The paper is concerned with positive-realness of sampled-data systems. We first review a few slightly different notions of positive-realness of sampled-data systems, and clarify their mutual relationships. We then focus on so-called strong positive-realness among them, and introduce a measure called the positive-realness gap index. We show that this index can be computed quite efficiently with a bisection method, and provide state space formulas for its computation. The importance of this index lies in that it is quite useful for stability analysis of sampled-data systems. This is demonstrated with the nonlinear stability analysis as well as the gain margin analysis, or a sort of stability radius analysis.
This paper is concerned with approximation of probability distributions behind discrete-time systems with stochastic dynamics. The aim is to facilitate advances in studies for control of such stochastic systems. An approach to approximating two-dimensional probability distributions using metaheuristics is suggested. Then, its effectiveness is demonstrated with a numerical example of stabilization synthesis.
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.
This paper begins by studying some spectral properties of the transfer operators of sampled-data systems described by applying the lifting technique. Through a "nonasymptotic" characterization of the transfer operator, its spectrum is determined in terms of finite-dimensional eigenvalue problems. Then, it is shown that a close connection with such eigenvalue problems and the exponential stability condition can be exploited to study the robust internal (exponential) stability problem of sampled-data systems. Since the transfer operator is relevant to input-output characteristics, the relationship between input-output stability and internal stability is also discussed in the context of sampled-data systems.
This paper provides a theoretical basis for discretization approaches to sampled-data systems in the L ∞ /L 2 optimal controller synthesis problem. Such approaches have been developed through the lifting treatment for the corresponding analysis problem and allowed us to compute the induced norm in an asymptotically exact fashion as the parameter M tends to ∞ in the discretization processes of the generalized plant. This paper aims at establishing that these approaches are actually meaningful equally in the synthesis problem. To this end, we introduce important inequalities independent of the discrete-time controller, which are constructed through the fast-lifted representation of sampled-data systems; this representation employs the aforementioned parameter M, by which the sampling interval [0; h) is divided intoM subintervals with an equal width without losing any information about the signals on the interval. Through these inequalities, we can verify that the discretization methods in our preceding study give a theoretical basis for tackling the optimal controller synthesis problem of minimizing the induced norm from L 2 to L ∞ in SISO LTI sampled-data systems.
This paper investigates a class of sampled-data systems in which at least one of the discrete-time zeros remains on the unit circle regardless of the sampling period. This type of zero is referred to as an "uniformly marginally stable" (UMS) zero. Utilizing this novel concept, we present necessary conditions as well as sufficient conditions for the continuous-time plant whose sampled-data model has an UMS zero. The significance of the symmetry regarding the zeros and the poles of the continuous-time plant is highlighted as it plays a vital role in enforcing a discrete-time zero of the sampled-data system to be UMS, for both discretization zeros and intrinsic zeros.We develop the theory for sampled-data systems employing a conventional sampler and a zero-order hold together with a single-input single-output continuous-time plant, and then observe that certain results can be extended to the generalized holds. As an application, we establish a condition for sampled-data systems that allow for discrete-time stable inversion-based control where the absence of such an UMS zero is crucial in the design of such control systems.
In this paper, we study H∞ performance limitation analysis for continuous-time SISO systems using LMIs. By starting from an LMI that characterizes a necessary and sufficient condition for the existence of desired controllers achieving a prescribed H∞ performance level, we represent lower bounds of the best H∞ performance achievable by any LTI controller in terms of the unstable zeros and the unstable poles of a given plant. The transfer functions to be investigated include the sensitivity function (1+PK)-1, the complementary sensitivity function (1+PK)-1PK, and (1+PK)-1P, the first and the second of which are well investigated in the literature. As a main result, we derive lower bounds of the best achievable H∞ performance with respect to (1+PK)-1P assuming that the plant has unstable zeros. More precisely, we characterize a lower bound in closed-form by means of the first non-zero coefficient of the Taylor expansion of the plant P(s) around its unstable zero.
