不确定非线性系统的Backstepping控制
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摘要
不确定非线性系统控制是近年来的研究热点之一,特别是带有执行器故障或时滞的非线性系统控制理论越来越受到人们的关注。控制系统在运行过程中不可避免地会发生执行器故障。不仅如此,时滞现象也普遍存在。到目前为止,线性系统自适应执行器故障补偿控制理论已有充分的研究,而非线性系统执行器故障补偿问题的研究并不深入。另一方面,线性时滞系统的研究也有非常丰富的成果,而非线性时滞系统的研究相对较少。本文在国内外已有研究成果的基础上,以Backstepping方法为基本工具,分别研究不确定非线性系统的自适应执行器故障补偿和不确定非线性时滞系统的控制问题。本文的主要工作如下。
1.针对一类带有未知定常执行器故障的输出反馈形式的多输入单输出非线性系统,提出一种修正的自适应执行器故障补偿方案。在该方案的设计过程中,首先建立一种新的带有未知系统参数和执行器故障参数的参数化模型,然后应用Backstepping技术设计自适应补偿控制器。所提出的控制方案保证了所有闭环信号的有界性,证明了输出跟踪误差渐近收敛于零。值得指出的是,在这种新的模型框架下,自适应控制设计简单,闭环系统的稳定性分析得到了很大的简化,并可使用较低阶的滤波器估计系统状态,从而更易于在工程实际中应用。以简化的双执行器球棒系统作为仿真对象,仿真结果表明了所提出的设计方法的有效性。
2.研究一类带有未知时变执行器故障的非线性系统的鲁棒自适应输出反馈控制问题。考虑执行器故障模型中的未建模项,提出一种新的线性参数化模型,该模型把所有未知系统参数和执行器故障参数集中起来,并与已知信号分离。基于此模型,递归地构造自适应控制器。为保证闭环系统的稳定性和鲁棒性,在自适应律中引入修正的切换σ信号。证明了所有闭环信号的有界性,通过适当选取设计参数保证了闭环系统的期望控制性能。最后,以带有依赖状态的非线性扰动的双执行器球棒系统为例,验证所提出的控制算法的有效性。
3.针对一类带有未知高频增益符号的非线性系统,考虑可参数化的时变执行器故障模型,基于Backstepping技术,提出一种自适应输出反馈故障补偿方案。引入Nuss-baum增益方法,去掉对高频增益符号的局限性假设。由于高频增益符号未知,一些新的常数变量需要重新定义,带有执行器故障信息的参数化模型需要重新建立,稳定性分析需要重新处理.所设计的补偿控制器在未知执行器故障存在的情况下保证了所有闭环信号的有界性和渐近的输出跟踪。最后,把所提出的控制方案应用于带有冗余执行器的两轴定位平台系统,仿真研究表明了控制算法的有效性。
4.研究一类单输入单输出严格反馈形式的非线性时滞系统的输出反馈跟踪控制问题。所考虑的系统不仅含有依赖于时滞的不确定函数,还包括外在的干扰项。为估计不可测量的系统状态,首先提出一种降阶观测器设计。使用降阶观测器的优势一方面可使形成的闭环系统更易于实现,另一方面还可降低求解设计参数的线性矩阵不等式的复杂度。然后,基于Backstepping方法递归地设计输出反馈控制器,并构造合适的Lyapunov-Krasovskii泛函补偿所有滞后的状态摄动函数。证明了所有闭环信号的有界性,通过适当选取设计参数保证了期望的跟踪性能。最后,给出一个仿真例子验证所提出的控制算法的有效性。
5.基于Backstepping方法解决了一类具有三角形结构的非线性时滞系统的自适应镇定问题。不仅被控系统参数而且时滞及其导数的上界都假设是未知的,它们通过设计自适应律而被在线估计。进一步放松了与时滞有关的非线性项的假设条件。为在这种弱化的条件下取得控制目标,首先引入一个函数分解引理,然后通过构造合适的Lyapunov-Krasovskii泛函递归地设计控制器。为消除参数估计对闭环稳定性带来的不利影响,从Backstepping设计的第二步开始,每一步的虚拟控制中都增加两个附加函数。值得指出的是,这些附加项应仔细选取以使递推设计顺利进行。设计方案不仅保证了所有闭环信号的有界性,而且使得系统状态渐近收敛于零。仿真研究进一步表明了所提出的控制策略的有效性。
6.针对一类带有状态时变时滞、不确定参数和外在干扰的非线性系统,提出了一种自适应Backstepping的跟踪控制方案。把调节函数方法引入到非线性时滞系统中,递归地构造自适应律和控制律。这种方法有效地避免了上一章中虚拟控制函数附加项选取的困难,而且自适应控制设计更加程序化。采用适当的Lyapunov-Krasovskii泛函补偿含有时滞状态的非线性函数.时滞及其导数的上界也假设是未知的,通过设计两套调节函数保证闭环系统所有信号的有界性,通过选取适当的设计参数保证闭环系统的跟踪性能。仿真结果验证了所提出的控制方案的有效性。
In recent years, the control problem of uncertain nonlinear systems becomes one of the research focuses. Particularly, the control theory for nonlinear systems with actuator failures or time delays has attracted increasing attention. It is well known that actuator failures are frequently encountered during operation of control systems. Also, time delay phenomenon widely exists. Up to now, many results have been achieved on adaptive control of linear systems with actuator failures. However, for nonlinear systems, adaptive actuator failure compensation has not been deeply investigated. On the other hand, despite the huge literature for linear time-delay systems, the control problem of nonlinear time-delay systems has been receiving relatively little attention until recently. Based on the current research findings, this dissertation uses the backstepping approach as a basic tool to address the problem of adaptive actuator failure compensation of uncertain nonlinear systems and uncertain nonlinear time-delay system control, respectively. The main contributions of the dissertation are as follows.
1. A modified adaptive actuator failure compensation scheme is proposed for a class of multi-input and single-output nonlinear systems in the output-feedback form. A con-stant actuator failure model is considered. In the control design, a new parametric model with unknown plant parameters and actuator failure parameters is firstly established. Then, an adaptive compensation controller is constructed by utilizing the backstepping technique. The boundedness of all the closed-loop signals is guaranteed. The output tracking error is proved to converge to zero asymptotically. It is worth pointing out that, within the new model framework, the adaptive control design is programmed, and the stability analysis can be sim-plified greatly. Moreover, the lower-order filters can be exploited to estimate the unmeasured plant states. Thus, the resulting control scheme is easier to be implemented. Finally, the presented scheme is applied to a simplified dual-actuator ball-beam system for the actuator failure compensation study. Simulation results demonstrate the effectiveness of the proposed design method.
2. The robust adaptive output-feedback control problem is addressed for a class of non-linear systems with unknown time-varying actuator failures. Additional un-modelled term in the actuator failure model is considered. A linearly parameterized model is proposed, where unknown plant parameters and actuator failure parameters are lumped up together and separated form known signals. Based on the new model, adaptive controller is recursively designed. The modified switching-σsignals are incorporated in the adaptive laws to ensure stability and robustness. The boundedness of all the closed-loop signals is established. The desired control performance of the closed-loop system is guaranteed by appropriately choos- ing the design parameters. Finally, a dual-actuator ball-beam system with state-dependent nonlinearities is used to illustrate the effectiveness of the proposed scheme.
3. For a class of nonlinear systems with unknown high-frequency gain signs and param-eterizable time-varying actuator failures, an adaptive output-feedback failure compensation scheme is proposed based on the backstepping technique. The Nussbaum gain approach is introduced to remove the restrictive assumption on the high-frequency gain signs. In view of unknown high-frequency gain signs, some constant variables are redefined, the para-metric model with actuator failure information is reestablished, and the stability analysis is retreated. The boundedness of all the closed-loop signals and the asymptotic output track-ing are guaranteed in spite of the unknown actuator failures. Finally, the proposed control scheme is applied to a two-axis positioning stage system with redundant actuators. The effectiveness of the control algorithm is verified by the simulation studies.
4. The problem of output-feedback tracking control is addressed for a class of single-input and single-output nonlinear systems in the strict-feedback form, which are subjected to both uncertain delay-related functions and external disturbances. A reduced-order observer is firstly introduced to provide the estimates of the unmeasured states. The advantage of exploiting the lower-dimension observer is two-fold. On the one hand, the resulting closed-loop system is easier to be implemented. On the other hand, since the design parameters are determined by a linear matrix inequality (LMI), the cost involved by the computation of the LMI may be much less. Then, an output-feedback controller is recursively designed based on the backstepping method. An appropriate Lyapunov-Krasovskii functional is constructed to compensate for all delayed state perturbation functions. All the signals in the closed-loop system are proved to be bounded. The desired tracking performance is guaranteed by suitably choosing the design parameters. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed control algorithm.
5. Based on the backstepping approach, the problem of adaptive stabilization for a class of nonlinear time-delay systems in triangular structure is addressed. Both the parameters of the system to be controlled and the upper bounds of the time delays and their deriva-tives are assumed to be unknown, which are estimated online by designing adaptive laws. The assumption on the delay-related nonlinearities is further relaxed. To achieve the con-trol objective under such weakened condition, a lemma on function factorization is firstly introduced. Then, an appropriate Lyapunov-Krasovskii functional is constructed and adap-tive controller is designed recursively. To counteract the influence of the parameter estimates on stability of the closed-loop system, from the second step of the backstepping procedure on, two terms are added in each virtual control function. It is worth pointing out that these additional functions should be selected carefully such that the backstepping design can be performed smoothly. It is shown that all the closed-loop signals are bounded, while the plant states converge to zero asymptotically. Simulation studies are provided to demonstrate the effectiveness of the proposed control strategy.
6. An adaptive backstepping tracking control scheme is proposed for a class of non-linear state time-varying delay systems, which are subject to parametric uncertainties and external disturbances. Tuning function method is exploited to construct the control law and adaptive laws recursively. This method not only effectively overcome the difficulty in the choice of additional terms in virtual control functions in the last chapter, but also make adap-tive control design more programmed. Unknown time-varying delays are compensated for by using appropriate Lyapunov-Krasovskii functional. The bounds of the time delays and their derivatives are also assumed to be unknown. Two set of tuning functions are designed to guarantee the boundedness of all the closed-loop signals. The tracking performance is ad-justed by choosing suitable design parameters. Simulation results illustrate the effectiveness of the proposed design procedure.
1.针对一类带有未知定常执行器故障的输出反馈形式的多输入单输出非线性系统,提出一种修正的自适应执行器故障补偿方案。在该方案的设计过程中,首先建立一种新的带有未知系统参数和执行器故障参数的参数化模型,然后应用Backstepping技术设计自适应补偿控制器。所提出的控制方案保证了所有闭环信号的有界性,证明了输出跟踪误差渐近收敛于零。值得指出的是,在这种新的模型框架下,自适应控制设计简单,闭环系统的稳定性分析得到了很大的简化,并可使用较低阶的滤波器估计系统状态,从而更易于在工程实际中应用。以简化的双执行器球棒系统作为仿真对象,仿真结果表明了所提出的设计方法的有效性。
2.研究一类带有未知时变执行器故障的非线性系统的鲁棒自适应输出反馈控制问题。考虑执行器故障模型中的未建模项,提出一种新的线性参数化模型,该模型把所有未知系统参数和执行器故障参数集中起来,并与已知信号分离。基于此模型,递归地构造自适应控制器。为保证闭环系统的稳定性和鲁棒性,在自适应律中引入修正的切换σ信号。证明了所有闭环信号的有界性,通过适当选取设计参数保证了闭环系统的期望控制性能。最后,以带有依赖状态的非线性扰动的双执行器球棒系统为例,验证所提出的控制算法的有效性。
3.针对一类带有未知高频增益符号的非线性系统,考虑可参数化的时变执行器故障模型,基于Backstepping技术,提出一种自适应输出反馈故障补偿方案。引入Nuss-baum增益方法,去掉对高频增益符号的局限性假设。由于高频增益符号未知,一些新的常数变量需要重新定义,带有执行器故障信息的参数化模型需要重新建立,稳定性分析需要重新处理.所设计的补偿控制器在未知执行器故障存在的情况下保证了所有闭环信号的有界性和渐近的输出跟踪。最后,把所提出的控制方案应用于带有冗余执行器的两轴定位平台系统,仿真研究表明了控制算法的有效性。
4.研究一类单输入单输出严格反馈形式的非线性时滞系统的输出反馈跟踪控制问题。所考虑的系统不仅含有依赖于时滞的不确定函数,还包括外在的干扰项。为估计不可测量的系统状态,首先提出一种降阶观测器设计。使用降阶观测器的优势一方面可使形成的闭环系统更易于实现,另一方面还可降低求解设计参数的线性矩阵不等式的复杂度。然后,基于Backstepping方法递归地设计输出反馈控制器,并构造合适的Lyapunov-Krasovskii泛函补偿所有滞后的状态摄动函数。证明了所有闭环信号的有界性,通过适当选取设计参数保证了期望的跟踪性能。最后,给出一个仿真例子验证所提出的控制算法的有效性。
5.基于Backstepping方法解决了一类具有三角形结构的非线性时滞系统的自适应镇定问题。不仅被控系统参数而且时滞及其导数的上界都假设是未知的,它们通过设计自适应律而被在线估计。进一步放松了与时滞有关的非线性项的假设条件。为在这种弱化的条件下取得控制目标,首先引入一个函数分解引理,然后通过构造合适的Lyapunov-Krasovskii泛函递归地设计控制器。为消除参数估计对闭环稳定性带来的不利影响,从Backstepping设计的第二步开始,每一步的虚拟控制中都增加两个附加函数。值得指出的是,这些附加项应仔细选取以使递推设计顺利进行。设计方案不仅保证了所有闭环信号的有界性,而且使得系统状态渐近收敛于零。仿真研究进一步表明了所提出的控制策略的有效性。
6.针对一类带有状态时变时滞、不确定参数和外在干扰的非线性系统,提出了一种自适应Backstepping的跟踪控制方案。把调节函数方法引入到非线性时滞系统中,递归地构造自适应律和控制律。这种方法有效地避免了上一章中虚拟控制函数附加项选取的困难,而且自适应控制设计更加程序化。采用适当的Lyapunov-Krasovskii泛函补偿含有时滞状态的非线性函数.时滞及其导数的上界也假设是未知的,通过设计两套调节函数保证闭环系统所有信号的有界性,通过选取适当的设计参数保证闭环系统的跟踪性能。仿真结果验证了所提出的控制方案的有效性。
In recent years, the control problem of uncertain nonlinear systems becomes one of the research focuses. Particularly, the control theory for nonlinear systems with actuator failures or time delays has attracted increasing attention. It is well known that actuator failures are frequently encountered during operation of control systems. Also, time delay phenomenon widely exists. Up to now, many results have been achieved on adaptive control of linear systems with actuator failures. However, for nonlinear systems, adaptive actuator failure compensation has not been deeply investigated. On the other hand, despite the huge literature for linear time-delay systems, the control problem of nonlinear time-delay systems has been receiving relatively little attention until recently. Based on the current research findings, this dissertation uses the backstepping approach as a basic tool to address the problem of adaptive actuator failure compensation of uncertain nonlinear systems and uncertain nonlinear time-delay system control, respectively. The main contributions of the dissertation are as follows.
1. A modified adaptive actuator failure compensation scheme is proposed for a class of multi-input and single-output nonlinear systems in the output-feedback form. A con-stant actuator failure model is considered. In the control design, a new parametric model with unknown plant parameters and actuator failure parameters is firstly established. Then, an adaptive compensation controller is constructed by utilizing the backstepping technique. The boundedness of all the closed-loop signals is guaranteed. The output tracking error is proved to converge to zero asymptotically. It is worth pointing out that, within the new model framework, the adaptive control design is programmed, and the stability analysis can be sim-plified greatly. Moreover, the lower-order filters can be exploited to estimate the unmeasured plant states. Thus, the resulting control scheme is easier to be implemented. Finally, the presented scheme is applied to a simplified dual-actuator ball-beam system for the actuator failure compensation study. Simulation results demonstrate the effectiveness of the proposed design method.
2. The robust adaptive output-feedback control problem is addressed for a class of non-linear systems with unknown time-varying actuator failures. Additional un-modelled term in the actuator failure model is considered. A linearly parameterized model is proposed, where unknown plant parameters and actuator failure parameters are lumped up together and separated form known signals. Based on the new model, adaptive controller is recursively designed. The modified switching-σsignals are incorporated in the adaptive laws to ensure stability and robustness. The boundedness of all the closed-loop signals is established. The desired control performance of the closed-loop system is guaranteed by appropriately choos- ing the design parameters. Finally, a dual-actuator ball-beam system with state-dependent nonlinearities is used to illustrate the effectiveness of the proposed scheme.
3. For a class of nonlinear systems with unknown high-frequency gain signs and param-eterizable time-varying actuator failures, an adaptive output-feedback failure compensation scheme is proposed based on the backstepping technique. The Nussbaum gain approach is introduced to remove the restrictive assumption on the high-frequency gain signs. In view of unknown high-frequency gain signs, some constant variables are redefined, the para-metric model with actuator failure information is reestablished, and the stability analysis is retreated. The boundedness of all the closed-loop signals and the asymptotic output track-ing are guaranteed in spite of the unknown actuator failures. Finally, the proposed control scheme is applied to a two-axis positioning stage system with redundant actuators. The effectiveness of the control algorithm is verified by the simulation studies.
4. The problem of output-feedback tracking control is addressed for a class of single-input and single-output nonlinear systems in the strict-feedback form, which are subjected to both uncertain delay-related functions and external disturbances. A reduced-order observer is firstly introduced to provide the estimates of the unmeasured states. The advantage of exploiting the lower-dimension observer is two-fold. On the one hand, the resulting closed-loop system is easier to be implemented. On the other hand, since the design parameters are determined by a linear matrix inequality (LMI), the cost involved by the computation of the LMI may be much less. Then, an output-feedback controller is recursively designed based on the backstepping method. An appropriate Lyapunov-Krasovskii functional is constructed to compensate for all delayed state perturbation functions. All the signals in the closed-loop system are proved to be bounded. The desired tracking performance is guaranteed by suitably choosing the design parameters. Finally, a simulation example is provided to demonstrate the effectiveness of the proposed control algorithm.
5. Based on the backstepping approach, the problem of adaptive stabilization for a class of nonlinear time-delay systems in triangular structure is addressed. Both the parameters of the system to be controlled and the upper bounds of the time delays and their deriva-tives are assumed to be unknown, which are estimated online by designing adaptive laws. The assumption on the delay-related nonlinearities is further relaxed. To achieve the con-trol objective under such weakened condition, a lemma on function factorization is firstly introduced. Then, an appropriate Lyapunov-Krasovskii functional is constructed and adap-tive controller is designed recursively. To counteract the influence of the parameter estimates on stability of the closed-loop system, from the second step of the backstepping procedure on, two terms are added in each virtual control function. It is worth pointing out that these additional functions should be selected carefully such that the backstepping design can be performed smoothly. It is shown that all the closed-loop signals are bounded, while the plant states converge to zero asymptotically. Simulation studies are provided to demonstrate the effectiveness of the proposed control strategy.
6. An adaptive backstepping tracking control scheme is proposed for a class of non-linear state time-varying delay systems, which are subject to parametric uncertainties and external disturbances. Tuning function method is exploited to construct the control law and adaptive laws recursively. This method not only effectively overcome the difficulty in the choice of additional terms in virtual control functions in the last chapter, but also make adap-tive control design more programmed. Unknown time-varying delays are compensated for by using appropriate Lyapunov-Krasovskii functional. The bounds of the time delays and their derivatives are also assumed to be unknown. Two set of tuning functions are designed to guarantee the boundedness of all the closed-loop signals. The tracking performance is ad-justed by choosing suitable design parameters. Simulation results illustrate the effectiveness of the proposed design procedure.
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