非线性系统的自适应学习控制
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摘要
学习控制的目的是在固定的有限时间区间或在一个无限时间上通过重复地直接更新控制输入,从而实现期望的系统性能。学习控制方法一般包括迭代学习控制(ILC)和重复学习控制(RLC),前者是基于压缩映射理论,后者是基于可视为压缩映射理论在无限时间区间延伸的小增益定理。
通过过引入参数的自适应机制,自适应控制系统在常参数不确定性存在的条件下,能够实现渐近跟踪收敛。然而,至今还没有自适应控制算法可以解决任意快的,非零变化的未知时变参数。当被控系统含有混合参数(时变和时不变参数)不确定性或时变增益系数时,如何充分利用优先的信息,通过自适应控制来设计迭代学习控制和重复学习控制,这是一个崭新的值得研究的课题.
本文主要从自适应控制的角度来研究迭代学习控制(ILC)和重复学习控制(RLC),基于Lyapunov-like能量函数方法,避免了传统的学习控制的一些缺陷和严格的假设。其主要工作概括如下:
1.针对一类含有结构不确定性和外部扰动的非线性系统,利用Backstepping方法,线性参数化和参数重构相结合,提出了一种自适应迭代学习控制方法。该方法由微分-差分型自适应律和学习控制律组成,保证对非一致目标的跟踪误差的平方在一个有限区间上的积分渐近收敛于零,克服了传统ILC对目标轨线的限制,可以跟踪非一致目标轨线。通过构造复合能量函数,给出了闭环系统收敛的一个充分条件。
2.针对含有未知时变控制系数和混合参数的非线性不确定系统,提出了一种自适应迭代学习控制方法。该方法利用改进的Backstepping方法和参数重组技巧相结合,可以处理目标轨线迭代可变的跟踪问题,通过引入微分-差分混合型参数自适应学习律,设计了一种自适应控制策略,使得跟踪误差在一个有限区间上的积分渐近收敛于零,通过构造Lyapunov-like函数,给出了闭环系统收敛的一个充分条件。
3.针对一类含有时变和时不变未知参数的非线性系统,利用分段积分机制,提出了一种新的自适应重复学习控制方法。该方法结合了反馈线性化,可以处理参数在一个未知紧集内周期性快时变的非线性系统,通过引进微分-差分参数自适应律,设计了一种自适应控制策略,使广义跟踪误差在误差平方范数意义下渐近收敛于零,通过构造Lyapunov函数,给出了闭环系统收敛的一个充分条件。
4.针对含有时变和时不变未知参数的高阶非线性系统,提出了一种自适应重复学习控制器的设计方案。该方案利用Backstepping方法和分段积分机制相结合,可以处理参数是非零变化的周期性快时变的非线性系统,通过引进周期参数自适应律,设计了一种自适应控制策略,使跟踪误差在误差平方积分范数意义下渐近收敛于零,通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件。
5.针对一类含有未知控制系数和混合参数的非线性不确定系统,设计了一种自适应重复学习控制方案。该方案利用分段积分机制,参数重组技巧和改进的Backstepping方法相结合,可以处理参数在一个未知紧集内周期性快时变的非线性系统的跟踪问题。通过引入微分-差分自适应学习律,设计了一种自适应控制策略,使跟踪误差在误差平方积分范数意义下渐近收敛于零;通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件。
6.针对含有周期时变参数和时不变参数的非线性参数化系统,利用参数重组技巧,提出了一种新的自适应重复学习控制方法。该方法结合反馈线性化,可以处理非线性参数是非零变化的周期性快时变的非线性系统,通过引进微分-差分耦合型参数自适应律,设计了一种自适应控制策略,使广义跟踪误差在误差平方范数意义下渐近收敛于零;通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件。结果被推广到一类含有未知时变控制系数和混合参数的非线性参数化系统的自适应学习控制。
7.针对具有一般形式的非线性参数化系统,提出了一种新的自适应重复学习控制方法。该方法利用系统转换思想,分段积分机制和参数重组技巧相结合,可以处理非线性参数在一个未知紧集内周期性快时变的非线性系统,通过引进微分-差分耦合型参数自适应律,设计了一种自适应控制策略,使广义跟踪误差在误差平方范数意义下渐近收敛于零,通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件。
8.提出了一种自适应学习控制方法,应用于不同混沌系统由于未知的周期时变参数的广义投影同步(GPS)。该方法利用Lyapunov-Krasovskii泛函稳定性理论,构造了微分-差分混合参数学习律和自适应学习控制律,使得两个不同混沌系统的状态误差在一个周期区间上平方范数的积分意义下渐近同步。方案成功地应用于Lorenz系统和Chen系统的广义投影同步,而且,数值仿真结果验证了所提方案的有效性。
Learning control aims at achieving the desired system performance through directly updating the control input, either repeatedly over a fixed finite time interval, or repetitively over an infinite time interval periodicity.Learning control approaches can be in general classified into iterative learning control(ILC) and repetitive learning control(RLC). The former is based on contraction mapping theory, and the latter is based on the small gain theorem which can be viewed as the extension of contraction mapping theory to the infinite time interval.
By introducing a parametric adaptation mechanism, the adaptive control system is able to achieve asymptotic tracking convergence in the presence of constant parametric uncertainties. However, no adaptive control algorithms developed hitherto can solve unknown parameters with arbitrarily fast and non-vanishing variations.When the plant has mixed parametric(time-varying and time-invariant parameters) uncertainties or the time-varying gain coefficient, how to make full use of the prior information and design ILC and RLC by adaptive control, have become a new subject worthwhile to research.
This dissertation considers the ILC and RLC from an adaptive control viewpoint, based on the Lyapunov-like energy function approaches, which avoids some drawbacks and restricted assumptions of traditional learning control. The main contributions included in the dissertation are summarized as follows.
1. A novel adaptive iterative learning control approach is proposed for high-order nonlinear systems with structure uncertainties and external distureances, by combining the Backstepping method, the linearly parametrized with the parameter regrouping technique. The approach consisted of a differential-deference type updating law and a learning control law, can deal with the non-uniform trajectory tracking problem,which avoids the restricted on the tracking trajectory in the traditional ILC. A sufficient condition of tracking error converging to zero in the means of mean-square on the finite interval is also given by constructing a novel composite energy function.
2. An adaptive iterative learning control of nonlinear systems with mixed-type parameters and unknown time-varying control coefficients is developed by combining the modifying Backstepping approach with the parameter regrouping technique, which can handle the iteration-varying trajectory tracking problem. A differential-difference type adaptive law and an adaptive iterative learning controller are constructed to ensure the asymptotic convergence of tracking error in the sense of square error norm on the finite interval, by introducing a Lyapunov-like function, a sufficient condition of the convergence of the method is also given.
3. Combining the pointwise integral mechanism with the feedback linearization approach, a novel adaptive repetitive learning control for nonlinear systems with time-varying and time-invariant parameters is proposed. It can be applied to the time-varying parametric uncertainty systems with unknown compact set, rapid time-varying, periodic and where the prior knowledge is the periodicity only. A differential-difference adaptive law and an adaptive repetitive learning control are constructed to ensure the asymptotic convergence of the extended tracking error in the sense of square error norm. Also, a sufficient condition of the convergence of the method is given.
4. A design approach to the adaptive repetitive learning controller is proposed for a class of nonlinear systems with time-varying and time-invariant parameters by combining the Backstepping technique with the pointwise integral mechanism. It can be applied in the time-varying parametric uncertainty systems with unknown compact set, rapid time-varying, periodic and only the prior knowledge is the periodicity. A periodic mixed adaptive law and an adaptive control law are constructed to ensure the asymptotic convergence of the tracking error in the sense of square error norm. And a sufficient condition of the convergence of the method is also given.
5. Combining the pointwise integral mechanism, the parameter regrouping technique with the modifying Backstepping approach, a novel adaptive repetitive learning control for high-order nonlinear systems with unknown control coefficients and mixed parameters is proposed. It can be applied to the time-varying parametric uncertainty systems with unknown compact set, rapid time-varying, periodic and where the prior knowledge is the periodicity only. A differential-difference mixed-type adaptive law and an adaptive repetitive learning control are constructed to ensure the asymptotic convergence of the tracking error in the sense of square error norm. Also, a sufficient condition of the convergence of the method is given.
6. A design approach to the adaptive repetitive learning controller is proposed for a class of nonlinearly parameterized systems with time-varying and time-invariant parameters by combining the parameter regrouping technique with the feedback linearization approach, It can be applied to the time-varying parametric uncertainty systems with rapid time-varying, periodic and where the prior knowledge is the periodicity only. A differential-difference coupled-type adaptive law and an adaptive repetitive learning control are constructed to ensure the asymptotic convergence of the extended tracking error in the sense of square error norm. Also, a sufficient condition of the convergence of the method is also given. The results are extended to a class of high-order nonlinearly parameterized systems with unknown time-varying control coefficients and mixed parameters.
7. A novel adaptive repetitive learning control for nonlinearly parameterized systems with unknown control coefficients and mixed parameters is proposed by combining the system replacement technique, the pointwise integral mechanism with the parameter regrouping approach, It can be applied to the time-varying parametric uncertainty systems with rapid time-varying, periodic and where the prior knowledge is the periodicity only. A differential-difference coupled-type adaptive law and an adaptive repetitive learning control are constructed to ensure the asymptotic convergence of the tracking error in the sense of square error norm. Also, a sufficient condition of the convergence of the method is given.
8. The problem of a learning control approach is applied to the generalized projective synchronization of different chaotic systems with unknown periodically time-varying parameters. Using the Lyapunov-Krasovskii functional stability theory, a differential-difference mixed-type parametric learning law and an adaptive learning control law are constructed to make the states of two different chaotic systems asymptotically synchronized. The scheme is successfully applied to the generalized projective synchronization between Lorenz system and Chen system. Moreover, numerical simulations results are used to verify the effectiveness of the proposed scheme.
通过过引入参数的自适应机制,自适应控制系统在常参数不确定性存在的条件下,能够实现渐近跟踪收敛。然而,至今还没有自适应控制算法可以解决任意快的,非零变化的未知时变参数。当被控系统含有混合参数(时变和时不变参数)不确定性或时变增益系数时,如何充分利用优先的信息,通过自适应控制来设计迭代学习控制和重复学习控制,这是一个崭新的值得研究的课题.
本文主要从自适应控制的角度来研究迭代学习控制(ILC)和重复学习控制(RLC),基于Lyapunov-like能量函数方法,避免了传统的学习控制的一些缺陷和严格的假设。其主要工作概括如下:
1.针对一类含有结构不确定性和外部扰动的非线性系统,利用Backstepping方法,线性参数化和参数重构相结合,提出了一种自适应迭代学习控制方法。该方法由微分-差分型自适应律和学习控制律组成,保证对非一致目标的跟踪误差的平方在一个有限区间上的积分渐近收敛于零,克服了传统ILC对目标轨线的限制,可以跟踪非一致目标轨线。通过构造复合能量函数,给出了闭环系统收敛的一个充分条件。
2.针对含有未知时变控制系数和混合参数的非线性不确定系统,提出了一种自适应迭代学习控制方法。该方法利用改进的Backstepping方法和参数重组技巧相结合,可以处理目标轨线迭代可变的跟踪问题,通过引入微分-差分混合型参数自适应学习律,设计了一种自适应控制策略,使得跟踪误差在一个有限区间上的积分渐近收敛于零,通过构造Lyapunov-like函数,给出了闭环系统收敛的一个充分条件。
3.针对一类含有时变和时不变未知参数的非线性系统,利用分段积分机制,提出了一种新的自适应重复学习控制方法。该方法结合了反馈线性化,可以处理参数在一个未知紧集内周期性快时变的非线性系统,通过引进微分-差分参数自适应律,设计了一种自适应控制策略,使广义跟踪误差在误差平方范数意义下渐近收敛于零,通过构造Lyapunov函数,给出了闭环系统收敛的一个充分条件。
4.针对含有时变和时不变未知参数的高阶非线性系统,提出了一种自适应重复学习控制器的设计方案。该方案利用Backstepping方法和分段积分机制相结合,可以处理参数是非零变化的周期性快时变的非线性系统,通过引进周期参数自适应律,设计了一种自适应控制策略,使跟踪误差在误差平方积分范数意义下渐近收敛于零,通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件。
5.针对一类含有未知控制系数和混合参数的非线性不确定系统,设计了一种自适应重复学习控制方案。该方案利用分段积分机制,参数重组技巧和改进的Backstepping方法相结合,可以处理参数在一个未知紧集内周期性快时变的非线性系统的跟踪问题。通过引入微分-差分自适应学习律,设计了一种自适应控制策略,使跟踪误差在误差平方积分范数意义下渐近收敛于零;通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件。
6.针对含有周期时变参数和时不变参数的非线性参数化系统,利用参数重组技巧,提出了一种新的自适应重复学习控制方法。该方法结合反馈线性化,可以处理非线性参数是非零变化的周期性快时变的非线性系统,通过引进微分-差分耦合型参数自适应律,设计了一种自适应控制策略,使广义跟踪误差在误差平方范数意义下渐近收敛于零;通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件。结果被推广到一类含有未知时变控制系数和混合参数的非线性参数化系统的自适应学习控制。
7.针对具有一般形式的非线性参数化系统,提出了一种新的自适应重复学习控制方法。该方法利用系统转换思想,分段积分机制和参数重组技巧相结合,可以处理非线性参数在一个未知紧集内周期性快时变的非线性系统,通过引进微分-差分耦合型参数自适应律,设计了一种自适应控制策略,使广义跟踪误差在误差平方范数意义下渐近收敛于零,通过构造Lyapunov泛函,给出了闭环系统收敛的一个充分条件。
8.提出了一种自适应学习控制方法,应用于不同混沌系统由于未知的周期时变参数的广义投影同步(GPS)。该方法利用Lyapunov-Krasovskii泛函稳定性理论,构造了微分-差分混合参数学习律和自适应学习控制律,使得两个不同混沌系统的状态误差在一个周期区间上平方范数的积分意义下渐近同步。方案成功地应用于Lorenz系统和Chen系统的广义投影同步,而且,数值仿真结果验证了所提方案的有效性。
Learning control aims at achieving the desired system performance through directly updating the control input, either repeatedly over a fixed finite time interval, or repetitively over an infinite time interval periodicity.Learning control approaches can be in general classified into iterative learning control(ILC) and repetitive learning control(RLC). The former is based on contraction mapping theory, and the latter is based on the small gain theorem which can be viewed as the extension of contraction mapping theory to the infinite time interval.
By introducing a parametric adaptation mechanism, the adaptive control system is able to achieve asymptotic tracking convergence in the presence of constant parametric uncertainties. However, no adaptive control algorithms developed hitherto can solve unknown parameters with arbitrarily fast and non-vanishing variations.When the plant has mixed parametric(time-varying and time-invariant parameters) uncertainties or the time-varying gain coefficient, how to make full use of the prior information and design ILC and RLC by adaptive control, have become a new subject worthwhile to research.
This dissertation considers the ILC and RLC from an adaptive control viewpoint, based on the Lyapunov-like energy function approaches, which avoids some drawbacks and restricted assumptions of traditional learning control. The main contributions included in the dissertation are summarized as follows.
1. A novel adaptive iterative learning control approach is proposed for high-order nonlinear systems with structure uncertainties and external distureances, by combining the Backstepping method, the linearly parametrized with the parameter regrouping technique. The approach consisted of a differential-deference type updating law and a learning control law, can deal with the non-uniform trajectory tracking problem,which avoids the restricted on the tracking trajectory in the traditional ILC. A sufficient condition of tracking error converging to zero in the means of mean-square on the finite interval is also given by constructing a novel composite energy function.
2. An adaptive iterative learning control of nonlinear systems with mixed-type parameters and unknown time-varying control coefficients is developed by combining the modifying Backstepping approach with the parameter regrouping technique, which can handle the iteration-varying trajectory tracking problem. A differential-difference type adaptive law and an adaptive iterative learning controller are constructed to ensure the asymptotic convergence of tracking error in the sense of square error norm on the finite interval, by introducing a Lyapunov-like function, a sufficient condition of the convergence of the method is also given.
3. Combining the pointwise integral mechanism with the feedback linearization approach, a novel adaptive repetitive learning control for nonlinear systems with time-varying and time-invariant parameters is proposed. It can be applied to the time-varying parametric uncertainty systems with unknown compact set, rapid time-varying, periodic and where the prior knowledge is the periodicity only. A differential-difference adaptive law and an adaptive repetitive learning control are constructed to ensure the asymptotic convergence of the extended tracking error in the sense of square error norm. Also, a sufficient condition of the convergence of the method is given.
4. A design approach to the adaptive repetitive learning controller is proposed for a class of nonlinear systems with time-varying and time-invariant parameters by combining the Backstepping technique with the pointwise integral mechanism. It can be applied in the time-varying parametric uncertainty systems with unknown compact set, rapid time-varying, periodic and only the prior knowledge is the periodicity. A periodic mixed adaptive law and an adaptive control law are constructed to ensure the asymptotic convergence of the tracking error in the sense of square error norm. And a sufficient condition of the convergence of the method is also given.
5. Combining the pointwise integral mechanism, the parameter regrouping technique with the modifying Backstepping approach, a novel adaptive repetitive learning control for high-order nonlinear systems with unknown control coefficients and mixed parameters is proposed. It can be applied to the time-varying parametric uncertainty systems with unknown compact set, rapid time-varying, periodic and where the prior knowledge is the periodicity only. A differential-difference mixed-type adaptive law and an adaptive repetitive learning control are constructed to ensure the asymptotic convergence of the tracking error in the sense of square error norm. Also, a sufficient condition of the convergence of the method is given.
6. A design approach to the adaptive repetitive learning controller is proposed for a class of nonlinearly parameterized systems with time-varying and time-invariant parameters by combining the parameter regrouping technique with the feedback linearization approach, It can be applied to the time-varying parametric uncertainty systems with rapid time-varying, periodic and where the prior knowledge is the periodicity only. A differential-difference coupled-type adaptive law and an adaptive repetitive learning control are constructed to ensure the asymptotic convergence of the extended tracking error in the sense of square error norm. Also, a sufficient condition of the convergence of the method is also given. The results are extended to a class of high-order nonlinearly parameterized systems with unknown time-varying control coefficients and mixed parameters.
7. A novel adaptive repetitive learning control for nonlinearly parameterized systems with unknown control coefficients and mixed parameters is proposed by combining the system replacement technique, the pointwise integral mechanism with the parameter regrouping approach, It can be applied to the time-varying parametric uncertainty systems with rapid time-varying, periodic and where the prior knowledge is the periodicity only. A differential-difference coupled-type adaptive law and an adaptive repetitive learning control are constructed to ensure the asymptotic convergence of the tracking error in the sense of square error norm. Also, a sufficient condition of the convergence of the method is given.
8. The problem of a learning control approach is applied to the generalized projective synchronization of different chaotic systems with unknown periodically time-varying parameters. Using the Lyapunov-Krasovskii functional stability theory, a differential-difference mixed-type parametric learning law and an adaptive learning control law are constructed to make the states of two different chaotic systems asymptotically synchronized. The scheme is successfully applied to the generalized projective synchronization between Lorenz system and Chen system. Moreover, numerical simulations results are used to verify the effectiveness of the proposed scheme.
引文
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