Regulation Problem For Linear Systems Research Articles

Further results on the regulation problem for linear systems with constraints on control and its increment

The regulator problem for both discrete-time and continuous-time linear systems is considered. The control and its increments are under non-symmetrical constraints and the domain of constraints includes the origin on its boundary. We derive necessary and sufficient conditions which ensure the satisfaction of all the constraints and also the asymptotic stability by a state feedback. An illustrative example shows the application of our method.

On the regulator problem for linear systems over rings and algebras

Abstract The regulator problem is solvable for a linear dynamical system Σ \Sigma if and only if Σ \Sigma is both pole assignable and state estimable. In this case, Σ \Sigma is a canonical system (i.e., reachable and observable). When the ring R R is a field or a Noetherian total ring of fractions the converse is true. Commutative rings which have the property that the regulator problem is solvable for every canonical system (RP-rings) are characterized as the class of rings where every observable system is state estimable (SE-rings), and this class is shown to be equal to the class of rings where every reachable system is pole-assignable (PA-rings) and the dual of a canonical system is also canonical (DP-rings).

Low-complexity learning of Linear Quadratic Regulators from noisy data

This paper considers the Linear Quadratic Regulator problem for linear systems with unknown dynamics, a central problem in data-driven control and reinforcement learning. We propose a method that uses data to directly return a controller without estimating a model of the system. Sufficient conditions are given under which this method returns a stabilizing controller with guaranteed relative error when the data used to design the controller are affected by noise. This method has low complexity as it only requires a finite number of samples of the system response to a sufficiently exciting input, and can be efficiently implemented as a semi-definite programme.

Output regulation of linear systems via sampled-data control

In this paper, we study the output regulation problem for linear systems by the aperiodic sampled-data control approach. Suppose the problem is solvable by a continuous-time static state feedback control law and a continuous-time dynamic output feedback control law. Then, we first show that, if the exogenous signal is constant, then the output regulation problem is solvable exactly by both the sampled-data static state feedback control law and the sampled-data dynamic output feedback control law. If the exogenous signal is time-varying with bounded derivative, then we further show that the problem is solvable practically by both the sampled-data static state feedback control law and the sampled-data dynamic output feedback control law.

Output regulation of unknown linear systems using average cost reinforcement learning

In this paper, we introduce an optimal average cost learning framework to solve output regulation problem for linear systems with unknown dynamics. Our optimal framework aims to design the controller to achieve output tracking and disturbance rejection while minimizing the average cost. We derive the Hamilton–Jacobi–Bellman (HJB) equation for the optimal average cost problem and develop a reinforcement algorithm to solve it. Our proposed algorithm is an off-policy routine which learns the optimal average cost solution completely model-free. We rigorously analyze the convergence of the proposed algorithm. Compared to previous approaches for optimal tracking controller design, we elevate the need for judicious selection of the discounting factor and the proposed algorithm can be implemented completely model-free. We support our theoretical results with a simulation example.

A Discrete-Time Recurrent Neural Network for Solving Rank-Deficient Matrix Equations With an Application to Output Regulation of Linear Systems.

This paper presents a discrete-time recurrent neural network approach to solving systems of linear equations with two features. First, the system of linear equations may not have a unique solution. Second, the system matrix is not known precisely, but a sequence of matrices that converges to the unknown system matrix exponentially is known. The problem is motivated from solving the output regulation problem for linear systems. Thus, an application of our main result leads to an online solution to the output regulation problem for linear systems.

Robust output regulation problem for discrete-time linear systems with both input and communication delays

Recently, the robust output regulation problem for continuous-time linear systems with both input and communication time-delays was studied. This paper will further present the results on the robust output regulation problem for discrete-time linear systems with input and communication delays. The motivation of this paper comes from two aspects. First, it is known that the solvability of the output regulation problem for linear systems is dictated by two matrix equations. While, for delay-free systems, these two matrix equations are same for both continuous-time systems and discrete-time systems, they are different for continuous-time time-delay systems and discrete-time time-delay systems. Second, the stabilization methods for continuous-time timedelay systems and discrete-time time-delay systems are also somehow different. Thus, an independent treatment of the robust output regulation problem for discrete-time time-delay systems will be useful and necessary.

Robust output regulation of linear time-delay systems: A state predictor approach

Summary This paper considers the robust output regulation problem for linear systems in the presence of state, input, and output delays. First, a state feedback control law is constructed from a state predictor, recently developed for systems with state and input delays. Necessary and sufficient conditions for the existence of a solution to the regulator equations are presented. Next, an error feedback control law for the output regulation problem is constructed by means of an error-based state predictor. Finally, the proposed error feedback solution is extended to solve the output regulation problem in the presence of model uncertainty. Numerical examples demonstrate the effectiveness of the proposed predictor-based solutions. Copyright © 2015 John Wiley & Sons, Ltd.

Robust output regulation problem for linear time-delay systems

In this paper, we study the robust output regulation problem for linear systems with input time-delay. By extending the internal model design method to linear time-delay systems, we have established solvability conditions for the problem by both dynamic state feedback control and dynamic output feedback control. The advantages of internal model approach over the feedforward design approach are that it can handle perturbations of the uncertain parameters in the plant and the control law, and it does not need to solve the regulator equations.

Output Regulation of Linear Systems with State, Input and Output Delays

This paper considers the output regulation problem for linear systems in the presence of state, input and output delays. A state feedback output regulation law is constructed from a state predictor, recently developed for systems with state and input delays, and the solution to a pair of regulator equations that transforms the output regulation problem into a stabilization problem. Necessary and sufficient conditions for the existence of a solution to the regulator equations are presented. Numerical examples demonstrate the effectiveness of the developed predictor-based solution.

Suboptimal stochastic linear feedback control of linear systems with state- and control-dependent noise: The incomplete information case

The stochastic regulation problem for linear systems with state- and control-dependent noise and a noisy linear output equation is considered. The optimal quadratic cost output-feedback control law in a class of linear controllers is found. This problem was first addressed in the early 1970s and solved, in the complete information case, by Wonham. In this paper we give the solution of the problem in the incomplete information case, that is, for a linear output equation corrupted by Gaussian noise. Moreover, a different method is used here, giving the solution in a more direct way even in the complete information case.

The Regulator Problem for Linear Systems With Saturations on the Control and its Increments or Rate: An LMI Approach

This paper solves the problem of designing stabilizing regulators for linear systems subject to control saturations and asymmetric constraints on its increment or rate, using reduced dimension linear matrix inequalities (LMIs) developed on a reduced-order state space. Compared with previous approaches, the proposed technique is valid for asymmetric constraints on the increment or rate of the control while the computing time is improved by resolving LMIs of reduced dimensions

The regulator problem for linear systems with constrained control: an LMI approach

The regulator problem for linear systems with constrained control: an LMI approach

A general formulation and solvability of the global robust output regulation problem

While the output regulation problem for linear systems accommodates both bounded and unbounded exogenous signals, the existing formulation of the output regulation problem for nonlinear systems only allows bounded exogenous signals produced by an exosystem with a neutrally stable equilibrium. In particular, when it comes to the robust output regulation problem, the only admissible exogenous signal is a finite combination of step and sinusoidal functions. In this paper, we give a general formulation of the robust output regulation problem that admits unbounded exogenous signals, and contains the previous formulations as special cases when the system is linear or when the exogenous signals are bounded. Then, we give conditions under which the problem can be solved by solving a robust stabilization problem of an augmented system. Finally, we apply our general result to obtain the solvability conditions of the general robust output regulation problem for the class of lower triangular nonlinear systems.

Regulator problem for linear systems with constraints on control and its increment or rate

This paper discusses the problem of constraints on both control and its rate or increment, for linear systems in state space form, in both the continuous and discrete-time domains. Necessary and sufficient conditions are derived for autonomous linear systems with constrained state increment or rate (for the continuous-time case), such that the system evolves respecting incremental or rate constraints. A pole assignment technique is then used to solve the inverse problem, giving stabilizing state feedback controllers that respect non-symmetrical constraints on both control and its increment or rate. An illustrative example shows the application of the method on the double integrator problem.

On output regulation for linear systems

For both continuous- and discrete-time systems, we revisit the output regulation problem for linear systems. We generalize the problem formulation in order � to expand the class of reference or disturbance signals, � to utilize the derivative or feedforward information of reference signals whenever it is available. Our study of the so-called generalized regulation problem depends on the classical notions of exact and almost disturbance decoupling. These classical notions are reexamined while considering a very broad category of controllers beyond the classical state and measurement feedback. Such controllers include classical state feedback and measurement feedbacks but also extensions where part of the disturbance signal can be used for feedback. We consider the case where, for a part of the disturbance signal, either all its derivatives up to a certain jth order or derivatives of arbitrary order are available. This is combined with state and with measurement feedback.

Duality and the regulator problem for linear systems over commutative rings

A commutative ring with identity is called a full quotient ring if every element of R is either a unit or a zero divisor. The Duality Principle, “Reachability is dual to observability,” holds for (stationary, discrete-time) linear systems over Noetherian full quotient rings. The solvability of the pole assignment problem is equivalent to reachability for these systems, and (dually) the solvability of the state-estimation problem is equivalent to observability. To summarize these facts, we say that: The regulator problem is solvable for linear systems over Noetherian full quotient rings.

Optimal stochastic control of linear systems with state- and control-dependent disturbances

A review of the solution of the linear regulator problem for linear systems with state- and control-dependent disturbances is presented. Both the finite and infinite terminal time cases are treated. The solution to the complete state feedback case is given in detail and that for the output feedback case is noted. The general conclusion is that control-dependent noise calls for conservative control (small gains) while state-dependent noise calls for vigorous control (large gains). Of course it is the degree of this behavior that is important and this is given explicitly by the algorithms in this paper.