Parameterized Linear Matrix Inequalities Research Articles

On the State-Feedback Controller Design for Polynomial Linear Parameter-Varying Systems with Pole Placement within Linear Matrix Inequality Regions

The present paper addresses linear parameter-varying systems with high-order time-varying parameter dependency known as polynomial LPV systems and their controller design. Throughout this work, a procedure ensuring a state-feedback controller from a parameterized linear matrix inequality (PLMI) solution is presented. As the main contribution of this paper, the controller is designed by considering the time-varying parameter rate as a tuning parameter with a continuous control gain in such a way that the closed-loop eigenvalues lie in a complex plane subset, with high-order time-varying parameters defining the system dynamics. Simulation results are presented, aiming to show the effectiveness of the proposed controller design.

A novel convex relaxation technique on affine transformed sampled-data control issue for fuzzy semi-Markov jump systems

This article investigates affine transformed sampled-data control problems for fuzzy semi-Markov jump systems (FSMJSs). First of all, in the novel fuzzy sampled-data control, an affine transformed membership function is introduced, which contributes to constructing the synchronous time scale grades of membership without any constraint condition. Then, by utilizing a mode-dependent Lyapunov function with the looped functions, a sufficient condition concerning the asymptotical stability of the closed-loop FSMJSs is established in the form of linear matrix inequality (LMI). Meanwhile, to solve parameterized LMI (PLMI), a novel convex relaxation technique is proposed, based on which less conservatism stabilization criteria of FSMJSs, and a maximum sampling interval with respect to sampled-data control are further derived. Finally, two examples are carried out to manifest numerically the validity of the raised method.

Admissibility and Design Issues for T-S Fuzzy Descriptor Systems With Perturbed Derivative Matrices in the Rules

This article mainly investigates the admissibility analysis and the parallel distributed compensator (PDC) for fuzzy descriptor systems with perturbed derivative matrices in the rules. The standard Takagi–Sugeno (T–S) fuzzy model is generalized into a fuzzy descriptor model with the perturbed derivative matrices existing in each rule, which can be used to represent a larger class of nonlinear and/or uncertain systems. Based on linear matrix inequalities (LMIs) technique, a new admissibility analysis condition for the considered fuzzy descriptor system is first presented, where the new results involve slack matrices for lessening the conservatism. Furthermore, we further cope with the fuzzy PDC synthesis associated with admissibility and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ρ</i> -admissibility assurance for the resulting closed-loop fuzzy descriptor systems. Since all the proposed criteria can be explicitly formulated in terms of LMIs or parametric LMIs, we can handily verify them via current LMI solvers. Finally, illustrative examples are given to demonstrate that the proposed approach is valid and effective.

Dissipative filtering for two-dimensional LPV systems: A hidden Markov model approach

In this article, the dissipative filtering problem is explored for two-dimensional (2-D) Markov jump linear parameter varying (MJLPV) systems. The underlying system is described based on the frequently-used Fornasini–Marchesini (FM) model with the measurement missing phenomenon. The hidden Markov model (HMM) is adopted to depict the partial accessibility between system and the designed filter modes. Based on this, a HMM-based filter is proposed, and sufficient conditions in the form of parameterized linear matrix inequalities (PLMIs) are established to guarantee that the filtering error system (FES) is asymptotically stable (AS) and strictly dissipative. Then, the corresponding filter synthesis problem is converted into a convex optimization problem. Ultimately, the simulation example is presented to demonstrate the utility of the obtained results.

Improvement of Slack Variable-Free Relaxation Method and its Application to Control Design of T-S Fuzzy Systems

The relaxation method of [1, Theorem 2.2] has been used in various studies to deal with parameterized linear matrix inequalities (PLMIs) without excessively increasing computational complexity. Inspired by this point, this paper proposes an improved relaxation method that can further reduce the conservatism of [1, Theorem 2.2] despite no additional slack variables, as well as reduce the number of LMIs (and LMI rows). To be specific, the reduction of both conservatism and computational complexity is verified through the well-known stabilization problems of both continuous-time and discrete-time T-S fuzzy systems, which shows the potential of the proposed method for its extension to other PLMI-based stabilization conditions.

Admissibility Analysis and Controller Design for Discrete Singular Time-Delay Systems Embracing Uncertainties in the Difference and Systems’ Matrices

This paper mainly investigates the admissibility analysis and the admissibilizing controller design for the uncertain discrete singular system with delayed state. Based on Lyapunov–Krasovskii stability theory, an original admissibility condition for the nominal singular delay system is first presented. By involving the uncertainties in both difference and system matrices simultaneously, we devote to analyzing the robust admissibility for the regarded uncertain discrete singular system with delayed state. Furthermore, by hiring the state feedback control law, we further discuss the admissibilizing controller design for the resulting closed-loop system. Since all the derived criteria are expressed in terms of strict linear matrix inequalities (LMIs) or parametric LMIs, we thus can handily verify them via current LMI solvers. Finally, two numerical examples are given to illustrate the effectiveness and validity of the proposed approach.

Leader-Following Consensus for Multi-Agent Systems With Asynchronous Control Modes Under Nonhomogeneous Markovian Jump Network Topology

Under a nonhomogeneous Markovian jump network topology, this paper addresses the problem of leader-following consensus for multi-agent systems with an asynchronous control mode. Specifically, based on an asynchronous mode-dependent Lyapunov function and a suitable relaxation process, sufficient conditions for leader-following consensus are derived in such a way that 1) non-convex terms multiplied by time-varying signed parameters can be addressed, 2) the emergence of an asynchronous control mode can be reflected in the control synthesis of the follower agents, and 3) the constraints of two combined parameters can be explicitly imposed on the process of transforming parameterized linear matrix inequalities (PLMIs) into LMIs. Finally, two illustrative examples are given to show the validity of the proposed method.

Smooth Switching LPV Dynamic Output-feedback Control

In this paper, we present an innovative design of smooth-switching LPV (Linear Parameter-Varying) dynamic output-feedback (DOF) controllers. For a given partition of scheduling parameter region, a family of LPV controllers are designed simultaneously with guaranteed system performance on each subregion and switching smoothness between adjacent subregions. The proposed control design, called smooth-switching mixed Input Covariance Constraint (ICC) and $${{\cal H}_\infty}$$ LPV control design, minimizes a combined cost of system output $${{\cal H}_2}$$ performance and smooth-switching index subject to $${{\cal H}_2}$$ constraints on control inputs and $${{\cal H}_\infty}$$ performance constraint. These stability and performance criteria are then formulated into a set of Parametric Linear Matrix Inequalities (PLMIs). In addition, a tunable coefficient is introduced in cost function to provide an optimal trade-off between system $${{\cal H}_2}$$ performance and switching smoothness, and therefore, the corresponding optimal LPV controllers can be derived iteratively by convex optimization. For illustration, an active magnetic bearing (AMB) example is used to show the effectiveness of the proposed simultaneous design approach by demonstrating significantly improved switching smoothness and an optimal trade-off between achievable output performance and switching smoothness.

Dynamic output-feedback control for singular T–S fuzzy systems using fuzzy Lyapunov functions

In light of fuzzy weighting-dependent Lyapunov functions, a dynamic output-feedback control for singular T–S fuzzy systems is introduced. Based on a set-equivalence technique and fuzzy weighting-dependent Lyapunov functions, this paper derives a sufficient condition ensuring the singular T–S fuzzy systems to be admissible (regular, impulse-free and stable) in the form of strict parametric linear matrix inequalities (PLMIs). Then, for the closed-loop systems with a dynamic, rather than static, output-feedback controller, an admissibility criterion is given with strict PLMIs by specially representing the block entries of matrices in Lyapunov function. By appropriately choosing the structures of PLMI variables, the strict LMIs are obtained, where slack variables are included in the relaxation process fully exploiting the properties of the fuzzy weighting functions. Three numerical examples are provided to show the effectiveness of the introduced dynamic control.

Robust L2-L∞ Filtering for LPV Systems with Distributed Delays

The linear parameter-varying(LPV) systems with distributed delays are studied by the robust filter design problem. Firstly, the parameter-dependent Lyapunov theory is used to propose the L2-L∞ performance criterion of time-delay correlation. Introducing the projection to understand the coupling between the Lyapunov parameter matrix and the system parameter matrix, and obtaining the L2-L∞ performance criterion which is easier to solve. Then the sufficient conditions for the existence of the filter are obtained. The parameterized linear matrix inequality(PLMIs) with infinite dimension is converted to LMIs with finite dimension by using approximate basis function and network technology. Finally, the feasibility of this method is verified by numerical simulation.

Synthesis Conditions for LPV Controller with Input Covariance Constraints

Abstract In this paper, novel synthesis conditions for state-feedback Linear Parameter Varying (LPV) controller with Input Covariance Constraints (ICC) are developed. The synthesis conditions achieve the following design requirements 1) some constraints need to be satisfied on the control energy and 2) optimizing the performance outputs for the entire parameter space of the LPV system. These conditions are formulated as convex optimization problem with Parameterized Linear Matrix Inequalities (PLMIs) constraints. The effectiveness of the proposed approach is illustrated through numerical examples.

Novel Stabilization Criteria for T–S Fuzzy Systems With Affine Matched Membership Functions

This paper presents a new parallel distributed compensation controller design approach for T–S (Takagi–Sugeno) fuzzy control systems with affine matched membership functions in the system and controller. In the new fuzzy control, affine transformed membership functions are adopted by scaling and biasing the original membership functions of the system. Stabilization and performance criterion of the closed-loop T–S fuzzy systems are obtained through a new parameterized linear matrix inequality, which is rearranged by affine matched membership functions. The conservativeness of stabilization condition for the T–S fuzzy system is significantly relaxed by utilizing the constraints condition of the controllers membership functions, which is determined from the difference of each transformed membership function. In addition, the controller gain is reconstructed by a decision variable separation technique with two different free weighting matrices without any scaling parameter. The superiority of proposed method is verified through numerical examples.

Smooth-switching LPV control for vibration suppression of a flexible airplane wing

In this paper, active vibration suppression of a Blended-Wing-Body flexible airplane wing is studied by utilizing a smooth-switching linear parameter-varying (LPV) dynamic output-feedback control. For the reduced-order LPV models, developed for each divided flight envelop subregion, a family of mixed Input Covariance Constraint and H∞ LPV controllers are designed to robustly suppress the wing bending displacement using hard-constrained control surfaces, while achieving smooth-switching between adjacent controllers. The proposed LPV controllers are developed by minimizing a combination of weighted H2 output performance and smoothness index, subject to a set of Parametric Linear Matrix Inequalities derived from stability and performance conditions. In addition, the weighting coefficient in the cost function is tuned to balance between H2 performance and switching smoothness by iteratively solving convex optimization problems. Simulation results demonstrate that simultaneous smooth-switching and improved performance can be achieved by the proposed LPV control.

H∞Control for LPV Discrete Systems with Random Time-Varying Network Delay

In this paper, we study theH∞control problem for Linear Parameter Varying (LPV) discrete systems with random time-varying network delay. The state matrices of LPV discrete systems are deterministic functions and changed with parameters; the range of parameters is measurable. Considering the characteristics of networks with random time-varying delay, we proposed a new parameter-dependentH∞performance criterion based on the Lyapunov stability theory. The coupling between Lyapunov functions and system matrices could be eliminated by introducing an additional matrix in this criterion, which made it easier for numerical implementation. On this basis, we designed a state feedback controller by virtue of linear matrix inequalities, which transformed the sufficient conditions into existence condition of solution of parametric linear matrix inequalities. The designed controller could keep the closed-loop system asymptotically stable under given time delay and probability and meet predefined performance metric. The validity of the proposed method is verified by numerical simulation.

Interval-Boundary-Dependent Control for Interval Type-2 T–S Fuzzy Systems

This paper introduces a state-feedback controller for interval type-2 Takagi–Sugeno fuzzy systems that are described as a convex combination of linear subsystems via normalized time-varying fuzzy weighting parameters, which are unknown but belong to intervals. The introduced structure employs lower and upper interval boundaries directly, rather than alternative parameters generated by the interval boundaries in the studies, which does lead to a stabilization condition of parameterized linear matrix inequalities (LMIs), associated only with the parameters and their interval boundaries. The exact relationship among the parameters and their interval boundaries can, therefore, be fully used to convert parameterized LMIs into LMIs, which provides a great feature distinguished from those in the studies. Two illustrative examples demonstrate the effectiveness and robustness of the derived controller.

Application of ICC LPV control to a blended-wing-body airplane with guaranteed [formula omitted] performance

This paper addresses the Input Covariance Constraint (ICC) control problem with guaranteed H∞ performance for continuous-time Linear Parameter-Varying (LPV) systems. The upper bound of the output covariance is minimized subject to the constraints on input covariance and H∞ output performance. This problem is an extension of the mixed H2/H∞ LPV control problem, in that the resulting gain-scheduling controllers guarantee not only closed-loop system robustness in terms of H∞ norm bound but also output covariance performance over the entire scheduling parameter space. It can be shown that this problem can be efficiently solved by utilizing the convex optimization of Parameterized Linear Matrix Inequalities (PLMIs). The main contributions of this paper are to characterize the mixed ICC/H∞ LPV control problem using PLMIs and to develop the optimal state-feedback gain-scheduling controllers, while satisfying both input covariance and H∞ constraints. The effectiveness of the proposed control scheme is demonstrated through vibration suppression of a blended-wing-body airplane model.

Formula omitted] control for Markovian jump fuzzy systems with partly unknown transition rates and input saturation

This paper considers less conservative conditions of H∞ control for Markovian jump fuzzy systems (MJFSs) with partly unknown transition rates and input saturation. To find H∞ control for level γ, a set invariance condition and the stabilization conditions are first formulated in terms of parameterized linear matrix inequalities (PLMIs) by considering the properties of transition rates. Then, to derive the less conservative stabilization conditions, all possible slack variables are incorporated into the relaxation process with fully considering the property of the fuzzy weights. Two numerical examples are provided to illustrate the effectiveness of the proposed method.

Static output-feedback robust gain-scheduling control with guaranteed [formula omitted] performance

This paper develops synthesis conditions for Static Output-Feedback (SOF) Gain-Scheduling (GS) control with guaranteed upper bound of H2 performance for continuous-time Linear Parameter Varying (LPV) systems, where measurements of scheduling parameters are affected by uncertainties or measurement noise. The control problem is solved through an iterative two-stage design procedure. In the first stage, parameter-dependent state-feedback controller is obtained to minimize upper-bound of the H2 performance. Then, this controller is used as input to the second stage to synthesize Robust Gain-Scheduling (RGS) static output-feedback gain with minimal H2 performance. In both stages, the synthesis conditions are given in terms of Parametrized Linear Matrix Inequalities (PLMIs). Robust SOF (parameter-independent) controller can be synthesized as special case of the developed synthesis conditions. Two examples have been presented to illustrate the benefit of the proposed approach. One is an academic numerical example from literature and the other one is a realistic LPV model of Electric Variable Valve Timing (EVVT) actuator for automotive engines that utilizes engine speed and vehicle battery voltage as time-varying scheduling parameters.

Exponential Pointwise Stabilization of Semilinear Parabolic Distributed Parameter Systems via the Takagi–Sugeno Fuzzy PDE Model

This paper deals with the problem of exponential stabilization for nonlinear parabolic distributed parameter systems using the Takagi–Sugeno (T–S) fuzzy partial differential equation (PDE) model, where a finite number of actuators are active only at some specified points of the spatial domain (these actuators are referred to as pointwise actuators). Three cases of state feedback are respectively considered in this study as follows: full state feedback, piecewise state feedback, and collocated pointwise state feedback. It is initially assumed that a T–S fuzzy PDE model obtained via the sector nonlinearity approach is employed to accurately represent the semilinear parabolic PDE system. Based on the obtained T–S fuzzy PDE model, Lyapunov-based design methodologies of fuzzy feedback control laws are subsequently derived for the above three state feedback cases by using the vector-valued Wirtinger's inequality to guarantee locally exponential pointwise stabilization of the semilinear PDE system, and presented in terms of standard linear matrix inequalities (LMIs). Moreover, the favorable property offered by sharing all the same premises in the T–S fuzzy PDE models and fuzzy controllers is not applicable for the case of collocated pointwise state feedback. A parameterized LMI is introduced for this case to enhance the stabilization ability of the fuzzy controller. Finally, the merit and effectiveness of the proposed design methods are demonstrated by numerical simulation results of two examples.

Parameterized LMI Based Diagonal Dominance Compensator Study for Polynomial Linear Parameter Varying System

For dynamic decoupling of polynomial linear parameter varying(PLPV) system, a robust dominance pre-compensator design method is given. The parameterized precompensator design problem is converted into an optimal problem constrained with parameterized linear matrix inequalities(PLMI) by using the conception of parameterized Lyapunov function(PLF). To solve the PLMI constrained optimal problem, the precompensator design problem is reduced into a normal convex optimization problem with normal linear matrix inequalities (LMI) constraints on a new constructed convex polyhedron. Moreover, a parameter scheduling pre-compensator is achieved, which satisfies robust performance and decoupling performances. Finally, the feasibility and validity of the robust diagonal dominance pre-compensator design method are verified by the numerical simulation on a turbofan engine PLPV model.