Globally stabilizing adaptive control design for nonlinearly-parameterized systems

In this note, a new adaptive control design is proposed for nonlinear systems that are possibly nonaffine and contain nonlinearly parameterized unknowns. The proposed control is not based on certainty equivalence principle which forms the foundation of existing and standard adaptive control designs. Instead, a biasing vector function is introduced into parameter estimate; it links the system dynamics to estimation error dynamics, and its choice leads to a new Lyapunov-based design so that affine or nonaffine systems with nonlinearly parameterized unknowns can be controlled by adaptive estimation. Explicit conditions are found for achieving global asymptotic stability of the state, and the convergence condition for parameter estimation is also found. The conditions are illustrated by several examples and classes of systems. Besides global stability and estimation convergence, the proposed adaptive control has the unique feature that it does not contains any robust control part which typically overpowers unknown dynamics, may be conservative, and also interferes with parameter estimation.     

Indirect adaptive fuzzy control for a class of nonaffine nonlinear systems with unknown control directions

This article presents an indirect adaptive fuzzy control scheme for a class of nonlinear uncertain nonaffine systems with unknown control directions. The nonlinear nonaffine system is first transformed into an affine form by using a Taylor series expansion, and then fuzzy systems are employed to approximate the equivalent affine system’s unknown nonlinearities. By modifying the estimated input control gain and using a novel smooth robust control term, a stable well-defined adaptive controller is proposed. Simulation results are provided to illustrate the efficiency of the proposed scheme.     

Adaptive neural control of high-order uncertain nonaffine systems: A transformation to affine systems approach

This brief investigates the adaptive neural network (NN) control of a class of high-order nonaffine nonlinear systems with completely unknown dynamics. Since the control terms appear within the unknown nonlinearity, traditional control schemes and stability analysis are usually rendered extremely complicated. Our main contribution includes a novel system transformation that converts the nonaffine system into an affine system through a combination of a low-pass filter and state transformation. As a result, the state-feedback control of the nonaffine system can be viewed as the output-feedback control of an affine system in normal form. The transformed system becomes linear with respect to the new input while the traditional backstepping approach is not needed thus allowing the synthesis to be extremely simplified. It is theoretically proven that all the signals in the closed-loop system are uniformly ultimately bounded (UUB). Simulation results are provided to demonstrate the performance of the developed controller.     

控制方向未知的SISO非仿射系统间接自适应模糊输出反馈控制

针对一类单输入单输出(SISO)非仿射非线性系统控制方向未知时出现的控制器奇异问题,提出了一种间接自适应模糊控制方案.利用中值定理将非仿射系统转化为仿射系统,通过模糊逻辑系统逼近该仿射系统中的未知函数,并构造模糊控制器,同时利用Lyapunov稳定性定理设计自适应律,最终克服了控制器的奇异问题;在此基础上,通过构造观测器估计跟踪误差,设计输出反馈自适应模糊控制器,解决了状态不可测时系统控制器设计难题,采用Lyapunov稳定性定理证明控制器能使得跟踪误差收敛同时闭环系统所有信号均有界.仿真结果验证了所设计控制方案的可行性与有效性.     

Robust estimation‐free decentralized prescribed performance control of nonaffine nonlinear large‐scale systems

In this paper, a low‐complexity robust estimation‐free decentralized prescribed performance control scheme is proposed and analyzed for nonaffine nonlinear large‐scale systems in the presence of unknown nonlinearity and external disturbance. To tackle the high‐order dynamics of each tracking error subsystem, a time‐varying stable manifold involving the output tracking error and its high‐order derivatives is constructed, which is strictly evolved within the envelope of user‐specialized prescribed performance. Sequentially, a robust decentralized controller is devised for each manifold, under which the output tracking error and its high‐order derivatives are proven to converge asymptotically to a small residual domain with prescribed fast convergence rate. Additionally, no specialized approximation technique, adaptive scheme, and disturbance observer are needed, which alleviates the complexity and difficulty of robust decentralized controller design dramatically. Finally, 3 groups of illustrative examples are used to validate the effectiveness of the proposed low‐complexity robust decentralized control scheme for uncertain nonaffine nonlinear large‐scale systems.     

Robust adaptive control of nonlinearly parametrized multivariable systems with unmatched disturbances

A new robust adaptive control scheme is developed for nonlinearly parametrized multivariable systems in the presence of parameter uncertainties and unmatched disturbances. The developed control scheme employs a new integrated framework of a functional bounding technique for handling nonlinearly parametrized system dynamics, an adaptive parameter estimation algorithm for dealing with parameter uncertainties, a nonlinear feedback controller structure for stabilization of interconnected system states, and a robust adaptive control design for accommodating unmatched disturbances. It is proved that such a new robust adaptive control scheme is capable of ensuring the global boundedness and mean convergence of all closed‐loop system signals. A complete simulation study on an air vehicle system with nonlinear parametrization in the presence of an unmatched wind disturbance is conducted, and its results verify the effectiveness of the proposed robust adaptive control scheme.     

Adaptive dynamic programming and optimal control of nonlinear nonaffine systems

In this paper, a novel optimal control design scheme is proposed for continuous-time nonaffine nonlinear dynamic systems with unknown dynamics by adaptive dynamic programming (ADP). The proposed methodology iteratively updates the control policy online by using the state and input information without identifying the system dynamics. An ADP algorithm is developed, and can be applied to a general class of nonlinear control design problems. The convergence analysis for the designed control scheme is presented, along with rigorous stability analysis for the closed-loop system. The effectiveness of this new algorithm is illustrated by two simulation examples.     

Robust adaptive trajectory tracking control of underactuated autonomous underwater vehicles with prescribed performance

In this paper, a novel robust adaptive trajectory tracking control scheme with prescribed performance is developed for underactuated autonomous underwater vehicles (AUVs) subject to unknown dynamic parameters and disturbances. A simple error mapping function is proposed in order to guarantee that the trajectory tracking error satisfies the prescribed performance. A novel additional control based on Nussbaum function is proposed to handle the underactuation of AUVs. The compounded uncertain item caused by the unknown dynamic parameters and disturbances is transformed into a linear parametric form with only single unknown parameter called virtual parameter. On the basis of the above, a novel robust adaptive trajectory tracking control law is developed using dynamic surface control technique, where the adaptive law online provides the estimation of the virtual parameter. Strict stability analysis indicates that the designed control law ensures uniform ultimate boundedness of the AUV trajectory tracking closed‐loop control system with prescribed tracking performance. Simulation results on an AUV in two different disturbance cases with dynamic parameter perturbation verify the effectiveness of our adaptive trajectory tracking control scheme.     

Linear adaptive control of a class of SISO nonaffine nonlinear systems

This paper addresses the problem of linear adaptive control for a class of uncertain continuous-time single-input single-output (SISO) nonaffine nonlinear dynamic systems. Using the implicit function theory, the existence of an ideal controller which can achieve control objectives is firstly demonstrated. However, this ideal controller cannot be known and computed even if the system model is well known. The aim of our work is to construct this unknown ideal controller using a simple linear controller with the free parameters updated online by a stable adaptation mechanism designed to minimise the error between the unknown ideal controller and the used linear controller. Since the mathematical model of the system is assumed unknown in this work, the proposed control scheme can be regarded as a simple model free controller for the studied class of nonaffine systems. We prove that the closed-loop system is stable and all the signals are bounded. An application of the proposed linear adaptive controller for a nonaffine system is illustrated through the simulation results to demonstrate the effectiveness of the proposed control scheme.     

时变时滞非仿射大系统的分散自适应控制

针对一类具有未知时变时滞的非仿射互联大系统基于神经网络的逼近能力, 提出了一种分散自适应神经网络控制方案。该方案利用中值定理对未知非仿射函数进行分离; 利用分离技术和Young's不等式放宽了对未知时滞及时滞互联不确定项的限制, 同时大大减少了在线调节参数的数量。此外, 利用Lyapunov Krasovskii 泛函补偿了未知时滞带来的不确定性。通过理论分析, 证明了闭环系统所有信号是有界的, 输出跟踪误差收敛到原点的一个小邻域内。最后, 仿真结果验证了所提控制方案的有效性。     

Robust adaptive control of nonlinear systems represented by input-output models

A robust adaptive control scheme is proposed for a class of nonlinear systems represented by input-output models with unmodeled dynamics. The scheme does not require the unknown parameters to satisfy the linear dependence condition and parameter estimation is not needed. With the proposed control scheme, all the variables in the closed-loop system are bounded in the presence of unmodeled dynamics and bounded disturbances. Moreover, the mean-square tracking error can be made arbitrarily small by choosing some design parameters appropriately.     

Robust adaptive prescribed performance control for a class of nonlinear pure‐feedback systems

An adaptive prescribed performance control design procedure for a class of nonlinear pure‐feedback systems with both unknown vector parameters and unmodeled dynamics is presented. The unmodeled dynamics lie within some bounded functions, which are assumed to be partially known. A state transformation and an auxiliary system are proposed to avoid using the cumbersome formula to handle the nonaffine structure. Simultaneously, a parameter‐type Lyapunov function and L function are designed to ensure the prescribed performance of the pure‐feedback system. As illustrated by examples, the proposed adaptive prescribed performance control scheme is shown to guarantee global uniform ultimate boundedness. At the same time, this method not only guarantees the prescribed performance of the system but also makes the tracking error asymptotically close to a certain value or stable.     

Adaptive dynamic surface control for uncertain nonaffine nonlinear systems

This paper focuses on the problem of adaptive control for uncertain nonaffine nonlinear systems. The original nonaffine systems are transformed into the augmented affine systems via adding an auxiliary integrator, which makes the explicit control design possible. By introducing a modified sliding mode filter in each step, a novel adaptive dynamic surface controller is proposed, where the ‘explosion of complexity’ problem inherent in the backstepping design is avoided. It is proven rigorously that for any initial control condition, the proposed adaptive scheme is able to ensure the semiglobal uniformly ultimately boundedness of all signals in the closed loop. An illustrative example is carried out to verify the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

一种自适应鲁棒控制方法及其闭环稳定性分析

研究了含有未建模动态的慢时变系统的自适应镇定问题.考虑的对象具有非最小相 位、含未建模动态和大范围时变参数等不良特性,且存在未知但有界外部扰动.这类对象很难 用时不变鲁棒控制器或传统自适应控制器进行镇定.利用l1优化设计结合参数估计的投影算 法,提出了一种自适应鲁棒控制策略.基于l1优化设计的连续性和投影算法的收敛性,证明了 这种控制策略能够持续适应慢时变对象并且保持闭环系统一致稳定性.鲁棒性分析表明这种 控制策略具有良好的鲁棒镇定性.     

Adaptive Robust Control for a Class of Uncertain MIMO Non-affine Nonlinear Systems

In this paper, the adaptive robust tracking control scheme is proposed for a class of multi-input and multioutput (MIMO) non-affine systems with uncertain structure and parameters, unknown control direction and unknown external disturbance based on backstepping technique. The MIMO nonaffine system is first transformed into a time-varying system with strict feedback structure using the mean value theorem, and then the bounded time-varying parameters are estimated by adaptive algorithms with projection. To handle the possible "controller singularity" problem caused by unknown control direction, a Nussbaum function is employed, and the dynamic surface control (DSC) method is applied to solve the problem of "explosion of complexity" in backstepping control. It is proved that the proposed control scheme can guarantee that all signals of the closed-loop system are bounded through Lyapunov stability theorem and decoupled backstepping method. Simulation results are presented to illustrate the effectiveness of the proposed control scheme.      

基于神经网络的非仿射块控非线性系统动态反演控制

即使已知非仿射非线性系统的逆存在,利用隐函数定理求解该显式逆仍然非常困难.为此,针对一类不确定块控非仿射系统,将动态反馈、反演、神经网络和反馈线性化技术相结合,提出一种自适应鲁棒控制器的设计方法.利用神经网络来逼近和消除未知函数,并证明了整个闭环系统在李雅普诺夫意义下是稳定的.仿真结果表明了所提出方法的有效性.     

Robust decentralized direct adaptive output feedback fuzzy control for a class of large-sale nonaffine nonlinear systems

In the previous work of Huang et al., a decentralized direct adaptive fuzzy H_∞ tracking controller of large-scale nonaffine nonlinear systems is obtained predicated upon the assumption that the mismatching error dynamics stay squared integrable. In this note, we focus in the absence of the conservative assumption upon developing a robust decentralized direct adaptive output feedback fuzzy controller. By combination of a state observer, a fuzzy inference system and robust control technique, the previous controller design is modified and no a priori knowledge of bounds on lumped uncertainties is required. All the signals of the closed-loop large-scale system are proved to be uniformly ultimately bounded. The effectiveness of the developed scheme is demonstrated through the simulation results of interconnected inverted pendulums.     

不确定非线性系统的鲁棒自适应输出反馈控制

基于自适应非线性阻尼,提出一种鲁棒自适应输出反馈控制方法。该方法适用于带有未建模动态、未知非线性、有界扰动、未知非线性参数和不确定控制系数的多输入多输出非线性系统。理论证明,在一定的假设条件下,该方法能保证闭环系统所有动态信号有界;不论有多少不确定非线性参数、多高阶的非线性系统,只需要一个自适应控制参数和观察参数;而且通过选择适当的控制器和观测器参数,能使控制误差和估计误差达到任意小。仿真结果表明了所提出方法的有效性。     

Robustness analysis of discrete-time indirect model reference adaptive control with normalized adaptive laws

For a class of discrete-time systems with unmodeled dynamics and bounded disturbance, the design and analysis of robust indirect model reference adaptive control (MRAC) with normalized adaptive law are investigated. The main work includes three parts. Firstly, it is shown that the constructed parameter estimation algorithm not only possesses the same properties as those of traditional estimation algorithms, but also avoids the possibility of division by zero. Secondly, by establishing a relationship between the plant parameter estimate and the controller parameter estimate, some similar properties of the latter axe also established. Thirdly, by using the relationship between the normalizing signal and all the signals of the closed-loop system, and some important mathematical tools on discrete-time systems, as in the continuous-time case, a systematic stability and robustness analysis approach to the discrete indirect robust MRAC scheme is developed rigorously.     

Robust adaptive self-structuring fuzzy control design for nonaffine,nonlinear systems

In this article, a robust adaptive self-structuring fuzzy control (RASFC) scheme for the uncertain or ill-defined nonlinear, nonaffine systems is proposed. The RASFC scheme is composed of a robust adaptive controller and a self-structuring fuzzy controller. In the self-structuring fuzzy controller design, a novel self-structuring fuzzy system (SFS) is used to approximate the unknown plant nonlinearity, and the SFS can automatically grow and prune fuzzy rules to realise a compact fuzzy rule base. The robust adaptive controller is designed to achieve an L ₂ tracking performance to stabilise the closed-loop system. This L ₂ tracking performance can provide a clear expression of tracking error in terms of the sum of lumped uncertainty and external disturbance, which has not been shown in previous works. Finally, five examples are presented to show that the proposed RASFC scheme can achieve favourable tracking performance, yet heavy computational burden is relieved.