Dissipativity Analysis for Linear Systems in the Behavioural Framework

This paper presents a robust control synthesis method for linear time-invariant (LTI) differential systems with parametric uncertainty. The uncertain dynamics is described by a parametric kernel representation. We use parametric quadratic differential forms (QDF) to represent the dissipativity properties (both storage function and supply rate) of the uncertain system. The control design utilizes sum-of-squares (SOS) programming to search for a QDF supply rate for the controller such that the supply rate of the closed-loop system satisfies robust stability and performance conditions. Such a supply rate is then applied for control synthesis based J-factorization.

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>

This paper studies the problem of stability analysis and controller design for discrete-time linear time invariant (LTI) systems with state saturation. Both full state saturation and partial state saturation are considered. Firstly, a new system model is constructed for solving the key problem. Then, a new method is presented for estimating the domain of attraction of the origin for a system with state saturation. Based on this method, an LMI-based ( linear matrix inequality) iterative algorithms is proposed for determining if a given ellipsoid is contractively invariant. Moreover, an LMI-based algorithm is developed for designing dynamic output-feedback controllers which guarantee that the domain of attraction of the origin for the closed-loop system is as large as possible. An example is given to illustrate the efficiency of the design method.

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.

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.

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.

In this paper, the cooperative stabilization problem of linear time-invariant (LTI) systems is studied based on a class of distributed Luenberger observers. A given LTI system is estimated by a network of observers. In case of observability configuration or external attack, each observer has to possess different measurement output of the LTI system. Classical distributed control strategies cannot be directly used because they are assumed to obtain identical information. In view of this problem, a class of distributed observers are designed under an undirected graph, which can recover the state of the LTI system by limited local information. By using any single channel of the estimated state, a control input is proposed to stabilize the LTI system. By analysing the stability of an augmented system which involves the LTI system and the estimation error system, some necessary and sufficient conditions are derived for the existence of control gains that guarantee the stabilization of the LTI system. In addition, some algorithms are proposed to specify the control gains. Lastly, some simulation examples are given to verify the effectiveness of the theoretical result.

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

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.

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

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.

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

For continuous-time linear time-invariant (LTI) systems with polytopic uncertainties, we develop a robust sampled-data state-feedback control design scheme in terms of linear matrix inequalities (LMIs). Truncated power series expansions are used to approximate a discretized model of the original continuous-time system. The system matrices obtained by using the power series approximations are then expressed as homogeneous polynomial parameter-dependent (HPPD) matrices of finite degrees, and conditions for designing the controller are formulated as a HPPD matrix inequality, which can be solved by means of a recent LMI relaxation technique to test the positivity of HPPD matrices with variables in the simplex. To take care of the errors induced by the remainder terms of the truncated power series, the terms are considered as norm bounded uncertainties and then incorporated into the proposed LMI conditions. Finally, examples are used to illustrate the approach.

A robust stability framework for LTI systems with time-varying sampling