Instability analysis and improvement of robustness of adaptive control

The effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed. Five possible types of instability mechanisms—parameter drift, ‘linear’ instability, ‘fast adaptation’ instability, ‘high frequency’ instability, and ‘throughput’ instability—are analyzed using simple examples. A procedure is used to construct Lyapunov-like functions for a modified adaptive controller applied to a dominant plant of relative degree one, in the presence of parasitics and disturbances, and obtain sufficient conditions under which none of the five types of instability can occur. The modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set. The size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast, the region of attraction becomes the whole space.     

Robust adaptive controller with zero residual tracking errors

The adaptive control of a plant whose dominant part has transfer function with relative degreen^{ast} = 1has been considered in the presence of parasitics and disturbances. A new adaptive law is proposed which guarantees the existence of a large region of attraction from which all signals are bounded and the tracking error converges to a small residual set. In contrast to the adaptive law used in [1], [2] the new adaptive law guarantees a smaller residual set for the tracking error, which reduces to zero when the parasitics and disturbances disappear.     

Robust redesign of adaptive control

Effects of unmodeled high frequency dynamics on stability and performance of adaptive control schemes are analyzed. In the regulation problem global stability properties are no longer guaranteed, but a region of attraction exists for exact adaptive regulation. The dependence of the region of attraction on unmodeled parasitics is examined first. Then the general case of model reference adaptive control is considered in which parasitics can destroy stability and boundedness properties. A modified adaptive law is proposed guaranteeing the existence of a region of attraction from which all signals converge to a residual set which contains the equilibrium for exact tracking. The size of this set depends on design parameters, the frequency range of parasitics, and the reference input signal characteristics.     

Decentralized adaptive control of interconnected systems with reduced-order models

The adaptive control of interconnected systems whose subsystems possess slow and fast modes is investigated in the presence of external disturbances. A simple example is first used to show that a decentralized adaptive controller can become unstable due to the unmodeled interconnections and/or neglected parasitics. An approach is then developed for stabilization and tracking using decentralized adaptive controllers with modified adaptive laws. Sufficient conditions are obtained under which a region of attraction exists for boundedness and exponential convergence of the state/parameter errors to a small residual set. The size of the region of attraction depends on the frequency range of the local parasitics in such a way that if all the local parasitics become infinitely fast, the region of attraction becomes the whole space.     

Reduced-order adaptive control scheme

The objective of this paper is to present a new adaptive law motivated by the work of Narendra et al. (1980) for the robust adaptive control of plants with unmodelled high frequency dynamics. In the regulation case the adaptive system has bounded solutions. Stable performance is still guaranteed when the effect of high frequency parasitics is considered. Simulation results are presented throughout the paper to complement the theoretical developments. It has been shown that the adaptive gain, the initial conditions, the mode-separation ratio and the magnitude and periodicity of the reference input sequence are important factors in the design of stable adaptive control schemes.     

A robust adaptive controller with minimal modifications fordiscrete time-varying systems

The goal of this paper is to show that an indirect adaptive controller with parameter projection as the only modification on the basis of conventional adaptive control algorithms can globally stabilize systems having fast parasitics, bounded external disturbances, and time-varying parameters without any restriction on signals in the closed-loop system such as persistence of excitation. Further, the controller can still retain the properties of earlier unmodified conventional adaptive controllers when the controlled plant satisfies so-called “ideal assumptions” or the rates at which the plant parameters change belong to the l₁ (or l₂) space     

已知非最小相位系统的跟踪问题研究

本文探讨了已知离散时间非最小相位系统的跟踪问题，证明了一大类针对非最小相位系统的线性控制器（其具体形式将在文中给出）不能实现对任意有界设定序列的渐近跟踪，这表明对未知离散非最小相位系统，不可能设计Ｃｌａｒｋｅ－Ｇａｗｔｈｒｏｐ型自动校正控制或自有因地制宜制品在去渐近跟踪任意有界设定序列。     

Finite-dimensional adaptive control for infinite-dimensional systems

This paper deals with the design of a finite-dimensional model reference adaptive controller for a distributed parameter system. The space of inputs and the space of outputs are both one dimensional. First an input—output representation is introduced, using filtered values generated from the inputs and outputs. Next it is shown that we can design a finite-dimensional adaptive control system by a new adaptive law. The new adaptive law guarantees the existence of a large region of attraction from which all signals are bounded and the tracking error converges to a small residual set.     

Convergence properties of LMS adaptive estimators with unbounded dependent inputs

This note presents limit theorems for the behavior of adaptive estimators using the LMS algorithm when the driving or input sequence is a member of a broad class of random processes which are not necessarily almost surely bounded and are dependent over time. Convergence in distribution of the estimates is established in the stationary case while general nonstationary tracking is characterized in the nonstationary case. These results follow from the exponential convergence of the homogeneous algorithm which in turn follows from a strong limit theorem for infinite products of ergodic and mixing sequences of matrices.     

Adaptive control of linear time-varying plants: a new modelreference controller structure

The problem of developing a control law which can force the output of a linear time-varying plant to track the output of a stable linear time-invariant reference model is discussed. It is shown that the standard model reference controller, used for linear time-invariant plants, cannot guarantee zero tracking error in general when the plant is time-varying. A new model reference controller is proposed which guarantees stability and zero tracking error for a general class of linear time-varying plants with known parameters. When the time-varying plant parameters are unknown but vary slowly with time, it is shown that the new controller can be combined with a suitable adaptive law so that all the signals in the closed loop remain bounded for any bounded initial conditions and the tracking error is small in the mean. The assumption of slow parameter variations in the adaptive case can be relaxed if some information about the frequency or the form of the fast varying parameters is available a priori. Such information can be incorporated in an appropriately designed adaptive law so that stability and improved tracking performance is guaranteed for a class of plants with fast varying parameters     

Reduced-order performance of parallel and series-parallel identifiers with weakly observable parasitics

The stability properties of discrete-time parallel and series-parallel identifiers with respect to a specific model-plant order mismatch are analyzed. While in a deterministic environment with no modeling error the two schemes give identical results, when used in a deterministic environment with modeling error their performance is different. We assume a singularly perturbed state representation for the plant where the modeling error consists of fast parasitics which are weakly observable in the plant output. Detailed bounds on parameter and output estimate errors are established and the robustness of the adaptive identifiers is established by showing that the error bound goes to zero as the modeling error goes to zero, i.e. as the parasitics become infinitely fast. The dependence of this residual identification error on the input signal, the neglected parasitics, and the initial error conditions is shown to be crucial. The bounds indicate possibilities for reducing the error by a proper choice of the input signal.     

Recursive least-squares identification algorithms with incompleteexcitation: convergence analysis and application to adaptivecontrol

The convergence properties of a fairly general class of adaptive recursive least-squares algorithms are studied under the assumption that the data generation mechanism is deterministic and time invariant. First, the (open-loop) identification case is considered. By a suitable notion of excitation subspace, the convergence analysis of the identification algorithm is carried out with no persistent excitation hypothesis, i.e. it is proven that the projection of the parameter error on the excitation subspace tends to zero, while the orthogonal component of the error remains bounded. The convergence of an adaptive control scheme based on the minimum variance control law is then dealt with. It is shown that under the standard minimum-phase assumption, the tracking error converges to zero whenever the reference signal is bounded. Furthermore, the control variable turns out to be bounded     

Robustness of absolute stability†

The stability of a class of single-input, single-output singularly perturbed systems formed by a linear time-invariant feedforward block with a sector bounded time varying feedback is considered. It is shown that if the reduced order ‘ slow ’ subsystem is absolutely stable and the parasitics are asymptotically stable and sufficiently fast then the full system is absolutely stable. Bounds on the singular perturbation parameter for uniform asymptotic stability and absolute stability are obtained.     

Adaptive rejection of multi-periodic disturbances for a class of linear systems

This paper deals with adaptive rejection of general multi-periodic disturbances for a class of linear systems via output feedback. The multi-periodic disturbance is a superposition of several general periodic disturbances. Assume that all periods of the multi-periodic disturbance are known. A method of estimating the multi-periodic disturbance and its derivatives is presented, which can ensure these estimate errors converge to zero. Based on these estimation methods, we present a new adaptive control method for disturbance rejection, which guarantees that in the closed-loop system, all the signals are bounded and the output converges to zero. A simulation example is presented to verify the effectiveness of the control scheme.     

Adaptive observers for nonlinearly parameterized class of nonlinear systems

In this paper, one proposes adaptive observers for a class of uniformly observable MIMO nonlinear systems with general nonlinear parameterizations. The state and the unknown parameters of the considered systems are supposed to lie in bounded domains which size can be arbitrarily large and the exponential convergence of the observers is shown to result under a well-defined persistent excitation condition. Moreover, the gain of the observers involves a design function that has to satisfy a simple condition which is given. Different expressions of such a function are proposed and it is shown that adaptive high gain like observers and adaptive sliding mode like observers can be derived by considering particular expressions of the design function. The theory is supported by simulation results related to the estimation of the biomass concentration and the Contois model parameters in a bioreactor.     

Observer-based adaptive neural dynamic surface control for a class of non-strict-feedback stochastic nonlinear systems

The problem of adaptive output feedback stabilisation is addressed for a more general class of non-strict-feedback stochastic nonlinear systems in this paper. The neural network (NN) approximation and the variable separation technique are utilised to deal with the unknown subsystem functions with the whole states. Based on the design of a simple input-driven observer, an adaptive NN output feedback controller which contains only one parameter to be updated is developed for such systems by using the dynamic surface control method. The proposed control scheme ensures that all signals in the closed-loop systems are bounded in probability and the error signals remain semi-globally uniformly ultimately bounded in fourth moment (or mean square). Two simulation examples are given to illustrate the effectiveness of the proposed control design.     

Robust Adaptive Synchronization Control for a Class of Perturbed and Delayed Neural Networks

This paper is concerned with the asymptotic synchronization problem of a general neural network using the robust adaptive control technique. It is considered a class of modified Cohen–Grossberg neural networks which is supposed to undergo unknown perturbations caused by state-independent nonlinearities and bounded mixed time-varying delays on neuron amplification and activation functions. An adaptive compensation control strategy is proposed to ensure the elimination of the perturbed and delayed effects by means of adaptive estimations of unknown controller parameters. Through Lyapunov stability theory, it is shown that the proposed adaptive compensation controllers can guarantee the asymptotic synchronization of neural networks without knowing the knowledge of bounds of nonlinearities and delays. A numerical example is provided to illustrate the effectiveness of the developed techniques.     

Adaptive control of robot manipulators in task space

Adaptive control of robotic manipulators in task space or Cartesian space is considered. A general Lyapunov-like concept is used to design an adaptive control law. It is shown that the global stability and convergence can be achieved for the adaptive control algorithm. The algorithm has the advantage that the inverse of Jacobian matrix is not required. The algorithm is further modified so that the requirement for the bounded inverse of estimated inertia matrix is also eliminated. In addition, two approaches are presented to achieve robustness to bounded disturbances     

含有负载扰动系统的多控制器自适应控制

对一类含有负载扰动的离散时间线性系统研究了多控制器自适应控制问题. 建立了能够改善扰动响应的二自由度极点配置控制器集合. 证明了度量无关的滞后切换逻辑经有限次切换后可将镇定控制器锁定在控制回路 中, 保证输入输出序列有界, 因而实现对负载扰动的抑制. 算例仿真表明了该算法的有效性.     

Superposition properties and performance bounds of stochastictimed-event graphs

This paper addresses the performance evaluation of stochastic timed-event graphs. The transition firing times are random variables with general distribution. We first consider a stochastic timed-event graph in which the firing times are generated by time superposition (or addition) of two sets of random variable sequences. Properties of this system are established. Chiefly, we prove that the average cycle time is subadditive, i.e., it is smaller than the sum of the average cycle times of the two stochastic timed-event graphs in which the firing times are generated by one of the two sets of random variable sequences, respectively. Based on these superposition properties, we derive various upper bounds of the average cycle time of a general stochastic timed-event graph. In particular, we obtain upper bounds which converge to the exact average cycle time as the standard deviations of the firing times decrease. Finally, we derive performance bounds for stochastic timed-event graphs with bounded firing times