochastic exponential stability in the mean square and the almost-surestability of It(o|^) stochastic differential systems of neutral type with Markov jumping areinvestigated. Based on the special structure of such systems, applying the generalizedIt(o|^) formula, the criterion for the stochastic exponential stability in the mean squareand the almost-sure stability of It(o|^) stochastic differential systems of neutral type withMarkov jumping are obtained by constructing Lyapunov functional in term of LMIs.
3. The stochastic exponential stability in the mean-square sense of nonlinear It(o|^)stochastic differential systems with Markov jumping are investigated. For the case ofnon-delay, delay and neutral type, some new criteria are given based on Grown-Wall inequality and generalized It(o|^) formula, which improve the existed results.
4. The robust control problem of delay It(o|^) stochastic differential systems withMarkov jumping is investigated. Based on the stability analysis, the approach of sta-bilization controller in the delay-independent and delay-dependent case is obtained.And based on the SLQ theory, the stabilization problem based on optimal SLQ con-troller is discussed. For norm-bounded uncertainty, the problem of robust H_∞analysisand the design approach of robust H_∞controller are investigated. And the problemof robust guaranteed cost control of such systems is proposed and solved.
引文
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