Quadratic Stability Research Articles

Output-feedback control design for Takagi-Sugeno fuzzy systems through Lyapunov functions depending polynomially on the states

This paper proposes a new strategy based on linear matrix inequalities (LMIs) for the synthesis of state- and output-feedback controllers through parallel distributed compensation for continuous-time Takagi-Sugeno fuzzy systems. The main novelty of the proposed design technique is the use of homogeneous polynomial Lyapunov functions of degree larger than two on the states, generalizing the results based on quadratic functions. Parametrized in terms of a constant matrix, those homogeneous polynomial Lyapunov functions provide less conservative results in terms of stability and decay rate of trajectories when dealing with membership functions whose time-derivatives have unknown bounds. The synthesis procedure is formulated in terms of a locally convergent iterative algorithm, where a set of LMIs defined in an augmented parameter space is solved at each iteration. Numerical examples are used to compare the proposed method with quadratic stabilizability techniques available in the literature, illustrating the advantages of higher degree Lyapunov functions.

A class of explicit second derivative general linear methods for non-stiff ODEs

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.

On the local quadratic stability of T–S fuzzy systems in the vicinity of the origin

The main goal of this paper is to introduce new local stability conditions for continuous-time Takagi-Sugeno (T-S) fuzzy systems. These stability conditions are based on linear matrix inequalities (LMIs) in combination with quadratic Lyapunov functions. Moreover, they integrate information on the membership functions at the origin and effectively leverage the linear structure of the underlying nonlinear system in the vicinity of the origin. As a result, the proposed conditions are proved to be less conservative compared to existing methods using fuzzy Lyapunov functions. Moreover, we establish that the proposed methods offer necessary and sufficient conditions for the local exponential stability of T-S fuzzy systems. Discussions on the inherent limitations associated with fuzzy Lyapunov approaches are also given. To illustrate the theoretical results, we provide comprehensive examples that demonstrate the core concepts and validate the efficacy of the proposed conditions.

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

A Frequency-Domain Criterion for the Quadratic Stability of Discrete-Time Systems with Switching between Three Linear Subsystems

Connected systems with switching between three linear discrete-time subsystems are considered, and a new frequency-domain criterion for the existence of a quadratic Lyapunov function ensuring the stability of such systems under arbitrary switching is proposed. The application of this criterion is demonstrated on an example of a third-order system.

Multi‐variable and multi‐objective gain‐scheduled control based on Youla‐Kucera parameterization: Application to autonomous vehicles

AbstractThis paper presents a Youla‐Kucera based interpolation between a set of Linear Parameter‐Varying (LPV) controllers, each one being a gain‐scheduled of Linear Time‐Invariant (LTI) controllers designed separately for different operating points. The gain‐scheduling is achieved based on Youla‐Kucera (YK) parameterization. A generalized LPV‐YK control structure is designed to interpolate between various LPV controllers. The closed‐loop system is proved to guarantee the quadratic stability for any continuous/discontinuous interpolating signals in terms of a set of Linear Matrix Inequalities (LMIs). The proposed method can help multi‐variable and multi‐objective systems to achieve high performances at different operating conditions and different critical situations regardless of the interpolation rate. A numerical example is simulated to show the importance of the proposed method to achieve different objectives for lateral control of autonomous vehicles. In addition, the approach has been tested on a real Renault ZOE vehicle to validate its real performance, and compare it with a standard polytopic LPV controller.

A Behavioral Approach to Data-Driven Control With Noisy Input–Output Data

This paper deals with data-driven stability analysis and feedback stabilization of linear input-output systems in auto-regressive (AR) form. We assume that noisy input-output data on a finite time-interval have been obtained from some unknown AR system. Data-based tests are then developed to analyse whether the unknown system is stable, or to verify whether a stabilizing dynamic feedback controller exists. If so, stabilizing controllers are computed using the data. In order to do this, we employ the behavioral approach to systems and control, meaning a departure from existing methods in data driven control. Our results heavily rely on a characterization of asymptotic stability of systems in AR form using the notion of quadratic difference form (QDF) as a natural framework for Lyapunov functions of autonomous AR systems. We introduce the concepts of informative data for quadratic stability and quadratic stabilization in the context of input-output AR systems and establish necessary and sufficient conditions for these properties to hold. In addition, this paper will build on results on quadratic matrix inequalities (QMIs) and a matrix version of Yakubovich's S-lemma.

Polytopic LPV modeling and gain scheduling [formula omitted] control of ball screw with position- and load-dependent variable dynamics

Ball screw drive (BSD) is a precision transmission mechanism widely used in high-precision positioning or tracking systems. The dynamic behavior of BSD varies with position and load, which causes tracking errors and poor robustness. Therefore, this paper proposes a polytopic linear parameter varying (LPV) model to express the varying dynamic behavior of BSD. The parameters of the LPV model are identified by the closed-loop frequency domain method and Levenberg–Marquardt iterative algorithm. Based on the polytopic LPV model, an LPV gain scheduling (GS) H∞ controller is proposed for the BSD with varying dynamics. Specifically, the controller is designed through polytope-based GS representation and mixed sensitivity synthesis. The most significant part is the proposal of a GS H∞ control algorithm to implement controller parameters that change with changing dynamics. Moreover, the stability of the closed-loop system is achieved by quadratic stabilization with state feedback. Finally, identification experiments and trajectory-tracking comparative experiments are carried out. The experimental results demonstrate that the proposed polytopic LPV modeling and GS H∞ control synthesis are effective in achieving accurate trajectory tracking and excellent robustness.

Quantized guaranteed cost dynamic output feedback control for uncertain nonlinear networked systems with external disturbance

In this article, the guaranteed cost control problem for a class of nonlinear networked control systems (NCSs) with dynamic quantization, external bounded disturbance, and data dropout is investigated via the dynamic output feedback. The Takagi–Sugeno (T-S) fuzzy model is used to represent the discrete-time nonlinear system discussed in this paper. Under the dynamic quantization strategy, two quantizers with dynamic parameters are used to quantize the measured output and control input, and the data dropout process is described by introducing the Bernoulli stochastic variable. By using the quadratic boundedness technique, the quadratic stability of the quantized NCS with external disturbance is described. Furthermore, the sufficient condition for the global design approach are proposed, that is, not only the dynamic output feedback controller but parameters of the quantizers are designed synchronously with strict linear matrix inequalities. Eventually, a simulation of nonlinear mass-spring-damper mechanical system is carried out to verify the effectiveness of the provided algorithm.

Parameter-Estimation-Based Gain-Scheduling Control of Weight on Bit in Drilling Process With Uncertain Penetration Resistance Coefficient.

It is inevitable to encounter through different formations in the drilling process for deep exploration, and the penetration resistance coefficient (PRC) is an uncertain parameter related to lithology. In this article, a parameter-estimation-based gain-scheduling controller is developed to eliminate undesired system performance deterioration due to the uncertain parameter. First, a drill-string axial finite element model with the uncertain PRC is established, and a control-oriented low-order model is derived via mode selection. A gain-scheduling controller is computed based on the quadratic stability condition of the closed-loop system, which can cope with the system's parameter uncertainty using variable low-frequency gain. An adaptive observer is designed to estimate the unmeasurable scheduling variable. Field data from a geothermal drilling well is obtained to validate our model. According to this drilling well scenario, both numerical and experimental results illustrate the effectiveness of our method. The explicit relationship between the controller gain and the uncertain parameter is presented. It is found that the performance of the closed-loop system is more sensitive to the controller gain when drilling in soft formations, requiring more attention in such scenarios.

Quadratic stability of uncertain large-scale networked systems

In this paper, the quadratic stability of continuous-time uncertain large-scale networked systems is considered. The systems are composed of many subsystems distributed in different locations in space and interconnected according to a certain network topology. A computationally efficient sufficient condition for the quadratic stability of large-scale networked systems is established, utilising effectively the block diagonal structure of the system parameter matrix and the sparseness of the subsystem connection matrix (SCM). Furthermore, a stability condition is presented, which is entirely contingent on the parameters of each subsystem. The simulation results show that the obtained conditions have obvious advantages in analysing the quadratic stability of large-scale networked systems.

DIGITAL STABILIZATION OF A SWITCHABLE LINEAR SYSTEM WITH COMMENSURATE DELAYS

An approach to the construction of a digital controller that stabilizes a non-continuous switchable linear system with commensurate delays in control is proposed. The approach to stabilization sequentially includes the construction of a switchable continuously discrete closed system with a digital controller, the transition to its discrete model, represented as a switchable system with modes of various orders and the construction of a discrete dynamic controller based on the quadratic stability condition of a closed switchable discrete system.

Ostensible Metzler Linear Uncertain Systems: Goals, LMI Synthesis, Constraints and Quadratic Stability

This paper deals with the design problem for a class of linear continuous systems with dynamics prescribed by the system matrix of an ostensible Metzler structure. The novelty of the proposed solution lies in the diagonal stabilization of the system, which uses the idea of decomposition of the ostensible Metzler matrix, preserving the incomplete positivity of the system during the synthesis. The proposed approach creates a unified framework that covers compactness of interval system parameter representation, Metzler parametric constraints, and quadratic stability. Combining these extensions, all of the conditions and constraints are expressed as linear matrix inequalities. Implications of the results, both for design and for research directions that follow from the proposed method, are discussed at the end of the paper. The efficiency of the method is illustrated by a numerical example.

Event-triggered non-fragile state estimator design for interval type-2 Takagi–Sugeno fuzzy systems with bounded disturbances

In this paper, the state estimator design problem of interval type-2 Takagi–Sugeno fuzzy systems suffering from bounded disturbances is studied. To enhance the resilience of the estimator, a non-fragile design scheme is proposed to tackle the estimator gain variations. Meanwhile, an event-triggered communication mechanism is introduced for relieving the transmission burden over networks. To settle down the non-fragile estimator design issue subject to bounded disturbances and event-induced error, we propose a new definition of quadratic boundedness via the multiple Lyapunov functions. Based on this definition, a novel co-design method of estimator and event generator for fuzzy system models in the presence of both measurable and immeasurable premise variables is presented. In virtue of quadratic boundedness framework, less conservative conditions of the existence and quadratic stability of the fuzzy estimators are obtained, and the upper bound of estimation error is given explicitly. The desired estimator gains are determined by convex optimization technique using slack matrices. Two illustrative examples are exploited to validate the availability and superiority of the addressed design approach.

Quadratic Stability of Switched a ne DAEs Via State-Dependent Switching Law

In this paper, we deal with quadratic stabilization for a particular class of impulse-free switched differential affine algebraic systems (switched affine DAEs) which consisted of a finite set of affine subsystems with the same algebraic constraint, where both subsystem matrices and affine vectors in the vector fields are switched independently and no single subsystem has desired quadratic stability. We show that if a convex combination of subsystem matrices such that the result LTI algebraic system is assymptotically stable, and another convex combination of affine vectors is zero, then we can design a state-dependent switching law such that the entire switched affine DAEs system is quadratically stable at the origin. But when the convex combination of affine vectors is not zero, we discuss the quadratic stabilization to a convergence set defined by the convex combination of subsystem matrices and the convex combination of affine vectors. &nbsp

Iterative Design Algorithm for Robust Disturbance-Rejection Control

An iterative design algorithm is developed for robust disturbance–rejection control of uncertain systems with time-varying parameter perturbations in this paper. For more design degrees of freedom, a generalized equivalent-input-disturbance estimator is adopted to approximate the effect of both disturbances and uncertainties. By the bound real lemma, the H∞ norm is used to evaluate the robust disturbance–rejection performance of the closed-loop uncertain system. To avoid the constraints introduced by the widely used commutative condition, the control gains are divided into two groups and calculated by steps. Further, two robust quadratic stability conditions are derived, and an iterative design algorithm is developed to optimize the robust H∞ disturbance–rejection performance. Finally, the effectiveness and advantages of the developed method are demonstrated by a case study of a suspension system of modern vehicles.

H2/H∞ Robust Observed-State Feedback Control Based on Slack LMI-LQR for LCL-Filtered Inverters

A robust controller for an LCL-filtered grid-connected inverter (GCI) is presented in this paper in the context of the system model uncertainties and non-ideal grid environment. Under the occurrence of exogenous inputs from parameter drifts affecting the controller and observer, the H₂/H∞ approach of the linear quadratic regulator (LQR) control is developed to obtain the state-feedback gain and observer gain with the system stability guarantee by means of Lyapunov stability theory. Furthermore, the use of slack-linear matrix inequalities (LMIs) to handle parameter-dependent Lyapunov functions in synthesizing the robust controller can achieve a less conservative condition than the conventional quadratic stability. System robustness and stability are assessed through the closed-loop pole map and the discrete-time frequency analysis approaches. Hardware experiments are conducted for the LCL-filtered GCI to validate the feasibility and effectiveness of the proposed control method. Fair comparisons with the conventional schemes are also performed to highlight the superior performance of the proposed scheme.

Data-driven quadratic stabilization and LQR control of LTI systems

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 H2 cost. The effectiveness of the proposed algorithm is illustrated through comparisons against existing data-driven methods that handle ℓ∞ bounded noise.

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.

Quadratic stability of flux limiters

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.