Control Laws For Linear Systems Research Articles

Aperiodic dynamic event-triggered control for linear systems: A looped-functional approach

The recent literature on event-triggered control has demonstrated the potential of dynamic periodic event-triggered control. Compared to continuous-time event-triggering rules, the benefit of considering periodic event-triggered control is to avoid the Zeno phenomenon, which refers to the situation when there are an infinite number of updates in a bounded interval of time. The idea of periodic event-triggered control is to trigger the control law only at known allowable periodic sampling instants. In this paper, our objective is to relax the constraint on the periodicity of the allowable sampling instants and to adapt this framework to the dynamic event-triggered control, which has not been considered in the literature, as far as we are aware of. Following the successful efforts to assess the stability of aperiodic sampled-data control, here we propose a generic framework to emulate aperiodic dynamic event-triggered control law for linear systems, for which the allowable sampling instants are not necessarily equidistant. Such an analysis is made possible thanks to the looped-functionals framework, which gives the flexibility to consider the periodic/aperiodic static/dynamic event-triggered control in a single formulation. Finally, the efficiency of the proposed results is illustrated through the study of two academic examples.

Combined Left and Right Temporal Robustness for Control Under STL Specifications

Many modern autonomous systems, particularly multi-agent systems, are time-critical and need to be robust against timing uncertainties. Previous works have studied left and right time robustness of signal temporal logic specifications by considering time shifts in the predicates that are either only to the left or only to the right. We propose a combined notion of temporal robustness which simultaneously considers left and right time shifts. For instance, in a scenario where a robot plans a trajectory around a pedestrian, this combined notion can now capture uncertainty of the pedestrian arriving earlier or later than anticipated. We first derive desirable properties of this new notion with respect to left and right time shifts and then design control laws for linear systems that maximize temporal robustness using mixed-integer linear programming. Finally, we present two case studies to illustrate how the proposed temporal robustness accounts for timing uncertainties.

Observer‐based iterative learning control for linear time‐varying systems using LMI

This study presents the design of an observer-based iterative learning control law for linear systems using the stability theory and optimal function. The observer is introduced for states condition monitoring and output feedback. The D-type iterative learning control law is utilised to update the control inputs. Properties of stability, convergence, robustness, and optimality are investigated and demonstrated by stability theory and optimal theory. The convergence properties of the observer-based iterative learning control design are obtained by linear matrix inequalities. A simulation example is used to illustrate the validity and effectiveness of the proposed method.

Improved Q-Learning Method for Linear Discrete-Time Systems

In this paper, the Q-learning method for quadratic optimal control problem of discrete-time linear systems is reconsidered. The theoretical results prove that the quadratic optimal controller cannot be solved directly due to the linear correlation of the data sets. The following corollaries have been made: (1) The correlation of data is the key factor in the success for the calculation of quadratic optimal control laws by Q-learning method; (2) The control laws for linear systems cannot be derived directly by the existing Q-learning method; (3) For nonlinear systems, there are some doubts about the data independence of current method. Therefore, it is necessary to discuss the probability of the controllers established by the existing Q-learning method. To solve this problem, based on the ridge regression, an improved model-free Q-learning quadratic optimal control method for discrete-time linear systems is proposed in this paper. Therefore, the computation process can be implemented correctly, and the effective controller can be solved. The simulation results show that the proposed method can not only overcome the problem caused by the data correlation, but also derive proper control laws for discrete-time linear systems.

Some Insights on Synthesizing Optimal Linear Quadratic Controllers Using Krotov Sufficient Conditions

This letter revisits the problem of synthesizing the optimal control laws for linear systems with a quadratic cost. Traditionally, these laws are computed using the Hamilton-Jacobi-Bellman method, where the solution to the original problem is obtained by solving the Riccati equation, which hinges upon a priori information of the optimal cost function. Within the general Krotov global optimal control framework, though being less explored in the literature, that information is no longer needed. However, utilizing this framework, the original optimization problem is translated into a non-convex problem, which is solved by using iterative methods. In this letter, we propose a new method to compute a direct (non-iterative) solution by transforming the resulting non-convex optimization problem into a convex problem. It turns out that the proposed method naturally leads to the Riccati inequality as the crucial intermediate step, of which the origin was not well understood, although it serves as a strong backbone to address linear quadratic problems and other significant linear system theoretic results. Numerical results and future directions, particularly for solving the optimal control problem for bilinear systems, are also provided to demonstrate the usability of the proposed method.

Linear Quadratic Optimal Control Design: A Novel Approach Based on Krotov Conditions

This paper revisits the problem of synthesizing the optimal control law for linear systems with a quadratic cost. For this problem, traditionally, the state feedback gain matrix of the optimal controller is computed by solving the Riccati equation, which is primarily obtained using calculus of variations- (CoV-) based and Hamilton–Jacobi–Bellman (HJB) equation-based approaches. To obtain the Riccati equation, these approaches require some assumptions in the solution procedure; that is, the former approach requires the notion of costates and then their relationship with states is exploited to obtain the closed-form expression of the optimal control law, while the latter requires a priori knowledge regarding the optimal cost function. In this paper, we propose a novel method for computing linear quadratic optimal control laws by using the global optimal control framework introduced by V. F. Krotov. As shall be illustrated in this article, this framework does not require the notion of costates and any a priori information regarding the optimal cost function. Nevertheless, using this framework, the optimal control problem gets translated to a nonconvex optimization problem. The novelty of the proposed method lies in transforming this nonconvex optimization problem into a convex problem. The convexity imposition results in a linear matrix inequality (LMI), whose analysis is reported in this work. Furthermore, this LMI reduces to the Riccati equation upon imposing optimality requirements. The insights along with the future directions of the work are presented and gathered at appropriate locations in this article. Finally, numerical results are provided to demonstrate the proposed methodology.

SDP-based approximation of stabilising solutions for periodic matrix Riccati differential equations

ABSTRACTNumerically finding stabilising feedback control laws for linear systems of periodic differential equations is a nontrivial task with no known reliable solutions. The most successful method requires solving matrix differential Riccati equations with periodic coefficients. All previously proposed techniques for solving such equations involve numerical integration of unstable differential equations and consequently fail whenever the period is too large or the coefficients vary too much. Here, a new method for numerical computation of stabilising solutions for matrix differential Riccati equations with periodic coefficients is proposed. Our approach does not involve numerical solution of any differential equations. The approximation for a stabilising solution is found in the form of a trigonometric polynomial, matrix coefficients of which are found solving a specially constructed finite-dimensional semidefinite programming (SDP) problem. This problem is obtained using maximality property of the stabilising solution of the Riccati equation for the associated Riccati inequality and sampling technique. Our previously published numerical comparisons with other methods shows that for a class of problems only this technique provides a working solution. Asymptotic convergence of the computed approximations to the stabilising solution is proved below under the assumption that certain combinations of the key parameters are sufficiently large. Although the rate of convergence is not analysed, it appeared to be exponential in our numerical studies.

Optimal control of linear systems with large and variable input delays

This paper proposes an optimal control law for linear systems affected by input delays. Specifically we prove that when the delay functions are known it is possible to generate the optimal control for arbitrarily large delay values by using a DDE without distributed terms. The solution can be seen as a chain of predictors whose size depends on the maximum delay.

Finite‐Horizon ε‐Optimal Tracking Control of Discrete‐Time Linear Systems Using Iterative Approximate Dynamic Programming

AbstractIn this paper, we propose a finite‐horizon neuro‐optimal tracking control strategy for a class of discrete‐time linear systems. In applying the iterative approximate dynamic programming (ADP) algorithm to determine the optimal tracking control law for linear systems, we need finite iterations to obtain the result in practical applications, instead of infinite iterations. An ε‐error bound is introduced into the ADP algorithm to determine the number of iteration steps. The approximation optimal tracking control law will approach the solution of the Hamilton‐Jacobi‐Bellman (HJB) equation through a self‐adaptive iteration within the given value of ε‐error bound. ε error bound is used to stop the iteration process. So, we can obtain the ε‐approximation tracking control law in a finite number of iterations. Nevertheless, different ε will produce different control performances. Furthermore, we will find an optimal ε error bound, which can obtain optimal performance of the ADP algorithm on the basis of the controlled system tracking the desired trajectory. One example is included to complete the ADP algorithm under different error bounds. From the simulation results, we can find the optimal ε error bound. Finally, the simulation validates the efficiency of the proposed algorithm.

An active set solver for input-constrained robust receding horizon control

An efficient optimization procedure is proposed for computing a receding horizon control law for linear systems with linearly constrained control inputs and additive disturbances. The procedure uses an active set approach to solve the dynamic programming problem associated with the min–max optimization of an H∞ performance index. The active constraint set is determined at each sampling instant using first-order necessary conditions for optimality. The computational complexity of each iteration of the algorithm depends linearly on the prediction horizon length. We discuss convergence, closed loop stability and bounds on the disturbance l2-gain in closed loop operation.

A compartmental model-based control strategy for NeuroMuscular Blockade level

AbstractThis paper presents a compartmental model-based control strategy to drive the NeuroMuscular Blockade level of patients undergoing general anesthesia to a predefined target. For that purpose, a compartmental realization of a minimally parameterized nonlinear Wiener model was derived and used on an adapted version of the standard compartmental control law for linear systems. The identification of the model parameters was recursively performed by one Extended Kalman Filter during the initial bolus induction period and stopped afterwards. To overcome the fact that this identification is stopped during the closed-loop control period, uncertainties in the parameters are assumed to be present and included in the control law. Information taken from the identification of real collected cases was used to tune the parameter uncertainties. The feasibility of the whole strategy was evaluated in a bank of simulated models, giving rise to good reference tracking results even in the presence of noise.

H<SUB align=right>∞ repetitive control of linear systems based on Two-Dimensional model

This paper concerns with the H ∞ repetitive control of linear system based on Two-Dimensional (2D) system theory. First, a 2D model is established to describe the control behaviour within a repetition period and the learning process between periods. Next, the problem of designing an H ∞ repetitive control law is formulated as an H ∞ state-feedback design problem for the 2D model. A new existence condition and method of designing an H ∞ state-feedback control law for the 2D model is presented. Moreover, this idea is extended to solve the problem of H ∞ repetitive control law for linear system. Finally, a numerical example was presented to illustrate the effectiveness of the presented method.

Optimal Control Systems

1R6. Optimal Control Systems. - DS Naidu (Idaho State Univ, Pocatello ID). CRC Press LLC, Boca Raton FL. 2003. 433 pp. ISBN 0-8493-0892-5. $99.95.Reviewed by I Kolmanovsky (Sci Res Lab, MD-2036, Ford Motor Co, 2101 Village Rd, Dearborn MI 48124).The book, Optimal Control Systems, by DS Naidu provides an introduction to key concepts and methods of the optimal control theory for deterministic continuous-time and discrete-time finite-dimensional dynamic systems. The material of the book is based on the author’s experience teaching graduate level optimal control courses and has grown out of the lecture notes used by the author in these courses. After an Introduction (the first chapter of the book), the second chapter describes in detail the main ideas in the calculus of variations (culminating in the study of Euler-Lagrange equations and second variation-based Legendre conditions) and seamlessly transitions to the treatment of the optimal control problems. The Hamiltonian function is introduced early on which subsequently features prominently in the treatment of the Pontryagin’s maximum principle. The end result of Chapter 2 is a solution to several types of optimal control problems (such as fixed end-time, with no control constraints) using the principles of the calculus of variations but stated in the formalism that will later be useful for more general optimal control problems. These developments are already sufficient for the in-depth treatment of Linear Quadratic Optimal Control problem for continuous-time systems, which is one of the central tools in the control theory. This treatment is a subject of Chapters 3 and 4 where the finite horizon and infinite horizon cases, set point and trajectory tracking, fixed-end point regulator system, Linear Quadratic Regulator with a specified degree of stability, and other topics are considered in detail. Frequency domain properties of the linear quadratic regulator are analyzed and the celebrated results on one half to infinity gain margin and 60 degrees phase margin are derived. Chapter 5 treats similar issues as in Chapters 2-4 but for discrete-time systems. In Chapter 6, the treatment of constrained control problems begins with the introduction of the Pontryagin maximum principle for these systems (using the notion of admissible variations) and dynamic programming techniques. The latter are also used to illustrate an alternative approach to deriving continuous-time and discrete-time linear quadratic regulator solutions. Time-optimal control problems are treated in Chapter 7 including the analysis of the number of switchings in the optimal “bang-bang” control laws for linear systems. Other special optimal control problems (energy and fuel-optimal control problems with integral costs) including the case of singular controls are touched upon as well. The developments are completed by considering the point wise-in-time state constraints and describing how they can be treated using either the penalty function method or the slack variable method. The appendices help make the book self-contained by reviewing some of the key results in linear algebra and state space analysis of linear systems, and by providing the listings of the relevant Matlab files (available electronically from the author). The book is very readable and careful attention has certainly been given to provide the reader with a comprehensive set of tools without overburdening with the complex mathematical details and notations (although the key mathematical ideas are, indeed, well exposed and the logic underlying the developments is transparent and thorough). It should be accessible to a wide audience including graduate students and practitioners from engineering and other fields. Optimal Control Systems can be utilized both as a textbook in the introductory courses in the optimal control theory, or as a quick reference by practicing engineers. The use of Matlab/Simulink to treat the examples (including the snippets of the actual code), historical remarks about the scientists behind the major discoveries in the optimal control theory, recipe-style summaries of the key methodologies, exercise problems at the end of each of the chapters are undoubtedly the strong points of this book which significantly contribute to its pedagogical value. On a somewhat more critical side, most of the examples in the book are, in fact, of relatively low order linear systems. This can actually be viewed as an advantage in that the in-depth treatment of these examples is possible so that the main ideas can be illustrated rather well. At the same time, the treatment of more of higher order nonlinear system examples in combination with a more extended introduction into the numerical methods of the optimal control could have been beneficial for the readers to gain an additional insight into how the underlying techniques can be applied.

Matching, linear systems, and the ball and beam

A recent approach to the control of underactuated systems is to look for control laws which will induce some specified structure on the closed loop system. In this paper, we describe one matching condition and an approach for finding all control laws that fit the condition. After an analysis of the resulting control laws for linear systems, we present the results from an experiment on a nonlinear ball and beam system.

Opened loop control of switching linear systems

The purpose of this paper is to elaborate an opened loop control law for linear systems with Boolean inputs and to show a way to find a convergence domain of those systems. First, we determine the convergence domain for the state vector by expressing it with respect to the time using linear systems theory. Afterwards, the algorithm of control is determined. We apply those results to a two-dimensional system and a geometric approach is then presented. Finally, we present simulation results as an example.

Observer-based robust-H-infinity control laws for uncertain linear systems

Based on the algebraic Riccati equation approach, this paper presents a simple and flexible method for designing observer-based robust-H-infinity control laws for linear systems with structured parameter uncertainty. The observer-based robust-H-infinity output-feedback control law, obtained by solving three augmented algebraic Riccati equations, provides both robust stability and disturbance attenuation with H-infinity-norm bound for the closed-loop uncertain linear system. Several tuning parameters are embedded into the augmented algebraic Riccati equations so that flexibility in finding the symmetric positive-definite solutions (and hence, the robust-H-infinity control laws) is significantly increased. A benchmark problem associated with a mass-spring system, which approximates the dynamics of a flexible structure, is used to illustrate the design methodologies, and simulation results are presented.

Design of observer-based robust stabilizing controllers

This paper is concerned with the problem of designing stabilizing control laws for linear systems with time-varying uncertainty. Since the entire state cannot be measured, the controller must operate on some estimate of the state. The approach used is to first obtain a linear robust controller based on the assumption that the complete state is available. Then an estimated state is generated via an observer and this estimate used in the implementation of the controller. Examples indicate that the use of an observer usually reduces the magnitude of the tolerable uncertainty. The amount of this reduction depends on the choice made for the measured state variables.

Methods for obtaining robust tracking control laws

This paper treats the problem of determining a tracking control law for linear systems with uncertain parameters. The tracking control is robust in the sense that it achieves asymptotic tracking of the command reference input for all admissible parameter values. Based on the available theoretical results for determining tracking controllers, numerical methods for obtaining the feedback gains for a robust control law are presented. The efficacy of these methods is demonstrated by applying them to two aircraft examples.

Design methodology for robust stabilizing controllers

This paper considers the problem of designing control laws for linear systems with time-varying uncertainty. A method for determining a linear feedback control which stabilizes the system for all possible uncertainty is presented. This control is robust in the sense that it guarantees asymptotic stability regardless of the disturbance. The results are applied to several aircraft examples .

Optimal Feedback Control for Linear Systems Having Time Delays

The optimal feedback control law for linear systems having time delay in both the state and control variables requires the solution of a coupled ordinary and partial Riccati differential equations. This requires excessive computer time. In this paper, we show how Fade approximants can be made use of in substantially reducing the computation load in finding the optimal control law. The use of Pade approximants up to third order has been indicated. Higher order approximants can be treated similarly.