基于模糊模型的非线性系统镇定控制和H_∞滤波研究
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摘要
随着科学技术的飞速发展,设计者们所研究的控制系统也越来越复杂,以至于很难对其建立足够精确的数学模型.在这种情况下,传统的控制理论便很难奏效.为了解决这类复杂系统的控制问题,人们发明了模糊控制理论并将其成功应用于工业过程控制、信号处理、智能机器、模式识别、医疗等领域.在过去的二十年里,模糊控制理论一直是一个热点研究领域；特别的,基于模糊模型的各种复杂非线性系统的镇定控制和H∞滤波问题吸引了众多研究者们的注意.研究者们致力于对已有方法的不断改进和在理论上的逐步完善等方面的研究且取得了许多优秀的研究成果.需要指出的是,随着控制性能要求的逐步提高,已有的模糊系统镇定控制和H∞滤波设计条件会因为其保守性较大而无法得到应用.因此,探索和研究更有效地模糊镇定控制和H∞滤波条件具有了重要的理论意义和应用价值.本文在总结已有方法的基础之上,进一步深入地研究了基于模糊模型的非线性系统的镇定控制和H∞滤波问题,主要创新工作如下：
1.钊对连续时间T-S模糊系统的二次镇定控制问题,提出了两种减少T-S模糊控制系统二次镇定条件保守性的新方法.第一种方法通过有效的考虑不同模糊隶属函数之间乘积界的信息,引入了更多的自由矩阵变量,放松了已有二次镇定条件；另外一种方法基于对模糊隶属函数空间进行边界划分的思想,通过提出一种对模糊隶属函数空间进行边界划分的新算法来将整个模糊隶属函数空间人为划分为若干个子空间,然后基于这些子空间基础之上设计一种新型的模糊切换控制器,获得了能够减少连续时间T-S模糊系统二次镇定条件保守性的一种新思路.
2.针对连续时间T-S模糊系统的稳定性分析和非二次镇定控制问题,提出了基于一种新颖的增广多指标矩阵方法的放松性稳定性条件和非二次镇定条件.在开环T-S模糊系统稳定性条件的推导过程中,先利用了著名的Finsler引理来扩展系统的整个设计空间;然后引入了一种高阶依赖于模糊隶属函数的齐次矩阵多项式型拉格朗日乘子,进而运用一种新颖的增广多指标矩阵方法推导出保守性更小的开环T-S模糊系统稳定性条件.在非二次镇定条件的设计时,提出了一种新型的非-PDC控制律,这种状态反馈控制律高阶依赖于模糊隶属函数,因而能够更好的利用模糊隶属函数的有用信息;然后结合齐次矩阵多项式技术和增广多指标矩阵方法推导出放松性的非二次镇定条件,大大减少了已有结果的保守性.
3.针对离散时间T-S模糊系统的非二次镇定控制问题,从改进李亚普诺夫函数和设计新型的模糊状态反馈控制律两个方面进行减少结果保守性的探索.设计了一种新型的非-PDC控制律(这里称之为齐次多项式非二次控制律,HPNQCL)来对闭环T-S模糊控制系统进行镇定控制；在控制综合过程中选取了一种模糊隶属函数高阶依赖型李亚普诺夫函数,再结合模糊隶属函数的代数特性将所得的参数化矩阵不等式条件分解成一组线性矩阵不等式条件；并且利用Polya's定理证明了当所选取两个设计变量值足够大时可以得到保证该参数化矩阵不等式成立的充分必要条件.另外,也在理论上与已有的一些相关研究成果进行了比较,从保守性的大小和计算效率的高低两个方面证明了本文所提方法的有效性.
4.针对一类离散时间T-S模糊系统的H∞滤波问题,提出了一种新型的模糊H∞滤波器的设计方案,即,齐次多项式参数依赖(HPPD)型模糊H∞滤波器.与已有的基于并行分布补偿控制思想下的模糊H∞滤波器不同,本文设计的这种HPPD型H∞滤波器高阶依赖于模糊隶属函数；可以引入了更多的滤波设计矩阵变量,因而具备很大的可以减少滤波设计保守性的潜力.在H∞滤波器设计过程中,还提出了一种可以保证齐次多项参数依赖式型李亚普诺夫矩阵严格正定的放松性线性矩阵不等式方法,有效的利用了模糊隶属函数的信息,进而得到比以往结果保守性更小的模糊H∞滤波条件.
5.针对一类连续时间非线性随机系统的H∞滤波问题,提出了一种基于模糊双曲正切模型的H∞滤波设计方案.由于模糊双曲正切模型的一些特点,避免了在利用T-S模糊李亚普诺夫函数对连续时间T-S模糊系统进行H∞滤波设计时需要涉及到模糊隶属函数对时间导数的问题,并且所设计的模糊双曲H∞滤波器具有结构简单和性能良好等优点,是当性能指标与所需计算量之间发生矛盾时可以选择的一个理想的折中化设计方案.
6.针对离散时间Roesser型非线性二维系统的镇定控制问题,提出了三种基于T-S模糊模型的非线性二维系统控制策略.分别设计了用来镇定Roesser型二维T-S模糊系统的PDC控制律、非-PDC控制律和齐次多项式非二次控制律.在控制设计中,还提出了几种针对二维T-S模糊系统镇定条件的松弛矩阵变量方法来减少了相应镇定条件的保守性.最后,又提出一种基于齐次多项式参数依赖型李亚普诺夫函数的Roesser型二维T-S模糊系统镇定方法.与常规的1-D系统不同,针对Roesser型二维系统进行齐次多项式参数依赖型李亚普诺夫函数的差分计算时会产生水平和垂直两个方向前向一步模糊隶属函数.我们在控制综合中克服了这个困难并且利用一些改进的齐次矩阵多项式技术得到了保守性很小的镇定条件.
最后,指出了基于模糊模型的非线性系统镇定控制和H∞滤波设计方法研究中一些尚未解决的问题,并对接下来的研究工作进行了展望.
With the fast development of the science and technology, control systems inves-tigated by the designers becomes more and more complex, so it is hard to establish its mathematical model with satisfied precisions. In this case, the conventional con-trol theory is not applicable. With the purpose of resolving this problem, people have invented the fuzzy control theory, and it has also been successfully applied to many fields, such as industrial process control, signal processing, intelligent machine, pattern recognition, and medicine. Over the past two decades, fuzzy control theory has been an active research field. In particular, stabilization and H∞filtering of nonlinear systems based on fuzzy models have attracted wide attention from many investigators. They have been applied themselves to the improvement and gener-alization of existing results and many valuable productions have been achieved. It is worth noting that the existing conditions of fuzzy stabilization and H∞filtering will fail to work while the requirement of the control performance index gradually increases. Therefore, the research on exploiting more effective conditions for fuzzy stabilization and H∞, filtering is significant on academic and practical aspects. Based on the precious work of other researchers, this thesis further investigates the prob-lem of stabilization and H∞filtering of nonlinear systems via fuzzy models, and the main contributions of this dissertation are as follows:
1. For the problem of quadratic stabilization of continuous-time T-S fuzzy sys-tems, two kinds of relaxed quadratic stabilization conditions are proposed. The first one is developed by considering the bounds of different fuzzy mem-bership functions'cross products. More free matrix variables are introduced and hence the conservatism is further reduced. The other one is based on the viewpoint of dividing the fuzzy membership functions space by applying the simplex edgewise subdivision approach. A new simplex edgewise subdivi-sion algorithm is proposed and thus the fuzzy membership functions space is divided into a lot of sub-spaces. Then, a novel kind of fuzzy switching con-troller is designed for relaxing the quadratic stabilization of continuous-time T-S fuzzy systems.
2. For the problem of stability analysis and non-quadratic stabilization of continuous-time T-S fuzzy systems, relaxed stability conditions and non-quadratic stabilization conditions are proposed by applying a novel kind of augmented multi-indexed matrix approach. In the derivation process of sta-bility analysis of opened-loop T-S fuzzy systems, the famous Finsler lemma is used for enlarging the design space, and homogeneous matrix polynomial-type Lagrange multipliers, which is parameter-dependent on fuzzy membership function with a higher degree, are introduced. Then a new augmented multi-indexed matrix approach is developed for achieving less conservative stability conditions of opened-loop T-S fuzzy systems. In the derivation process of non-quadratic stabilization of closed-loop T-S fuzzy systems, a new kind of non-PDC control scheme, which is parameter-dependent on fuzzy member-ship function with a higher degree, is proposed for making good use of the information of fuzzy membership functions; thus relaxed non-quadratic stabi-lization conditions are obtained by applying both the homogenous polynomial technique and the augmented multi-indexed matrix approach while the con-servatism is further reduced.
3. For the problem of non-quadratic stabilization of discrete-time T-S fuzzy sys-tems, less conservative non-quadratic stabilization conditions are developed by both improving the form of Lyapunov function and designing new fuzzy state feedback control scheme. A new kind of non-PDC control scheme(named as homogenous polynomial non-quadratic control scheme (HPNQCL) here) is proposed to stabilizing the closed-loop T-S fuzzy control systems; then a Lya-punuov function which is parameter-dependent on fuzzy membership functions with an higher degree is selected for implementing the control synthesis. The algebraic properties of fuzzy membership functions are exploited to convert the attained parameterized matrix inequality into a sequence of linear matrix inequalities. As the values of two design variables tend to sufficiently large, the exactness of ensuring the attained parameterized matrix inequality is cer-tificated by using the Polya's theorem. Moreover, the advantages over those existing ones are also certificated in theory. It shows that the one in this thesis is both less conservative and more efficient in computations.
4. For the problem of H∞filtering of a class of discrete-time T-S fuzzy systems, a novel fuzzy H∞filter is developed, i.e., homogenous polynomial parameter-dependent (HPPD) H∞filter. Unlike to the usual fuzzy H∞, filters which is motivated by the viewpoint of parallel distributed compensation, the one pro- vided in this thesis is parameter-dependent on fuzzy membership functions with a higher degree; more filtering matrix variables are introduced, and hence it has the potential of further reducing the conservatism. In the process of H^filtering, the positivity of the underlying homogenous polynomial parameter-dependent Lyapunov matrix is ensured by proposing a relaxed linear matrix inequality method; the information of fuzzy membership functions is effectively utilized and less conservative fuzzy H∞filtering condition is obtained.
5. For the problem of H∞filtering of a class of continuous-time nonlinear stochas-tic systems, a kind of H∞filtering is proposed based on the fuzzy hyperbolic tangent model. Due to some characteristics of the fuzzy hyperbolic tangent model, the problem that the fuzzy membership functions'time derivatives are involved in the process of H∞, filtering via the T-S fuzzy Lyapunov functions are avoided. The obtained fuzzy H∞filter is with a simple structure and nicer performance. This is a good tradeoff between the performance and the required computational burden.
6. For the problem of stabilization of discrete-time Roesser nonlinear two-dimensional(2-D) system, three kinds of control schemes are proposed based on the T-S fuzzy model, i.e., the PDC control scheme, the non-PDC con-trol scheme and the homogenous polynomial non-quadratic control scheme for2-D T-S fuzzy systems respectively. In the process of control synthesis, sev-eral kinds of new slack matrix variable approaches are proposed for relaxing the stabilization conditions of the Roesser2-D T-S fuzzy systems. Finally, a homogenous polynomial parameter-dependent Lyapunov function is proposed for conceiving relaxed stabilization of the the Roesser2-D T-S fuzzy systems. Unlike to the usual1-D T-S fuzzy system, there will be two different one-step-ahead fuzzy membership functions along the horizonal direction and the vertical direction respectively when the variation of the underlying Lyapunov function is calculated. This obstacle is overcome in control synthesis and some improved homogenous polynomial techniques are also developed for further re-ducing the conservatism.
Finally, some unsolved problems for control synthesis and H∞filtering of non-linear systems based on fuzzy models are pointed out. Furthermore, the prospects of the future study are also given.
1.钊对连续时间T-S模糊系统的二次镇定控制问题,提出了两种减少T-S模糊控制系统二次镇定条件保守性的新方法.第一种方法通过有效的考虑不同模糊隶属函数之间乘积界的信息,引入了更多的自由矩阵变量,放松了已有二次镇定条件；另外一种方法基于对模糊隶属函数空间进行边界划分的思想,通过提出一种对模糊隶属函数空间进行边界划分的新算法来将整个模糊隶属函数空间人为划分为若干个子空间,然后基于这些子空间基础之上设计一种新型的模糊切换控制器,获得了能够减少连续时间T-S模糊系统二次镇定条件保守性的一种新思路.
2.针对连续时间T-S模糊系统的稳定性分析和非二次镇定控制问题,提出了基于一种新颖的增广多指标矩阵方法的放松性稳定性条件和非二次镇定条件.在开环T-S模糊系统稳定性条件的推导过程中,先利用了著名的Finsler引理来扩展系统的整个设计空间;然后引入了一种高阶依赖于模糊隶属函数的齐次矩阵多项式型拉格朗日乘子,进而运用一种新颖的增广多指标矩阵方法推导出保守性更小的开环T-S模糊系统稳定性条件.在非二次镇定条件的设计时,提出了一种新型的非-PDC控制律,这种状态反馈控制律高阶依赖于模糊隶属函数,因而能够更好的利用模糊隶属函数的有用信息;然后结合齐次矩阵多项式技术和增广多指标矩阵方法推导出放松性的非二次镇定条件,大大减少了已有结果的保守性.
3.针对离散时间T-S模糊系统的非二次镇定控制问题,从改进李亚普诺夫函数和设计新型的模糊状态反馈控制律两个方面进行减少结果保守性的探索.设计了一种新型的非-PDC控制律(这里称之为齐次多项式非二次控制律,HPNQCL)来对闭环T-S模糊控制系统进行镇定控制；在控制综合过程中选取了一种模糊隶属函数高阶依赖型李亚普诺夫函数,再结合模糊隶属函数的代数特性将所得的参数化矩阵不等式条件分解成一组线性矩阵不等式条件；并且利用Polya's定理证明了当所选取两个设计变量值足够大时可以得到保证该参数化矩阵不等式成立的充分必要条件.另外,也在理论上与已有的一些相关研究成果进行了比较,从保守性的大小和计算效率的高低两个方面证明了本文所提方法的有效性.
4.针对一类离散时间T-S模糊系统的H∞滤波问题,提出了一种新型的模糊H∞滤波器的设计方案,即,齐次多项式参数依赖(HPPD)型模糊H∞滤波器.与已有的基于并行分布补偿控制思想下的模糊H∞滤波器不同,本文设计的这种HPPD型H∞滤波器高阶依赖于模糊隶属函数；可以引入了更多的滤波设计矩阵变量,因而具备很大的可以减少滤波设计保守性的潜力.在H∞滤波器设计过程中,还提出了一种可以保证齐次多项参数依赖式型李亚普诺夫矩阵严格正定的放松性线性矩阵不等式方法,有效的利用了模糊隶属函数的信息,进而得到比以往结果保守性更小的模糊H∞滤波条件.
5.针对一类连续时间非线性随机系统的H∞滤波问题,提出了一种基于模糊双曲正切模型的H∞滤波设计方案.由于模糊双曲正切模型的一些特点,避免了在利用T-S模糊李亚普诺夫函数对连续时间T-S模糊系统进行H∞滤波设计时需要涉及到模糊隶属函数对时间导数的问题,并且所设计的模糊双曲H∞滤波器具有结构简单和性能良好等优点,是当性能指标与所需计算量之间发生矛盾时可以选择的一个理想的折中化设计方案.
6.针对离散时间Roesser型非线性二维系统的镇定控制问题,提出了三种基于T-S模糊模型的非线性二维系统控制策略.分别设计了用来镇定Roesser型二维T-S模糊系统的PDC控制律、非-PDC控制律和齐次多项式非二次控制律.在控制设计中,还提出了几种针对二维T-S模糊系统镇定条件的松弛矩阵变量方法来减少了相应镇定条件的保守性.最后,又提出一种基于齐次多项式参数依赖型李亚普诺夫函数的Roesser型二维T-S模糊系统镇定方法.与常规的1-D系统不同,针对Roesser型二维系统进行齐次多项式参数依赖型李亚普诺夫函数的差分计算时会产生水平和垂直两个方向前向一步模糊隶属函数.我们在控制综合中克服了这个困难并且利用一些改进的齐次矩阵多项式技术得到了保守性很小的镇定条件.
最后,指出了基于模糊模型的非线性系统镇定控制和H∞滤波设计方法研究中一些尚未解决的问题,并对接下来的研究工作进行了展望.
With the fast development of the science and technology, control systems inves-tigated by the designers becomes more and more complex, so it is hard to establish its mathematical model with satisfied precisions. In this case, the conventional con-trol theory is not applicable. With the purpose of resolving this problem, people have invented the fuzzy control theory, and it has also been successfully applied to many fields, such as industrial process control, signal processing, intelligent machine, pattern recognition, and medicine. Over the past two decades, fuzzy control theory has been an active research field. In particular, stabilization and H∞filtering of nonlinear systems based on fuzzy models have attracted wide attention from many investigators. They have been applied themselves to the improvement and gener-alization of existing results and many valuable productions have been achieved. It is worth noting that the existing conditions of fuzzy stabilization and H∞filtering will fail to work while the requirement of the control performance index gradually increases. Therefore, the research on exploiting more effective conditions for fuzzy stabilization and H∞, filtering is significant on academic and practical aspects. Based on the precious work of other researchers, this thesis further investigates the prob-lem of stabilization and H∞filtering of nonlinear systems via fuzzy models, and the main contributions of this dissertation are as follows:
1. For the problem of quadratic stabilization of continuous-time T-S fuzzy sys-tems, two kinds of relaxed quadratic stabilization conditions are proposed. The first one is developed by considering the bounds of different fuzzy mem-bership functions'cross products. More free matrix variables are introduced and hence the conservatism is further reduced. The other one is based on the viewpoint of dividing the fuzzy membership functions space by applying the simplex edgewise subdivision approach. A new simplex edgewise subdivi-sion algorithm is proposed and thus the fuzzy membership functions space is divided into a lot of sub-spaces. Then, a novel kind of fuzzy switching con-troller is designed for relaxing the quadratic stabilization of continuous-time T-S fuzzy systems.
2. For the problem of stability analysis and non-quadratic stabilization of continuous-time T-S fuzzy systems, relaxed stability conditions and non-quadratic stabilization conditions are proposed by applying a novel kind of augmented multi-indexed matrix approach. In the derivation process of sta-bility analysis of opened-loop T-S fuzzy systems, the famous Finsler lemma is used for enlarging the design space, and homogeneous matrix polynomial-type Lagrange multipliers, which is parameter-dependent on fuzzy membership function with a higher degree, are introduced. Then a new augmented multi-indexed matrix approach is developed for achieving less conservative stability conditions of opened-loop T-S fuzzy systems. In the derivation process of non-quadratic stabilization of closed-loop T-S fuzzy systems, a new kind of non-PDC control scheme, which is parameter-dependent on fuzzy member-ship function with a higher degree, is proposed for making good use of the information of fuzzy membership functions; thus relaxed non-quadratic stabi-lization conditions are obtained by applying both the homogenous polynomial technique and the augmented multi-indexed matrix approach while the con-servatism is further reduced.
3. For the problem of non-quadratic stabilization of discrete-time T-S fuzzy sys-tems, less conservative non-quadratic stabilization conditions are developed by both improving the form of Lyapunov function and designing new fuzzy state feedback control scheme. A new kind of non-PDC control scheme(named as homogenous polynomial non-quadratic control scheme (HPNQCL) here) is proposed to stabilizing the closed-loop T-S fuzzy control systems; then a Lya-punuov function which is parameter-dependent on fuzzy membership functions with an higher degree is selected for implementing the control synthesis. The algebraic properties of fuzzy membership functions are exploited to convert the attained parameterized matrix inequality into a sequence of linear matrix inequalities. As the values of two design variables tend to sufficiently large, the exactness of ensuring the attained parameterized matrix inequality is cer-tificated by using the Polya's theorem. Moreover, the advantages over those existing ones are also certificated in theory. It shows that the one in this thesis is both less conservative and more efficient in computations.
4. For the problem of H∞filtering of a class of discrete-time T-S fuzzy systems, a novel fuzzy H∞filter is developed, i.e., homogenous polynomial parameter-dependent (HPPD) H∞filter. Unlike to the usual fuzzy H∞, filters which is motivated by the viewpoint of parallel distributed compensation, the one pro- vided in this thesis is parameter-dependent on fuzzy membership functions with a higher degree; more filtering matrix variables are introduced, and hence it has the potential of further reducing the conservatism. In the process of H^filtering, the positivity of the underlying homogenous polynomial parameter-dependent Lyapunov matrix is ensured by proposing a relaxed linear matrix inequality method; the information of fuzzy membership functions is effectively utilized and less conservative fuzzy H∞filtering condition is obtained.
5. For the problem of H∞filtering of a class of continuous-time nonlinear stochas-tic systems, a kind of H∞filtering is proposed based on the fuzzy hyperbolic tangent model. Due to some characteristics of the fuzzy hyperbolic tangent model, the problem that the fuzzy membership functions'time derivatives are involved in the process of H∞, filtering via the T-S fuzzy Lyapunov functions are avoided. The obtained fuzzy H∞filter is with a simple structure and nicer performance. This is a good tradeoff between the performance and the required computational burden.
6. For the problem of stabilization of discrete-time Roesser nonlinear two-dimensional(2-D) system, three kinds of control schemes are proposed based on the T-S fuzzy model, i.e., the PDC control scheme, the non-PDC con-trol scheme and the homogenous polynomial non-quadratic control scheme for2-D T-S fuzzy systems respectively. In the process of control synthesis, sev-eral kinds of new slack matrix variable approaches are proposed for relaxing the stabilization conditions of the Roesser2-D T-S fuzzy systems. Finally, a homogenous polynomial parameter-dependent Lyapunov function is proposed for conceiving relaxed stabilization of the the Roesser2-D T-S fuzzy systems. Unlike to the usual1-D T-S fuzzy system, there will be two different one-step-ahead fuzzy membership functions along the horizonal direction and the vertical direction respectively when the variation of the underlying Lyapunov function is calculated. This obstacle is overcome in control synthesis and some improved homogenous polynomial techniques are also developed for further re-ducing the conservatism.
Finally, some unsolved problems for control synthesis and H∞filtering of non-linear systems based on fuzzy models are pointed out. Furthermore, the prospects of the future study are also given.
引文
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