Robust control of discrete time uncertain dynamical systems

This paper investigates the problem of robust control for a class of systems with both Markovian jumping parameters and parametric uncertainty. The jumping parameters considered here form a continuous-time discrete-state homogeneous Markov process, and the parameter uncertainty appearing in the state equation is real, time-dependent, and normbounded. A method is proposed for the design of robust state-feedback controllers. Both the cases of finite and infinite horizon are addressed. We show that the problem of robust control for linear systems with Markovian jumping parameters can be solved in terms of the solutions to a set of either coupled Riccati differential equations, or algebraic Riccati equations.

H In this paper we address the problems of robust stability and robust perform2 ance for uncertain discrete time systems with nonlinear parametric uncertainties. We consider two families of systems with parametric uncertainties described by state-space models which offer a fairly general representation of most uncertain systems with one or two parameters (the approach can be extended to more parameters). For these two families we obtain explicit expressions for the Schur stability H radius and for the robust performance radius in the case of uncertainties with a 2 single parameter. Moreover, we provide a line search algorithm for these two problems in the case of two parameters. Both for the robust stability and the robust performance problem, explicit necessary and sufficient conditions are derived.

Control of Nonlinear Systems With Unknown Actuator Nonlinearities

The basic idea behind infinite horizon state feedback k-cost-cumulant (kCC) control is to seek a constant state feedback law which minimizes the value of a finite linear combination of the first k cost cumulants of a traditional integral quadratic cost associated with a linear stochastic system, when there is no specification of a large terminal time. The paper begins with the development of matrix algebraic equations for the cost cumulants. These equations permit the incorporation of classes of linear feedback controllers for linear dynamical systems defined on infinite horizon. The infinite horizon control problem with the optimization goal of a finite linear combination of the first k cost cumulants is then stated. Because the control optimization problem at hand involves matrix equality constraints, the optimal feedback solution is investigated by utilizing a Lagrange multipliers technique. The efficacy and applicability of this control paradigm, based upon the first three cost cumulants, are demonstrated on the second earthquake generation benchmark for response control of cable-stayed bridges (Caicedo, J.M., et al., 2003). The simulation results indicate that the cost cumulants in the infinite horizon case offer significant improvements in robust stability margin while keeping comparable levels of structural performance when comparing to those of the baseline LQG in the benchmark.

Currently, the main control strategy of the modular multi‐level converter (MMC) is a combination of vector control (VC) and proportional‐integral (PI) control, which allows decoupling active and reactive power control and maintaining the power balance within the high voltage direct current (HVDC) system. Although the existing PI control or conventional sliding mode control (SMC) shows a satisfactory control performance under steady state, its robust performance is poor when parameter uncertainty occurs, which greatly affects the stable operation of MMC. Therefore, a novel interval sliding mode control (ISMC) for MMC‐based HVDC systems is proposed in this paper, along with the stability criterion and design procedure. The proposed method can attenuate the influence of the parameter uncertainty, make the system robust and stable, and achieve both satisfactory transient and steady‐state performance under parameter uncertainty. The comparative simulation experiments of PI control, conventional SMC, and ISMC for MMC were conducted in PSCAD/EMTDC, demonstrating that the proposed ISMC shows better robustness and tracking performance than either PI control or SMC under different disturbance working conditions.

The traditional models of distributed computing systems employ the mathematical formalisms of discrete event dynamical systems along with Petri Nets. However, such models are complex to analyze and computationally expensive. Interestingly, the evolving distributed computing systems closely resemble the discrete dynamical systems. This paper constructs and analyzes the model of distributed computing systems by applying mathematical formalisms of the discrete dynamical systems. The proposed model embeds the partial ordering of states under happened-before relation in the set of globally observable states. The stability of the proposed model is analyzed using the first order linear approximation.

This textbook provides a broad introduction to continuous and discrete dynamical systems. With its hands-on approach, the text leads the reader from basic theory to recently published research material in nonlinear ordinary differential equations, nonlinear optics, multifractals, neural networks, and binary oscillator computing. Dynamical Systems with Applications Using Python takes advantage of Python’s extensive visualization, simulation, and algorithmic tools to study those topics in nonlinear dynamical systems through numerical algorithms and generated diagrams. After a tutorial introduction to Python, the first part of the book deals with continuous systems using differential equations, including both ordinary and delay differential equations. The second part of the book deals with discrete dynamical systems and progresses to the study of both continuous and discrete systems in contexts like chaos control and synchronization, neural networks, and binary oscillator computing. These later sections are useful reference material for undergraduate student projects. The book is rounded off with example coursework to challenge students’ programming abilities and Python-based exam questions. This book will appeal to advanced undergraduate and graduate students, applied mathematicians, engineers, and researchers in a range of disciplines, such as biology, chemistry, computing, economics, and physics. Since it provides a survey of dynamical systems, a familiarity with linear algebra, real and complex analysis, calculus, and ordinary differential equations is necessary, and knowledge of a programming language like C or Java is beneficial but not essential.

We describe the destabilizing (in the sense of a decrease in the reserve of mean-square asymptotic stability) effect of random parametric perturbations of the white-noise type in quasilinear continuous and discrete dynamical systems (Lur’e-Postnikov systems of automatic control with nonlinear feedback). We use stochastic Lyapunov functions in the form of linear combinations of the types “a quadratic form of phase coordinates plus the integral of a nonlinearity” (continuous systems) and “a quadratic form of phase coordinates plus the integral sum for a nonlinearity” (discrete systems) and the matrix algebraic Sylvester equations associated with stochastic Lyapunov functions of this form.

This paper shows how value sets and the zero exclusion principle can be used to obtain results on the robust stability and design of controllers against a special infinite dimensional l/sub 1/ norm bounded parametric uncertainty in both numerator and denominator of scalar discrete-time plants. This parametric uncertainty has properties of both parametric and unstructured uncertainty and allows standard H/sup /spl infin// design tools to be used without conservatism.

Shows how value sets and the zero exclusion principle can be used to obtain results on the robust stability and design of controllers against a special infinite-dimensional l/sub 1/ norm bounded parametric uncertainty in both numerator and denominator of scalar discrete-time plants. This parametric uncertainty has properties of both parametric and unstructured uncertainty and allows standard H/sup /spl infin// design tools to be used without conservatism.

With the rapid development of computer science, there are more and more kinds of discrete dynamic systems. Computer integrated system CIMS, network communication database administrator system, and human behavior analysis system are all discrete dynamic systems. At present, many researchers have studied by adding human behavior data to discrete dynamic systems. This paper aims to study the behavior data of English learners by using the discrete dynamic modeling technology of complex systems and the discrete dynamic system modeling method of Petri nets. By adding the behavior data of learners to the discrete dynamic system of fuzzy Petri nets, the system is diagnosed and optimized. The experimental results show that the complex discrete dynamic system in this paper has achieved good experimental results according to the performance indicators selected in theory. Based on the combination of the above technologies and systems, the fuzzy Petri net discrete dynamic system studied in this paper improves the processing speed of English learners' behavior data.

This paper deals with uncertain model structures, model validation and robust performance analysis of active magnetic bearings. The dynamics of active magnetic bearing systems are characterized by their instability and complex dynamics of rotor and electromagnets. The feedback control is indispensable to stabilize the system, further the closed-loop systems of magnetic bearings should have robustness for stability and performance against model uncertainties. First we derive a nominal mathematical model of AMBs as a linear state-space model under some assumption and idealization, then we consider the discrepancy between the real physical systems and the obtained nominal design model. This discrepancy can be expressed as the structured uncertainties by Linear Fractional Transformation. These uncertainties include linearization error, parametric uncertainties, unmodeled dynamics, and gyroscopic effect. Then we set the interconnection structure which contains the above structurally represented uncertainties. Next we design a robust controller which achieves robust performance condition. Finally, we validate the interconnection structure with the nominal model and uncertainties, and analyze the robustness of stability and performance of the closed-loop system via the mixed structured singular value.

The problem of robust stabilisation is considered for uncertain systems using linear static output feedback (SOF) controllers and linear dynamical output feedback (DOF) controllers with fixed order. The existence conditions of output feedback controllers are given by matrix inequalities and result in some nonlinear optimisation problems. A class of uncertain plants whose nominal systems are minimal phase with a full row rank can be robustly stabilised by an SOF controller and DOF controller with any order, and controllers can be obtained by solving some linear matrix inequalities (LMIs). The method is demonstrated by two numerical examples

The synthesis of an internal model control (IMC) that is equable to the inverse of the model is extremely important in order to ensure perfect set-point tracking. This inversion presents the fundamental problem of the IMC structure for multivariable (MIMO) under-actuated systems with parametric uncertainty. However, the proposed IMC strategy is extended in the case of multi-input multi-output systems such that the number of control inputs is less than the outputs one. In the controller design procedure, a new method is presented to add to the model matrix underactuation a number of transfer function in order to make it a square matrix. Furthermore, the internal model control structure is changed in order to eliminate the excess control inputs by using the usual arithmetic operators. The problem of stability of controller is solved by applying a linear matrix inequality (LMI) approach, which can be easily solved using numerical standard software such as MATLAB. Robust stability of final closed control loops is tested using the value set concept and zero exclusion condition. In a set of experiments, the controlled system is approximated by a MIMO matrix with parametric uncertainty, the controllers are designed, the robust stability is verified, and the final control responses are tested and evaluated. Simulation results have proved the effectiveness and reliability of the proposed method.

The robust stability and stabilization, and H-infinity control problems for discrete-time Markovian jump singular systems with parameter uncertainties are discussed. Based on the restricted system equivalent (r.s.e.) transformation and by introducing new state vectors, the singular system is transformed into a discrete-time Markovian jump standard linear system, and the linear matrix inequality (LMI) conditions for the discrete-time Markovian jump singular systems to be regular, causal, stochastically stable, and stochastically stable with γ-disturbance attenuation are obtained, respectively. With these conditions, the robust state feedback stochastic stabilization problem and H-infinity control problem are solved, and the LMI conditions are obtained. A numerical example illustrates the effectiveness of the method given in the paper.