Sufficient Dilated LMI Conditions for Static Output Feedback Robust Stabilization of Linear Continuous-Time Systems

New sufficient dilated linear matrix inequality (LMI) conditions for the ∞ static output feedback control problem of linear continuous-time systems with no uncertainty are proposed. The used technique easily and successfully extends to systems with polytopic uncertainties, by means of parameter-dependent Lyapunov functions (PDLFs). In order to reduce the conservatism existing in early standard LMI methods, auxiliary slack variables with even more relaxed structure are employed. It is shown that these slack variables provide additional flexibility to the solution. It is also shown, in this paper, that the proposed dilated LMI-based conditions always encompass the standard LMI-based ones. Numerical examples are given to illustrate the merits of the proposed method

H∞ and L2–L∞ filtering for two-dimensional linear parameter-varying systems

This is the post print version of the article. The official published version can be obtained from the link below - Copyright 2007 Wiley-BlackwellIn this paper, the H∞ and l2–l∞ filtering problem is investigated for two-dimensional (2-D) discrete-time linear parameter-varying (LPV) systems. Based on the well-known Fornasini–Marchesini local state-space (FMLSS) model, the mathematical model of 2-D systems under consideration is established by incorporating the parameter-varying phenomenon. The purpose of the problem addressed is to design full-order H∞ and l2–l∞ filters such that the filtering error dynamics is asymptotic stable and the prescribed noise attenuation levels in H∞ and l2–l∞ senses can be achieved, respectively. Sufficient conditions are derived for existence of such filters in terms of parameterized linear matrix inequalities (PLMIs), and the corresponding filter synthesis problem is then transformed into a convex optimization problem that can be efficiently solved by using standard software packages. A simulation example is exploited to demonstrate the usefulness and effectiveness of the proposed design method

Fixed-Order H-infinity Controller Design via Convex Optimization Using an Alternative to Youla Prameterization

All H-infinity controllers of a SISO LTI system are parameterized thanks to the relation between Bounded Real Lemma and Positive Real Lemma and a new concept of strict positive realness of two transfer functions with the same Lyapunov matrix in the matrix inequality of the Kalman-Yakubovic-Popov lemma. This new parameterization shares the same features with Youla parameterization, namely on the convexity of H-infinity norm constraints for the closed-loop transfer functions. However, in contrary to Youla parameterization, it can deal with any controller order and any controller structure such as e.g. PID. The main feature of the proposed method is that it can be extended easily for the systems with polytopic uncertainty. This way, a convex inner approximation of all H-infinity controllers for polytopic systems is given, which can be enlarged by increasing the controller order. In order to design a low-order robust H-infinity controller with less conservatism, rank of the k-th Sylvester resultant matrix of the controller is made to be deficient via a convex approximation of the rank minimization problem. The effectiveness of the proposed method is shown via simulation results

Frequency-Domain Control Design in Power Systems

The scope of this thesis encompasses two main subjects: fixed-structure data-driven control design on one side, and control design in power systems on the other. The overall goal is to identify challenging and relevant problems in power systems, to express them as rigorous specifications from the viewpoint of control systems, and to solve them by developing and applying advanced methods in robust control. This work aims to combine expertise from both fields to open up a holistic perspective and bridge the gap between control and power systems. First, the derivation of a novel fixed-structure, data-driven frequency-domain control design method for multivariable systems is described. A key feature of the method is that only the frequency response of the plant is required for the design, and no parametric model is required. The designed controllers are fully parametrized in terms of matrix polynomial functions and can take a centralized, decentralized or distributed structure. The controller performance is formulated as H_2 and H_infinity constraints on any loop transfer function. A convex formulation of the optimization problem is derived, and it is shown that the solution converges to a locally optimal solution of the original problem. The versatility of the design method is demonstrated in various simulation examples, as well as in experiments on two electromechanical setups. Next, a frequency-domain modeling approach for power grids is discussed. A model based on dynamic phasors is developed that represents the electromagnetic and electromechanic dynamics of lines, inverters, synchronous machines and constant power loads. It also offers a modular structure that makes it straightforward to combine white-, grey- and blackbox models in a single framework. Then, the control design method and dynamic phasor model are applied in two relevant power systems case studies. First, the design of a decentralized current controller for parallel, grid-connected voltage source inverters in a typical distribution grid is considered. It is shown how performance specifications can be formulated as frequency-domain constraints in order to attenuate the resonances introduced by the output filters and coupling effects, and to provide robustness against model uncertainties and grid topology changes. The controllers for all VSIs are designed in a single step, and stability and performance is guaranteed by design. Furthermore, an approach for plug-and-play control design is presented. The results are validated in numerical simulation as well as in power-hardware-in-the-loop experiments. The second study concerns the design of a distributed controller that combines primary and secondary frequency and voltage control for an islanded, meshed low-voltage grid with any number of voltage source inverters and synchronous generators in a single framework. No assumption on the R/X-ratio of the lines is made, and it is shown how advanced control specifications such as proportional active power sharing, zero frequency steady-state error and decoupling can be formulated as constraints on the norm of weighted sensitivity functions. Furthermore, the communication delays of the distributed controller are considered exactly during the design. The controller is implemented in numerical simulation, and results show significantly improved performance as compared to the classical hierarchical structure

Robust State Feedback Lmi Methods For Continuous-time Linear Systems: Discussions, Extensions And Numerical Comparisons

This paper provides a brief survey on the subject of LMI (Linear Matrix Inequality) methods for robust state feedback control design. The focus is on continuous-time linear systems with time-invariant uncertain parameters belonging to a polytope. Several LMI conditions from the literature are reviewed and discussed. The relationship between quadratic stabilizability (i.e. constant Lyapunov matrix) and LMI conditions based on parameter-dependent Lyapunov functions is highlighted. As a contribution, a generalization of a family of parameter-dependent conditions is proposed. Discussions, possible extensions and interpretations are provided along the presentation. Finally, the numerical efficacy of the LMI conditions in finding robust controllers when one stabilizing gain is known to exist is investigated. The methods have been tested against a set of hard uncertain systems that are guaranteed to be stabilized by some robust state feedback controller, including a large subset of problems which are known to be stabilized by some robust controller but not to be quadratically stabilizable by any controller. © 2011 IEEE.10381043Horisberger, H.P., Belanger, P.R., Regulators for linear, time invariant plants with uncertain parameters (1976) IEEE Trans. Autom. Control, 21, pp. 705-708Barmish, B.R., Stabilization of uncertain systems via linear control (1983) IEEE Trans. Autom. Control, 28 (8), pp. 848-850. , AugustBarmish, B.R., Necessary and sufficient conditions for quadratic stabilizability of an uncertain system (1985) J. Optim. 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H∞ And H2 Control Design For Polytopic Continuous-time Markov Jump Linear Systems With Uncertain Transition Rates

This paper investigates the problems of H∞ and H2 state feedback control design for continuous-time Markov jump linear systems. The matrices of each operation mode are supposed to be uncertain, belonging to a polytope, and the transition rate matrix is considered partly known. By appropriately modeling all the uncertain parameters in terms of a multi-simplex domain, new design conditions are proposed, whose main advantage with respect to the existing ones is to allow the use of polynomially parameter-dependent Lyapunov matrices to certify the mean square closed-loop stability. Synthesis conditions are derived in terms of matrix inequalities with a scalar parameter. The conditions, which become LMIs for fixed values of the scalar, can cope with H∞ and H2 state feedback control in both mode-independent and mode-dependent cases. Using polynomial Lyapunov matrices of larger degrees and performing a search for the scalar parameter, less conservative results in terms of guaranteed costs can be obtained through LMI relaxations. 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