Shinji Hara

Dear Editor, In this letter, we focus on the algebraic relationship between the coefficient matrices and the solution of the stochastic algebraic Riccati equation. It is revealed that, if the coefficient matrices are in an algebra, then the solution (and also the control gain in many cases) is also in the same algebra. The main result is verified by a numerical simulation.

As a closed-loop learning control method, repetitive control has been widely used in a variety of areas from appliances to aviation. A repetitive control system features perfect reference tracking and disturbance rejection in the steady state for periodic signals with a fixed period. This characteristic is important not only for conventional technologies and conventional industries but also for advanced technologies and emerging industries. This paper first explains the concept of repetitive control from its original idea. Next, it describes the structure of a repetitive controller as an internal model and shows the respective points of continuous- and discrete-time repetitive control. It presents a categorized list of practical applications of repetitive control. Moreover, two concrete applications, namely the control of a robotic manipulator and a rotating system, demonstrate the validity of the method with experimental results. Several current studies in this field are also reviewed, and some challenges and future studies for repetitive control are provided.

We introduce the notions of negative imaginariness input and output indices (NI I/O-indices) for describing systems that do not manifest the negative imaginary property on certain frequency bands, thereby characterising classes of linear time-invariant systems that are larger than those of negative imaginary systems without poles at the origin. We show that if the total frequency-dependent negative imaginariness in a feedback interconnection is positive as measured by the NI indices, whereby any deficiency in negative imaginariness in one open-loop component can be compensated for by a surplus of negative imaginariness in another, then the feedback interconnection is stable if and only if the static (a.k.a. DC) loop gain is less than unity. The result covers the feedback interconnection of a negative imaginary system and a strictly negative imaginary system, which naturally gives rise to positive total negative imaginariness. Importantly, we derive the NI indices based condition from a more general robust stability result established herein involving quadratic frequency dependent inequalities that may be used to characterise system properties beyond negative imaginariness. The proof relies on the multivariable Nyquist stability criterion and does not make use of state-space realisations of the underlying systems.

The robustness of instability is investigated for feedback interconnections of rational transfer functions, motivated by the relevance of such in the study of oscillatory dynamics. An analysis is presented in terms of normalized coprime factorizations and the $\mathscr {L}_{2}$ -gap metric. The main result is a necessary and sufficient condition for feedback instability to persist across a particular uncertainty set for one open-loop component of the interconnection, with the other component fixed. This uncertainty set is a subset of an ${\mathscr {L}}_{2}$ -gap ball. It includes the associated $\nu$ -gap metric ball, and other winding-number based partitions that enable necessity to be established for sufficiently large ball radius.

The purpose of this paper is to develop a general framework of mean-square robust control in the presence of stochastic multiplicative uncertainties. We consider block diagonally structured stochastic uncertainties of a variety of structures including diagonal scalar, element-by-element, repeated scalar and full-block stochastic processes with prescribed variance bounds, which may arise from multiple sources and can be used to model various communication losses. In an important distinction from the previous work, we allow the stochastic uncertainties to be statistically correlated. A general mean-square robustness measure, termed mean-square stability radius (MSSR), is introduced as the metric to quantify stability robustness under the mean-square criterion. Explicit expressions of the MSSR are derived, and a small-gain type necessary and sufficient condition is obtained for mean-square robust stability. Based on the MSSR expressions, mean-square stabilization problems are studied by minimizing the MSSR over all possible stabilizing controllers, leading to stabilizability conditions for mean-square feedback stabilization. Moreover, as an analogy to its counterpart in the robust control theory, a mean-square optimal performance problem is shown to be equivalent to one of mean-square stabilization, by introducing a fictitious stochastic uncertainty and augmenting the system appropriately.

This paper is concerned with a new optimization problem named “phase change rate maximization” for single-input-single-output linear time-invariant systems. The problem relates to two control problems, namely robust instability analysis against stable perturbations and minimum-norm strong stabilization. We define an index of the instability margin called “robust instability radius (RIR)” as the smallest $\mathcal{H}_\infty$ norm of a stable perturbation that stabilizes a given unstable system. This paper has two main contributions. It is first shown that the problem of finding the exact RIR via the small-gain condition can be transformed into the problem of maximizing the phase change rate at the peak frequency with a phase constraint. Then, we show that the maximum is attained by a constant or a first-order allpass function and derive conditions, under which the RIR can be exactly characterized, in terms of the phase change rate. Two practical applications are provided to illustrate the utility of our results.

This paper is concerned with robust instability analysis for linear multi-agent dynamical systems with cyclic structure. This relates to interesting and important periodic oscillation phenomena in biology and neuronal science, since the nonlinear phenomena often occur when the linearized model around an equilibrium point is unstable. We first make a problem setting on the analysis and define the notion of robust instability radius (RIR) as a quantitative measure for maximum allowable stable dynamic perturbation in terms of the H-infinity norm. After showing lower bounds of the RIR, we derive the exact RIR, which is analytic and scalable, for first order time-lag agents. Finally, we make a remark on the potential applicability to some classes of higher order systems.