Linear Matrix Inequality Feasibility Problem Research Articles

T‐S fuzzy observer and actuator fault‐tolerant controller designs for descriptor systems

SummaryThis work focuses on designing an observer to measure states and actuator faults as well as fault‐tolerant control (FTC) for fuzzy Takagi–Sugeno (T–S) singular systems. An enhanced system is built by creating a new vector from the system state and the actuator faults. The first task is simultaneously to estimate system states and actuator faults with an adaptive observer. Then, in order to get a stabilized faulty singular system, we conceive an observer based on FTC. Using Lyapunov's approach, a linear matrix inequalities (LMIs) feasibility problem is formulated to provide the controller and the observer gains. Compared with previous studies, the proposed method significantly simplifies the design and simultaneously provides the controller and the observer gains in one step. Finally, three examples are given to demonstrate the effectiveness of the proposed control.

Formal Synthesis of Neural Barrier Certificates for Continuous Systems via Counterexample Guided Learning

This paper presents a novel approach to safety verification based on neural barrier certificates synthesis for continuous dynamical systems. We construct the synthesis framework as an inductive loop between a Learner and a Verifier based on barrier certificate learning and counterexample guidance. Compared with the counterexample-guided verification method based on the SMT solver, we design and learn neural barrier functions with special structure, and use the special form to convert the counterexample generation into a polynomial optimization problem for obtaining the optimal counterexample. In the verification phase, the task of identifying the real barrier certificate can be tackled by solving the Linear Matrix Inequalities (LMI) feasibility problem, which is efficient and makes the proposed method formally sound. The experimental results demonstrate that our approach is more effective and practical than the traditional SOS-based barrier certificates synthesis and the state-of-the-art neural barrier certificates learning approach.

Flight and Vibration Control of Flexible Air-Breathing Hypersonic Vehicles Under Actuator Faults.

The issue of modeling and fault-tolerant control (FTC) design for a class of flexible air-breathing hypersonic vehicles (FAHVs) with actuator faults is investigated in this article. Different from previous research, the shear deformation of the fuselage is considered, and an ordinary differential equations-partial differential equations (ODEs-PDEs) coupled model is established for the FAHVs. A feedback control is proposed to ensure flight stable and an adaptive FTC method is designed to deal with actuator faults while suppressing the system's vibrations. Besides, the stability analysis of the closed-loop system is given via the Lyapunov direct method and an algorithm that transfers the bilinear matrix inequalities (BMIs) feasibility problem to the linear matrix inequalities (LMIs) feasibility problem is provided for determining the control gains. Finally, the numerical simulation results show that the proposed controller can stabilize the flight states and suppresses the vibration of the fuselage efficiently.

Counter-Example Guided Inductive Synthesis of Control Lyapunov Functions for Uncertain Systems

We propose a counter-example guided inductive synthesis (CEGIS) scheme for the design of control Lyapunov functions and associated state-feedback controllers for linear systems affected by parametric uncertainty with arbitrary shape. In the CEGIS framework, a learner iteratively proposes a candidate control Lyapunov function and a tailored controller by solving a linear matrix inequality (LMI) feasibility problem, while a verifier either falsifies the current candidate by producing a counter-example to be considered at the next iteration, or it certifies that the tentative control Lyapunov function actually enjoys such feature. We investigate the Lipschitz continuity of the objective function of the global optimization problem solved by the verifier, which is key to establish the convergence of our method in a finite number of iterations. Numerical simulations confirm the effectiveness of the proposed approach.

Formula omitted]-gain filtering of semi-Markov jump systems subjected to persistent bounded disturbances

In this work, the problem of L∞-gain filtering of semi-Markov jump systems (S-MJSs) subjected to persistent bounded disturbances is investigated. We first provide a sufficient condition for input-to-state stability (ISS) of the systems under consideration. The analytical result of ISS is then recast as a linear matrix inequality (LMI) feasibility problem by using the S-procedure. Based on the ISS analysis, a novel L∞-gain filter design scheme is developed. The resulting filter ensures the trajectories of the estimation error at steady state to be bounded in the mean sense and makes the peak of the estimation error as small as possible under the effect of the external disturbances. Simulation results demonstrate the effectiveness of our proposed design scheme.

On PID Design Constraints in Relation to Control of Strictly Metzler Linear MIMO Systems

For time-invariant Metzler linear MIMO systems, this paper proposes an original approach reflecting necessary matching conditions, specifically structural system constraints and necessary positiveness in solving the problem of MIMO PID control. Covering the matching conditions by the supporting structure of measurement, refining the controller and system parameter constraints and introducing enhanced equivalent system descriptions, the reformulated design task is consistent with PID control law parameter representation and is formulated as a linear matrix inequality feasibility problem. Characterization of the PID control law parameters is permitted to highlight dynamical properties of the closed-loop system and the structural influence of the control derivative gain value in the design step. For the first time, the paper comprehensively sets the synthesis standard for PID control of MIMO Metzler systems because others for the given task have not been created at present.

Formula omitted] filtering for nonlinearly coupled complex networks subjected to unknown varying delays and multiple fading measurements

In this paper, the robust filtering problem for uncertain complex networks with time-varying state delay and stochastic nonlinear coupling based on H∞ performance criterion is studied. The random connections of coupling nodes are represented by utilizing independent random variables and the multiple fading measurements phenomenon is characterized by introducing diagonal matrices with independent stochastic elements. Moreover, the probabilistic time-varying delays in the measurement outputs are described by white sequences with the Bernoulli distributions. Furthermore, All system’s matrices are supposed to have uncertainty and a quadratic bound is assumed for nonlinear part of the network. This bound can be obtained by solving a sum of squares (SOS) optimization problem. By applying the Lyapunov theory, we design a robust filter for each node of the network so that the filtering error system is asymptomatically stable and the H∞ performances are met. Then, the parameters of the filters are achieved by solving a linear matrix inequality (LMI) feasibility problem. Finally, the applicability and performance of the proposed H∞ filtering approach are demonstrated via a practical example.

PID Control Design for SISO Strictly Metzlerian Linear Systems

For linear time-invariant Metzlerian systems, this paper proposes an original approach reflecting specific structural system constraints and positiveness in solving the problem of PID control. Refining parameter constraints and introducing enhanced equivalent system descriptions, the reformulated design task is consistent with the control law representation and is formulated as a linear matrix inequality feasibility problem. Taking into account structural restriction of Metzlerian positive systems, a characterization of PID control law parameters is permitted, to highlight dynamical properties of the closed-loop system solutions and the significant structural influence of derivative gain value of the control law parameters in design.

Sampled-Data Fuzzy Control for Nonlinear Delayed Distributed Parameter Systems

Under spatially local averaged measurements (SLAMs), this article introduces a sampled-data fuzzy control (SDFC) for nonlinear delayed distributed parameter systems (DDPSs). First, we use a Takagi–Sugeno (T–S) fuzzy parabolic partial differential-difference equation (PDDE) to accurately describe the nonlinear DDPS. Then, on basis of the T–S fuzzy PDDE model, an SDFC design under SLAMs via space-dependent linear matrix inequalities (SDLMIs) is subsequently developed to ensure the exponential stability of the closed-loop nonlinear DDPSs by using inequality techniques and Lyapunov functional, where slow-varying and fast-varying delays are respected. Furthermore, to solve SDLMIs, the SDFC design problem for nonlinear DDPS under SLAMs is formulated as a linear matrix inequality feasibility problem. Finally, numerical simulations of two examples are presented to support the given SDFC strategy.

Mixed H2/H∞ Control Synthesis for Discrete-time Linear Positive Systems using Enhanced Set of Linear Matrix Inequalities

This paper provides an idea, resulting from linear matrix inequality representation of parameter constraints of discrete-time linear positive systems, to formulate state-feedback synthesis for this class of systems. The design conditions are imposed to obtain control respecting existing strictly positivity or non-negativity in the system matrix description. Formulated as a linear matrix inequality feasibility problem it is reiterated that approach leads iteratively to estimation of norms of closed-loop system

Analysis and design of quadratically bounded QPV control systems

A nonlinear system is said to be quadratically bounded (QB) if all its solutions are bounded and this is guaranteed using a quadratic Lyapunov function. This paper considers the QB analysis and state-feedback controller design problems for quadratic parameter varying (QPV) systems. The developed approach, which relies on a linear matrix inequality (LMIs) feasibility problem, ensures that the QB property holds for an invariant ellipsoid which contains a predefined polytopic region of the state space. An example is used to illustrate the main characteristics of the proposed approach and to confirm the validity of the theoretical results.

Robust model predictive control for switched linear systems

Developed is a robust model predictive control scheme for a class of discrete-time switched linear systems. The system is described in linear fractional transformation form in the presence of model uncertainty and induced norm bounded disturbances. The objective is to minimize the upper bound of an infinite horizon cost function subject to a terminal inequality. A Lyapunov function analysis for the switched system shows guaranteed closed-loop stability. Taking into account the switching structure of the system, the predictive control design problem along with sufficient conditions for the existence of a solution is expressed in terms of Riccati–Metzler inequalities. Then, these inequalities are turned into a linear matrix inequality feasibility problem. Three cases are analyzed to demonstrate the performance and effectiveness of the proposed robust model predictive controller for switched discrete-time linear systems.

Small signal stability criteria for descriptor form power network model

ABSTRACT This work considers the problem of small signal stability of a lossless power network by representing its mathematical model in descriptor form. A stability criterion is derived by constructing two orthogonal projectors corresponding to the system model. These projectors are then used in (i) constructing a Lyapunov function and (ii) deriving another criterion to identify the presence of poles (finite) of the system to the right of a given vertical line in the complex plane. The latter criterion is required to analyse whether the dominant low-frequency electro-mechanical modes, that cause oscillatory instability in the power network, settle down within a specified time limit. Using the derived criteria, linear matrix inequality feasibility problems are formulated to analyse small signal instability and oscillatory instability. Power network examples, including an IEEE benchmark system, are considered to demonstrate the developed approach.

Robust stochastic stabilization of attitude motion

This study proposes robust stochastic stabilization of rigid body attitude motion within the framework of geometric mechanics, which can globally represent the attitude dynamics model. The system is subject to a stochastic input torque with an unknown variance parameter and an unknown nonlinear diffusion coefficient matrix. Our development starts with introducing a general notion of the stochastic stability in probability within the framework of geometric mechanics. Then, the Morse–Lyapunov (M–L) technique is employed to design a nonlinear continuous stochastic feedback control law. Finally, the asymptotic stability of the system is guaranteed in probability and the control gain parameters are obtained via solving a linear matrix inequality feasibility problem. An estimate of the region of attraction of the system is calculated to provide a better insight for tuning the control gain parameters. Two illustrative examples are performed based on the discretized model of the closed-loop system to demonstrate the effectiveness of the proposed control scheme.

Robust Controller Design for Attitude Dynamics Subjected to Time-Delayed State Measurements

Attitude control and time-delay systems are well-developed fields in the control theory, but only a modicum of papers have explored control systems that fall within the intersection of the two. Indeed, combining kinematics and dynamics nonlinearities with sensor and actuator delays reinvigorates the original attitude control problem, typically leading to involved stability arguments based on nonlinear analysis techniques. This paper instead proposes solving the attitude stabilizer design problem by formulating it as a linear matrix inequality feasibility problem. The proposed approach simplifies the stability arguments, without loosing generality; the obtained conditions cope with the general case of rigid bodies that suffer from unknown, heterogeneous, time-varying state measurement delays, and have inertia uncertainties. This methodology is particularly well suited to resource-limited applications, because controllers can be designed offline using computationally efficient tools. Although simple, numerical evidence shows the stability criterion derived in this paper largely outperforms previous results.

An adaptive sliding mode approach for Markovian jump systems with uncertain mode-dependent time-varying delays and partly unknown transition probabilities

This paper deals with the problems of stochastic stability and sliding mode control for a class of continuous-time Markovian jump systems with mode-dependent time-varying delays and partly unknown transition probabilities. The design method is general enough to cover a wide spectrum of systems from those with completely known transition probability rates to those with completely unknown transition probability rates. Based on some mode-dependent Lyapunov-Krasovski functionals and making use of the free-connection weighting matrices, new delay-dependent conditions guaranteeing the existence of linear switching surfaces and the stochastic stability of sliding mode dynamics are derived in terms of linear matrix inequalities (LMIs). Then, a sliding mode controller is designed such that the resulted closed-loop system's trajectories converge to predefined sliding surfaces in a finite time and remain there for all subsequent times. This paper also proposes an adaptive sliding mode controller design method which applies to cases in which mode-dependent time-varying delays are unknown. All the conditions obtained in this paper are in terms of LMI feasibility problems. Numerical examples are given to illustrate the effectiveness of the proposed methods.

Analysis and design of quadratic parameter varying (QPV) control systems with polytopic attractive region

This paper proposes a gain-scheduling approach for systems with a quadratic structure. Both the stability analysis and the state-feedback controller design problems are considered for quadratic parameter varying (QPV) systems. The developed approach assesses/enforces the belonging of a polytopic region of the state space to the region of attraction of the origin, and relies on a linear matrix inequality (LMI) feasibility problem. The main characteristics of the proposed approach are illustrated by means of examples, which confirm the validity of the theoretical results.

Kalman Filtering Over Gilbert–Elliott Channels: Stability Conditions and Critical Curve

This paper investigates the stability of Kalman filtering over Gilbert-Elliott channels where random packet drop follows a time-homogeneous two-state Markov chain whose state transition is determined by a pair of failure and recovery rates. First of all, we establish a relaxed condition guaranteeing peak-covariance stability described by an inequality in terms of the spectral radius of the system matrix and transition probabilities of the Markov chain. We further show that that condition can be interpreted using a linear matrix inequality feasibility problem. Next, we prove that the peak-covariance stability implies mean-square stability, if the system matrix has no defective eigenvalues on the unit circle. This connection between the two stability notions holds for any random packet drop process. We prove that there exists a critical curve in the failure-recovery rate plane, below which the Kalman filter is mean-square stable and no longer mean-square stable above, via a coupling method in stochastic processes. Finally, a lower bound for this critical failure rate is obtained making use of the relationship we establish between the two stability criteria, based on an approximate relaxation of the system matrix.

Robust H∞ Proportional-Integral Observer for Fault Diagnosis: Application to Vehicle Suspension

The main contribution of this paper is the design of robust H∞ proportional-integral (PI) observer for fault detection, isolation and magnitude estimation, applied for uncertain linear time-invariant (LTI) system. It is based on a combination of PI observer and H∞ norm in order to deal with system uncertainty and disturbance attenuation. This multi-objective problem is tackled by the solution of linear matrix inequality (LMI) feasibility problem, obtained from the analysis of estimation errors and the use of majoration lemma. The application to a real suspension system is then presented to highlight the performances of the proposed observer.

Robust Observers for a Class of Nonlinear Systems Using PEM Fuel Cells as a Simulated Case Study

This brief presents an unknown input observer design method for a class of nonlinear systems in the presence of disturbances in both the system dynamics and the output. The theoretical result proposed can be considered as an extension of those established in [1] and [2] , which are dedicated to observer design for estimating both the state of the system and an unknown input. By means of regular transformations, the robust observer design problem is turned into a linear matrix inequality feasibility problem. One of the main motivations of this brief is to develop a state estimator for proton exchange membrane fuel cell stack systems. More precisely, this brief addresses the problem of observer design for estimating the main variables of the fuel cell systems. Simulation results are presented to highlight the good convergence properties of the observers proposed.