Tomomichi Hagiwara

Design of a stable state feedback controller based on the multirate sampling of the plant output

A new type of controller is proposed which detects the ith plant output N/sub i/ times during a period of T/sub 0/ and changes the plant inputs once during T/sub 0/. It is shown that an arbitrary state feedback can be realized by such controllers if the plant is observable. This implies, for instance, that arbitrary symmetric pole assignment is possible if the plant is controllable. It is also shown that, if the plant has no zeros at the origin, the state transition matrix of the controller itself can be set arbitrarily without changing the state feedback to be realized. That is to say, inversely expressed, any state feedback can be equivalently realized by a controller with any prescribed degree of stability. >

Stability of the limiting zeros of sampled-data systems with zero- and first-order holds

This paper is concerned with the zeros of sampled-data systems resulting from continuous-time systems preceded by a hold and followed by a sampler. The holds we shall consider are a zero-order hold and a first-order hold. For sufficiently small or large sampling periods, such zeros are called limiting zeros. For sufficiently small sampling periods, they are known to consist of two different types of zeros: the zeros of the first type correspond to the zeros of the original continuous-time system, while those of the second type have no continuous-time counterparts. We first show basic properties of the zeros of sample-data systems for sufficiently small sampling periods. Next, we clarify, in more detail, the correspondence between the former-type zeros of the sampled-data systems and the zeros of the original continuous-time system, including the stability property of these zeros. We also study stability properties of the latter-type zeros. In addition, we study limiting zeros for sufficiently large sampli...

FR-operator approach to the H/sub 2/ analysis and synthesis of sampled-data systems

Recently, a frequency-domain operator called frequency response (FR) operator was defined and shown to represent the transfer characteristics of a stable sampled-data system. Using this novel frequency-domain notion and introducing its extended notion called hybrid FR-operator, we define an H/sub 2/-norm for sampled-data systems in this paper. Then, sampled-data H/sub 2/ control problems are formulated and solved, whereby the usefulness of these frequency-domain notions is demonstrated both in the analysis and synthesis aspects of sampled-data systems. For the case of sampled-data systems with hybrid (i.e., both continuous-time and discrete-time) input and output signals, the H/sub 2/-norm defined by a hybrid FR-operator turns out to be slightly different from that defined in previous studies. The source of the discrepancy is also identified. >

Robust Performance Analysis of Linear Time-Invariant Uncertain Systems by Taking Higher-Order Time-Derivatives of the State

In this paper, we propose new LMI-based conditions for robust stability/performance analysis of linear time-invariant (LTI) uncertain systems. To get around the conservatism of existing conditions resulting from Lyapunov’s stability theory, we first consider to employ Lyapunov functions that can be associated with higher-order derivatives of the state vectors. This motivates us to introduce a redundant system description so that we can take the behavior of the higher-order derivatives of the state into consideration. Indeed, by considering suitable redundant system descriptions, the existence conditions of those Lyapunov functions can be reduced into constrained inequality conditions, to which we can apply Finsler’s Lemma. Thus we can readily obtain new LMI-based conditions for (robust) stability/performance analysis of LTI systems in a constructive way. It turns out that the proposed LMI conditions can be regarded as a natural extension of those known as extended or dilated LMIs in the literature.

Exact Stability Analysis of 2-D Systems Using LMIs

In this note, we propose necessary and sufficient conditions for the asymptotic stability analysis of two-dimensional (2-D) systems in terms linear matrix inequalities (LMIs). By introducing a guardian map for the set of Schur stable complex matrices, we first reduce the stability analysis problems into nonsingularity analysis problems of parameter-dependent complex matrices. Then, by means of the discrete-time positive real lemma and the generalized S-procedure, we derive LMI-based conditions that enable us to analyze the asymptotic stability in an exact (i.e., nonconservative) fashion. It turns out that, by employing the generalized S-procedure, we can derive smaller size of LMIs so that the computational burden can be reduced

Analytic study on the intrinsic zeros of sampled-data systems

This paper investigates the properties of the mapping from the simple zero /spl gamma/ of a scalar continuous-time system to the corresponding zero /spl Gamma/(T) of the sampled-data system that results by its discretization using a zero-order hold, where T is the sampling period. It is shown that /spl Gamma/(T) admits a Taylor expansion with respect to T, and that it coincides with that of exp(/spl gamma/T) at least up to the second-order term, in general, and at least up to the third-order term if the relative degree of the continuous-time system is greater than or equal to two. The result is applied to derive a new stability condition of /spl Gamma/(T) for sufficiently small sampling periods.

New dilated LMI characterizations for continuous-time control design and robust multiobjective control

It has been recognized that the dilation of the LMI characterizations has new potentials in dealing with such involved problems as multiobjective control, robust performance analysis or synthesis for real polytopic uncertainty and so on. Contrary to the success in this direction in the discrete-time setting, analogous characterizations in the continuous-time setting are still open and challenging. The main contribution of the paper is to propose a general procedure to construct such dilated LMI characterizations for continuous-time control design. Because of our particular procedure, the dilated LMI characterizations are proved to have some very nice and interesting features that are to some extent analogous to the ones already obtained in the discrete-time setting.

On H/sub /spl infin// model reduction using LMIs

In this note, we deal with the problem of approximating a given th-order linear time-invariant system by an th-order system where . It is shown that lower bounds of the norm of the associated error system can be analyzed by using linear matrix ineqaulity (LMI)-related techniques. These lower bounds are given in terms of the Hankel singular values of the system and coincide with those obtained in the previous studies where the analysis of the Hankel operators plays a central role. Thus, this note provides an alternative proof for those lower bounds via simple algebraic manipulations related to LMIs. Moreover, when we reduce the system order by the multiplicity of the smallest Hankel singular value, we show that the problem is essentially convex and the optimal reduced-order models can be constructed via LMI optimization.In this note, we deal with the problem of approximating a given nth-order linear time-invariant system G by an rth-order system G/sub r/ where r<n. It is shown that lower bounds of the H/sub /spl infin// norm of the associated error system can be analyzed by using linear matrix ineqaulity (LMI)-related techniques. These lower bounds are given in terms of the Hankel singular values of the system G and coincide with those obtained in the previous studies where the analysis of the Hankel operators plays a central role. Thus, this note provides an alternative proof for those lower bounds via simple algebraic manipulations related to LMIs. Moreover, when we reduce the system order by the multiplicity of the smallest Hankel singular value, we show that the problem is essentially convex and the optimal reduced-order models can be constructed via LMI optimization.

Robust Performance Analysis of Uncertain LTI Systems: Dual LMI Approach and Verifications for Exactness

This paper addresses robust performance analysis problems of 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 role. Nevertheless, since LMI-based conditions are expected to be conservative in general, it is often the case that we cannot conclude anything directly 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, which does exist if the primal LMI is infeasible. By pursuing this direction, in this paper, we provide a rank condition 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 readily be obtained. The key idea to derive these results comes from simultaneous diagonalizability property of commuting diagonalizable matrices. The block-moment matrix structure of the dual variable plays an essential role to make good use of this property.

Robust controller synthesis with parameter-dependent Lyapunov variables: a dilated LMI approach

This paper enhances our previous results on dilated LMIs so that we can address robust controller synthesis problems for continuous-time LTI systems subject to real polytopic uncertainties. The replacement of a constant scalar involved in the previous dilated LMIs by an adjustable parameter is the key to accomplish this extension. The particular form of the extension enables the use of parameter-dependent Lyapunov variables in such a sound way that an advantage of the dilated LMI approach is ensured explicitly in attacking such robust controller synthesis problems, namely the dilated LMI approach is shown to achieve better (no worse) results than the conventional one which is restricted to parameter-independent Lyapunov variables, provided that the adjustable parameter is taken to meet a certain simple condition.