Adaptive Control Of Linear Systems Research Articles

Memory Augmented Neural Network Adaptive Controllers: Performance and Stability

In this article, we propose a novel control architecture, inspired from neuroscience, for adaptive control of continuous-time systems. The proposed architecture, in the setting of standard neural network (NN)-based adaptive control, augments an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">external working memory</i> to the NN. The controller, through a write operation, writes the hidden layer feature vector of the NN to the external working memory and can also update this information with the observed error in the output. Through a read operation, the controller retrieves information from the working memory to modify the final control signal. First, we consider a simpler estimation problem to theoretically study the effect of an external memory and prove that the estimation accuracy can be improved by incorporating memory. We, then, consider a model reference NN adaptive controller for linear systems with matched uncertainty to implement and illustrate our ideas. We prove that the resulting controller leads to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">uniformly bounded</i> stable closed-loop system. Through extensive simulations and specific metrics, such as peak deviation and settling time, we show that memory augmentation improves learning significantly. Importantly, we also provide evidence for and insights on the mechanism by which this specific memory augmentation improves learning.

A New Adaptive Controller for Linear Systems: A Model Reference Approach

In this paper, a finite-time model reference adaptive controller is proposed to solve the tracking problem for a class of linear time-invariant systems with parameter uncertainties. The convergence to zero in a finite time for the tracking and parameter identification errors is ensured. The convergence proofs are developed based on Lyapunov function approach. Finally, some simulation results show the feasibility of the proposed approach.

Lyapunov-Based Real-Time and Iterative Adjustment of Deep Neural Networks

A real-time Deep Neural Network (DNN) adaptive control architecture is developed for general uncertain nonlinear dynamical systems to track a desired time-varying trajectory. A Lyapunov-based method is leveraged to develop adaptation laws for the output-layer weights of a DNN model in real-time while a data-driven supervised learning algorithm is used to update the inner-layer weights of the DNN. Specifically, the output-layer weights of the DNN are estimated using an unsupervised learning algorithm to provide responsiveness and guaranteed tracking performance with real-time feedback. The inner-layer weights of the DNN are trained with collected data sets to increase performance, and the adaptation laws are updated once a sufficient amount of data is collected. Building on the results in (Joshi and Chowdhary, 2019) and (Joshi et al., 2020), which focus on deep model reference adaptive control for linear systems with known drift dynamics and control effectiveness matrices, this letter considers general control-affine uncertain nonlinear systems. The real-time controller and adaptation laws enable the system to track a desired time-varying trajectory while compensating for the unknown drift dynamics and parameter uncertainties in the control effectiveness. A nonsmooth Lyapunov-based analysis is used to prove semi-global asymptotic tracking of the desired trajectory. Numerical simulation examples are included to validate the results, and the Levenberg-Marquardt algorithm is used to train the weights of the DNN.

H∞ finite-time control of unknown uncertain systems with actuator failure

Abstract This paper investigates the finite-time H∞ adaptive control of linear systems with unknown mismatched uncertainties. To deal with outage and stuck failures on the actuator and the uncertainties, a new finite-time adaptive H∞ control strategy is proposed. With the established strategy, the resultant closed-loop system is finite-time bounded with the required H∞ level. The validity of the proposed method is demonstrated by an existing rocket fairing model.

Dual Control for Stochastic Linear MIMO Systems With Parameter Uncertainty

This paper considers a dual control problem for stochastic linear MIMO (Multiple Input Multiple Output) systems with parameter uncertainty. A novel dual adaptive control law for MIMO systems is proposed. The design is based on the innovation of the dual control cost function, which was originally developed for conventional adaptive control of linear systems. However, the design process is modified and developed to cater to the stochastic MIMO case. This is a more challenging problem because the superposition of parameter uncertainty and the MIMO property makes the problem more intractable. As in all dual adaptive strategies, it leads to a control law that balances out the need for caution, due to parameter uncertainty, with the conflicting requirement of probing that acts to quickly reduce parameter uncertainty, which is the nature of dual control. The proposed control law has two parts. One reflects the goal of regulating the output, and the second reflects the ability to handle uncertain parameters. A learning factor is introduced to balance these two parts to obtain a control law that can be applied to the original system. In the simulation examples, the uncertain parameters can be estimated quickly and accurately for the unknown but constant case and the abrupt parameter case. Furthermore, it is shown that the novel dual control law is superior to other control strategies by comparing the performance of the cost function in a statistical sense.

Distributed MPC-based adaptive control for linear systems with unknown parameters

This paper is concerned with the adaptive control problem for a class of linear discrete-time systems with unknown parameters based on the distributed model predictive control (MPC) method. Instead of using the system state, the state estimate is employed to model the distributed state estimation system. In this way, the system state does not have to be measurable. Furthermore, in order to improve the system performance, both the output error and its estimation are considered. Moreover, a novel Lyapunov functional, comprised of a series of distributed traces of estimation errors and their transposes, has been presented. Then, sufficient conditions are obtained to guarantee the exponential ultimate boundedness of the system as well as the asymptotic stability of the error system by solving a nonlinear programming (NP) problem subject to input constraints. Finally, the simulation examples is given to illustrate the effectiveness and the validity of the proposed technique.

Global Adaptive Controller for Linear Systems With Unknown Input Delay

In this note, a switching-type delay-adaptive predictor feedback controller is proposed to globally stabilize linear systems with unknown input delay. The idea of our method is to take advantage of the robustness of feedback laws to delay estimate error that is sufficiently small. We first prove that, when the delay estimate is sufficiently close to the true delay, there is a Lyapunov–Krasovskii functional that is strictly decreasing, and asymptotic regulation is achievable. Then, with the help of the Lyapunov–Krasovskii functional as a supervisor, the switching logic plays the role of adjusting the estimated delay online to make it sufficiently close to the true delay. The result of this note shows that, global adaptive stabilization of linear systems with unknown input delay is achievable, with a logic-based switching control law.

Dual adaptive extremum control of Hammerstein systems

A novel dual adaptive controller for extremum control of stochastic and uncertain nonlinear Hammerstein systems is proposed. The design is based on the innovations dual control cost function originally developed for conventional adaptive control of linear systems. However, the design process is extensively modified and developed so as to cater for the extremum control scenario. This is a more challenging problem because the reference input is itself a nonlinear function of the unknown system parameters, rather than an independent and predefined external reference signal. As in all dual adaptive schemes, it leads to a control law that balances out the need for caution, due to parameter uncertainty, with the conflicting requirement of probing that acts to quickly reduce parameter uncertainty. The proposed controller's performance is analysed through extensive Monte Carlo simulation trials and compared with several other non-dual adaptive extremum controllers. It is shown that the novel extremum innovations dual controller is superior to other types of adaptive control systems that are based on a certainty equivalence assumption. In addition, a novel statistical measure is introduced that yields a more objective evaluation of the Monte Carlo simulation results.

Direct model reference adaptive control of linear systems with input/output delays

In this paper, we develop a Direct Model Reference Adaptive Tracking Controller for linear systems with unknown time varying input delays. This controller can also reject bounded disturbances of known waveform but unknown amplitude, e.g. steps or sinusoids. In this paper a robustness result is developed for DMRAC of linear systems with unknown small constant or time varying input delays using the concept of un-delayed ideal trajectories. We will show that the adaptively controlled system is globally stable, but the adaptive tracking error is no longer guaranteed to approach the origin. However, exponential convergence to a neighborhood can be achieved as a result of the control design. A simple example will be provided to illustrate this adaptive control method.

Adaptive Tracking Control of Linear Systems With Binary-Valued Observations and Periodic Target

This technical note studies the adaptive control for linear systems with set-valued observations to track a given periodic target. Based on the system parameters, accessorial parameters with the same order as that of the tracking targets are introduced and estimated. Considering the system parameters are unknown and set-valued observations can supply only limited information each time, a two-scale adaptive control algorithm is designed. Each control input is designed at the large time scale and lasts for a holding time (small scale), during which the parameter estimation algorithm is constructed. From the estimate of accessorial parameters, the control signal is updated at the large time scale by the certainty equivalence principle. As the holding time goes to infinity, the algorithm can be proved to be asymptotically efficient in a certain sense. Meanwhile, the adaptive tracking algorithm is shown to be asymptotically optimal. A numerical example is given to demonstrate the effectiveness of the algorithms and the main results obtained.

Concurrent learning adaptive control of linear systems with exponentially convergent bounds

SUMMARYConcurrent learning adaptive controllers, which use recorded and current data concurrently for adaptation, are developed for model reference adaptive control of uncertain linear dynamical systems. We show that a verifiable condition on the linear independence of the recorded data is sufficient to guarantee global exponential stability. We use this fact to develop exponentially decaying bounds on the tracking error and weight error, and estimate upper bounds on the control signal. These results allow the development of adaptive controllers that ensure good tracking without relying on high adaptation gains, and can be designed to avoid actuator saturation. Simulations and hardware experiments show improved performance. Copyright © 2012 John Wiley &amp; Sons, Ltd.

LMI-based analysis of robust adaptive control for linear systems with time-varying uncertainty

Passification-based adaptive control, also known as simple adaptive control, is studied with respect to its robustness to time-varying uncertainties. Results are formulated in terms of LMIs and are therefore testable in polynomial time using semi-definite programming solvers. The main result shows that the adaptive strategy allows, without measurement nor estimation of the uncertain parameters to guarantee asymptotic stability for a wide rage of these parameters. To achieve this result, the stability property is relaxed: convergence is proved to a small neighborhood of the origin, and attractive domain. It is also demonstrated that this attractor can be made as small as required the only limitation being implementation constraints.

Adaptive control of linear systems with coordinatewise-parametric perturbations of white noise type

Conditions for the mean-square dissipativity of adaptive stabilization systems for a linear object under coordinate-parametric perturbations of white noise type are obtained. A linear adaptive regulator with adjustable coefficients is chosen. For adjusting parameters, an adaptation algorithm is synthesized by the passification method. The number of inputs in objects under consideration may differ from that of outputs. The proof is based on the construction of a quadratic stochastic Lyapunov function. (In the case of purely parametric perturbations, the obtained conditions are known to be necessary and sufficient for the existence of a Lyapunov function with these properties.) Dissipativity conditions for the constructed closed system are obtained; it is shown that, in some special cases, the dissipativity of the closed system is preserved under white-noise perturbations of any intensity.

Passification‐based adaptive control of linear systems: Robustness issues

AbstractPassivity is a widely used concept in control theory having led to many significant results. This paper concentrates on one characteristic of passivity, namely passification‐based adaptive control. This concept applies to multi‐input multi‐output systems for which exists a combination of outputs that renders the open‐loop system hyper‐minimum phase. Under such assumptions, the system may be passified by both high‐gain static output feedback and by a particular adaptive control algorithm. This last control law is modified here to guarantee its coefficients to be bounded. The contribution of this paper is to investigate its robustness with respect to parametric uncertainty. Time response characteristics are illustrated on examples including realistic situations with noisy output and saturated input. Theoretical results are formulated as linear matrix inequalities and can hence be readily solved with semi‐definite programming solvers. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Two Fuzzy Adaptive Controllers for Linear Systems with Linear Disturbances

In this paper two adaptive fuzzy control algorithms are developed. The first fuzzy logic controller, which we call a fuzzy adaptive-proportional-α-integral (FAPαI)controller, has the same form as an adaptive-proportional-α-integral (APαI)except that the controller parameters α, β, and γ are time-varying parameters generated directly by a fuzzy logic controller. The second fuzzy controller, which we call a fuzzy adaptive variable structure (FAVS)controller, combines the performance of an adaptive-proportional-α-integral (APαI) and an adaptive-proportional-integral-integral (API2). The FAPαI tunes the controller parameters α, β, and γ and the FAVS controller schedules the final control signals under different antecedents such that it inherits the merits of the two conventional controllers. For both controllers the objective is to design an adaptive feedback law that gives good output regulation and disturbance rejection in any controllable linear time-invariant system. In addition, the problem of linear disturbance in linear time-invariant lumped single-input/single-output unstable minimum phase plants with relative degree one is studied. All simulation results prove that the performance of each of the FAPαI and FAVS controllers is better than the performance of the other controllers. On the other hand, the performance of FAVS is better than that of the FAPαI. Also, we show that, for our problem, the performance of the conventional PI controller (CPI) is completely unacceptable.

Robust stable pole-placement adaptive control of linear systems with multiestimation

This paper is concerned with a robust pole placement-based adaptive control of continuous linear systems in the presence of bounded noise and a common class of unmodeled dynamics with the use of multiple estimation schemes working in parallel. The multiestimation scheme consists of the above set of various single estimation schemes while being a tool used to minimize the plant identification error by building an estimate which is a convex combination of the estimates at all time. The weighting functions of the individual estimates are provided at each time by a suboptimization scheme for a quadratic loss function The robust stability of the overall adaptive scheme is ensured by an adaptation relative dead zone which takes into account the contribution of the unmodeled dynamics and bounded noise.

Adaptive Control of Linear Systems with Unknown Memoryless Input Nonlinearities Using ANN

In this paper a new adaptive scheme for control of linear systems with unknown memoryless input nonlinearities shall be presented. To do this, we define a useful combination of an artificial neural network (ANN) and a linear system which can represent a large number of nonlinear systems. The input nonlinearities of real plants are not only varying in their parameters but also in their structure. Furthermore, in general, it is expected that the input nonlinearity is not only an isolated static effect, like a dead-zone, but a combination of different types of nonlinear effects. A key feature of our new scheme is that the ANN can describe several different types of nonlinear functions without structural changes. Thus, this cascade connection will be very flexible and powerful for different regulating plants. To control the linear part of the system, an adaptive LQ-controller is used with the ANN compensating the static nonlinearity.

Adaptive control of linear systems with Markov perturbations

The stochastic model considered is a linear jump diffusion process X for which the coefficients and the jump processes depend on a Markov chain Z with finite state space. First, we study the optimal filtering and control problem for these systems with non-Gaussian initial conditions, given noisy observations of the state X and perfect measurements of Z. We derive a new sufficient condition which ensures the existence and the uniqueness of the solution of the nonlinear stochastic differential equations satisfied by the output of the filter. We study a quadratic control problem and show that the separation principle holds. Next, we investigate an adaptive control problem for a state process X defined by a linear diffusion for which the coefficients depend on a Markov chain, the processes X and Z being observed in independent white noises. Suboptimal estimates for the process X, Z and approximate control law are investigated for a large class of probability distributions of the initial state. Asymptotic properties of these filters and this control law are obtained. Upper bounds for the corresponding error are given.

Robust Adaptive Control for Linear Systems with Unknown Parameters

In this paper, a robust adaptive control problem for a class of uncertain linear systems on a finite-time interval is formulated as a minimax game problem. For this class of systems, with uncertain control coefficients augmenting the state space, the system under consideration has a bilinear structure. Although the minimax solution allows for a finite-dimensional estimator, the resulting compensator structure remains much more complex than the linear-quadratic problem. For this class of partially observed systems, the minimax problem is reduced to an equivalent full information control problem where the state-space dynamics are composed of an estimator equation and its associated Riccati equation. The equivalence between the minimax adaptive controller and a saddle-point certainty equivalence adaptive controller is shown via Hamilton-Jacobi-Bellman theory where the optimal return function is differentiable. Points of discontinuity in the partial derivatives of the optimal return function are shown to form a manifold of Darboux points, from which multiple global optimal trajectories emanate. Therefore, the Dynamic Programming approach can be extended to be valid over the entire phase space, and the uniqueness of the value of the optimal return function is guaranteed. We finally show that with additional assumptions the finite-time problem can be extended to infinite time. © 1997 Elsevier Science Ltd.

Adaptive Control with Improved Asymptotic Performance in the Presence of Deterministic Disturbances

The problem of adaptive control of linear systems subjected to both small parameter and bounded additive deterministic disturbances is considered. The controlled systems are not identifiable in view of poor a priori information about disturbances. It is supposed that for system with known parameters a certain regulator is given that ensures a desired non-adaptive asymptotic control performance. It is proven that this nonadaptive parameter-dependent control performance is achievable under adaptive control in spite of non-identifiability of the system. The solution is based on control-oriented estimation algorithm and uses set estimates of unknown parameters.