Robust adaptive stabilization for time-varying systems with unmodelled dynamics and disturbances

In this paper, robust adaptive stabilization is discussed for time-varying discrete time systems with disturbances and unmodelled dynamics. Both bounded and unbounded stochastic disturbances are considered. It is assumed that the parameters of the nominal model belong to a bounded convex set and that the ‘frozen time’ nominal model is stabilizable for all possible parameter values. Requiring neither external excitation nor stable invertibility of the nominal model, an adaptive regulator is constructed on the basis of the solution to a finite time Riccati equation and a projected gradient estimator. It is shown that the closed-loop system is stable if both the time average of the parameter variations and the model error are sufficiently small.     

A new decentralized model reference adaptive control for a class of interconnected dynamic systems using variable structure design

This paper presents a new decentralized model reference adaptive control for a class of large-scale interconnected dynamic systems. Interconnections among subsystems may be time-invariant or time-varying and linear or non-linear. The scheme proposed here only takes input and output measurements from each subsystem for input synthesis. Using a variable structure design concept, we show that the tracking errors will converge to zero in finite time despite the interconnections with any possible strengths.     

Global adaptive finite‐time stabilization of a class of switched nonlinear systems with parametric uncertainty

This paper investigates the global adaptive finite‐time stabilization of a class of switched nonlinear systems, whose subsystems are all in p (p≤1) normal form with unknown control coefficients and parametric uncertainties. The restrictions on the power orders and the nonlinear perturbations are relaxed. By using the parameter separation technique, the uncertain parameters are separated from nonlinear functions. A systematic design procedure for a common state feedback controller and a switching adaptive law is presented by employing the backstepping methodology. It is proved that the closed‐loop system is finite‐time stable under arbitrary switching by utilizing the common Lyapunov function. Finally, with the application to finite‐time control of chemical reactor systems, the effectiveness of the proposed method is demonstrated. Copyright © 2015 John Wiley & Sons, Ltd.     

Input-output theory of high-gain adaptive stabilization of infinite-dimensional systems with non-linearities

This paper develops an input-output theory of high-gain adaptive stabilization of certain infinite-dimensional processes with actuator and sensor non-linearities. It is shown that there is a wide range of gain adaption rules achieving stability for the class of processes under consideration if proportional output feedback is used. The abstract input-output results are applied to retarded systems and Volterra integrodifferential systems. The paper shows that the scope of applicability of universal adaptive stabilization ideas extends far beyond finite-dimensional linear systems.     

Decentralized indirect adaptive control of interconnected systems

Continuous time decentralized indirect adaptive controllers are proposed for a large-scale system composed of N interconnected linear subsystems with unknown parameters. Each local adaptive law utilizes a normalizing signal which is generated using the local input and the outputs of all the subsystems. This requires the exchange of output signals between the different subsystems, so that the scheme is no longer totally decentralized. Nevertheless, the degree of computational complexity is about the same as that of a totally decentralized scheme and this makes the scheme very attractive.     

Decentralized stochastic adaptive control of interconnected systems

This paper considers the stochastic adaptive control problem for a class of large-scale systems formed by arbitrary interconnection of subsystems with unknown parameters and non-linearities. For the estimation of the unknown parameters of the local controllers, stochastic approximation algorithms are used. Conditions sufficient for global stability of the overall system are established. It is shown that the overall tracking error is bounded by a quantity depending on the size of interconnections.     

Online optimal and adaptive integral tracking control for varying discrete-time systems using reinforcement learning

Conventional closed-form solution to the optimal control problem using optimal control theory is only available under the assumption that there are known system dynamics/models described as differential equations. Without such models, reinforcement learning (RL) as a candidate technique has been successfully applied to iteratively solve the optimal control problem for unknown or varying systems. For the optimal tracking control problem, existing RL techniques in the literature assume either the use of a predetermined feedforward input for the tracking control, restrictive assumptions on the reference model dynamics, or discounted tracking costs. Furthermore, by using discounted tracking costs, zero steady-state error cannot be guaranteed by the existing RL methods. This article therefore presents an optimal online RL tracking control framework for discrete-time (DT) systems, which does not impose any restrictive assumptions of the existing methods and equally guarantees zero steady-state tracking error. This is achieved by augmenting the original system dynamics with the integral of the error between the reference inputs and the tracked outputs for use in the online RL framework. It is further shown that the resulting value function for the DT linear quadratic tracker using the augmented formulation with integral control is also quadratic. This enables the development of Bellman equations, which use only the system measurements to solve the corresponding DT algebraic Riccati equation and obtain the optimal tracking control inputs online. Two RL strategies are thereafter proposed based on both the value function approximation and the Q-learning along with bounds on excitation for the convergence of the parameter estimates. Simulation case studies show the effectiveness of the proposed approach.     

A new decentralized stabilization method with application to power system stabilizer design for multimachine systems

A new decentralized stabilization method is presented, in which l single-input/single-output local feedback controllers are designed to stabilize or to improve the stability of a large-scale system consisting of N subsystems with l<N negative or poorly damped system modes. The method is shown to be simple and feasible to apply to power system stabilizer design for multimachine systems. Further research work is suggested.     

Construction of a robust adaptive regulator for time-varying discrete time systems

A robust adaptive regulator is constructed for single-input/single-output discrete time systems modelled by a linear time-varying difference equation that includes an error term to incorporate model errors and/or disturbances. It is assumed that the parameters of the nominal model belong to a known bounded convex set and that the ‘frozen time’ nominal model is stabilizable for all possible parameter values. The estimation of the parameters of the nominal model is carried out using a standard gradient-type algorithm with a projection operation. An adaptive regulator is then constructed from the solution to a finite time Riccati equation. It is shown that the resulting closed-loop system is globally stable if the mean of the parameter time variations is sufficiently small and if the model error is sufficiently small, but where the disturbances applied to the plant may be arbitrarily large.     

On necessary assumptions in discrete-time model reference adaptive control

In this paper we investigate the discrete-time model reference adaptive control problem, ascertain the extent to which the classical assumptions are necessary and provide several suitably modified inviolable requirements. In particular, we show that under a closed-loop causality constraint the problem is solvable only if there is an upper bound on the plant relative degree and the plant zeros outside the open unit disc lie in a finite set. We also derive a bound on the achievable asymptotic performance in the event that these requirements are not met.     

An indirect adaptive control for fully feedback linearizable discrete-time non-linear systems

An indirect adaptive control for a discrete-time non-linear system that is fully input–output linearizable is developed. The unknown parameters of the system are identified by using a multi-output RLS algorithm. Based on the certainty equivalence principle, the estimated parameters are then utilized in the controller design. Stability of the adaptively controlled closed-loop system is shown by using the Lyapunov method. Numerical simulations are included to illustrated the performance of the proposed strategy. © 1997 John Wiley & Sons, Ltd.     

Decentralized adaptive linear control of complex systems using relative deadzones

This paper aims to analyse the system stability when decentralized adaptive controllers are applied to multi-input/multi-output non-linear interconnected systems. The local adaptive controllers are designed based on linear models by employing relative deadzones. Using a small-gain-type argument, we can derive an M-matrix test condition for local stability. If the system to be controlled can be described by global Lipschitz functions, a global stable closed-loop system is obtained. © 1997 by John Wiley & Sons, Ltd.     

Improving transient performance of discrete-time model reference adaptive control architectures

Discrete-time adaptive control algorithms can be executed directly in embedded code unlike their continuous-time counterparts, which require discretization. However, their designs predicated on quadratic Lyapunov-based frameworks are quite intricate due to the resulting complexity in the Lyapunov difference expressions. Therefore, a wide array of available continuous-time results addressing transient performance issues using adaptive control algorithms cannot be applied or readily extended to the discrete-time case. In this article, we present a new model reference adaptive control architecture for discrete-time uncertain dynamical systems. Specifically, the proposed architecture consists of a command governor mechanism that adjusts the trajectory of a given command during the closed-loop transient response. It is shown that this mechanism is effective in improving transient performance of discrete-time model reference adaptive control architectures. Using a logarithmic Lyapunov function, we prove Lyapunov stability of the closed-loop system as well as asymptotic convergence of the system error states involving the difference between the states of the uncertain dynamical system and the states of the reference model, as well as driving the command governor signal to zero.     

Robust pole placement indirect adaptive control

The problem of designing a robustly stable pole placement indirect adaptive controller in the presence of output disturbances and unmodelled dynamics is addressed. The key features of such a design are the following. (1) The unknown parameters are estimated by a normalized least-squares algorithm with a dead zone to provide the stability robustness with respect to bounded disturbances and ‘small’ unmodelled dynamics. (2) The estimated model controllability is ensured by modifying the control law over a finite time. The modification involved consists of adding an internal impulse excitation and ‘freezing’ the controller parameters.     

An adaptive model predictive control strategy for a class of discrete-time linear systems with parametric uncertainty

In this study, an adaptive model predictive control (MPC) strategy is proposed for a class of discrete-time linear systems with parametric uncertainty. In the presented adaptive MPC, an updating law is firstly designed to update the estimated parameters online. By utilizing the estimated parameters, a standard MPC optimization problem for unconstrained systems is established. Then, to deal with constrained systems, the min–max MPC technique is developed under the set-based approach for the estimated parameters. Furthermore, it is shown theoretically that the recursive feasibility and closed-loop stability can be rigorously proved, respectively. Finally, numerical simulations and comparative analysis are presented to illustrate the superiority of the proposed algorithms in control performance.     

Robust and simple adaptive control system design

Recently, the simple adaptive control (SAC) method has attracted considerable interest because of the simple structure of its adaptive controller. The method can only be applied to plants with so-called almost strictly positive real (ASPR) characteristics. Unfortunately, most real plants do not satisfy this condition. Furthermore, real plants contain disturbances such as uncertain elements. This paper deals with such problems and proves that the implementation of a parallel feedforward compensator makes it possible to apply the SAC method to non-ASPR SISO plants with plant uncertainties. Furthermore, a robust SAC algorithm in the presence of a class of external disturbances is also considered. The effectiveness of the parallel feedforward compensator and robust adaptive controller designed in this way is examined through several numerical simulations.     

Decentralized adaptive backstepping stabilization of interconnected systems with input time delays in dynamic interactions

In this paper, we provide a solution to the problem by considering input time delays in dynamic interactions. Each local controller, designed simply on the basis of the model of each subsystem by using filtered transformation and standard backstepping technique, only employs local information to generate control signals. The robustness of decentralized adaptive controllers is established. It is shown that the designed decentralized adaptive backstepping controllers can globally stabilize the overall interconnected system asymptotically. The L₂ and L_∞ norms of the system outputs are also established as functions of design parameters. This implies that the transient system performance can be adjusted by choosing suitable design parameters. Copyright © 2011 John Wiley & Sons, Ltd.     

A new decentralized retrofit adaptive fault‐tolerant flight control design

In this paper, we develop a new decentralized retrofit adaptive fault‐tolerant control design for a class of nonlinear models arising in flight control. The proposed adaptive fault‐tolerant controller is designed to accommodate loss‐of‐effectiveness (LoE) failures in flight control actuators and achieve accurate estimation of failure‐related parameters. The design is based on local estimation of LoE parameters and generation of local retrofit control signals to accommodate the failures. Using state‐dependent closed‐loop estimation errors, we show the overall system to be stable and demonstrate the tracking error to converge to zero asymptotically for any combination of actuator failures. Through computer simulation of F/A‐18 aircraft under actuator LoE failures, the proposed approach is also shown to achieve better parameter estimation performance compared to the fully centralized design and the design employing local observers and a centralized adaptive controller. Copyright © 2012 John Wiley & Sons, Ltd.     

Adaptive input‐to‐output stabilization of nonlinear systems

The problem of adaptive asymptotic stabilization of a set for nonlinear convexly parameterized dynamical systems is considered. The main result provides a framework for adaptive input‐to‐output stabilization theory in the presence of disturbances. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of robust and adaptive pole placement

The previous investigation showed the usefulness of a two-degree-of-freedom configuration. Such a controller can also be designed using pole placement. An adaptive controller based on this concept is developed and applied to the problem with good results. Particular features, such as selection of model structure, design parameters and excitation, are also discussed.