LMI-based robust sampled-data stabilization of polytopic LTI systems: A truncated power series expansion approach

Establishing robust stability of discrete-time systems with time-varying uncertainty: The Gram-SOS approach

This paper reviews robust feedback control applications in power system. Robust control concept in power system control area evolves by the way of uncertainty modeling. Uncertainties can be caused by variation of loading, load parameters, network topology and machine parameters. One direct way to take into consideration of uncertainties is to enumerate various system operating conditions and use a set of linear time invariant systems to model the power systems. Enumeration method is used in robust control signal selection and robust controller design. Another way is to model uncertainties as structured or unstructured uncertainties. Controllers are designed based on nominal system condition to tolerate uncertainty as much as possible. The controller synthesis problem is solved by either Algebraic Ricatti Equation (ARE) or Linear Matrix Inequality (LMI). LMI is shown to be more flexible in controller design since it provides a way to specify system dynamic performance. Most recently, linear parameter varying (LPV) concept expands the use of enumeration method to express a set of linear time invariant (LTI) systems with state space dependent on linear parameters. LPV based control methods extend gain scheduling to design linear control for each operating conditions and overall a nonlinear controller for the varying parameter system.

It is well-known that determining the Mahler measure is important in networked control systems. Indeed, this measure allows one to derive stabilizability conditions in such systems. This paper investigates the Mahler measure in networked control systems linearly affected by a single uncertain parameter constrained into an interval, i.e. systems described by a matrix pencil. It is shown that conditions for establishing an upper bound of the largest Mahler measure over the matrix pencil can be formulated through linear matrix inequalities (LMIs). In particular, two LMI conditions are proposed, one based on the construction of a parameter-dependent Lyapunov function, and another based on eigenvalue analysis through the determinants of augmented matrices. The proposed LMI conditions have the advantage to be exact, i.e. they are sufficient for any size of the LMIs and they are also necessary for a certain size of the LMIs which is known a priori.

This article addresses the admissibility analysis and state-feedback control synthesis problems of discrete-time polytopic descriptor systems with possibly time-varying parameters. First, we present a new necessary and sufficient strict linear matrix inequality (LMI) condition for the admissibility analysis of linear time-invariant (LTI) descriptor systems. Then, based on the concept of poly-quadratic admissibility, introduced in this article, we extend this result to the admissibility analysis of possibly time-varying parameter-dependent descriptor systems. This extension, which applies to both uncertain and measurable parameters, uses parameter-dependent Lyapunov functions and employs two slack variables. The extended conditions are also expressed as strict LMI conditions which are easily tractable numerically compared to the non-strict ones often encountered when dealing with descriptor systems. Note that we have proposed two separate admissibility analysis conditions: one directly exploitable for state estimation and the other for state-feedback control. The need for different admissibility analysis conditions for each synthesis problem is motivated by the fact that the duality between state estimation and state-feedback control, which hold in the case of measurable parameters, does not hold when dealing with uncertain ones. In our approach, we overcome the absence of such duality by considering a dilation only on the dynamical part of the descriptor system. The application of our analysis result to both robust state-feedback control and polytopic state-feedback control are also presented in this article. Our analysis results extend to descriptor systems, some existing results developed for regular systems.

Convergent upper bounds of peak response of LTI and polytopic LTV systems through LMIs

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

Dissipative linear time-invariant (LTI) systems represent a large class of systems, which include bounded real, positive real and sector bounded systems as special cases, In this paper, a sufficient condition is developed for stability of the feedback interconnection of two dissipative LTI systems. The stability result generalizes and unifies previous stability results such as small gain, passivity and sector-boundedness theorems. State-space characterization of dissipative LTI systems is presented in terms of a "dissipativity lemma", which extends the positive realness (Kalman-Yakubovich), bounded realness and sector-boundedness lemmas. These conditions are equivalently expressed as linear matrix inequalities.

This paper is concerned with the problem of distributed multi-objective filters (DMFs) design for a class of linear time-invariant (LTI) continuous-time systems with sensor networks (SNs). According to the topology of the SN, a set of distributed filters are given as Luenberger-like with consensus terms in order to estimate the state in a fully distributed manner. Then, a less conservative sufficient condition is proposed in terms of linear matrix inequalities (LMIs) to such that the filtering error systems of all local DMFs are stable with an H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> performance constraint and a quadratic cost function is minimized in the absence of external disturbances. Moreover, a suboptimal distributed filter design is also proposed. Finally, a simulation example is used to demonstrate the effectiveness and merit of the proposed DMFs design scheme.

A class of linear, time-invariant (LTI) systems, referred to as β-bounded LTI systems, which constrain both the gain and the phase of an LTI system, is introduced in this paper. This characterization corresponds to gain bounded systems, or bounded real systems, for β = 1, and it corresponds to phase bounded, positive real systems for β = 0. State space characterization of β-bounded LTI systems is presented in terms of linear matrix inequalities (LMIs), similar to the LMIs for bounded realness and positive realness. Robust stability results are developed in this paper for β-bounded systems, including stability of the feedback interconnection of β-bounded LTI systems and stability of uncertain systems with β-bounded uncertainties. Small gain and passivity conditions for robust stability are seen as special cases of these results for β-bounded systems. Finally, a convex optimization procedure is presented to select parameters for a tight characterization of uncertain LTI systems in terms of β-boundedness, and the technique is illustrated with a spring-mass-damper example.

Robust stability analysis of polytopic linear time-invariant (LTI) systems is a basic problem in engineering. This paper analyzes the complexity of three fundamental methods used to address this problem, all of them providing a sufficient and necessary condition (with finite and known sizes) for robust stability. The first method is based on the use of a polynomially parameter-dependent Lyapunov function. The second method is a simplified version of the Routh-Hurwitz stability criterion. Lastly, the third method is based on eigenvalue combinations. It is explained that the robust stability conditions provided by these three methods require to establish positive definiteness of symmetric matrix forms (SMFs) over the simplex. Also, it is explained that a sufficient condition for the latter problem can be given in terms of a linear matrix inequality (LMI) feasibility test. Hence, the complexity of the three methods is analyzed and compared by deriving the number of scalar variables in the LMI feasibility tests. A numerical example is also presented to investigate the computational time of these tests.

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This paper addresses robust performance analysis problems of linear time-invariant (LTI) systems affected by real parametric uncertainties. These problems, known also as a special class of structured singular value computation problems, are inherently intractable (NP-hard problems). As such intensive research effort has been made to obtain computationally tractable and less conservative analysis conditions, where linear matrix inequality (LMI) plays an important. Nevertheless, since LMI-based conditions are expected to be conservative in general, it is often the case that we cannot conclude anything if the LMI at hand turns out to be infeasible. This motivates us to consider the dual of the LMI and examine the structure of the dual solution. By pursuing this direction, in this paper, we provide rank conditions on the dual solution matrix under which we can conclude that the underlying robust performance is never attained. In particular, a set of uncertain parameters that violates the specified performance can be computed. These results come from block-moment matrix structure of the dual variable, which is consistent with the recent results on polynomial optimization. This particular structure enables us to make good use of simultaneous diagonalizability property of commuting diagonalizable matrices so that the sound rank conditions for the exactness verification can be obtained. </para>

Discrete-Time H∞ Preview Feedforward Controller Using Uncertain Prior External Input Information via GKYP Lemma

This study investigates characteristic root equivalency relations between commensurate order and integer order Linear Time Invariant (LTI) systems. Author introduces some useful properties of a special class of commensurate order systems, which is called characteristic root equivalency class of LTI systems. These properties present potential to facilitate design and analysis efforts of this class of commensurate order systems. In this sense, straightforward stability checking procedures and design approaches for commensurate order root equivalent systems of the first and second order LTI systems are demonstrated. Findings of the study are validated by illustrative examples.

An LMI approach to design robust fault detection filter for uncertain LTI systems

The Tensor Product (TP) model transformation is a recently proposed technique for transforming given Linear Parameter-Varying (LPV) state-space models into polytopic model form, namely, to parameter-varying convex combination of Linear Time Invariant (LTI) systems. The main advantage of the TP model transformation is that it is executable in a straightforward way and the Linear Matrix Inequality (LMI) based control design frameworks can immediately be applied to the resulting polytopic models to yield controllers with tractable and guaranteed performance. The main contribution of this paper is that it represents the friction of a DC servo drive in TP model form, automatically designs LMI based controller for friction compensation. The reference signal compensation is also based on the TP model transformation. The results were simulated for speed control of the DC servo drive and were compared to other control techniques.