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In this paper, we construct explicit second derivative general linear methods (SGLMs) with quadratic stability and a large region of absolute stability for the numerical solution of non-stiff ODEs. The methods are constructed in two different cases: SGLMs with p = q = r = s and SGLMs with p = q and r = s = 2 in which p, q, r and s are respectively the order, stage order, the number of external stages and the number of internal stages. Examples of the methods up to order five are given. The efficiency of the constructed methods is illustrated by applying them to some well-known non-stiff problems and comparing the obtained results with those of general linear methods of the same order and stage order.

We propose a novel approach to study the quadratic stability of 2D flux limiters for non expansive transport equations. The theory is developed for the constant coefficient case on a cartesian grid. The convergence of the fully discrete nonlinear scheme is established in 2D with a rate not less than O(Δx½) in quadratic norm. It is a way to bypass the Goodman–Leveque obstruction Theorem. A new nonlinear scheme with corner correction is proposed. The scheme is formally second-order accurate away from characteristics points, satisfies the maximum principle and is proved to be convergent in quadratic norm. It is tested on simple numerical problems. [ABSTRACT FROM AUTHOR]

This paper proposes a nonlinear parameter varying (NLPV) observer to estimate, in real‐time, the damper force of an electrorheological (ER) damper in a road vehicle suspension system. The objectives are to minimize the effects of bounded unknown road profile disturbances and measurement noises on the estimation errors in the ℋ∞ framework. Furthermore, the nonlinearity coming from the damper model (and considered in the observer formulation) is handled through a Lipschitz condition. The observer inputs are data given by two low‐cost sensors (two accelerometers data from the sprung mass and the unsprung mass). For performance assessment, the observer is implemented on a 1/5‐scaled real vehicle testbench of GIPSA‐lab, namely INOVE (see www.gipsa‐lab.fr/projet/inove). Both simulation and experimental results demonstrate the effectiveness of the proposed observer in terms of the ability to estimate the damper force in real‐time and to minimize the effects of measurement noises and road disturbances. [ABSTRACT FROM AUTHOR]

This article proposes a black-box behavioral modeling framework for analog circuit blocks (CBs) operating under small-signal conditions around nonstationary operating points. Such variations may be induced either by changes in the loading conditions or by event-driven updates of the operating point for system performance optimization, e.g., to reduce power consumption. An extension of existing data-driven parameterized reduced-order modeling techniques is proposed, which considers the time-varying bias components of the port signals as nonstationary parameters. These components are extracted at runtime by a low-pass filter and used to instantaneously update the matrices of the reduced-order state-space model realized as a SPICE netlist. Our main result is a formal proof of quadratic stability of such linear parameter varying (LPV) models, enabled by imposing a specific model structure and representing the transfer function in a basis of positive functions whose elements constitute a partition of unity. The proposed quadratic stability conditions are easily enforced through a finite set of small-size linear matrix inequalities (LMIs), used as constraints during model construction. Numerical results on various CBs, including voltage regulators, confirm that our approach not only ensures the model stability but also provides speedup in runtime up to two orders of magnitude with respect to full transistor-level circuits. [ABSTRACT FROM AUTHOR]

The active suspension system of a vehicle manipulated using electro-hydraulic actuators is a challenging nonlinear control problem. In this research work, a novel Linear Parameter Varying (LPV) State-Space (SS) model with a fictional input is proposed to represent a nonlinear half-car active suspension system. Four different scheduling parameters are used to embed the nonlinearities of both the suspension and the electro hydraulic actuators to represent its nonlinear behavior. A recursive least squares (RLS) algorithm is used to predict the future behavior of the scheduling parameters along the prediction horizon. A Model Predictive Control-Linear Quadratic Regulator (MPC-LQR) is implemented as the control strategy and, to ensure stability, Quadratic Stability conditions are imposed as Linear Matrix Inequalities (LMI) constraints. Furthermore, the inclusion of attraction sets to overcome the conservative performance imposed by the Quadratic Stability conditions is included, as well as a terminal set were the switching between the MPC and the LQR controller is made. Simulations results for the half-car active suspension model over a typical road disturbance are tested to show the effectiveness of the proposed MPC-LQR-LPV controller with quadratic stability conditions in terms of comfort and road-holding.

Construction of strong stability preserving second derivative diagonally implicit multistage integration methods (SDIMSIMs) as a subclass of second derivative general linear methods (SGLMs) with and without quadratic stability (QS) has been investigated. Examples of such methods with p = q = r = s = 2 , 3 and 4 have been constructed, where p is the order, q is the stage order, r is the number of external stages, and s is the number of internal stages of the method. To show the efficiency of the proposed methods together with verification of the order of convergence and capability of these methods in preserving some nonlinear stability properties such as total variation and positivity, several one dimensional numerical experiments are performed. [ABSTRACT FROM AUTHOR]

This paper presents the functional observer-based Takagi–Sugeno fuzzy controller to enhance the dynamic response of the synchronous generator infinite bus power system under low-frequency oscillations. The unmeasurable states of the synchronous generator are estimated by using minimum order functional observer. The T–S fuzzy controller rules are a function of the estimated states in which the functional observer based on T–S fuzzy conditions presented in rank equality form. The Lyapunov theory in the form of linear matrix inequalities (LMIs) represented here in this paper to synthesis the functional observer stability. Small disturbances are taken to simulate the synchronous generator oscillations. The results clearly show that our scheme is a better response when compared with the full observer-based T–S fuzzy system. [ABSTRACT FROM AUTHOR]

The control of an automotive suspension system by means of a hydraulic actuator is a complex nonlinear control problem. In this work, a linear parameter varying (LPV) model is proposed to reduce the complexity of the system while preserving the nonlinear behavior. In terms of control, a dual controller consisting of a model predictive control (MPC) and a Linear Quadratic Regulator (LQR) is implemented. To ensure stability, quadratic stability conditions are imposed in terms of Linear Matrix Inequalities (LMI). Simulation results for quarter-car model over several disturbances are tested in both frequency and time domain to show the effectiveness of the proposed algorithm.

In this paper we study a novel approach to the design of a robust switched controller for continuous-time systems described by a novel robust plant model using quadratic stability and multi parameter dependent quadratic stability approaches. In the proposed design procedure with an output feedback a novel quadratic cost function is proposed which allows to obtain different performance dependence on the working points. Finally a numerical examples are investigated.

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system.

We study a networked control problem of uncertain systems based on quantized signals sent over unreliable, lossy communication channels. The coarsest quantization is characterized for attaining the objective of quadratic stability of the overall closed-loop system in a stochastic sense. Our result indicates that the coarsest quantization is given by the logarithmic type and that more information is required through finer quantization for more uncertain systems and more unreliable channels. The characterization provides an analytic upper bound for the coarseness, which is tight for some special cases and generalizes known results for systems without uncertainties and for channels without losses. [ABSTRACT FROM AUTHOR]

The problem of robust output regulation is studied for a class of parameter uncertain systems under unity output feedback control. The objective is tracking of the desired reference trajectory in the presence of the disturbance, both generated by a common exosystem. Because of the anti‐stability of the exosystem and potentially unbounded reference trajectory, the output tracking error is employed as the measurement signal and used as the input to the feedback controller. The method of p ‐copy of the internal model is utilised to augment the plant dynamics. Assuming that the output regulation condition is satisfied for all the parameter uncertainties, it is shown that the problem of robust output regulation is equivalent to the problem of robust output stabilisation. Furthermore, for quadratically bounded parameter uncertainties, an application of the notion of the quadratic stability leads to H∞ ‐based robust control, and the maximum allowable uncertainty bound can be computed, below which the robust output regulation can be achieved. [ABSTRACT FROM AUTHOR]

This paper deals with the design and analysis of robust model reference control systems for a class of linear parameter‐varying (LPV) systems with disturbances. Two types of robust model reference control systems are proposed to achieve a prescribed tracking performance. The first type is the LPV H∞ state‐feedback model reference control system and the second type is the LPV H∞ output‐feedback model reference control system. Based on the notion of quadratic stability, necessary and sufficient conditions for the existence of LPV reference models are given. Also, a parameterisation of LPV reference models is derived in this study. A practical example of a single‐link flexible‐joint robot is given to illustrate the design steps and to show the effectiveness of the proposed LPV H∞ model reference controllers. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to ensure a tracking control of output state especially the active and reactive powers of doubly fed induction generator (DFIG) by using the approach of Parallel Distributed Compensation (PDC) of the fuzzy control type Takagie-Sugeno (T-S) which determines the control laws containing fuzzy tracking and return state. This tracking has been built to converge the output vectors of the DFIG to a desired state. In this work, the quadratic function of lyaponov and a linear matrix inequality (LMI) are used to obtain the gains of the tracking control and the controller. The simulations results of law control based on the tracking and controller gains allow the active and reactive powers obtained must follow the reference proposed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a framework to solve the data-driven quadratic stabilization (DDQS) and the data-driven linear quadratic regulator (DDLQR) problems for both continuous and discrete-time systems. Given noisy input/state measurements and a few priors, we aim to find a state feedback controller guaranteed to quadratically stabilize all systems compatible with the a-priori information and the experimental data. In principle, finding such a controller is a non-convex robust optimization problem. Our main result shows that, by exploiting duality, the problem can be recast into a convex, albeit infinite-dimensional, functional Linear Program. To address the computational complexity entailed in solving this problem, we show that a sequence of increasingly tight finite dimensional semi-definite relaxations can be obtained using sum-of-squares and Putinar's Positivstellensatz arguments. Finally, we show that these arguments can also be used to find controllers that minimize a worst-case (over all plants in the consistency set) closed-loop H 2 cost. The effectiveness of the proposed algorithm is illustrated through comparisons against existing data-driven methods that handle ℓ ∞ bounded noise. [ABSTRACT FROM AUTHOR]

Our paper deals with robust gain-scheduled controller design for uncertain LPV systems which ensures closed-loop stability and guaranteed cost for all scheduled parameter changes. The novel procedure is based on LPV paradigm, Lyapunov theory of stability and guaranteed cost from LQ theory. The feasible design procedures are obtained in the form of BMI or LMI. The class of control structure includes centralized or decentralized fixed order output feedbacks like PI controller. Numerical examples illustrate the effectiveness of the proposed approach.

Stability and stabilisation problems for fractional‐order linear (FOL) uncertain systems with fractional order α(0<α<1) are difficult and interesting. This study focuses on observer‐based robust control for FOL uncertain system with fractional order 0<α<1. Recently, a new method based on linear matrix inequality is presented to solve the stability and stabilisation problems of FOL systems with fractional order 0<α<1. Based on the new method and the matrix's singular value decomposition, a necessary and sufficient condition for quadratic stability of observer‐based FOL uncertain systems is presented in this study. Finally, two examples are given to illustrate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

Theory of general linear methods (GLMs) for the numerical solution of autonomous system of ordinary differential equations of the form y ′ = f ( y ) is extended to include the second derivative y ″ = g ( y ) : = f ′ ( y ) f ( y ) . This extension of GLMs is called second derivative general linear methods (SGLMs). In this paper we will construct two-stage A - and L -stable SGLMs of order p and stage order q = p up to six with low error constants. We will show the efficiency of the proposed methods by implementing on some well-known stiff problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a novel approach to the design of an output feedback switched controller with an arbitrary switching algorithm for continuous-time invariant systems that is described by a novel plant model as a gain-scheduled plant using the multiple quadratic stability and quadratic stability approaches. In the proposed design procedure, there is no need to use the notion of the "dwell time". The obtained results are in the form of bilinear matrix inequalities (BMI). Numerical examples show that, in the proposed method, the design procedure is less conservative and gives more possibilities than that described in the papers published previously. [ABSTRACT FROM AUTHOR]