Class Of Linear Dynamical Systems Research Articles

A Novel relaxation of the static output feedback problem for a class of plants

The static output feedback control problem is important, as it is concerned with the case when one cannot measure all state variables. It seeks to obtain a stabilising control under these conditions and is, often, difficult to solve. For a certain class of linear dynamical systems, we provide a novel approach to obtain a stabilising control gain matrix. The class of plants considered is of those, for which we can measure at least half of all state variables and where those measurements affect at least half of all state variables. Moreover, when we write the (linearly transformed) system matrix in block form, we require that either the two lower block matrices are nonsingular, or, if the inputs affect the measured state variables directly, at least the lower left one is nonsingular. We then show that we can determine the stabilising control gain matrix from the solution of a linear matrix inequality. Finally, we apply the approach to different benchmark problems, where it performs well, and confirm its good performance through additional numerical experiments.

APPLICATION OF SPLINE FUNCTIONS AND WALSH FUNCTIONS IN PROBLEMS OF PARAMETRIC IDENTIFICATION OF LINEAR NONSTATIONARY SYSTEMS

Context. In this article, a generalized parametric identification procedure for linear nonstationary systems is proposed, which uses spline functions and orthogonal expansion in a series according to the Walsh function system, which makes it possible to find estimates of the desired parameters by minimizing the integral quadratic criterion of discrepancy based on solving a system of linear algebraic equations for a wide class of linear dynamical systems. The accuracy of parameter estimation is ensured by constructing a spline with a given accuracy and choosing the number of terms of the Walsh series expansion when solving systems of linear algebraic equations by the A. N. Tikhonov regularization method. To improve the accuracy of the assessment, an algorithm for adaptive partitioning of the observation interval is proposed. The partitioning criterion is the weighted square of the discrepancy between the state variables of the control object and the state variables of the model. The choice of the number of terms of the expansion into the Walsh series is carried out on the basis of adaptive approximation of non-stationary parameters in the observation interval, based on the specified accuracy of their estimates. The quality of the management of objects with variable parameters is largely determined by the accuracy of the evaluation of their parameters. Hence, obtaining reliable information about the actual nature of parameter changes is undoubtedly an urgent task.
 Objective. Improving the accuracy of parameter estimation of a wide class of linear dynamical systems through the joint use of spline functions and Walsh functions.
 Method. A generalized parametric identification procedure for a wide class of linear dynamical systems is proposed. The choice of the number of terms of the expansion into the Walsh series is made on the basis of the proposed algorithm for adaptive partitioning of the observation interval.
 Results. The results of modeling of specific linear non-stationary systems confirm the effectiveness of using the proposed approaches to estimating non-stationary parameters.
 Conclusions. The joint use of spline functions and Walsh functions makes it possible, based on the proposed generalized parametric identification procedure, to obtain analytically estimated parameters, which is very convenient for subsequent use in the synthesis of optimal controls of real technical objects. This procedure is applicable to a wide class of linear dynamical systems with concentrated and distributed parameters.

Multiple Faults Estimation in Dynamical Systems: Tractable Design and Performance Bounds

In this article, we propose a tractable nonlinear fault estimation filter along with explicit performance bounds for a class of linear dynamical systems in the presence of both additive and nonlinear multiplicative faults. We consider the case, where both faults may occur simultaneously and through an identical dynamical relationship, a setting that is relevant to several application domains, including automotive driving, aviation, and chemical plants. The proposed filter architecture combines tools from model-based approaches in the control literature and regression techniques from machine learning. To this end, we view the regression operator through a system-theoretic perspective to develop operator bounds that are then utilized to derive performance bounds for the proposed estimation filter. In the case of constant, simultaneously, and identically acting additive and multiplicative faults, it can be shown that the estimation error converges to zero with an exponential rate. The performance of the proposed estimation filter in the presence of incipient faults is validated through an application on the lateral safety systems of SAE level 4 automated vehicles. The numerical results show that the theoretical bounds of this study are indeed close to the actual estimation error.

Connection of Signed and Unsigned Networks Based on Solving Linear Dynamic Systems

In signed networks, the cooperation and antagonism cause great difficulties for their behavior analysis, especially, when they are subject to time-varying topologies. This is different from unsigned networks involving only cooperations, of which behavior analysis can be feasibly achieved based on the nonnegative matrix theory. With these facts, this article first bridges a relation between signed and unsigned networks and then takes advantage of the relation for the behavior analysis of signed networks under directed switching topologies. In particular, a solution is provided for the connection of signed and unsigned networks via solving a class of linear dynamic systems, which is obtained by separating antagonisms from cooperations. The solution makes it possible to employ the convergence results for unsigned networks to address the convergence issues for signed networks. If the joint spanning tree condition is met, then switching signed networks can achieve the quasi-interval bipartite consensus. Moreover, the established results can be applied to signed networks with both continuous-time and discrete-time dynamics.

ADEQUACY EVALUATION OF THE SYNTHESIS OF MATHEMATICAL MODELS FOR ONE CLASS OF LINEAR DYNAMIC SYSTEMS WITH PARAMETERS, VARIABLE IN TIME AND IN SPACE

The object of research: the authors of this work have synthesized in previous publications a mathematical model of one class of linear dynamic systems with variable parameters in time and space, which includes the system for lifting and lowering a load with a crane in the case of taking into account the influence of longitudinal vibrations that arise in the cable of the lifting mechanism, and shift in time of the moment of inertia of the rotating masses reduced to the motor shaft. Investigated problem: simulation models are synthesized in the Simulink graphical software environment of the MATLAB application package to assess the adequacy of the specified mathematical model. A calculation was carried out using the passport data and characteristics of all components of the crane structure to find the numerical values of the parameters of simulation models, including the electric motor, cable and gearbox. The main scientific results: the results of modeling the dynamics of the electric drive system during acceleration, deceleration, the prevailing velocity are given. By comparing the results obtained on the simulation model with the results of experimental studies of the same class of dynamical systems carried out by other authors, the adequacy of the mathematical models synthesized by the authors of this work is proved. It is proved that the use of the classical equation of the dynamics of the electric drive system, which contains a constant moment of inertia and a constant moment of load, leads to errors in the estimates of the processes accompanying the lifting and lowering of loads by cranes. And this, in turn, does not allow predicting the behavior of the system in various operating modes with sufficient accuracy. The area of practical use of the research results: enterprises of the machine-building industry, specializing in the production, modernization of lifting equipment. Innovative technological product: a simulation model of a crane mechanism, which makes it possible to take into account the influence of longitudinal vibrations that arise in the cable of the hoisting mechanism, and the shift in time of the moment of inertia of the rotating masses reduced to the motor shaft. Scope of application of the innovative technological product: design and development of control systems for electric drives of lifting equipment.

Finite-time Peak-to-peak Gain Minimization

The finite-time peak-to-peak filtering problem is studied for a class of linear dynamic systems. By reconstructing the system, the dynamic filtering error system is obtained. Our aim is to design a peak-to-peak filter such that the induced L∞ gain from the unknown disturbance to the estimated errors is minimized with respect to the finite-time interval. By using a proper Lyapunov function, sufficient conditions are established on the existence of peak-to-peak filter which also guarantees the finite-time boundedness of the filtering error dynamic systems. The design criteria are presented in the form of linear matrix inequalities and then described as an optimization problem. Simulation results are given to illustrate the validity of the proposed approaches.

ADAPTIVE NEURAL NETWORK PID CONTROLLER

The problem of constructing an adaptive PID controller based on the Hopfield neural network for a linear dynamic plant of the second order is considered. A description of the plant in the form of a discrete transfer function is used, the coefficients of which are determined with the help of a neural network that minimizes the discrepancy between the outputs of the plant and the model. The neural network processes the current and delayed input and output signals of the plant, forming an output for estimating the coefficients of the model. Another neural network determines the PID regulator coefficients at which the dynamics of the system approach the dynamics of the reference process. The calculation of weights and displacements of neurons in Hopfield networks used for identification and control is based on the construction of Lyapunov functions. The proposed methodology can be used to organize adaptive control of a wide class of linear dynamic systems with variable parameters. The results of the simulation in the article show the effectiveness of the proposed method.

Universality in the coarse-grained fluctuations for a class of linear dynamical systems

The overdamped dynamics of linearly coupled elements are studied in the stochastic force-field of a reservoir. The stable equilibrium state is fixed by the reservoir, and the stochastic force field induces fluctuations around the stable state. The dynamics of the system is investigated in the case when the stochastic force-field is a multivariate Gaussian noise. For a class of practically relevant problems we prove that the coarse-grained properties of the fluctuations are not influenced by the interactions among the elements. Compact results are given for such coarse-grained quantities and an application for the price fluctuation of an economic product is discussed.

Метод ідентифікації математичної моделі одного класу лінійних динамічних систем зі змінними параметрами

The article presents a method for identifying a mathematical model of a single class of linear dynamic systems with variable parameters, which includes a traction electric drive of a crane with variable parameters, caused by the change of the equivalent mass moment of inertia and the load moment of the traction motor when lifting (lowering) the load. The article also proposes and justifies the calculated relations to determine changes of the equivalent mass moment of inertia of the electric drive, caused by winding and unwinding of the cable, to which the load is suspended, on the reel connected through the reduction drive to the motor shaft. It also proposes and justifies calculated relations to determine changes of the load moment due to shortening and lengthening of the cable to which the load is suspended because of lifting or lowering the load. The article presents a step-by-step algorithm for determining all the parameters of a mathematical model of the dynamics of a crane electric drive with the time-varying moment of inertia of rotating masses and the load moment. The article presents the results of modeling the dynamics of the electric drive system during acceleration, braking, steady-state speed, idling and reverse of the traction motor considering variable parameters and using a mathematical model identified by specific passport characteristics of the traction motor, reduction drive and cable in the graphical programming environment Simulink of MATLAB application program package. The article also presents a comparative analysis of the obtained results and shows that the use of the classical equation of dynamics of an electric drive system with the constant moment of inertia and load moment leads to significant errors. And this is even despite the smooth speed control using frequency regulators for asynchronous motors and using power transistors for DC motors to change the voltage applied to the motor armature.

Positive invariant set estimation of adaptive controller in the presence of input saturation

SUMMARYFor a class of linear dynamical systems with constant unknown parameters, an adaptive control scheme is developed that provides stable adaptation in the presence of input magnitude constraints. Whereas for open‐loop stable systems the results are global, for open‐loop unstable systems, the problem of nonconservative estimation of the nonempty positive invariant set is cast into an LMI framework, which can be efficiently solved numerically via convex optimization. To achieve this, a standard result toward invariant set characterization is appropriately extended to accommodate bounded disturbance and model uncertainties. In addition to closed‐loop stability, performance bounds of the adaptive closed‐loop system are analyzed, and the degradation due to the possible control deficiency is quantified. Simulation examples of aerospace applications are included to illustrate the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.

Stable adaptation in the presence of input constraints

For a class of linear dynamical systems with unknown parameters a direct model reference adaptive control framework is developed that provides stable adaptation in the presence of input constraints. The proposed design methodology, termed “positive μ -modification”, has the ability to protect the control law from actuator position saturation. The reference system is adaptively modified to ensure global asymptotic tracking for open-loop stable systems. For unstable systems an estimate for the domain of attraction is derived based on the input saturation magnitude and system parameters. Simulation of a benchmark example verifies the theoretical statements.

Modeling Sensorimotor Learning with Linear Dynamical Systems

Recent studies have employed simple linear dynamical systems to model trial-by-trial dynamics in various sensorimotor learning tasks. Here we explore the theoretical and practical considerations that arise when employing the general class of linear dynamical systems (LDS) as a model for sensorimotor learning. In this framework, the state of the system is a set of parameters that define the current sensorimotor transformation-the function that maps sensory inputs to motor outputs. The class of LDS models provides a first-order approximation for any Markovian (state-dependent) learning rule that specifies the changes in the sensorimotor transformation that result from sensory feedback on each movement. We show that modeling the trial-by-trial dynamics of learning provides a substantially enhanced picture of the process of adaptation compared to measurements of the steady state of adaptation derived from more traditional blocked-exposure experiments. Specifically, these models can be used to quantify sensory and performance biases, the extent to which learned changes in the sensorimotor transformation decay over time, and the portion of motor variability due to either learning or performance variability. We show that previous attempts to fit such models with linear regression have not generally yielded consistent parameter estimates. Instead, we present an expectation-maximization algorithm for fitting LDS models to experimental data and describe the difficulties inherent in estimating the parameters associated with feedback-driven learning. Finally, we demonstrate the application of these methods in a simple sensorimotor learning experiment: adaptation to shifted visual feedback during reaching.

Modeling Sensorimotor Learning with Linear Dynamical Systems

Recent studies have employed simple linear dynamical systems to model trial-by-trial dynamics in various sensorimotor learning tasks. Here we explore the theoretical and practical considerations that arise when employing the general class of linear dynamical systems (LDS) as a model for sensorimotor learning. In this framework, the state of the system is a set of parameters that define the current sensorimotor transformation— the function that maps sensory inputs to motor outputs. The class of LDS models provides a first-order approximation for any Markovian (state-dependent) learning rule that specifies the changes in the sensorimotor transformation that result from sensory feedback on each movement. We show that modeling the trial-by-trial dynamics of learning provides a sub-stantially enhanced picture of the process of adaptation compared to measurements of the steady state of adaptation derived from more traditional blocked-exposure experiments. Specifically, these models can be used to quantify sensory and performance biases, the extent to which learned changes in the sensorimotor transformation decay over time, and the portion of motor variability due to either learning or performance variability. We show that previous attempts to fit such models with linear regression have not generally yielded consistent parameter estimates. Instead, we present an expectation-maximization algorithm for fitting LDS models to experimental data and describe the difficulties inherent in estimating the parameters associated with feedback-driven learning. Finally, we demonstrate the application of these methods in a simple sensorimotor learning experiment: adaptation to shifted visual feedback during reaching.

Optimal control for a class of systems subjected to disturbances

A special class of linear dynamical systems acted upon by controls and bounded disturbances is considered. It is assumed that the disturbances are due to an error in the control implementation; there are no disturbances in the case of the zero control, but the range of the disturbances increases with the control intensity. The problem of constructing a control which provides a minimax to a specified optimality criterion for an arbitrary permissible form of the disturbances is formulated. The solution of the minimax problem is reduced to the solution of transcendental equations. Under certain conditions the solution is obtained in explicit form. An example is considered in which the optimal control is constructed both in the form of a program and in the form of a synthesis.

CHAOS FOR BACKWARD SHIFT OPERATORS

Backward shift operators provide a general class of linear dynamical systems on infinite dimensional spaces. Despite linearity, chaos is a phenomenon that occurs within this context. In this paper we give characterizations for chaos in the sense of Auslander and Yorke [1980] and in the sense of Devaney [1989] of weighted backward shift operators and perturbations of the identity by backward shifts on a wide class of sequence spaces. We cover and unify a rich variety of known examples in different branches of applied mathematics. Moreover, we give new examples of chaotic backward shift operators. In particular we prove that the differential operator I + D is Auslander–Yorke chaotic on the most usual spaces of analytic functions.

Learnability and Adaptability from the Viewpoint of Passivity Analysis

Abstract This paper attempts to give a mathematical and physical interpretation of practice-based learning (so-called ‘learning control’) from the passivity viewpoint for a class of linear dynamical systems with passivity and general nonlinear differential equations of robotic motion. It is shown from an axiomatic argument that the passivity of a pair of input and output plays a crucial role in the ability of learning. More precisely in case of robot dynamics it is shown that the passivity between a residual input torque vector and an output as a linear combination of the angular velocity error and the saturated joint angle error enables trajectory tracking errors of the robot motion to converge to zero with repeating practices. For a class of robot tasks when the endpoint is holonomically constrained on a surface, the problem of convergence of residual position and force trajectory tracking errors is also discussed under the joint-space orthogonalization principle.

Design of Memoryless Controllers for Robust Tracking and Model Following of Uncertain Systems with Multiple Time Delays

The problem of robust tracking and model following is considered for a class of linear dynamical systems with multiple delayed state perturbations, time–varying uncertain parameters, and disturbance. A class of continuous memoryless state feedback controllers is proposed for robust tracking of dynamical signals. The proposed robust tracking controllers can guarantee that the tracking error decreases asymptotically to zero in the presence of multiple delayed state perturbations, time–varying uncertain parameters, and disturbance. A procedure for designing such a class of zero–error tracking state feedback controllers is also introduced. Finally, a numerical example is given to demonstrate the validity of the results.

Robust Tracking and Model Following Control with Zero Tracking Error for Uncertain Dynamical Systems

The problem of robust tracking and model following for a class of linear dynamical systems with time-varying uncertain parameters and disturbances is considered. A class of continuous (nonlinear) state feedback controllers is proposed for robust tracking of dynamical signals. The proposed robust tracking controllers can guarantee that the tracking error decreases asymptotically to zero in the presence of uncertain parameters and disturbances. A procedure for designing such a zero tracking state feedback controller is also introduced. Finally, a numerical example is given to demonstrate the validity of the results.

Description of a set of solutions of a dissipation inequality for third-order relaxation systems

For the class of linear dynamic systems indicated in the title, a variety of the third degree is determined and it is proved that the set of solutions of a dissipation inequality is characterized by a limited convex component of this variety. Singular varieties are determined, and the relationship between them and the maximum and minimum elements of the set of solutions for the dissipation inequality is established. The results obtained can find application in the theory of linear viscoelasticity and the theory of generalized thermodynamic systems with memory.

Iterative learning of impedance control from the viewpoint of passivity

This paper proposes an iterative learning control scheme for impedance control of robotic tasks when the tool endpoint covered by soft and deformable material presses a rigid object or environment at a prescribed periodic force pattern. To this end, an iterative learning control scheme for a class of linear dynamical systems with a negative feedback structure is analysed and convergence of the proposed learning update law after a sufficient number of repetitions is proved. It is shown that this convergence realizes impedance matching in a sense of electric circuit theory if the feedback system can be expressed as a lumped-parameter electric circuit. The iterative learning control scheme is then applied for a case of impedance control of robotic tasks when the characteristics of reproducing force of the deformable material is nonlinear in its displacement and unknown and the tool mass is uncertain. Simulation results are also presented, which show effectiveness of the proposed learning control scheme. Copyright © 2000 John Wiley & Sons, Ltd.