Sufficient Linear Matrix Inequality Research Articles

Fixed-Order Linear Parameter-Varying Feedback Control of a Lab-Scale Overhead Crane

This brief presents a numerically attractive approach to design fixed-order $ \mathcal {H}_{2}/ \mathcal {H}_{\infty }$ controllers for discrete-time linear parameter-varying (LPV) systems. In this approach, the controller order that is completely determined by the number of states and the parameter dependence is selected in advance. For a prefixed controller order, parameter-dependent sufficient linear matrix inequalities (LMIs) are presented, relying on an a priori computed full-order LPV controller that stabilizes the LPV system for all the possible parameter trajectories. Polya’s theorem and polynomial approximations are used to obtain numerically tractable LMI problems that guarantee the feasibility of the parameter-dependent synthesis conditions. The practical viability of the approach is demonstrated by experimental validations on a lab-scale overhead crane with varying cable lengths.

Robust static output feedback synthesis for platoons under leader and predecessor feedback

SummaryA multi‐objective static output feedback synthesis problem is considered for the control of vehicle platoons under leader and predecessor feedback. Sufficient linear matrix inequality conditions are derived for the solvability of the problem in a way to facilitate static feed‐forward as well. A novel velocity‐dependent spacing policy is integrated into the control scheme together with a platoon model in which the emphasis on the predecessor information can be adjusted by a normalized scalar weight. It is shown that the string stability of the spacing errors and the acceleration signals can always be guaranteed by choosing this weight sufficiently small. Moreover, provided that the time headway is chosen sufficiently large, the synthesis can be performed in a way to avoid the amplification of acceleration energies if compared with the leader. As a particularly convenient feature, the target spacing between the vehicles becomes smaller when moving backward along the platoon. The total increase in the platoon length caused by the introduction of the velocity‐dependent scheme is shown to be bounded and decreasing with decreasing predecessor weight. It is also established that the predecessor weight can be adjusted smoothly over time without endangering the formation stability. In addition to the optimization of the parameters of common fixed‐structure controllers for general vehicle models, the proposed synthesis procedure provides various tools for improving robustness against measurement noise, communication delay, and model uncertainty. Copyright © 2016 John Wiley & Sons, Ltd.

A Convex Characterization of Robust Stability for Positive and Positively Dominated Linear Systems

We provide convex necessary and sufficient conditions for the robust stability of linear positively dominated systems. In particular, we show that the structured singular value is always equal to its convex upper bound for nonnegative matrices and we use this result to derive necessary and sufficient Linear Matrix Inequality (LMI) conditions for robust stability that involve only the system's static gain. We show how this approach can be applied to test the robust stability of the Foschini–Miljanic algorithm for power control in wireless networks in presence of uncertain interference.

ROBUST STABILITY, STABILISATION AND H-INFINITY CONTROL FOR PREMIUM-RESERVE MODELS IN A MARKOVIAN REGIME SWITCHING DISCRETE-TIME FRAMEWORK

AbstractThe premium pricing process and the medium- and long-term stability of the reserve policy under conditions of uncertainty present very challenging issues in relation to the insurance world. Over the last two decades, applications of Markovian regime switching models to finance and macroeconomics have received strong attention from researchers, and particularly market practitioners. However, relatively little research has so far been carried out in relation to insurance. This paper attempts to consider how a linear Markovian regime switching system in discrete-time could be applied to model the medium- and long-term reserves and the premiums (abbreviated here as the P-R process) for an insurer. Some recently developed techniques from linear robust control theory are applied to explore the stability, stabilisation and robustH∞-control of a P-R system, and the potential effects of abrupt structural changes in the economic fundamentals, as well as the insurer's strategy over a finite time period. Sufficient linear matrix inequality conditions are derived for solving the proposed sub-problems. Finally, a numerical example is presented to illustrate the applicability of the theoretical results.

Robust reliable dissipative filtering for Markovian jump nonlinear systems with uncertainties

SummaryThis paper investigates the problem of robust reliable dissipative filtering for a class of Markovian jump nonlinear systems with uncertainties and time‐varying transition probability matrix described by a polytope. Our main attention is focused on the design of a reliable dissipative filter performance for the filtering error system such that the resulting error system is stochastically stable and strictly dissipative. By introducing a novel augmented Lyapunov–Krasovskii functional, a new set of sufficient conditions is obtained for the existence of reliable dissipative filter design in terms of linear matrix inequalities (LMIs). More precisely, a sufficient LMI condition is derived for reliable dissipative filtering that unifies the conditions for filtering with passivity and H∞ performances. Moreover, the filter gains are characterized in terms of solution to a set of linear matrix inequalities. Finally, two numerical examples are provided to demonstrate the effectiveness and potential of the proposed design technique. Copyright © 2016 John Wiley & Sons, Ltd.

Fault detection for switched T-S fuzzy systems in finite frequency domain

This paper is concerned with the problem of the fault detection for switched T-S fuzzy systems with actuator faults. The system faults and unknown disturbances are considered to be in a finite frequency domain. An effective fault detection filter is designed to measure the fault sensitivity and disturbance robustness. By using Parseval’s theorem and S-procedure, a new finite frequency method of fault detection for switched T-S fuzzy systems is formulated. Sufficient linear matrix inequalities (LMIs) conditions are proposed to design the fault detection filter, which can guarantee the finite frequency $H_{-}$ and $H_{\infty}$ performance. The effectiveness of the given finite frequency method for switched T-S fuzzy systems is illustrated through two numerical simulation examples.

Optimal H2 Output-Feedback Control of Sampled Systems

Abstract: This paper addresses the problem of output feedback control with limited data-rate constraints. These communication constraints may affect the transmission of control signals or measurements, giving rise to three different structures: LSRA (local sensor - remote actuator), RSLA (remote sensor - local actuator) and RSRA (remote sensor - remote actuator). The interconnection between plant and controller is a Hybrid System whose stability and H2 performance can be determined in terms of necessary and sufficient Linear Matrix Inequalities. However, for the design of the appropriate feedback gains, those conditions become non-convex and we propose an approach to obtain suboptimal solutions based on a Cone Complementarity algorithm. Numerical examples are included.

Reset strategy for consensus in networks of clusters

This paper addresses the problem of consensus in networks structured in several clusters. The clusters are represented by fixed, directed and strongly connected graphs. They are composed by a number of agents which are able to interact only with other agents belonging to the same cluster. To every agent we associate a scalar real value representing its state. The states continuously evolve following a linear consensus protocol and approach local agreements specific to each cluster. In order to enforce a global agreement over the whole network, we consider that each cluster contains an agent that can be exogenously controlled. The state of this agent, called leader, will be quasi-periodically reseted by a local master controller that receives information from some neighboring leaders. In order to control the consensus value we have to firstly characterize it. Precisely we show that it depends only on the initial condition and the interaction topologies. Secondly, we provide sufficient Linear Matrix Inequality (LMI) conditions for the global uniform exponential stability of the consensus in presence of a quasi-periodic reset rule. The study of the network behavior is completed by a decay rate analysis. Finally we design the interaction network of the leaders which allows to reach a prescribed consensus value. Numerical implementation strategy is provided before illustrating the results by some simulations.

<inline-formula> <tex-math notation="TeX">${\cal H}_{\infty}$</tex-math></inline-formula> and <inline-formula> <tex-math notation="TeX">${\cal H}_{2}$</tex-math></inline-formula> Norms of 2-D Mixed Continuous-Discrete-Time Systems via Rationally-Dependent Complex Lyapunov Functions

This paper addresses the problem of determining the ${\cal H}_{\infty}$ and ${\cal H}_{2}$ norms of 2-D mixed continuous-discrete-time systems. The first contribution is to propose a novel approach based on the use of complex Lyapunov functions with even rational parametric dependence, which searches for upper bounds on the sought norms via linear matrix inequalities (LMIs). The second contribution is to show that the upper bounds provided are nonconservative by using Lyapunov functions in the chosen class with sufficiently large degree. The third contribution is to provide conditions for establishing the tightness of the upper bounds. The fourth contribution is to show how the numerical complexity of the proposed approach can be significantly reduced by proposing a new necessary and sufficient LMI condition for establishing positive semidefiniteness of even Hermitian matrix polynomials. This result is also exploited to derive an improved necessary and sufficient LMI condition for establishing exponential stability of 2-D mixed continuous-discrete-time systems. Some numerical examples illustrate the proposed approach. It is worth remarking that nonconservative LMI methods for determining the ${\cal H}_{\infty}$ and ${\cal H}_{2}$ norms of 2-D mixed continuous-discrete-time systems have not been proposed yet in the literature.

Robust Control of Switched Linear Systems with Output Switching Strategy

This paper investigates the robust control problem of continuous-time uncertain switched linear systems, using only a switching strategy depending on the plant output. The proposed method is based on linear matrix inequalities (LMIs). A set of slack variables is introduced to reduce the design conservatism, and new sufficient LMI conditions for the synthesis of the controllers are presented. Two examples show that the proposed method has an adequate performance even in situations when the matrices of the linear subsystems are not Hurwitz and offers a simple and efficient solution for this control problem.

State-space [formula omitted] controller design for descriptor systems

This paper proposes a new linear matrix inequality (LMI) method to design state-space H∞ controllers for linear time-invariant descriptor systems. Unlike preceding studies, where descriptor-type controllers are first computed and then numerically transformed to state-space controllers, the proposed method carries out the transformation analytically in the parameter domain. We derive a necessary and sufficient LMI condition for the existence of a state-space controller with the same dynamic order of the descriptor system to be controlled, which makes the closed-loop system regular, impulse-free, stable, and guarantees the H∞ norm bound imposed on the closed-loop transfer function. Furthermore, we present parameterization of all such state-space controllers by variables satisfying the LMI condition and an arbitrary nonsingular matrix. The LMIs utilized in this paper are strict ones, that is, those containing no equality, while LMIs with equality constraints have been extensively used in the analysis and design for descriptor systems. The strict LMIs play key roles in deriving the results of this paper.

Tracking of REMUS autonomous underwater vehicles with actuator saturations

This paper addresses on a horizontal waypoint tracking problem for a class of nonlinear autonomous underwater vehicles (AUVs) with actuator saturations while keeping its constant surge speed. The concerned problem is formulated as an exponential stabilization of the error dynamics with respect to the desired surge speed and the desired yaw angle determined from the line of sight. It is shown that the feedback linearization technique makes the separate design of the available control inputs, the propeller thrust and the rudder angle, possible. By using the sector nonlinearity, the error dynamics is modeled as a polytopic linear parameter varying (LPV) system. Then, sufficient linear matrix inequality (LMI) conditions for its regional exponential stabilizability are derived in the sense of Lyapunov stability criterion. An example is provided to illustrate the effectiveness of the proposed methodology.

Sufficient LMI conditions for reduced-order multi-objective [formula omitted] control of LTI systems

This paper presents a novel projection lemma based linear matrix inequality (LMI) framework to design reduced-order multi-objective H2/H∞ controllers for linear time-invariant systems. This framework relies on a set of full-order H2/H∞ controllers, which are used as parameters in sufficient LMIs for the reduced-order controller design. Continuous-time and discrete-time controller designs are treated in a unified fashion. It is theoretically and numerically demonstrated that the approach allows the computation of reduced-order controllers that are potentially less conservative than full-order designs resulting from well-known LMI approaches. Various comparisons with existing reduced-order controller design approaches illustrate the potential of the proposed framework of sufficient LMIs.

Observer-based controller for nonlinear analytical systems

In this paper, we propose to design a polynomial observer-based control for nonlinear systems and to determine sufficient linear matrix inequality (LMI) global stabilisation conditions of the polynomial controlled system augmented by its observer. The design of the observer-based control leverages some notations from the Kronecker product and the power of matrices properties for the state space description of polynomial systems. The stability study of the polynomial controlled system augmented by its observer is based on the Lyapunov stability direct method. Intensive simulations are performed to illustrate the validity and the effectiveness of the polynomial approach used to design the control.

Exponential H∞ filtering for discrete-time switched neural networks with random delays.

This paper addresses the exponential H∞ filtering problem for a class of discrete-time switched neural networks with random time-varying delays. The involved delays are assumed to be randomly time-varying which are characterized by introducing a Bernoulli stochastic variable. Effects of both variation range and distribution probability of the time delays are considered. The nonlinear activation functions are assumed to satisfy the sector conditions. Our aim is to estimate the state by designing a full order filter such that the filter error system is globally exponentially stable with an expected decay rate and a H∞ performance attenuation level. The filter is designed by using a piecewise Lyapunov-Krasovskii functional together with linear matrix inequality (LMI) approach and average dwell time method. First, a set of sufficient LMI conditions are established to guarantee the exponential mean-square stability of the augmented system and then the parameters of full-order filter are expressed in terms of solutions to a set of LMI conditions. The proposed LMI conditions can be easily solved by using standard software packages. Finally, numerical examples by means of practical problems are provided to illustrate the effectiveness of the proposed filter design.

Delay‐dependent Stability and Static Output Feedback Control of 2‐D Discrete Systems with Interval Time‐varying Delays

AbstractThis paper is concerned with the problems of delay‐dependent stability and static output feedback (SOF) control of two‐dimensional (2‐D) discrete systems with interval time‐varying delays, which are described by the Fornasini‐Marchesini (FM) second model. The upper and lower bounds of delays are considered. Applying a new method of estimating the upper bound on the difference of Lyapunov function that does not ignore any terms, a new delay‐dependent stability criteria based on linear matrix inequalities (LMIs) is derived. Then, given the lower bounds of time‐varying delays, the maximum upper bounds in the above LMIs are obtained through computing a convex optimization problem. Based on the stability criteria, the SOF control problem is formulated in terms of a bilinear matrix inequality (BMI). With the use of the slack variable technique, a sufficient LMI condition is proposed for the BMI. Moreover, the SOF gain can be solved by LMIs. Numerical examples show the effectiveness and advantages of our results.

Necessary and Sufficient LMI Conditions for Stability and Performance Analysis of 2-D Mixed Continuous-Discrete-Time Systems

This paper proposes necessary and sufficient conditions for stability and performance analysis of 2-D mixed continuous-discrete-time systems that can be checked with convex optimization, in particular linear matrix inequalities (LMIs). Specifically, the first contribution of the paper is a condition for exponential stability based on the introduction of a complex Lyapunov function depending polynomially on a parameter and on the use of the Gram matrix method. It is shown that this condition is sufficient for any chosen degree of the complex Lyapunov function, and necessary for an a priori known degree. The second contribution is a non-Lyapunov condition for exponential stability based on eigenvalue products. This condition is necessary and sufficient, and has the advantage of requiring a significantly smaller computational burden for achieving necessity. Lastly, the third contribution is to show how upper bounds on the H∞ norm of 2-D mixed continuous-discrete-time systems can be obtained through a semidefinite program based on complex Lyapunov functions. A necessary and sufficient condition is provided for establishing the tightness of the found upper bounds.

Optimal H2 and H∞ Mode-Independent Control for Generalized Bernoulli Jump Systems

This paper deals with state-feedback control of discrete-time linear jump systems. The random variable representing the system modes has a generalized Bernoulli distribution, which is equivalent to a Markov chain where the transition probability matrix has identical rows. Another assumption is about the availability of the mode to the controller. We derive necessary and sufficient linear matrix inequalities (LMI) conditions to design optimal H2 and H∞ state-feedback controllers for the particular class of transition probabilities under consideration and subject to partial mode availability constraints or equivalently cluster availability constraints, which include mode-dependent and mode-independent designs as particular cases. All design conditions are expressed in terms of LMIs. The results are illustrated through a numerical example.

Optimal and mode-independent filters for generalised Bernoulli jump systems

This paper provides the optimal solution of the filtering design problem for a special class of discrete-time Markov jump linear systems whose transition probability matrix has identical rows. In the two-mode case, this is equivalent to saying that the random variable has a Bernoulli distribution. For that class of dynamic systems we design, with the help of new necessary and sufficient linear matrix inequality conditions, and optimal mode-independent filters with the same order of the plant. As a first proposal available in the literature, for partial information characterised by cluster availability of the mode, we also show it is possible to design optimal full-order linear filters. If some plant matrices do not vary within the same cluster, we show that the optimal filter exhibits the internal model structure. We complete the results with illustrative examples. A realistic practical application considering sensors connected to a network using a communication protocol such as the Token Ring is included in order to put in evidence the usefulness of the theoretical results.

Sufficient and Necessary LMI Conditions for Robust Stability of Rationally Time-Varying Uncertain Systems

This technical note addresses robust stability of uncertain systems with rational dependence on unknown time-varying parameters constrained in a polytope. First, the technical note proves that a sufficient linear matrix inequality (LMI) condition that we previously proposed, based on homogeneous polynomial Lyapunov functions (HPLFs) and on the introduction of an extended version of Polya's theorem, is also necessary. Second, the technical note proposes a new sufficient and necessary LMI condition by exploiting properties of the simplex and sum-of-squares (SOS) parameter-dependent polynomials. Lastly, the technical note investigates relationships among these conditions and conditions based on the linear fractional representation (LFR). It is worth remarking that sufficient and necessary LMI conditions for this problem have not been proposed yet in the literature.