Linear Autonomous Systems Research Articles

About the exact determination of the solution to linear completely regular differential-algebraic systems with delay

We study the problem of exact reconstruction of the solution from measurements of the observed output for linear autonomous completely regular differential-algebraic systems with commensurate delays. The reconstruction process is implemented using a finite observer, which is the output of a linear autonomous retarded system type with commensurate concentrated and distributed delays and a predetermined finite spectrum. The criterion for the existence of such an observer is obtained.

Explainable data-driven Q-learning control for a class of discrete-time linear autonomous systems

Explaining what a reinforcement learning (RL) control agent learns play a crucial role in the safety critical control domain. Most of the approaches in the state-of-the-art focused on imitation learning methods that uncover the hidden reward function of a given control policy. However, these approaches do not uncover what the RL agent learns effectively from the agent-environment interaction. The policy learned by the RL agent depends in how good the state transition mapping is inferred from the data. When the state transition mapping is wrongly inferred implies that the RL agent is not learning properly. This can compromise the safety of the surrounding environment and the agent itself. In this paper, we aim to uncover the elements learned by data-driven RL control agents in a special class of discrete-time linear autonomous systems. Here, the approach aims to add a new explainable dimension to data-driven control approaches to increase their trust and safe deployment. We focus on the classical data-driven Q-learning algorithm and propose an explainable Q-learning (XQL) algorithm that can be further expanded to other data-driven RL control agents. Simulation experiments are conducted to observe the effectiveness of the proposed approach under different scenarios using several discrete-time models of autonomous platforms.

On Some Problems with Multivalued Mappings

We consider some problems with a set-valued mapping, which can be reduced to minimization of a homogeneous Lipschitz function on the unit sphere. Latter problem can be solved in some cases with a first order algorithm—the gradient projection method. As one of the examples, the case when set-valued mapping is the reachable set of a linear autonomous controlled system is considered. In several settings, the linear convergence is proven. The methods used in proofs follow those introduced by B.T. Polyak for the case where Lezanski– Polyak–Lojasiewicz condition holds. Unlike algorithms that use approximation of the reachable set, the proposed algorithms depend far less on dimension and other parameters of the problem. Efficient error estimation is possible. Numerical experiments confirm the effectiveness of the considered approach. This approach can also be applied to various set-theoretical problems with general set-valued mappings.

On the issue of control of neutral-type systems by hybrid controllers with state feedback

For linear autonomous systems of neutral type, an approach to control design problems in the form of feedback based on a new class of hybrid controllers is proposed. The structure of hybrid regulators necessarily includes a differential equation, so the closed system becomes differential-algebraic. A distinctive feature of hybrid controllers is the existence of elementary transformation of equations of a closed-loop system, which make it possible to obtain an independent subsystem of neutral type. In this case, the specified subsystem of neutral type uniquely determines the behavior of the solution $x(t)$ of the original open-loop system (possibly as a vector component of the solution vector of the closed-loop system). The main advantages of using hybrid controllers include the possibility of their application to systems that do not satisfy the “traditional” controllability properties. The properties of hybrid controllers have been studied. The author gives an example of using these controllers to solve a new problem of controlling the coefficients of a characteristic quasi-polynomial in the case where the approaches known in the literature are not applicable. The possibility of using hybrid controllers to solve the $0$-controllability problem using the finite control method is demonstrated.

On the Feedback Synthesis for an Autonomous Linear System with Perturbations

On the Feedback Synthesis for an Autonomous Linear System with Perturbations

Safety-Critical Stabilizing Design of Switched Linear Autonomous Systems

For continuous-time switched linear autonomous systems, this work addresses the stabilization problem under safety-critical constraints. Given an origin-symmetric region that any state trajectory should be stayed, we seek to design a switching law to achieve both stability and safety. The main approach is to characterize, both theoretically and algorithmically, the safe initial set that any initial state within the set could stay in the given safety region with a properly designed switching law. Technically, we develop a design procedure that could approximate the safe initial set with the help of the pathwise state feedback switching strategy. A third-order example is presented to validate the effectiveness of the proposed methodology.

Optimization of transient performance for continuous-time switched linear autonomous systems

For the general class of continuous-time switched linear autonomous systems, this work presents a constructive design scheme to estimate and optimize the transient response in terms of state overshoot and settling time. Based on the pathwise state feedback switching strategy, we present theoretical characterizations of the transient performance, and develop a computational design procedure that optimizes the state overshoot and settling time in a constructive manner. A numerical example is presented to show the effectiveness of the proposed design scheme.

Oriented Rotatability Exponents of Solutions to Homogeneous Autonomous Linear Differential Systems

Oriented Rotatability Exponents of Solutions to Homogeneous Autonomous Linear Differential Systems

Equations of Motion of Mechanical Systems with Switchable Constraints

The article deals with a numerical and analytical approach to solving the equations of motion, which enables to treat the considered problems with the change of system structure or number of degrees of freedom without interrupting the numerical integration process. The described methodology allows effectively incorporate switchable constraints in the systems in accordance with their flexible structures. The crucial idea is based on the formulation of the resulting differentialalgebraic equations into a saddle point system, where the switchable constraints are represented by a sign matrix with variable rank. In connection with this property, a pseudoinversion is applied to eliminate algebraic variables and transform the problem to the first order system of ordinary differential equations. Moreover, the time independent case leads to linear autonomous systems with non-diagonalizable matrices, as is proved. The relevant numerical scheme is based on Runge-Kutta methods, that correspond to the power series of the resulting matrix exponential for time independent problems. The methodology presented is illustrated on the idealized two-mass oscillator with a switchable constraint. The numerical experiments performed range from initial stages, through simple transient cases to damped intentional control. The advanced applications can be found in robotics, active and controlled systems, and in the simulations of complex systems in biology and related areas. Moreover, the methodology can also be applied in the simulation of transport systems, especially in relation to vehicle technology, a quarter car suspension system, a vibration control mechanism, a torsion system with a clutch, and machine balancing and storage should to be highlighted.

Explicit convergence rate parameters for linear autonomous systems

In this paper, we propose a change of coordinates that brings the state matrix of an autonomous linear system into a modified Jordan Block form. Such a change of coordinates allows us to obtain exact values of the scaling factor and the convergence rate of the exponential stability bound for linear systems. The analysis is then applied to a control design with requirements on the evolution of the state norm. Numerical examples are also provided to illustrate the effectiveness of the proposed approach.

CANONICAL APPROXIMATIONS IN IMPULSE STABILIZATION FOR A SYSTEM WITH AFTEREFFECT

For optimal stabilization of an autonomous linear system of differential equations with aftereffect and impulse controls, the formulation of the problem in the functional state space is used. For a system with aftereffect, approximating systems of ordinary differential equations proposed by S.N. Shimanov and J. Hale are used. A method for constructing approximations for optimal stabilizing control of an autonomous linear system with aftereffect and impulse controls is proposed. Matrix Riccati equations are used to find approximating controls.

Ob opredelenii koeffitsientov kvadratichnoy funktsii Lyapunova s zadannymi svoystvami v sluchae kratnykh korney kharakteristicheskogo uravneniya

For continuous and discrete linear autonomous systems, we discuss the possibility of choosing the coefficients of the quadratic Lyapunov function that ensure the validity of the condition of sign-negativity of its first derivative (first difference) with a given margin in the case of multiple roots of the characteristic equation.

Asymptotic expansion of the solution to an optimal control problem for a linear autonomous system with a terminal convex quality index depending on slow and fast variables

In this paper, we investigate a problem of optimal control over a finite time interval for a linear autonomous system with slow and fast variables in the class of piecewise continuous controls with smooth geometric constraints in the form of a ball. We consider a terminal convex quality index that depends on slow and fast variables. We substantiate a limit relation for the vector determining the optimal control as the small parameter tends to zero. We refine the limit relation for the case of an indirect control problem with a terminal quality index, which is the sum of values of two strictly convex continuously differentiable functions, the first of which depends only on slow variables, and the second one depends only on fast variables and has a minimum at zero. In doing so, we show that the first component of the determining vector converges to the determining vector of the limit problem while the second component tends to zero. In the problem of indirect control of a system of material points in a medium with resistance, we obtain the complete asymptotics of the determining vector in powers of a small parameter.

Stabilization of a Differential-Difference System of Delay Type

For a linear autonomous differential-difference system with commensurate delays, algorithms for constructing controllers that provide asymptotic, finite-time, or complete stabilization of this system are substantiated. A distinctive feature of the proposed approach is that it does not require a priori information about the location of the roots of the characteristic quasipolynomial of the original system. The results are illustrated with examples.

The behavior of hidden bifurcation in 2D scroll via saturated function series controlled by a coefficient harmonic linearization method

Abstract In this article, the behavior of hidden bifurcation in a two-dimensional (2D) scroll via saturated function series controlled by the coefficient harmonic linearization method is presented. A saturated function series approach for chaos generation. The systematic saturated function series methodicalness improved here can make multi-scroll and grid scroll chaotic attractors from a 3D linear autonomous system with a plain saturated function series supervisor. We have used a hidden bifurcation method in grid scroll., where the method of hidden bifurcation presented by Menacer, et al. in (2016) for Chua multi-scroll attractors. This additional parameter, which is absent from the initial problem, is perfectly adapted to unfold the structure of the multispiral chaotic attractor. The novelty of this article is twofold: first, the saturated function series model for hidden bifurcation in a 2 – D scroll; and second, the control of hidden bifurcation behavior by the value of the harmonic coefficient k 3 {k}_{3} .

Stability Criteria for Time-Delay Systems From an Insightful Perspective on the Characteristic Equation

This article provides a delay-dependent and a necessary and sufficient delay-independent stability criterion for linear autonomous continuous-time systems with a discrete delay. We take a simple perspective on the two-variable formulation of the characteristic equation, which leads to the following advantageous but not widespread delay-independent criterion: The sum of the coefficient matrix of the delay-free term and the elementwise unitarily rotated matrix of the delay term must remain Hurwitz for all rotation angles. A graphical test for the latter is shown to require no more than three lines of code. Concerning delay-independent stability, our main contribution is to extend this sufficient criterion to a necessary and sufficient one. Concerning delay-dependent stability, the focus is on the critical delay that bounds the initial delay interval of stability. We formulate a constrained minimization problem that gives the exact value of this critical delay. The taken perspective is especially insightful in terms of how the coefficient matrices may look like, and, for scalar systems, the delay-dependent stability chart becomes obvious at a first glance. The presented criteria are complementary to the well-known frequency-sweeping test, which results from another of three possible perspectives on the two-variable criterion. Besides of a unified treatment of these different perspectives, the article also discusses corollaries of the one taken, which include Mori’s famous criterion.

On the Preservation of a Quadratic Lyapunov Function of a Linear Differential Autonomous System under Constant Perturbations of the Coefficients

For an autonomous linear homogeneous asymptotically stable differential system, we obtain sufficient conditions for the smallness of perturbations in the class of autonomous linear homogeneous systems under which a quadratic form that is a Lyapunov function of the original system remains a Lyapunov function of the perturbed system.

On the Preservation of a Quadratic Lyapunov Function of a Linear Differential Autonomous System under Constant Perturbations of the Coefficients

On the Preservation of a Quadratic Lyapunov Function of a Linear Differential Autonomous System under Constant Perturbations of the Coefficients

Criterion for Lyapunov Reducibility of a Linear Autonomous Differential System to a Linear Autonomous Equation

Criterion for Lyapunov Reducibility of a Linear Autonomous Differential System to a Linear Autonomous Equation

THE ASYMPTOTIC STABILITY OF THE SIMPLEST HYBRID SYSTEMS

The linear autonomous discrete-continuous systems (also known as hybrid systems) with uncertain coefficients is considered. We consider the class of hybrid systems, that consist of difference and ordinary differential equations. We obtain a stability domain for hybrid systems from this class. Two illustrative examples are presented