Robust adaptive control for unmatched systems with guaranteed parameter estimation convergence

This paper provides a modified model reference adaptive control (MRAC) scheme to achieve better transient control performance for systems with unknown unmatched dynamics, where an adaptive law with guaranteed convergence is introduced. We first revisit the standard MRAC system and analyze the tracking error bound by using L₂‐norm and Cauchy‐Schwartz inequality. Based on this analysis, we suggest a feasible way to compensate the undesired transient dynamics induced by the gradient descent–based adaptive laws subject to sluggish convergence or even parameter drift. Then, a modified adaptive law with an alternative leakage term containing the parameter estimation error is developed. With this adaptive law, the convergence of both the estimation error and tracking error can be proved simultaneously. This enhanced convergence property can contribute to deriving smoother control signal and improved control response. Moreover, this paper provides a simple and numerically feasible approach to online verify the well‐known persistent excitation condition by testing the positive definiteness of an introduced auxiliary matrix. Comparative simulations based on a benchmark 3‐DOF helicopter model are given to validate the effectiveness of the proposed MRAC approach and show the improved performance over several other MRAC schemes.     

Novel model reference adaptive control architecture using semi‐initial excitation‐based switched parameter estimator

This paper is a generalization of the recently developed techniques of initial excitation (IE)–based adaptive control with an introduction to the definition of semi‐initial excitation (semi‐IE), a still more relaxed notion than IE. Classical adaptive controllers typically ensure Lyapunov stability of the extended error dynamics (tracking error + parameter estimation error) and asymptotic tracking, while requiring a stringent condition of persistence of excitation (PE) for parameter convergence. Of late, the authors have proposed a new adaptive control architecture, which guarantees parameter convergence under the online‐verifiable IE condition leading to exponential stability of the extended error dynamics. In earlier works, it has been established that the IE condition is significantly milder than the classical PE condition. The current work further slackens the excitation condition by proposing the concept of semi‐IE. The proposed adaptive controller is proved to ensure convergence of the parameter estimation error to a lower‐dimensional manifold under the weaker semi‐IE condition, while the stronger condition of IE guarantees convergence of the parameter estimation error to zero. The designed algorithm is shown to improve transient response of tracking error sufficiently in contrast to conventional adaptive controllers.     

Indirect model reference adaptive control with dynamic adjustment of parameters

The paper discusses in detail a new method for indirect model reference adaptive control (MRAC) of linear time-invariant continuous-time plants with unknown parameters. The method involves not only dynamic adjustment of plant parameter estimates but also those of the controller parameters. Hence the overall system can be described by a set of non-linear differential equations as in the case of direct control. Many of the difficulties encountered in the conventional indirect approach, where an algebraic equation is solved to determine the control parameters, are consequently bypassed in this method. The proof of stability of the equilibrium state of the overall system is found to be different from that used in direct control. Using Lyapunov's theory, it is first shown that the parameter errors between the parameter estimates of the identifier and the true parameters of the plant, as well as those between the actual parameters of the controller and their desired values, are bounded. Following this, using growth rates of signals in the adaptive loop as well as order arguments, it is shown that the error equations are globally uniformly stable and that the tracking (control) error tends to zero asymptotically. This in turn establishes the fact that both direct and indirect model reference adaptive schemes require the same amount of prior information to achieve stable adaptive control.     

Algorithms of adaptive tracking of unknown multisinusoidal signals in linear systems with arbitrary input delay

The problem of adaptive tracking control is addressed for the class of linear time‐invariant plants with known parameters and arbitrary known input delay. The reference signal is a priori unknown and is represented by a sum of biased harmonics with unknown amplitudes, frequencies, and phases. Asymptotic tracking is provided by predictive adjustable control with parameters generated by one of three designed adaptation algorithms. The first algorithm is based on a gradient scheme and ensures zero steady‐state tracking error with all signals bounded. The other two algorithms additionally involve the scheme with fast parametric convergence improving the closed‐loop system performance. In all the algorithms, the problem of delay compensation is resolved by special augmentation of tracking error. The adjustable control law proposed do not require identification of the reference signal parameters.     

Model reference adaptive iterative learning control for linear systems

In this paper, we propose a model reference adaptive control (MRAC) strategy for continuous‐time single‐input single‐output (SISO) linear time‐invariant (LTI) systems with unknown parameters, performing repetitive tasks. This is achieved through the introduction of a discrete‐type parametric adaptation law in the ‘iteration domain’, which is directly obtained from the continuous‐time parametric adaptation law used in standard MRAC schemes. In fact, at the first iteration, we apply a standard MRAC to the system under consideration, while for the subsequent iterations, the parameters are appropriately updated along the iteration‐axis, in order to enhance the tracking performance from iteration to iteration. This approach is referred to as the model reference adaptive iterative learning control (MRAILC). In the case of systems with relative degree one, we obtain a pointwise convergence of the tracking error to zero, over the whole finite time interval, when the number of iterations tends to infinity. In the general case, i.e. systems with arbitrary relative degree, we show that the tracking error converges to a prescribed small domain around zero, over the whole finite time interval, when the number of iterations tends to infinity. It is worth noting that this approach allows: (1) to extend existing MRAC schemes, in a straightforward manner, to repetitive systems; (2) to avoid the use of the output time derivatives, which are generally required in traditional iterative learning control (ILC) strategies dealing with systems with high relative degree; (3) to handle systems with multiple tracking objectives (i.e. the desired trajectory can be iteration‐varying). Finally, simulation results are carried out to support the theoretical development. Copyright © 2006 John Wiley & Sons, Ltd.     

Adaptive robust control of a class of nonlinear systems in semi‐strict feedback form with non‐uniformly detectable unmeasured internal states

This paper proposes a novel control method for a special class of nonlinear systems in semi‐strict feedback form. The main characteristic of this class of systems is that the unmeasured internal states are non‐uniformly detectable, which means that no observer for these states can be designed to make the observation error exponentially converge to zero. In view of this, a projection‐based adaptive robust control law is developed in this paper for this kind of system. This method uses a projection‐type adaptation algorithm for the estimation of both the unknown parameters and the internal states. Robust feedback term is synthesized to make the system robust to uncertain nonlinearities and disturbances. Although the estimation error for both the unknown parameters and the internal states may not converge to zero, the tracking error of the closed‐loop system is proved to converge to zero asymptotically if the system has only parametric uncertainties. Furthermore, it is theoretically proved that all the signals are bounded, and the control algorithm is robust to bounded disturbances and uncertain nonlinearities with guaranteed output tracking transient performance and steady‐state accuracy in general. The class of system considered here has wide engineering applications, and a practical example—control of mechanical systems with dynamic friction—is used as a case study. Simulation results are obtained to demonstrate the applicability of the proposed control methodology. Copyright © 2010 John Wiley & Sons, Ltd.     

An indirect adaptive pole‐placement control for MIMO discrete‐time stochastic systems

In this paper, an indirect adaptive pole‐placement control scheme for multi‐input multi‐output (MIMO) discrete‐time stochastic systems is developed. This control scheme combines a recursive least squares (RLS) estimation algorithm with pole‐placement control design to produce a control law with self‐tuning capability. A parametric model with a priori prediction outputs is adopted for modelling the controlled system. Then, a RLS estimation algorithm which applies the a posteriori prediction errors is employed to identify the parameters of the model. It is shown that the implementation of the estimation algorithm including a time‐varying inverse logarithm step size mechanism has an almost sure convergence. Further, an equivalent stochastic closed‐loop system is used here for constructing near supermartingales, allowing that the proposed control scheme facilitates the establishment of the adaptive pole‐placement control and prevents the closed‐loop control system from occurring unstable pole‐zero cancellation. An analysis is provided that this control scheme guarantees parameter estimation convergence and system stability in the mean squares sense almost surely. Simulation studies are also presented to validate the theoretical findings. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust MRAC to disturbance using VS‐type adaptive identifier

It is well known that variable structure (VS)‐type model reference adaptive controls (MRAC) are robust to disturbances. However, almost all conventional design schemes of VS‐MRAC may cause chattering because of the switching function used in the control synthesis input. In this paper, we propose a new design scheme of MRAC which uses a VS‐type adaptive identifier. This design scheme can avoid occurrence of chattering. Furthermore, this design scheme is robust to disturbances. © 1999 Scripta Technica, Electr Eng Jpn, 130(2): 80–87, 2000     

Output‐feedback co‐ordinated decentralized adaptive tracking: The case of MIMO subsystems with delayed interconnections

Exact decentralized output‐feedback Lyapunov‐based designs of direct model reference adaptive control (MRAC) for linear interconnected delay systems with MIMO subsystems are introduced. The design process uses a co‐ordinated decentralized structure of adaptive control with reference model co‐ordination which requires an exchange of signals between the different reference models. It is shown that in the framework of the reference model co‐ordination zero residual tracking error is possible, exactly as in the case with SISO subsystems. We develop decentralized MRAC on the base of a priori information about only the local subsystems gain frequency matrices without additional a priori knowledge about the full system gain frequency matrix. To achieve a better adaptation performance we propose proportional, integral time‐delayed adaptation laws. The appropriate Lyapunov–Krasovskii type functional is suggested to design the update mechanism for the controller parameters, and in order to prove stability. Two different adaptive DMRAC schemes are proposed, being the first asymptotic exact zero tracking results for linear interconnected delay systems with MIMO subsystems. Copyright © 2005 John Wiley & Sons, Ltd.     

Filtered adaptive constrained sampled-data control for uncertain multivariable nonlinear systems

A filtered adaptive constrained sampled-data controller for uncertain multivariable nonlinear systems in the presence of various constraints is synthesized in this paper. A piecewise constant adaptive law drives that estimation error dynamics to zero at each sampling time instant yields adaptive parameters. The filtered control scheme consists of two components. Based on an estimation/cancellation strategy, a disturbance rejection control law is designed to compensate the nonlinear uncertainties within the bandwidth of low-pass filters, whereas a constraint violation avoidance control law is designed to solve an online constrained optimization problem. Although a reduced sampling time helps to minimize the estimation error caused by the neglect of unknowns, the resulting aggressive signals put more restrictions on the control law. Greater sacrifice of tracking performance is required to satisfy the constraints. The constraints violation avoidance control law is in favor of a larger sampling time. Sufficient conditions are given to guarantee the stability of the closed-loop system with the sampled-data controller, where the input/output signals are held constant over the sampling period. Numerical examples are provided to validate the theoretical results, comparisons between the constrained sampled-data controller and unconstrained adaptive controller with the implementation of different sampling times are carried out.     

Model reference adaptive control system design for linear time‐varying systems with unstructured and bounded varying functions

In recent years, many methods of model reference adaptive control system (MRACS) for a linear time‐varying (LTV) plant have been proposed. These methods assumed that the structure of plant parameters is known in advance. However, it is difficult to get a priori information of plant parameters. In this paper, an MRACS design for an LTV system based on high‐order estimator (HOE) is proposed. By applying dynamic certainty equivalence (DyCE) to LTV plants, a new MRAC law of LTV system is derived without knowing the structure of the plant parameters. The MRACS law is generated by using high‐order derivatives of an estimated parameter, so that robust HOE with a normalization signal and σ modification for the system introduced. Our proposed method can attain better performance than conventional methods, such as estimation with variable forgetting factor (VF) and the gradient projection method (GPM). The robust HOE establishes the boundedness of all of the estimated parameters under the condition that the estimated parameter and the first derivative of the parameter are bounded. It is shown that all signals in the adaptive loop are bounded and the output error converges to a closed set. The proposed method is compared to the familiar schemes, the gradient projection method and the estimation based on forgetting factor through numerical simulations, and the effectiveness of our proposed method is shown. © 2000 Scripta Technica, Electr Eng Jpn, 130(4): 87–98, 2000     

A new approach to model reference adaptive control

A new approach to model reference adaptive control, based on a combination of direct and indirect control methods, is introduced in this paper. The controller structure is identical to that used in the direct method, but the algorithm used to update the controller parameters depends both on the output error as in direct control and on the plant parameter estimates as in indirect control. The global stability of the overall system is assured by the existence of a Lyapunov function. In the ideal case discussed here, the combined approach results in improved transient response with smaller amplitude of the control input as compared to the constituent methods.     

An adaptive control scheme using multiple reference models

This paper develops an extended model reference adaptive control scheme to expand the capacity of state feedback state tracking adaptive control to handle the plant‐model matching uncertainties for single‐input LTI systems. The extended scheme is developed, using multiple reference model systems (only one of which is required to be able to match the controlled plant), and multiple controllers (which are updated from adaptive laws generated from multiple reference model systems based estimation errors), as two key features of such design to relax a plant‐model matching condition. A switching mechanism is constructed using those multiple estimation errors, capable of selecting the suitable control input from the multiple control signals, to achieve the desired system performance. An aircraft flight control example is presented to show the capacity of such design in relaxing a practical design condition. Copyright © 2013 John Wiley & Sons, Ltd.     

On the need of projections in input‐error model reference adaptive control

The main objective of this note is to contribute, if modestly, toward the understanding of the input‐error model reference adaptive control scheme revealing an instability mechanism that arises if the projection of the plant high‐frequency gain coefficient estimate is omitted. In addition, a self‐contained proof of global convergence of the scheme with the projections for a simple first‐order plant is given.     

Behavior of the least mean square algorithm with a periodically time‐varying input power

The paper analyzes the transient and steady‐state performances of a least mean square algorithm in the rarely‐studied situation of a time‐varying input power. A scenario of periodic pulsed variation of the input power is considered. The analysis is carried out in the context of tracking a Markov plant with a white Gaussian input. It is shown that the mean square deviation (MSD) converges to a periodic sequence having the same period as that of the variation of the input power. Expressions are derived for the convergence time and the steady‐state peak MSD. Surprisingly, it is found that neither the transient performance nor the steady‐state performance degrades with rapid variation of the input power. On the other hand, slow input power variation causes degradation in both the transient and steady‐state performances for given amplitude of variation of the input power. In the case of a time‐invariant plant, neither rapid nor slow variation of the input power causes degradation in the steady‐state performance. On the other hand, there is degradation in the transient performance for slow variation of the input power. Copyright © 2012 John Wiley & Sons, Ltd.     

An adaptive digital filter for noise attenuation in sampled control systems

A recursive smoothing filter employing a bank of fading‐memory polynomial sub‐filters is presented. Variance estimates are used to mix the outputs of the sub‐filters, imparting variable gain and phase characteristics that permit it to automatically adapt to signal parameter changes. The proposed adaptive technique does not involve the estimation of plant parameters; therefore, it may be used in both open‐loop and closed‐loop configurations. In open‐loop estimation problems, variable gain/bandwidth allows it to reduce the impact of random errors caused by sensor noise and the impact of bias errors caused by model mismatch during ‘maneuvers’. In feedback control problems, variable phase/delay allows it to act as a lag filter for an improved steady‐state response (i.e. greater noise attenuation) and act as a lead filter, or a proportional‐derivative controller, for an improved transient response (i.e. increased closed‐loop damping) for input discontinuities. Copyright © 2015 John Wiley & Sons, Ltd.     

基于软计算的感应电机无速度传感器控制

文章提出了一种基于软计算的感应电机无速度传感器控制方案，用以提高无速度传感器控制的低速性能，采用T－S模糊估计器，通过定子电流和定子频率估计定子电阻，以Elman网络为电流转速模型，采用模型参考自适应的方法估计转速和暂态时的转子电阻。令稳态运行的转子电阻以估计的定子电阻同比率变化。文末仿真了采用文章提出的控制方案竽转子磁场定向的无速度传感器控制的低速较长时间运行的情况。说明了这种控制方案可以在一定程度上提高系统的低速性能。     

A clarifying analysis of feedback error learning in an LTI framework

Feedback error learning (FEL) is a proposed technique for reference‐feedforward adaptive control. FEL in a linear and time‐invariant (LTI) framework has been studied recently; the studies can be seen as proposed solutions to a ‘feedforward MRAC’ problem. This paper reanalyzes two suggested schemes with new interpretations and conclusions. It motivates the suggestion of an alternative scheme for reference‐feedforward adaptive control, based on a certainty‐equivalence approach. The suggested scheme differs from the analyzed ones by a slight change in both the estimator and the control law. Boundedness and error convergence are then guaranteed when the estimator uses normalization combined with parameter projection onto a convex set where stability of the estimated closed‐loop system holds. Copyright © 2008 John Wiley & Sons, Ltd.     

Transients in output feedback adaptive systems with observer‐like reference models

This note presents analysis and quantification of transient dynamics in Model Reference Adaptive Control (MRAC) with output feedback and observer‐like reference models. A practical design methodology for this class of systems was first introduced in 1 , 2 , where an output error feedback was added to the reference model dynamics. Here, this design is complemented with an analysis of the corresponding transients. Specifically, it is shown that employing observer‐like reference models in MRAC leads to a trade‐off between achieving fast transient dynamics and using large error feedback gains in the modified reference model. For clarity sake, only systems with matched uncertainties are analyzed, yet the reported results can be extended to a broader class of uncertainties by utilizing MRAC modifications for robustness 3 , 4 . The note ends with a summary of the derived results and a discussion on practical design guidelines for adaptive output feedback controllers with observer‐like reference models. Copyright © 2015 John Wiley & Sons, Ltd.     

The innovation algorithms for multivariable state‐space models

This paper derives the input‐output representation of the dynamical system described by a linear multivariable state‐space model and the corresponding multivariate linear regressive model (ie, multivariate equation‐error model). A projection identification algorithm, a multivariate stochastic gradient identification algorithm, and a multi‐innovation stochastic gradient (MISG) identification algorithm are proposed for multivariate equation‐error systems by using the negative gradient search and the multi‐innovation identification theory. The convergence analysis of the MISG algorithm indicates that the parameter estimation errors converge to zero under the persistent excitation condition. Finally, a numerical example illustrates the effectiveness of the proposed algorithms.