Linear Matrix Inequality Characterization Research Articles

Robust consensus of uncertain multi agent systems with one‐sided Lipschitz nonlinearity

AbstractThis article develops protocols for the consensus of an uncertain nonlinear multi‐agent system that satisfies the one‐sided Lipschitz and quadratic‐inner bounded conditions. Fully distributed consensus protocols with observer‐based and observer‐less schemes are investigated. For the observer‐based system, a full order Luenberger observer is designed. To overcome the difficulties of designing the controller and observer‐based controller gains, sufficient conditions are provided with a complete linear matrix inequality characterization. Finally, two simulation examples are provided to illustrate the effectiveness of the proposed approach.

Adaptive Threshold Generation for Vehicle Fault Detection Using Switched T–S Interval Observers

This paper is concerned with the robust passive fault detection problem for switched continuous-time linear parameter-varying systems with mensurable and unmeasurable scheduling parameters. A switched Takagi–Sugeno (TS) interval observer is designed to estimate the set of admissible state values. Using multiple fuzzy input to state stability-Lyapunov function and average dwell time (ADT) concept, sufficient conditions to guarantee the convergence and the robustness of the proposed observer are obtained. These conditions are formulated as a linear matrix inequality (LMI) problem. In contrast to the existing results based on multiple Lyapunov function and ADT switching, the derived conditions lead to less conservative LMIs characterization. Subsequently, the residual intervals are generated using the designed interval observer and used directly for fault detection (FD) decision making. Finally, the proposed methodology is tested using two examples. First, an academic example is used to illustrate the obtained relaxation. Second, a nonlinear vehicle model corrupted by faults is considered. Longitudinal velocity and cornering stiffness coefficients are treated, respectively, as the measurable and unmeasurable scheduling parameters. Simulation results based on experimental data show the effectiveness of the proposed FD schema.

Some Extensions on the Bounded Real Lemma for Positive Systems

Recent advances have shown that the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain of positive systems is fully determined by the static gain matrix, and hence the linear matrix inequality (LMI) characterizing their H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance admits a diagonal Lyapunov matrix variable. In this technical note, we aim to extend this property for systems with input, state and output defined on general cones. Although it is known that the L <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -gain is no longer governed by the static gain matrix for systems defined on general proper cones, we show that this property is still valid when the cone in the state space is symmetric, and the cones in the input and output spaces are self-dual. Meanwhile, it is proved that the Lyapunov matrix appearing in the LMI characterization for the H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> performance can be constructed by the quadratic representation of the Jordan algebra associated with the symmetric cone in the state space. Finally, the established results are illustrated via a linear system defined on second-order cones.

Dilated LMI characterization for the robust finite time control of discrete-time uncertain linear systems

This paper provides new dilated linear matrix inequalities (LMIs) characterizations for the finite time boundedness (FTB) and the finite time stability (FTS) analysis of discrete-time uncertain linear systems. The dilated LMIs are later used to design a robust controller for the finite time control of discrete-time uncertain linear systems. The relevant feature of the proposed approach is the decoupling between the Lyapunov and the system matrices, that allows considering a parameter-dependent Lyapunov function. In this way, the conservativeness with respect to previous results is decreased. Numerical examples are used to illustrate the results.

On Stabilization of Switched Linear MI Systems Using in Part Common Left Eigenvector Assignment Based on LMIs

The problem of assigning in part common left eigenvector assignment by the state feedback control of linear multi-input systems is considered in the paper. Based on linear matrix inequalities characterization, the exposition of the problem is formulated to handle a common left eigenvector for switched modes. Limited by preexisting degrees of freedom, which factually makes mutually contingent subsets of the closed-loop eigenvalue spectrum of each mode, the design conditions for in part common left eigenvector assignment of all modes are derived in the terms of linear matrix inequalities. The procedure provides a way available in linear switched control system design while the state-space control structures take standard forms. An example is included to assess the properties of the technique and its application.

Distributed Fault Detection and Isolation Filter Design for a Network of Heterogeneous Multiagent Systems

In this brief a distributed fault detection and isolation (FDI) methodology for a network of heterogeneous multiagent systems with different dynamics and order from one another is proposed. An FDI filter is designed such that the effects of disturbances and control inputs on the residual signals are minimized (for accomplishing the fault detection task) subject to the constraint that the transfer matrix function from the faults to the residuals is equal to a preassigned diagonal transfer matrix (for accomplishing the fault isolation task). Moreover, by utilizing the proposed methodology, isolation of simultaneous occurring faults can also be handled. Sufficient conditions for solvability of the problem are obtained in terms of linear matrix inequality (LMI) feasibility conditions. The extended LMI characterization is then used to reduce the conservativeness of the solution by eliminating the couplings between the Lyapunov matrices and the agents' matrices. Simulation results presented demonstrate the effectiveness and capabilities of our proposed design methodology.

On Extended LMI Conditions for ℋ2/ℋ∞ Control of Discrete-Time Linear Systems

This paper provides new insights into the currently available extended linear matrix inequality (LMI) conditions for control of discrete-time linear systems, motivating the use of two intrinsically different extended LMI characterizations for ℋ2 performance. While these conditions are equivalent for ℋ2 analysis and many ℋ2 control problems related to precisely known linear time-invariant (LTI) systems, they generally yield different results when employed for multi-objective ℋ2/ℋ∞ control and ℋ2 analysis and control of uncertain linear systems. The advantage of considering both the ℋ2 LMIs for robust ℋ2 state feedback and multi-objective ℋ2/ℋ∞ control design is demonstrated by means of exhaustive numerical comparisons.

Robust sliding control for mismatched uncertain fuzzy time-delay systems using linear matrix inequality approach

In this article, we propose a robust sliding control design method for uncertain time-delay systems that can be represented by Takagi–Sugeno fuzzy models. The uncertain fuzzy time-delay systems under consideration have mismatched parameter uncertainties in the state matrix and external disturbances. We derive existence conditions of linear sliding surfaces guaranteeing the asymptotic stability in terms of constrained linear matrix inequalities (LMIs). We present an LMI characterization of such sliding surfaces. Also, an LMI-based algorithm is given to design the switching feedback control term so that a stable sliding motion is induced in finite time. Finally, we give two numerical design examples to show the effectiveness of the proposed method.

Controller design for affine fuzzy systems via characterization of dilated linear matrix inequalities

We study the design of a state feedback controller for a class of nonlinear systems described by continuous-time affine fuzzy models. By introducing extra slack variables, the Lyapunov matrix and the system matrix are decoupled such that the controller parametrization is independent of the Lyapunov matrix. A novel quadratic stability analysis condition for affine fuzzy systems is derived in the formulation of linear matrix inequalities (LMIs), which is equivalent to existing results. Using the analytical results and a diffeomorphic state transformation, a stabilizing condition under which the affine fuzzy system is quadratically stabilizable is derived and can be solved by means of an LMI technique in conjunction with a search for scaling parameters. In contrast to existing work, the stabilizability condition we derive leads to less conservative LMI characterizations. The result is also extended to H∞ state feedback synthesis. Finally, a numerical example illustrates the merits of the new results.

Piecewise controller design for affine fuzzy systems via dilated linear matrix inequality characterizations

This paper studies the problem of state feedback controller design for a class of nonlinear systems, which are described by continuous-time affine fuzzy models. A convex piecewise affine controller design method is proposed based on a new dilated linear matrix inequality (LMI) characterization, where the system matrix is separated from Lyapunov matrix such that the controller parametrization is independent of the Lyapunov matrix. In contrast to the existing work, the derived stabilizability condition leads to less conservative LMI characterizations and much wider scope of the applicability. Furthermore, the results are extended to H∞ state feedback synthesis. Finally, two numerical examples illustrate the superiority and effectiveness of the new results.

Formula omitted] robust and networked control of discrete-time MJLS through LMIs

This paper addresses the H∞ state-feedback control of Markov Jump Linear Systems (MJLS) through Linear Matrix Inequalities (LMIs). First, under the assumption that the Markov parameter is measured, the main contribution is on the LMI characterization of all linear feedback controllers such that the closed loop output remains bounded by a given norm level. This result allows the robust controller design to deal with convex bounded parameter uncertainty, probability uncertainty and cluster availability of the Markov mode. For partly unknown transition probabilities, the proposed design problem is proved to be less conservative than the one already available. For the case where each transition probability is considered unknown and belonging to a given interval, we also prove that our conditions are less conservative than a previous result from the literature. Under cluster availability of the Markov modes, we incorporate several models for measurement transmission network channels, such as the generalized Gilbert–Elliot, Fritchman or McCullough ones, into the MJLS framework. An example is solved for illustration and comparisons.

New Stabilization Schemes for Linear Hybrid Systems With Time-Varying Delays

In this paper, we provide new stabilization schemes for a class of linear hybrid time-delay systems under arbitrary switching. These schemes are delay-independent and delay-dependent H∞ stabilization based on proportional-plus-derivative (PPD) feedback strategy. By adopting a selective Lyapunov–Krasovskii functional, new criteria are constructed in a systematic way in terms of feasibility testing of linear matrix inequalities (LMIs). When the time delay is a continuous bounded function, we derive the solution for nominal and polytopic models and identify several existing results as special cases. In case the time delay is a differentiable time-varying function satisfying some bounding relations, we establish a new parametrized LMI characterization for PPD feedback stabilization. The theoretical developments are illustrated on examples of combustion in rocket motor chambers, river pollution control, and resilience analysis, and the ensuing results are compared with the conventional feedback stabilization.

Optimal Design of FIR Triplet Halfband Filter Bank and Application in Image Coding

This correspondence proposes an efficient semidefinite programming (SDP) method for the design of a class of linear phase finite impulse response triplet halfband filter banks whose filters have optimal frequency selectivity for a prescribed regularity order. The design problem is formulated as the minimization of the least square error subject to peak error constraints and regularity constraints. By using the linear matrix inequality characterization of the trigonometric semi-infinite constraints, it can then be exactly cast as a SDP problem with a small number of variables and, hence, can be solved efficiently. Several design examples of the triplet halfband filter bank are provided for illustration and comparison with previous works. Finally, the image coding performance of the filter bank is presented.

Integral sliding mode control of time-delay systems with mismatching uncertainties

A linear matrix inequality (LMI)-based sliding surface design method for integral sliding mode control of uncertain time-delay systems with mismatching uncertainties is proposed. The uncertain time-delay system under consideration may have mismatching norm bounded uncertainties in the state matrix as well as the input matrix. A sufficient condition for the existence of a sliding surface is given to guarantee asymptotic stability of the full order sliding mode dynamics. An LMI characterization of the sliding surface is given, together with an integral sliding mode control law guaranteeing the existence of a sliding mode from the initial time. Finally, a simulation is given to show the effectiveness of the proposed method.

On Delay-Dependent Global Asymptotic Stability for Pendulum-Like Systems

This paper deals with global asymptotic stability for the delayed nonlinear pendulum-like systems with polytopic uncertainties. The delay-dependent criteria, guaranteeing the global asymptotic stability for the pendulum-like systems with state delay for the first time, are established in terms of linear matrix inequalities (LMIs) which can be checked by resorting to recently developed algorithms solving LMIs. Furthermore, based on the derived delay-dependent global asymptotic stability results, LMI characterizations are developed to ensure the robust global asymptotic stability for delayed pendulum-like systems under convex polytopic uncertainties. The new extended LMIs do not involve the product of the Lyapunov matrix and the system matrices. It enables one to check the global asymptotic stability by using parameter-dependent Lyapunov methods. Finally, a concrete application to phase-locked loop (PLL) shows the validity of the proposed approach.

${\cal H}_{\infty}$ Filtering of Discrete-Time Markov Jump Linear Systems Through Linear Matrix Inequalities

This technical note addresses the discrete-time Markov jump linear systems H infin filtering design problem. First, under the assumption that the Markov parameter is measurable, the main contribution is the linear matrix inequality (LMI) characterization of all linear filters such that the estimation error remains bounded by a given H infin norm level, yielding the complete solution of the mode-dependent filtering design problem. Based on this result, a robust filter design able to deal with polytopic uncertainty is considered. Second, from the same LMI characterization, a design procedure for mode-independent filtering is proposed. Some examples are solved for illustration and comparisons.

Design of Exactly Linear Phase $K$-Regular IIR Half-Band Filter

This paper proposes a novel method to design exactly linear phase infinite impulse response half-band filters with arbitrary regularity. Broadly speaking, the design problem is formulated as a semi-infinite program, which is then turned into a semidefinite program of minimal order via a new linear matrix inequality characterization of convex hulls of trigonometric polynomials. In contrast to maximally flat approach, the proposed method allows direct control of various design parameters, which in turn enables the synthesis of filters with better transition response. The viability of the proposed method is demonstrated through several numerical examples.

Robust Stabilization of Uncertain Fuzzy Systems Using Variable Structure System Approach

Based on the variable structure system (VSS) theory, we develop a fuzzy control system design method for a class of uncertain nonlinear multivariable systems that can be represented by a Takagi-Sugeno fuzzy model. We make the first attempt to relax the restrictive assumption that each nominal local system model shares the same input channel, which is required in the traditional VSS-based fuzzy control design methods. As the local controller we use a sliding mode controller with a switching feedback control term. In terms of linear matrix inequalities (LMIs), we derive a sufficient condition for the existence of linear sliding surfaces guaranteeing asymptotic stability of the reduced-order equivalent sliding mode dynamics. We present an LMI characterization of such sliding surfaces. We also give an LMI-based algorithm to design the switching feedback control term so that a stable sliding motion is induced in finite time. Finally, we give a numerical design example.

Efficient Large-Scale Filter/Filterbank Design via LMI Characterization of Trigonometric Curves

Many filter and filterbank design problems can be posed as the optimization of linear or convex quadratic objectives over trigonometric semi-infinite constraints. Recent advances in design methodology are based on various linear matrix inequality (LMI) characterizations of the semi-infinite constraints, and semidefinite programming (SDP) solutions. Despite these advances, the design of filters of several hundredth order, which typically arise in multicarrier communication and signal compression, cannot be accommodated. This hurdle is due mainly to the large number of additional variables incurred in the LMI characterizations. This paper proposes a novel LMI characterization of the semi-infinite constraints that involves additional variables of miminal dimensions. Consequently, the design of high-order filters required in practical applications can be achieved. Examples of designs of up to 1200-tap filters are presented to verify the viability of the proposed approach.

Complex Matrix Decomposition and Quadratic Programming

This paper studies the possibilities of the linear matrix inequality characterization of the matrix cones formed by nonnegative complex Hermitian quadratic functions over specific domains in the complex space. In its real-case analog, such studies were conducted in Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246–267]. In this paper it is shown that stronger results can be obtained for the complex Hermitian case. In particular, we show that the matrix rank-one decomposition result of Sturm and Zhang [Sturm, J. F., S. Zhang. 2003. On cones of nonnegative quadratic functions. Math. Oper. Res. 28 246–267] can be strengthened for the complex Hermitian matrices. As a consequence, it is possible to characterize several new matrix co-positive cones (over specific domains) by means of linear matrix inequality. As examples of the potential application of the new rank-one decomposition result, we present an upper bound on the lowest rank among all the optimal solutions for a standard complex semidefinite programming (SDP) problem, and offer alternative proofs for a result of Hausdorff [Hausdorff, F. 1919. Der Wertvorrat einer Bilinearform. Mathematische Zeitschrift 3 314–316] and a result of Brickman [Brickman, L. 1961. On the field of values of a matrix. Proc. Amer. Math. Soc. 12 61–66] on the joint numerical range.