Linear Time-invariant Control Systems Research Articles

Controller and Triggering Mechanism Co-design for Control Over Time-Slotted Networks

This article considers the control of linear time-invariant systems over a time-slotted communication network, which supports infrequent deterministic transmission slots and additional opportunistic transmission slots that can only be used if the network traffic satisfies a desired traffic specification. For two scenarios, controllers as well as combined time-based and event-based triggering mechanisms are designed such that three goals can be jointly achieved. The first goal is to guarantee asymptotic stability by using the deterministic slots, independent of any knowledge about the use of opportunistic transmissions. Second, the triggering mechanisms are designed such that the opportunistic transmissions satisfy a given token bucket traffic specification. As a third goal, the opportunistic transmission decisions should be dependent on the possible performance improvement attributed to the transmitted message. This property is formalized in the newly defined concept of performance-oriented triggering. The two derived controllers and triggering mechanisms are demonstrated using a numerical example.

Matrix eigenvalue spectrum assignment for linear control systems by static output feedback

For a linear time-invariant control system defined by a linear differential equation of the n-th order with a multidimensional state, input and output, we set and study the problem of arbitrary matrix eigenvalue spectrum assignment by linear static output feedback. We obtain controllability-like rank conditions that are necessary and sufficient for the problem of arbitrary matrix eigenvalue spectrum assignment and are sufficient for the problem of arbitrary eigenvalue spectrum assignment by linear static output feedback. It is proved that, in particular cases, when the system has block scalar matrix coefficients, these conditions can be relaxed. The obtained results generalize the known results for the corresponding problem by static state feedback and by static output feedback in the case of the one-dimensional equation. Examples are presented to illustrate the results.

On the observability of conformable linear time-invariant control systems

<p style='text-indent:20px;'>In this paper, we analyze the concept of observability in the case of conformable time-invariant linear control systems. Also, we study the Gramian observability matrix of the conformable linear system, its rank criteria, null space, and some other conditions. We also discuss some properties of conformable Laplace transform.</p>

Efficient Representation and Approximation of Model Predictive Control Laws via Deep Learning.

We show that artificial neural networks with rectifier units as activation functions can exactly represent the piecewise affine function that results from the formulation of model predictive control (MPC) of linear time-invariant systems. The choice of deep neural networks is particularly interesting as they can represent exponentially many more affine regions compared to networks with only one hidden layer. We provide theoretical bounds on the minimum number of hidden layers and neurons per layer that a neural network should have to exactly represent a given MPC law. The proposed approach has a strong potential as an approximation method of predictive control laws, leading to a better approximation quality and significantly smaller memory requirements than previous approaches, as we illustrate via simulation examples. We also suggest different alternatives to correct or quantify the approximation error. Since the online evaluation of neural networks is extremely simple, the approximated controllers can be deployed on low-power embedded devices with small storage capacity, enabling the implementation of advanced decision-making strategies for complex cyber-physical systems with limited computing capabilities.

On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

Impulsive observer and impulsive control for time-delay systems

In this paper, we study the problems of impulsive observer-based impulsive control for linear time-invariant systems with time-varying delays. Compared with traditional observer-based control schemes, our observer and controller are modelled by impulsive differential equations, where both the states of observer and controller are updated abruptly at impulse times and moreover, the information of time-varying delays is not required in both of observer and controller. By Lyapunov function method and impulsive delay differential inequality, some sufficient conditions for globally exponential stability of the closed-loop system are derived in terms of linear matrix inequalities (LMIs). Moreover, the gain matrices could be designed by solving the corresponding LMIs. Finally, a numerical example is pretended to illustrate the effectiveness of the proposed approach.

ON EQUIVALENCE OF LINEAR CONTROL SYSTEMS AND ITS USAGE TO VERIFICATION OF THE ADEQUACY OF DIFFERENT MODELS FOR A REAL DYNAMIC PROCESS

A problem of description of algebraic invariants for a linear control system that determine its structure is considered. With the help of these invariants, the equivalence problem of two linear time-invariant control systems with respect to actions of some linear groups on the spaces of inputs, outputs, and states of these systems is solved. The invariants are used to establish the necessary equivalence conditions for two nonlinear systems of differential equations generalizing the well-known Hopfield neural network model. Finally, these conditions are applied to establish the adequacy of two neural network models designed to describe the behavior of a real dynamic process given by two different sets of time series.

A new Solution of the Fractional-order Singular Linear Time-Invariant System

A new solution, the distributional solution of the fractional-order singular linear time-invariant control system (FOSLTIS) is studied in the paper. On the basis of proposing the definition of the Caputo fractional derivatives of Dirac function δ(t), the Laplace transform was obtained, then was derived. With regard to FOSLTIS, the system was decomposed into fast subsystem and slow subsystem by restricted equivalent transformation. Using Laplace transformation and taking into account, distributional solution for the fast subsystem was obtained. Two examples illustrated the correctness of the distributional solution of the FOSLTIS.

Output Control of Linear Time-invariant Systems Under Input and Output Disturbances

The novel control algorithm for linear time-invariant systems under disturbances and measurement noises is proposed. The designed control law, based on the noise and disturbance estimation, ensures the accuracy in steady state depending on the disturbance, only one component of noise vector and its first derivatives. Sufficient conditions in terms of linear matrix inequality (LMI) providing stability of the closed-loop system are obtained. The simulations show efficiency of the proposed method compared with existing ones.

Finite-time stabilization of linear systems by bounded linear time-varying feedback

Time-varying features are generally considered to be detrimental to the analysis and design of control systems. This paper establishes methods to design bounded linear time-varying (LTV) controllers such that the control performance of a linear time-invariant (LTI) system can be improved, that is, the finite-time stability of the closed-loop system can be obtained. Specifically, for an LTI control system, by using the solution to a parametric Lyapunov equation (PLE), a bounded LTV controller containing a suitable time-varying parameter is designed. By fully exploiting properties of the solution to the PLE, it is shown that the closed-loop system is finite-time stable. Both state feedback and observer based output feedback, in which both the observer gain and the state feedback gain are time-varying, are considered. As a consequence, the finite-time semi-global stabilization and the fixed-time (prescribed finite-time) stabilization problems for linear systems by bounded controls are solved. The established method is utilized to the design of the spacecraft rendezvous control system and its effectiveness is verified by simulations.

On Model-Free Adaptive Control and Its Stability Analysis

In this paper, the main issues of model-based control methods are first reviewed, followed by the motivations and the state of the art of the model-free adaptive control (MFAC). MFAC is a novel data-driven control method for a class of unknown nonaffine nonlinear discrete-time systems since neither explicit physical model nor Lyapunov stability theory or key technical lemma is used in the controller design and theoretical analysis except only for the input/output (I/O) measurement data. The basis of MFAC is the dynamic linearization data modeling method at each operating point of the closed-loop system. The established dynamic linearization data model is a virtual equivalent data relationship in the I/O sense to the original nonlinear plant by means of a novel concept called pseudo-partial derivative (PPD) or pseudo-gradient (PG) vector. Based on this virtual equivalent dynamic linearization data model and the time-varying PPD or PG estimation algorithm designed merely using the I/O measurements of a controlled plant, the MFAC system is constructed. The main contribution of this paper is that the theoretical analysis of the bounded-input bounded-output stability, the monotonic convergence of the tracking error dynamics, and the internal stability of the full form dynamic linearization based MFAC scheme are rigorously presented by the contraction mapping principle; the well known PID control and the traditional adaptive control for linear time-invariant systems are explicitly shown as the special cases of this MFAC. The simulation results verify the effectiveness of the proposed approach.

Discrete Time Pontryagin Maximum Principle Under State-Action-Frequency Constraints

We establish a Pontryagin maximum principle for discrete time optimal control problems under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, and c) constraints on the frequency spectrum of the optimal control trajectories. While the first two types of constraints are already included in the existing versions of the Pontryagin maximum principle, it turns out that the third type of constraints cannot be recast in any of the standard forms of the existing results for the original control system. We provide two different proofs of our Pontryagin maximum principle in this article, and include several special cases fine-tuned to control-affine nonlinear and linear system models. In particular, for minimization of quadratic cost functions and linear time invariant control systems, we provide tight conditions under which the optimal controls under frequency constraints are either normal or abnormal.

АДАПТИВНОЕ ОЦЕНИВАНИЕ И УПРАВЛЕНИЕ ДЛЯ СЛОЖНЫХ СИСТЕМ:ПРОШЛЫЕ ДОСТИЖЕНИЯ И НЕРЕШЕННЫЕ ЗАДАЧИ

Engineering systems’ adaptability to the unknown and unexpectedly changing equip-ment condition and environmental data is provided artificially by equipping the main loop of control with the adaptor loop. The paper characterizes the current status of re-search and practical studies in the field. As shown for estimation and control systems, the adaptor’s task is to ensure the efficient and conforming solution of the following three problems. Problem 1 is the quickest change point detection in the acting model relative to the previous model. Problem 2 is the unbiased and adequately quick identification of the acting model. Problem 3 is the computationally efficient modifying the feedback in the main loop of control. The paper characterizes the current status of research and prac-tical studies in the field over many publications and some author’s developments. Among the latter, there is the active principle of adaptation for model identification in the class of linear time-invariant control systems.

Stabilization of linear control systems. Pole assignment problem. A survey

This paper reviews the problems of stabilization and pole assignment in the control of linear time-invariant systems using the static state and output feedback. The main results are presented, along with a literature review. Stationary and nonstationary stabilizations with the static feedback are considered. The algorithms of low- and high-frequency stabilization of linear systems for solving Brockett’s stabilization problem are provided. The effective necessary and sufficient conditions for stabilization of two- and three-dimensional controllable linear systems are given in terms of the system parameters. The pole assignment problem and the related issues for linear systems are considered.

Time-optimal hands-off control for linear time-invariant systems

In this article, we investigate theoretical properties of the time-optimal hands-off control for linear time-invariant systems. The purpose of the control is to maximize the time duration on which the control value is exactly zero (maximum hands-off control) and also to minimize the response time to achieve a given state transition (time-optimal control). For this, we introduce a cost function described by a linear combination of the L0 measure and the response time of the control. Since the L0 measure is non-convex and discontinuous, we adopt the L1 relaxation technique for the analysis of the optimal control. By using this relaxation, we show the existence of the time-optimal hands-off control, and the equivalence between L0 andL1 optimal controls under the normality assumption.

A Framework Utilizing Augmented Reality to Enhance the Teaching–Learning Experience of Linear Control Systems

In recent years, engineering education has become more complex, more uncertain and more interdisciplinary. It is therefore important to supplement the theoretical introduction of technical topics and concepts with practical, learning by doing experiences, so as to maximize the learning outcome. When introduced to challenging technical concepts and topics through conventional diagrams and equations, such as those found in textbooks, many students have difficulty with visualization; limiting their ability to gain a deep understanding of the concept. The computer visualization (using Virtual Reality or Augmented Reality (AR)) has apparently become a problem-solving tool in engineering education. AR can overlay 3D virtual objects to a particular real-world context and is an effective way to enhance the visualization and as a result, the understanding of technical concepts. One topic commonly challenging for students to understand is the stability analysis of linear time-invariant control systems. To address this, this paper introduces the framework of a tabletop AR learning environment based on the tangible user interface (TUI) and consisting of a fixed VGA camera with a display screen.

Relaxing the high-frequency gain sign assumption in direct model reference adaptive control

A new, high performance, solution to the classical problem of direct model reference adaptive control for linear time-invariant systems with unknown sign of the high frequency gain is reported in the paper. The proposed algorithm directly estimates this parameter with the only required prior knowledge of a lower bound on its absolute value. To avoid the possible appearance of singularities in the controller calculation a switched projection mechanism is introduced to change, if needed, the sign of the estimate. The recently introduced dynamic regressor extension and mixing estimator is used to ensure monotonicity of the estimation error of the high frequency gain, guaranteeing that the switching appears (at most) once and avoiding the possible appearance of chattering—that may happen in classical gradient-based algorithms. Comparative simulations with the Nussbaum gain-based and gradient estimators illustrate the dramatic performance improvement of the proposed controller.

Application of stochastic fractal search in approximation and control of LTI systems

The present work deals with the application of evolutionary computation in approximation and control of linear time invariant (LTI) systems. Stochastic fractal search algorithm (SFS) has been proposed to obtain low order system (LOS) from LTI higher order system (HOS) as well as in speed control of DC motor with PID controller. SFS is quite simple to use in control system and employs the diffusion property present in random fractals to discover the search space. In approximation of LTI systems, the integral square error (ISE) while in control of DC motor, the integral of time multiplied absolute error has been taken as an objective/fitness functions. In system’s approximation, the results show that the proposed SFS based LOS preserves both the transient and steady state properties of original HOS. The simulation results have also been compared in terms of; ISE, integral absolute error and impulse response energy with well known familiar and recently published works in the literature which shows the superiority of SFS algorithm. In control of DC motor, the obtained results are satisfactory having no overshoot and less rise and settling times in comparison to existing techniques.

Mixed sensitivity H∞ control for LTI systems with varying time delays

Designing a robust controller for a system with time-varying delays poses a major challenge. In this paper, we propose a method based on mixed sensitivity H ∞ for the control of linear time invariant (LTI) systems with varying time delays. The time delay is assumed bounded and the upper bound is known. In the technique we propose, the delay affecting the plant to be controlled is treated as an unmodeled uncertainty (in form of multiplicative uncertainty). That uncertainty is approximated and then an H ∞ based controller, for the plant represented by the multiplicative uncertainty and the nominal model, is calculated. The obtained H ∞ controller is used to control the LTI systems with varying time delays. Simulation examples are given to illustrate the effectiveness of the proposed method.

Relaxed LMI Conditions for the Design of Robust Static Output Feedback Controllers

Abstract In this paper, we propose new sufficient conditions in terms of linear matrix inequalities (LMIs) for the design of robust static output feedback (SOF) controllers for linear time-invariant systems. With this approach, it is possible to deal with the practical problem of not having all state variables available to be directly measured. Additionally, this strategy considers systems subject to the presence of uncertain parameters in its structure. The proposed LMIs were obtained using the Finsler’s Lemma. We also present the employment of two relaxation strategies to improve the feasibility region in the controller design. Moreover, this paper presents the results of a practical implementation in an active suspension system susceptible to a fault in the actuator control signal to evaluate the technique’s efficiency.