System Of Linear Matrix Inequalities Research Articles

Generalized homogeneous control with integral action

AbstractA generalized homogeneous control with integral action for a multiple‐input plant operating under uncertainty conditions is designed. The stability analysis is essentially based on a special version of the nonsmooth Lyapunov function theorem for differential equations with discontinuous right‐hand sides. A Lyapunov function for analysis of the closed‐loop system is presented. For negative homogeneity degree, this Lyapunov function becomes a strict Lyapunov function allowing an advanced analysis to be provided. In particular, the maximum control magnitude and the settling‐time of the closed‐loop system are estimated and a class of disturbances to be rejected by the control law is characterized. The control parameters are tuned by solving a system of Linear Matrix Inequalities (LMIs), whose feasibility is proved at least for small (close to zero) homogeneity degrees. The theoretical results are illustrated by numerical simulations.

Optimization of Invariant Ellipsoid Technique for Sparse Controllers Design

The paper deals with the method for the design of linear controllers with sparse state feedback matrices for control the plants under conditions of unknown and bounded disturbances. The importance of the sparsity property in feedback can be explained by two factors. First, by minimizing the columnar norm of the feedback matrix in the control it becomes pos- sible to use a minimum number of measuring devices. Secondly, by minimizing the row norm of the feedback matrix, the required number of executive (control) devices is minimized. Both properties, if they are achievable in the synthesis of the controller, reduce the cost of the system and improve the fault tolerance and quality of regulation by reducing the structural complexity. The search algorithm for sparse matrices is based on the method of invariant ellipsoids and is formulated as a solution to a system of linear matrix inequalities with additional constraints. A special set of optimization conditions is proposed which for a disturbed system minimizes overshoot and overshoots in transient processes of the disturbed closed- loop system simultaneously with minimizing errors in the steady state. The proposed method also assumes the possibility of minimizing both the row norm of the feedback matrix and the column one, while preserving the robustness properties, which makes it possible to solve the sparse control problem (a sparse control is understood as a linear controller with a sparse feedback matrix). The efficiency of the proposed control scheme is confirmed by the results of computer modeling and comparison with some existing ones.

Fault-tolerant control system for the formation of carbon products

The production of carbon products is largely resource- and energy-intensive. That is why increasing the efficiency of this production is an urgent scientific and practical task, especially in modern conditions of constant growth of energy costs. An effective way to solve this problem is to create a modern process control system, taking into account possible failures of system components. A method for the synthesis of a fault-tolerant control system for the cyclic formation of carbon products has been developed, which takes into account control errors that are caused by malfunctions of controllers under conditions of unknown disturbances. According to the cyclic nature of the technological process under consideration, a control method with iterative learning was used in the synthesis of the control system. This method considers cyclic processes based on a two-dimensional model (2D model). The proposed control algorithm ensures the convergence of the control process with the task both in time and in each work cycle in order to promote the required quality of control even in the event of unknown disturbances and errors in the performance of controllers. The synthesis of the control system is based on the solution of a system of linear matrix inequalities. Based on the combination of a control method with iterative learning and a control method that takes into account failures in controllers, a method of constructing a fault-tolerant control system for the cyclic formation of carbon products has been synthesized to ensure acceptable operation of the control object in abnormal conditions. The control system has been synthesized by solving a system of linear matrix inequalities with the MATLAB software. In the future, it is necessary to consider optimal settings of the proposed control system and examine its effectiveness in comparison with conventional fault-tolerant systems for non-cyclic processes.

On homogeneous controllability functions

The controllability function method, introduced by V. I. Korobov in late 1970s, is known to be an efficient tool for control systems design. It is developed for both linear/nonlinear and finite/infinite dimensional systems. This paper bridges the method with the homogeneity theory popular today. The standard homogeneity known since 18th century is a symmetry of function with respect to uniform scaling of its argument. Some generalizations of the standard homogeneity were introduced in 20th century. This paper shows that the so-called homogeneous norm is a controllability function of the linear autonomous control system and the corresponding closed-loop system is homogeneous in the generalized sense. This immediately yields many useful properties known for homogeneous systems such as robustness (Input-to-State Stability) with respect to a rather large class of perturbations, in particular, with respect to bounded additive measurement noises and bounded additive exogenous disturbances. The main theorem presented in this paper slightly refines the design of the controllability function for a multiply-input linear autonomous control systems. The design procedure consists in solving subsequently a linear algebraic equation and a system of linear matrix inequalities. The homogeneity itself and the use of the canonical homogeneous norm essentially simplify the design of a controllability function and the analysis of the closed-loop system. Theoretical results are supported with examples. The further study of homogeneity-based design of controllability functions seems to be a promising direction for future research.

Stability analysis of mechanical systems with distributed delay via decomposition

The article analyzes a linear mechanical system with a large parameter at the vector of velocity forces and a distributed delay in positional forces. With the aid of the decomposition method, conditions are obtained under which the problem of stability analysis of the original system of the second-order differential equations can be reduced to studying the stability of two auxiliary first-order subsystems. It should be noted that one of the auxiliary subsystems does not contain a delay, whereas for the second subsystem containing a distributed delay, the stability conditions are formulated in terms of the feasibility of systems of linear matrix inequalities. To substantiate this decomposition, the Lyapunov direct method is used. Special constructions of Lyapunov—Krasovskii functionals are proposed. The developed approach is applied to the problem of monoaxial stabilization of a rigid body. The results of a numerical simulation are presented confirming the conclusions obtained analytically.

Nonlinear modelling and optimal control via Takagi-Sugeno fuzzy techniques: A quadrotor stabilization

AbstractUsing the principles of Takagi-Sugeno fuzzy modelling allows the integration of flexible fuzzy approaches and rigorous mathematical tools of linear system theory into one common framework. The rule-based T-S fuzzy model splits a nonlinear system into several linear subsystems. Parallel Distributed Compensation (PDC) controller synthesis uses these T-S fuzzy model rules. The resulting fuzzy controller is nonlinear, based on fuzzy aggregation of state controllers of individual linear subsystems. The system is optimized by the linear quadratic control (LQC) method, its stability is analysed using the Lyapunov method. Stability conditions are guaranteed by a system of linear matrix inequalities (LMIs) formulated and solved for the closed loop system with the proposed PDC controller. The additional GA optimization procedure is introduced, and a new type of its fitness function is proposed to improve the closed-loop system performance.

Robust stability and evaluation of the quality functional for linear control systems with matrix uncertainty

New methods of robust stability analysis for equilibrium states and optimization of linear dynamic systems are developed. Sufficient stability conditions of the zero state are formulated for a linear control systems with uncertain coefficient matrices and measurable output feedback. In addition, a general quadratic Lyapunov function and ellipsoidal set of stabilizing matrices for the feedback amplification coefficients are given. Application of the results is reduced to solving the systems of linear matrix inequalities.

Tracking Control for Wheeled Mobile Robot Based on Delayed Sensor Measurements

This paper proposes a novel Takagi-Sugeno fuzzy predictor observer to tackle the problem of the constant and known delay in the measurements. The proposed observer is developed for a trajectory-tracking problem of a wheeled mobile robot where a parallel-distributed compensation control is used to control the robot. The L2-stability of the proposed observer is also proven in the paper. Both, the control and the observer gains are obtained by solving the proposed system of linear matrix inequalities. To illustrate the efficiency of the proposed approach, an experimental comparison with another predictor observer was done.

Linear Matrix Inequalities in Stability Problems: Retrospective and Theoretical Aspects

Some aspects of the development of the theory of linear matrix inequalities are considered. A number of results obtained at the initial stage of the development of this theory, both in the development of numerical methods and in obtaining analytical conditions for their solvability, are highlighted. The main attention is focused on the system of linear matrix inequalities arising in solving the absolute stabi lity problem. E. S. Pyatnitskiy and his followers showed that the solvability of this system is a criterion for the existence of a quadratic Lyapunov function and a sufficient condition for absolute stability. The prerequisites leading to this result are considered here. The use of the considered system of inequalities for studying the stability of hybrid systems described by differential inclusions and switching systems is shown. An analysis is given of citing some works of Pyatnitskiy’s school on the theory of stability and the theory of systems of linear matrix inequalities, from which the relevance of the results of these works at the present time follows.In developing numerical methods, it was first shown in the work of Pyatnitskiy and Skorodinskiy that the solvability problem for a system of linear matrix inequalities reduces to a convex programming problem. An interesting gradient algorithm for finding solutions to such a system is also presented. In analyzing analytical conditions of solvability, an unsolvability criterion for the system of our interest obtained by Kamenetskiy and Pyatnitskiy is noted. In modern terms, this result can be considered as a description of an admissible set in the dual semidefinite programming problem. A similar result is given in the famous book by S. Boyd et al. The paper shows that the result of Boyd et al. is a simple corollary of the unsolvability criterion. Here the unsolvability criterion is generalized and refined.

A Switching Controller for a Class of MIMO Bilinear Systems With Time Delay

In this paper, we propose a state-dependent switching controller for multiple-input multiple-output (MIMO) bilinear systems with constant delays in both the state and the input. The control input is assumed to be restricted to take only a finite number of values. The stability analysis of the closed loop is based on a Lyapunov–Krasovskii functional, and the design is reduced to solve a system of linear matrix inequalities. The controller can be designed by considering (state) delay-dependent or delay-independent conditions.

Estimation of the Attraction Domain for an Affine System with Constrained Vector Control Closed by the Linearizing Feedback

Nonlinear affine systems with constrained vector control that are represented in a canonical (normal) form and are closed by feedbacks linearizing the system in a neighborhood of the origin, are considered. For the nonlinear closed-loop system, the problem is set to construct an estimate of the attraction domain of an equilibrium position. A method for constructing an estimate of the attraction domain, which is based on results of absolute stability theory, is suggested. The estimate is sought as a Cartesian product of positive invariant sets of the subsystems composing the system. In the case of ellipsoidal invariant sets, construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

Synthesis of a Minimum Functional State Observer Approach for Unperturbed/Perturbed Dynamical Systems

In this paper, we investigate the synthesis of minimum functional state observer for unperturbed and perturbed linear time invariant systems. The principal contribution is the design of a minimum functional state observer, which can estimate directly the state feedback control law. Hence, for the linear time invariant systems, the existence conditions of a minimum functional state observer are obtained by verification of a special dimension condition on system matrices. As a matter of fact, the exact solution of the proposed approach is determined, and the minimum functional state observer that has the same dimension as the control vector is derived by solving a set of linear matrix inequality (LMI) constraints. Whereas, for perturbed linear systems, the proposed minimum functional state observer scheme is developed to ensure the robust quadratic stability of the augmented system. The robustness issue is given via the reconstructed control law designed using an LMI based H∞ method; so that the desired design matrices are derived through the resolution of an optimum LMI system. The effectiveness and usefulness of the proposed approach are validated through a flexible link robot example.

Estimation of the Attraction Domain for Nonlinear Single-Input System with Constrained Control * *This work was supported by Program I.31 “Actual problems of robotics” of the Presidium of Russian Academy of Sciences.

Abstract Nonlinear single-input affine systems represented in a canonical (normal) form are considered. The control resource is assumed to be constrained. For a closed-loop system obtained by applying a linearizing feedback, the problem of construction of an estimate of the attraction domain of zero solution is studied. A method for constructing an ellipsoidal estimate that is based on results of the absolute stability theory is suggested. Construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

Attraction domain estimate for single-input affine systems with constrained control

Nonlinear single-input affine systems represented in a canonical (normal) form are considered. The control resource is assumed to be constrained. For a closed-loop system obtained by applying a linearizing feedback, the problem of finding an estimate of the attraction domain is set. A method for constructing an ellipsoidal estimate that is based on results of absolute stability theory is suggested. Construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

A MATLAB Toolbox for Estimating the Second Moments of Earthquake Ruptures

ABSTRACT I present a set of routines in MATLAB for estimating the second‐degree moments of an earthquake’s rupture from far‐field body waves. The second moments describe the length, width, duration, and directivity of a rupture. The second‐moments approach is particularly useful when a seismic dataset is dense enough to resolve the primary finite‐source properties but the geodetic data needed for a well‐resolved finite‐fault inversion are not available. In particular for M w 3–6 earthquakes, this approach can be a useful way to estimate rupture area without the assumptions of typical corner‐frequency approaches. The provided software utilizes empirical Green’s function deconvolution to isolate the apparent source time function (ASTF) for each station and phase. The spatial variations in the duration of the ASTF are quantified and inverted for the second moments. The inverse problem is solved using MATLAB’s convex optimization routines for systems of linear matrix inequalities. An error analysis using the jackknife and bootstrap methods is included. An example M w 4.7 earthquake from the San Jacinto fault is used to demonstrate the method.

Output Stabilization and Weighted Suppression of Disturbances in Discrete-Time Control Systems

Output stabilizability conditions for some classes of discrete-time linear and nonlinear control systems are formulated by using dynamic controllers. Methods for constructing static and dynamic controllers that provide a specified evaluation of the weighted damping level of external and initial disturbances are proposed for systems with controllable and observable outputs. The implementation of these methods with the use of static state feedback or complete order dynamic controllers is based on the solution of systems of linear matrix inequalities. The example of a controller synthesis for a two-mass mechanical system is given.

Dissipative Switched Linear Differential Systems

The authors develop a dissipativity theory for switched systems whose dynamical modes are described by systems of higher order linear differential equations. They give necessary and sufficient conditions for dissipativity based on systems of linear matrix inequalities (LMIs), constructed from the coefficient matrices of the differential equations describing the modes. The relationship between dissipativity and stability is also discussed, and an application to the stabilization of power converters is provided.

Functional observer design using linear matrix inequalities

It is shown that the problem of observer design for estimating a set of linear combinations of state variables of a plant can be formulated in terms of linear matrix inequalities. An algorithm for constructing functional observers is proposed based on a nonsingular transformation of a plant model in the state space by matrix canonization with subsequent solution of the system of linear matrix inequalities.

A novel scheme for current sensor faults diagnosis in the stator of a DFIG described by a T-S fuzzy model

Abstract Diagnosis of current sensor faults (CSF) for doubly fed induction generators (DFIGs) is of paramount importance for the reliable power generation of DFIG-based wind turbines (WT). In this paper, a new scheme is developed for current sensors faults diagnosis in the stator of a DFIG-based WT. The nonlinear model of the DFIG is first transformed into an equivalent Takagi-Sugeno (T-S) fuzzy model. Secondly, using this model, a novel fault detection and isolation (FDI) algorithm is proposed. This algorithm is based on a bank of Luenberger observers for residuals generation combined with a new proposed residual vector. Furthermore, a new binary decision logic is used for CSF isolation. Stability analysis of the observer bank is analyzed using a Lyapunov theorem, which allows deriving sufficient stability conditions by solving a system of Linear Matrix Inequalities (LMIs). A simulation study is carried out to assess the performance and the effectiveness of the new FDI scheme.

A computational approach to steady-state convergence of fluid limits for Coxian queuing networks with abandonment

Many-server queuing networks with general service and abandonment times have proven to be a realistic model for scenarios such as call centers and health-care systems. The presence of abandonment makes analytical treatment difficult for general topologies. Hence, such networks are usually studied by means of fluid limits. The current state of the art, however, suffers from two drawbacks. First, convergence to a fluid limit has been established only for the transient, but not for the steady state regime. Second, in the case of general distributed service and abandonment times, convergence to a fluid limit has been either established only for a single queue, or has been given by means of a system of coupled integral equations which does not allow for a numerical solution. By making the mild assumption of Coxian-distributed service and abandonment times, in this paper we address both drawbacks by establishing convergence in probability to a system of coupled ordinary differential equations (ODEs) using the theory of Kurtz. The presence of abandonments leads in many cases to ODE systems with a global attractor, which is known to be a sufficient condition for the fluid and the stochastic steady state to coincide in the limiting regime. The fact that our ODE systems are piecewise affine enables a computational method for establishing the presence of a global attractor, based on a solution of a system of linear matrix inequalities.