非线性系统控制理论若干问题研究及其应用
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摘要
近年来,控制领域范围内出现了很多具有挑战性问题。在这些新的问题中,有两个基本的特性,第一,在控制器的设计中,必须考虑非线性的影响,因为根据线性系统设计的控制器已经不能满足实际的需要；第二,在物理系统建模时不可能做到完全精确,必然存在不确定性,这种不确定性包括参数不确定性、结构不确定性和各种干扰等。由于存在不确定性,设计的控制器必须能够处理这些不确定性,使之对系统的动态性能不会有太大的影响。
因此针对带有不确定性的系统,利用无源化控制、自适应控制和滑模控制相结合的方法设计控制器,并且分析系统的稳定性和性能,是本论文主要的研究方法和研究内容。它具体包括以下几个方面,
在第三章中,将一种非线性控制方法——基于无源化的互联和阻尼控制,应用到线性时不变系统中,得到一些线性矩阵不等式,这些线性矩阵不等式是可解的充要条件是线性时不变系统是可镇定的。和已有的方法相比,我们简化了证明过程,并且不需要假设系统在s=0时没有不可控的极点。此外,我们将这种控制方法也应用到线性的机械系统中,找到这种方法应用于机械系统的充要条件是包含输入矩阵和弹簧劲度系数矩阵的线性矩阵不等式条件得到满足。
在第四章中,将另外一种非线性控制方法——基于无源化的功率整形控制从静态反馈控制推广到动态反馈控制,这种控制方法是通过调整系统中的功率而不是能量来实现系统镇定的,由于加入了动态扩展,可以增加设计的自由度以简化求解利用基于无源化的功率整形控制得到的偏微分方程。此外,这种方法克服了基于无源化的能量平衡控制中存在的耗散性问题,并且在这种控制方法在设计过程中会给出Lyapunov函数的设计方法。
在第五章中,提出了一种新的自适应滑模控制器设计方法,这种方法可以处理带有不匹配参数摄动的系统。这种方法将浸入和不变自适应控制策略和线性滑模控制结合起来,与将Backstepping自适应控制策略与滑模控制结合的方法类似,但是他们不同的地方在于参数估计律的设计上。Backstepping自适应估计律来自于Lyapunov函数的设计,而浸入和不变自适应估计律是来自于参数估计误差的动态。应用这种自适应滑模控制器,不仅可以使得系统达到和Backstepping自适应控制策略与滑模控制结合的同样效果,并且不需要系统满足严格反馈的形式。
在第六章中,针对带参数不确定性和外界干扰的非线性系统,提出了基于Hamiltonian结构的自适应积分滑模控制方法。应用这种方法,首先要将原始的非线性系统通过基于无源化的互联和阻尼控制转变成具有端口受控的Hamiltonian结构,然后再利用自适应积分滑模控制,这里我们将浸入和不变自适应控制策略和积分滑模结合在一起。这种新的自适应滑模积分控制保持了浸入和不变自适应控制和积分滑模控制两种方法的优点,当系统只有参数不确定性时,系统可以达到渐近稳定的效果；当系统存在参数不确定性和外界扰动时,系统具有很好的鲁棒性。这种方法和将Backstepping自适应控制策略与滑模控制结合方法相比的不同点在于自适应估计律上。
最后,本论文针对非线性系统的无源化控制和鲁棒自适应控制进行了总结,并且对今后的研究进行了展望。
In the last few years we have witnessed the appearance of a series of challenging con-trol engineering problems. There are two common features of these new control problems. First, the interesting range of operation of the system is not necessarily close to an equi-librium, hence nonlinear effects have to be explicitly taken into account for a successful controller design. Second, even though physical modeling allows to accurately identify cer-tain well-defined nonlinear effects, the controller has to deal with high level of uncertainty, mainly due to lack of knowledge of the system parameters and the external disturbance.
This situation justifies the need for the development of tools for controller design for uncertain nonlinear systems, which is the main topic of the dissertation. The main contents are outlined as follows.
In Chapter 3, it is shown that the linear matrix inequalities that result from the appli-cation of interconnection and damping assignment passivity-based control (IDA-PBC) to general linear time-invariant systems are feasible if and only if the system is stabilizable. A very simple proof of this fact is given that, in contrast to previous results, does not require the assumption that the system has no uncontrollable pole at s=0. A second contribution of the chapter is the proof that, for the case of mechanical systems, the method is applica-ble if and only if a new linear matrix inequality involving the input matrix and the springs stiffness coefficients matrix is satisfied.
It is well known that energy-balancing passivity-based control is stymied by the pres-ence of pervasive dissipation. To overcome this problem in electrical circuits, some authors have used power-shaping techniques, where stabilization is achieved by shaping a function akin to power instead of energy. Some extensions of the techniques to general nonlinear systems, yielding static state-feedback control laws, have also been reported. In Chapter 4, we extend these techniques to dynamic feedback control and apply them to nonlinear chemical processes. The stability analysis is carried out using the shaped power function as Lyapunov function.
In Chapter 5, a design method on adaptive sliding mode control of systems with mis-matched parametric perturbations is proposed. The method combines immersion and in-variance (I&I) adaptive control with linear sliding mode control (LSMC), which resembles the well-known combining adaptive backstepping with sliding mode control, but is differ-ent from the parameter estimation law, which allows for prescribed dynamics to be assigned to the estimation error and is easier to tune than the backstepping adaptive obtained from Lyapunov redesign. By applying I&I adaptive law and sliding mode to design the con-trollers, not only the mismatched parametric perturbations are automatically overcoming during the sliding mode, but also the property of asymptotical stability of controlled sys-tems is achieved at the same time. Moreover, the knowledge of the upper bound of partial parametric perturbations is not required and the systems do not have to be in strict-feedback form.
In Chapter 6, Hamiltonian-based adaptive integral sliding mode control is proposed to deal with the regulation problem of uncertain nonlinear systems, which may possess both parametric uncertainties and unknown nonlinear functions that may represent modeling errors and external disturbances. By utilizing the proposed method, the original nonlinear system is first converted into the port-controlled Hamiltonian (PCH) form by Interconnec-tion and Damping Assignment Passivity-based Control (IDA-PBC), and then a adaptive integral sliding mode control is designed to control the system. The proposed method com-bines immersion and invariance (I&I) adaptive scheme with integral sliding mode control (ISMC), which preserves the advantages of the two methods, namely asymptotic stability of adaptive control in the presence of parametric uncertainties, and robustness with inte-gral sliding mode control for both parametric uncertainties and unknown bounded nonlin-ear functions. The method is different from the approach combining backstepping adaptive scheme and sliding mode control in the parameter estimation law, which allows for pre-scribed dynamics to be assigned to the estimation error and is easier to tune.
Finally, the passivity-based control and robust adaptive control of nonlinear systems are outlined, and the perspective of the future studies is also referred in the dissertation.
因此针对带有不确定性的系统,利用无源化控制、自适应控制和滑模控制相结合的方法设计控制器,并且分析系统的稳定性和性能,是本论文主要的研究方法和研究内容。它具体包括以下几个方面,
在第三章中,将一种非线性控制方法——基于无源化的互联和阻尼控制,应用到线性时不变系统中,得到一些线性矩阵不等式,这些线性矩阵不等式是可解的充要条件是线性时不变系统是可镇定的。和已有的方法相比,我们简化了证明过程,并且不需要假设系统在s=0时没有不可控的极点。此外,我们将这种控制方法也应用到线性的机械系统中,找到这种方法应用于机械系统的充要条件是包含输入矩阵和弹簧劲度系数矩阵的线性矩阵不等式条件得到满足。
在第四章中,将另外一种非线性控制方法——基于无源化的功率整形控制从静态反馈控制推广到动态反馈控制,这种控制方法是通过调整系统中的功率而不是能量来实现系统镇定的,由于加入了动态扩展,可以增加设计的自由度以简化求解利用基于无源化的功率整形控制得到的偏微分方程。此外,这种方法克服了基于无源化的能量平衡控制中存在的耗散性问题,并且在这种控制方法在设计过程中会给出Lyapunov函数的设计方法。
在第五章中,提出了一种新的自适应滑模控制器设计方法,这种方法可以处理带有不匹配参数摄动的系统。这种方法将浸入和不变自适应控制策略和线性滑模控制结合起来,与将Backstepping自适应控制策略与滑模控制结合的方法类似,但是他们不同的地方在于参数估计律的设计上。Backstepping自适应估计律来自于Lyapunov函数的设计,而浸入和不变自适应估计律是来自于参数估计误差的动态。应用这种自适应滑模控制器,不仅可以使得系统达到和Backstepping自适应控制策略与滑模控制结合的同样效果,并且不需要系统满足严格反馈的形式。
在第六章中,针对带参数不确定性和外界干扰的非线性系统,提出了基于Hamiltonian结构的自适应积分滑模控制方法。应用这种方法,首先要将原始的非线性系统通过基于无源化的互联和阻尼控制转变成具有端口受控的Hamiltonian结构,然后再利用自适应积分滑模控制,这里我们将浸入和不变自适应控制策略和积分滑模结合在一起。这种新的自适应滑模积分控制保持了浸入和不变自适应控制和积分滑模控制两种方法的优点,当系统只有参数不确定性时,系统可以达到渐近稳定的效果；当系统存在参数不确定性和外界扰动时,系统具有很好的鲁棒性。这种方法和将Backstepping自适应控制策略与滑模控制结合方法相比的不同点在于自适应估计律上。
最后,本论文针对非线性系统的无源化控制和鲁棒自适应控制进行了总结,并且对今后的研究进行了展望。
In the last few years we have witnessed the appearance of a series of challenging con-trol engineering problems. There are two common features of these new control problems. First, the interesting range of operation of the system is not necessarily close to an equi-librium, hence nonlinear effects have to be explicitly taken into account for a successful controller design. Second, even though physical modeling allows to accurately identify cer-tain well-defined nonlinear effects, the controller has to deal with high level of uncertainty, mainly due to lack of knowledge of the system parameters and the external disturbance.
This situation justifies the need for the development of tools for controller design for uncertain nonlinear systems, which is the main topic of the dissertation. The main contents are outlined as follows.
In Chapter 3, it is shown that the linear matrix inequalities that result from the appli-cation of interconnection and damping assignment passivity-based control (IDA-PBC) to general linear time-invariant systems are feasible if and only if the system is stabilizable. A very simple proof of this fact is given that, in contrast to previous results, does not require the assumption that the system has no uncontrollable pole at s=0. A second contribution of the chapter is the proof that, for the case of mechanical systems, the method is applica-ble if and only if a new linear matrix inequality involving the input matrix and the springs stiffness coefficients matrix is satisfied.
It is well known that energy-balancing passivity-based control is stymied by the pres-ence of pervasive dissipation. To overcome this problem in electrical circuits, some authors have used power-shaping techniques, where stabilization is achieved by shaping a function akin to power instead of energy. Some extensions of the techniques to general nonlinear systems, yielding static state-feedback control laws, have also been reported. In Chapter 4, we extend these techniques to dynamic feedback control and apply them to nonlinear chemical processes. The stability analysis is carried out using the shaped power function as Lyapunov function.
In Chapter 5, a design method on adaptive sliding mode control of systems with mis-matched parametric perturbations is proposed. The method combines immersion and in-variance (I&I) adaptive control with linear sliding mode control (LSMC), which resembles the well-known combining adaptive backstepping with sliding mode control, but is differ-ent from the parameter estimation law, which allows for prescribed dynamics to be assigned to the estimation error and is easier to tune than the backstepping adaptive obtained from Lyapunov redesign. By applying I&I adaptive law and sliding mode to design the con-trollers, not only the mismatched parametric perturbations are automatically overcoming during the sliding mode, but also the property of asymptotical stability of controlled sys-tems is achieved at the same time. Moreover, the knowledge of the upper bound of partial parametric perturbations is not required and the systems do not have to be in strict-feedback form.
In Chapter 6, Hamiltonian-based adaptive integral sliding mode control is proposed to deal with the regulation problem of uncertain nonlinear systems, which may possess both parametric uncertainties and unknown nonlinear functions that may represent modeling errors and external disturbances. By utilizing the proposed method, the original nonlinear system is first converted into the port-controlled Hamiltonian (PCH) form by Interconnec-tion and Damping Assignment Passivity-based Control (IDA-PBC), and then a adaptive integral sliding mode control is designed to control the system. The proposed method com-bines immersion and invariance (I&I) adaptive scheme with integral sliding mode control (ISMC), which preserves the advantages of the two methods, namely asymptotic stability of adaptive control in the presence of parametric uncertainties, and robustness with inte-gral sliding mode control for both parametric uncertainties and unknown bounded nonlin-ear functions. The method is different from the approach combining backstepping adaptive scheme and sliding mode control in the parameter estimation law, which allows for pre-scribed dynamics to be assigned to the estimation error and is easier to tune.
Finally, the passivity-based control and robust adaptive control of nonlinear systems are outlined, and the perspective of the future studies is also referred in the dissertation.
引文
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