Differential Linear Matrix Inequalities Research Articles

Sampled-data controller synthesis using dissipative linear periodic jump-flow systems with design applications

In this paper, we will propose linear-matrix-inequality-based techniques for the design of sampled-data controllers that render the closed-loop system dissipative with respect to quadratic supply functions, which includes passivity and an upper-bound on the system’s H∞-norm as a special case. To arrive at these results, we model the sampled-data control system as a linear periodic jump-flow system, study dissipativity in terms of differential linear matrix inequalities (DLMIs) and then convert these DLMIs into a single linear matrix inequality. We will present three applications of these synthesis techniques: (1) passivity-based controller synthesis, as found in teleoperations, (2) input–output-response matching of a continuous-time filter with a discrete-time filter (by minimizing the H∞-norm of a generalized plant) and (3) a sampled-data controller redesign problem, where the objective is to find the best sampled-data controller, in the H∞-norm sense, for a given continuous-time controller. We will show that synthesizing sampled-data controllers leads to better closed-loop system behaviour than using a Tustin discretization of a continuous-time controller.

Trajectory‐based robustness analysis for nonlinear systems

AbstractThis article considers the robustness of an uncertain nonlinear system along a finite‐horizon trajectory. The uncertain system is modeled as a connection of a nonlinear system and a perturbation. The analysis relies on three ingredients. First, the nonlinear system is approximated by a linear time‐varying (LTV) system via linearization along a trajectory. This linearization introduces an additional forcing input due to the nominal trajectory. Second, the input/output behavior of the perturbation is described by time‐domain, integral quadratic constraints (IQCs). Third, a dissipation inequality is formulated to bound the worst‐case deviation of an output signal due to the uncertainty. These steps yield a differential linear matrix inequality (DLMI) condition to bound the worst‐case performance. The robustness condition is then converted to an equivalent condition in terms of a Riccati differential equation. This yields a computational method that avoids heuristics often used to solve DLMIs, for example, time gridding. The approach is demonstrated by a two‐link robotic arm example.

Passivity and passification of fractional-order memristive neural networks with time delays

A class of fractional-order memristive neural networks (FMNNs) with time delays is studied. At first, the original network system is converted to fractional-order uncertain one to simplify the analysis by a variable transformation. Successively, some new LMIs-based passivity criteria are derived by differential inclusions, set-valued maps, inequality techniques and linear matrix inequality approach. Furthermore, a feedback control protocol is designed to solve the passification problem for the considered system, whose feedback control effect on different neurons can be changed artificially, which can be better applied to neural networks. The obtained results include some existing ones as special cases. A numerical example is proposed to illustrate the theoretical results.

Differential Linear Matrix Inequalities in Optimal and Robust Sampled-Data Filtering

This paper tackles the design of a robust filter for sampled-data linear time-invariant systems subject to parameter uncertainties. After the optimal solution is obtained, the two major classes of parameter uncertainty are considered, namely, convex bounded and norm bounded. For each of them, the filter associated to the minimum H2 guaranteed performance is determined. The design conditions are expressed through differential linear matrix inequalities (DLMIs), a mathematical device well adapted to cope with sampled-data systems because it naturally avoids matrix exponential and similar mappings. In this framework, the optimal nominal filter is determined, and it is shown that it exhibits a time-invariant observer-based structure. Illustrative examples are presented and discussed in order to put in clear evidence the theoretical results.

Finite-time Boundedness of Polytopic Uncertain Systems with Guaranteed Cost Output Feedback Control

A problem of designing robust guaranteed cost reduced-order output feedback controllers for finite-time boundedness of linear time-varying uncertain systems is considered in this paper. Assuming the system parameters to lie in a polytopic set, differential linear matrix inequality (DLMI) conditions for the design of reduced-order dynamic controllers to handle worst-case exogenous disturbances are derived. The designed controller guarantees the cost on considered disturbance attenuation performance on the system to be below an upper bound, while simultaneously satisfying the conditions for finite-time boundedness and finite L2 gain. Numerical simulations are carried out to demonstrate the applicability of the proposed method in bounding the system trajectories to a predefined set.

Bipartite Synchronization of Multi-Agent Systems With Signed Graph Under Deception Attacks and Application in Speech Communication

This paper addresses the bipartite synchronization of multi-agent systems with time-varying delays and signed graph under deception attacks, where the systems contain both the self-structure delayed term and the coupling delayed term. The deception attacks are assumed to be randomly launched at each impulse instant. By using the gauge transformation, Lyapunov function method, Halanay differential inequality and linear matrix inequality techniques, several sufficient conditions are newly established to guarantee the achievement of bipartite synchronization, in which some special cases of multi-agent systems without delay or deception attacks are further considered. Two numerical examples are provided to illustrate the effectiveness of the proposed results. Finally, a new multi-agent speech communication system is constructed based on the bipartite synchronization of the multi-agent systems, where two agents for point-to-point communication can generate synchronous chaotic encryption/decryption key streams. The experimental results show that the proposed multi-agent speech communication system has the advantages of high security against some classical attacks.

Reduced-Order Min-Max Guaranteed Cost Control for Finite-Time Boundedness of Uncertain Linear Systems Using Output Feedback

This letter considers a problem of designing reduced-order dynamic output feedback controllers for finite-time boundedness of uncertain linear time-varying systems subjected to exogenous disturbances. Differential linear matrix inequality conditions are derived for finite-time boundedness with output feedback, such that the uncertain system simultaneously satisfies finite <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathscr {L}_{2}$ </tex-math></inline-formula> gain in the presence of worst-case exogenous disturbances. The designed min-max controller minimizes an upper bound on the considered performance index, providing a guaranteed cost for disturbance attenuation performance. Finite-time bounded state feedback controller design is shown to be a special case of the proposed method. The performance efficacy and applicability of the controller designed using the proposed iterative algorithm are compared with some of the recent methods from the literature.

A polynomial eigenvalue test for checking DLMI constraints

In order to obtain an optimization problem with a finite number of variables and constraints, Differential Linear Matrix Inequalities (DLMIs) are usually recast into the form of a system of LMIs. To this end, a typical approach consists in parameterizing the solution as a linear combination of known functions, with weights to be optimized. The constraints are then imposed on a finite set of discrete time points. This technical communiqué is concerned with the problem of checking whether a matrix polynomial solution obtained in this manner satisfies the original DLMI constraints. To this end, a novel test based on the concept of polynomial eigenvalues is presented. The proposed test is illustrated through a numerical example based on the H∞ synthesis of a sampled-data controller.

Finite-Time Stabilization of Linear Systems With Unknown Control Direction via Extremum Seeking

In this article, the finite-time stabilization problem is solved for a linear time-varying system with unknown control direction by exploiting a modified version of the classical extremum-seeking algorithm. We propose to use a suitable oscillatory input to modify the system dynamics, at least in an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">average</i> sense, so as to satisfy a differential linear matrix inequality condition, which in turn guarantees that the system’s state remains inside a prescribed time-varying hyperellipsoid in the state space. The finite-time stability (FTS) of the averaged dynamics implies the FTS of the original system, as the distance between the original and the averaged dynamics can be made arbitrarily small by choosing a sufficiently high value of the dithering frequency used by the extremum-seeking algorithm. The main advantage of the proposed approach resides in its capability of dealing with systems with unknown control direction, and/or with a control direction that changes over time. Being FTS a quantitative approach, this article also gives an estimate of the necessary minimum dithering/mixing frequency provided, and the effectiveness of the proposed finite-time stabilization approach is analyzed by means of numerical examples.

Input–Output Finite-Time Stabilization of Linear Time-Varying Discrete-Time Systems

This article deals with the input–output finite-time stability (IO-FTS) of linear discrete-time systems. In current control science, discrete-time systems play a very important role in many engineering contexts. Moreover, many discrete-time control problems are defined over a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">finite</i> -interval of time; therefore, the development and application of finite-time control methodologies is of particular relevance. The first contribution of this article is a pair of necessary and sufficient conditions for IO-FTS. The former involves the solution of a set of generalized difference Lyapunov equations; the latter allows one to establish the feasibility of an optimization problem by solving a set of difference linear matrix inequalities (DLMIs). The second contribution of this article is a theorem for IO finite-time stabilization via state feedback, followed by a more general one for stabilization via output feedback. Both conditions are necessary and sufficient, and lead to optimization problems cast in the form of DLMIs. The applicability of the devised results is illustrated through a numerical example.

Sampled-Data Model Predictive Control

This paper focus on model predictive control (MPC) design in the context of sampled-data control systems with full-state measurements. It is shown that recent results on this area can be successfully generalized to cope with sampled-data MPC. The open-loop plant is subjected to polytopic parameter uncertainty and at sampling times a controlled output variable satisfies a set of convex constraints. A guaranteed H2 performance index with infinity horizon is minimized such as the feedback control preserves asymptotic stability and feasibility. The design conditions are expressed though differential linear matrix inequalities (DLMIs). Continuous-time systems are treated with no kind of discrete-time modeling approximation. Comparisons with classical methods from the literature dealing with continuous-time systems are presented and discussed. Examples are included for illustration.

Decentralized finite‐time control for a class of large‐scale systems with symmetrically interconnected subsystems

AbstractThe finite‐time stability (FTS) of a symmetrically interconnected system is studied under the consideration of utilizing the system structural properties to reduce the complexity of analysis. The FTS of this kind of large‐scale system can be guaranteed by that of two lower dimensional systems. The concept of mixed stability, taking care of both transient and steady‐state performances of a system, is introduced. Based on the FTS analysis, a decentralized finite‐time controller via state feedback is given to stabilize the large‐scale system. Meanwhile, an observer‐based decentralized output feedback controller is provided to make the closed‐loop system finite‐time stable. The FTS conditions and related decentralized stabilization controller parameters are both derived from solving some differential or algebraic linear matrix inequalities.

Robust Stability of Fractional Order Memristive BAM Neural Networks with Mixed and Additive Time Varying Delays

This paper is concerned with the problem of the robust stability of fractional-order memristive bidirectional associative memory (BAM) neural networks. Based on Lyapunov theory, fractional-order differential inequalities and linear matrix inequalities (LMI) are applied to obtain a robust asymptotical stability. Finally, numerical examples are presented.

Impulsive effects on stochastic bidirectional associative memory neural networks with reaction-diffusion and leakage delays

In this paper, the problem of global asymptotic stability analysis for stochastic reaction-diffusion bidirectional associative memory neural networks (BAMNNs) with mixed delays and impulsive effects are investigated. The mixed delays consists of time delays in the leakage terms and continuously distributed delays. Based on the Lyapunov–Krasovskii functional (LKF), It's differential formula and linear matrix inequality (LMI) method, some sufficient conditions for global asymptotic stability in mean square of the equilibrium point of the systems are derived. The feasibility of the conditions are verified using the MATLAB LMI toolbox. Finally, two examples are provided to illustrate the effectiveness and validity of the derived main results.

Research for axial power distribution control of a space nuclear reactor based on nonlinear model

During the operating process of a space nuclear reactor, once the local overload of heat energy occurs in the core, the corresponding fuel elements would be damaged and the lifespan of the reactor would be seriously reduced. In fact, the heat flux at any point in the core is proportional to the neutron flux density. A sufficiently flat axial neutron flux density distribution can protect the fuel assembly from frequent load changes. It is so clear that the axial power distribution control is crucial for the safe and stable operation of the space nuclear reactor. However, the space nuclear reactor is a distributed parameter system, which brings great challenges to the traditional lumped parameter control methods. In this paper, a distributed parameter control method is proposed to suppress the initial neutron flux peaks and realize the exponential stability of the neutron flux distribution control system. First of all, the spatiotemporal model of the reactor core based on the neutron diffusion equation is established. Then, the stability analysis of the control law for the neutron flux distributed parameter control problem is performed by constructing the Lyapunov function. The optimization technique based on the spatial differential linear matrix inequality is used to approximate the solution of the suboptimal integral control law. The control law is implemented by rotating drum, and the flat distribution of the axial power of the core is realized by the distributed parameter control. Finally, the numerical results are provided to validate the feasibility of the proposed control scheme.

Finite-time Annular Domain Stability and Stabilization of Itô-type Stochastic Time-Varying Systems with Wiener and Poisson noises

This paper investigates finite time annular domain (FTAD) stability and stabilization for Itô-type stochastic time-varying systems with continuous Wiener and discontinuous Poisson noises (STVSWPNs). First, using Itô-Levy formula and time-varying multiple quadratic Lyapunov functions, two less conservative FTAD-stability conditions based generalised differential Lyapunov equations (GDLEs) and differential linear matrix inequalities (DLMIs) are obtained. Second, the FTAD-stabilization is studied and some new sufficient conditions for the existence of state feedback and static output feedback controllers are presented by tractable differential linear matrix inequalities. Moreover, a new numerical algorithm is given. Finally, a numerical example and a real-world example are utilized to show the effectiveness of the proposed methods.

Feedback Finite-Time Stabilization of Impulsive Linear Systems over Piecewise Quadratic Domains

This paper investigates the state feedback stabilization problem for a class of impulsive linear time-varying systems over specified time intervals and piecewise quadratic domains (PQDs). First, concepts related to finite-time stability and PQDs are given. Second, finite-time stability analysis over PQDs is implemented, and a variety of stability conditions involving differential linear matrix inequalities are investigated. Then, computationally tractable stability conditions are established for the control design. Finally, an illustrative example is presented to show the effectiveness of the designed state feedback control.

Annular Stochastic Finite-Time Stability Using Piecewise Quadratic Lyapunov Functions

In this letter we present some new sufficient conditions for the annular stochastic finite-time stability of a class of stochastic linear time-varying systems. These new conditions are obtained adopting time-varying piecewise quadratic Lyapunov functions rather than the classical quadratic ones. The proposed approach allows us to extend the class of consider domains, which are typically limited to ellipsoidal domains. The proposed finite-time stability conditions can be converted into a feasibility problem based on a set of differential linear matrix inequalities. Two numerical examples are considered to perform a comparison with the previous results, and they show that the new proposed conditions are less conservative than the previous ones.

Sampled-data control of Lur’e systems

The main purpose of this paper is to fill a gap in the literature concerning the problem of designing a sampled-data control law for continuous-time Lur’e systems. The goal is to design a state feedback sampled-data control for this class of nonlinear systems preserving global asymptotic stability and minimizing a guaranteed quadratic cost. The main challenge towards the solution of the proposed problem is to handle this class of nonlinear system in order to propose less conservative design conditions expressed through differential linear matrix inequalities - (DLMIs). Bellman’s Principle of Optimality applied together with the Popov–Lyapunov function that emerges from the celebrated Popov Stability Criterion is the key issue to obtain the reported results. Two examples are solved for illustration.

L2 norm-based finite-time stability and boundedness of singular distributed parameter systems with parabolic type

In this study, the problem of finite-time stability and boundedness for parabolic singular distributed parameter systems in the sense of [Formula: see text] norm is investigated. First, two new results on [Formula: see text] norm-based finite-time stability and finite-time boundedness for above-mentioned systems, inspired by the light of partial differential equations theory and Lyapunov functional method, are presented. Then, some sufficient conditions of [Formula: see text] norm-based finite-time stability and boundedness are established by virtue of differential inequalities and linear matrix inequalities. Furthermore, the distributed state feedback controllers are constructed to guarantee the [Formula: see text] norm-based finite-time stable and bounded of the closed-loop singular distributed parameter systems. Finally, numerical simulations on a specific numerical example and the building temperature control system equipped with air conditioning are given to demonstrate the validity of the proposed methods.