Leonid Mirkin

Some Remarks on the Use of Time-Varying Delay to Model Sample-and-Hold Circuits

This note revises the so-called input delay approach to the control of sampled-data systems with nonuniform sampling, in which the sample-and-hold circuit is embedded into an analog system with a time- varying input delay. It is shown that the conservatism in calculating the maximal admissible sampling period is reduced by about 57% if the time- varying delay embedding step is omitted.

Brief Every stabilizing dead-time controller has an observer-predictor-based structure

This paper considers the stabilization problem for linear time-invariant systems with a single delay h in the feedback loop. Two parametrizations of all stabilizing controllers are derived in terms of a state-space realization of the rational part of the plant. These parametrizations have simple structures and clear interpretations: the first is in the form of a generalized dead-time compensator (Smith predictor) and the second is in the observer-predictor form. Applications of the proposed parametrization to the H^2 optimization and the robust stabilization problems are discussed.

H/sup /spl infin// control of systems with multiple I/O delays via decomposition to adobe problems

The standard (four-block) H/sup /spl infin// control problem for systems with multiple input-output delays in the feedback loop is studied. The central idea is to see the multiple delay operator as a special series connection of elementary delay operators, called the adobe delay operators. The adobe delay case is solved and thereby the general case is solved as a nested set of solutions to adobe delay problems.

On the Approximation of Distributed-Delay Control Laws ⋆

This note studies lumped-delay approximations of distributed-delay control laws. It is shown that approximation problems reported in some recent studies are caused by a brutal combination of poor approximation accuracy in the high-frequency range (a non-strictly proper approximation of a strictly proper transfer function) and excessive sensitivity of the design method to high-frequency additive plant uncertainties. It is also shown that a safe implementation can be achieved by eliminating either of these two factors. Some remedies toward this end are proposed.

On the extraction of dead-time controllers and estimators from delay-free parametrizations

A novel approach to control design for dead-time (DT) systems is proposed. The underlying idea is to treat the DT element not as a part of the generalized plant, but rather as a (causality) constraint imposed upon the controller (estimator). This enables one to use well-understood parametrizations of all delay-free controllers in the DT design. In particular, DT con- trollers can be extracted from such delay-free parametrizations. In this paper, the extraction procedures are developed in both and settings. It is shown that the proposed approach yields simple solution procedures and new transparent and intuitively appealing structures for the resulting controllers and estimators.

H/sup /spl infin// control and estimation with preview-part I: matrix ARE solutions in continuous time

Preview control and fixed-lag smoothing allow a noncausal component in the controller/estimator. Time domain variational analysis is used in a reduction to an open loop differential game, leading to a complete, necessary and sufficient characterization of suboptimal values and an explicit state space design, in terms of a parameterized (nonstandard) algebraic matrix Riccati equation in a general continuous time linear system setting. The solution offers insight into the appropriate structure of the associated Hamiltonian, where the state and co-state are not the usual state of the original dynamic system and that of its adjoint. Rather, the state and co-state are selected to capture the respective lumped effects of initial data and future input selection in the allied game.

H/sup /spl infin// control and estimation with preview-part II: fixed-size ARE solutions in discrete time

H/sup /spl infin// preview control and fixed-lag smoothing problems are solved in general discrete time linear systems, via a reduction to equivalent open-loop differential games. To prevent high order Riccati equations, found in some solutions, the state of the Hamilton-Jacobi system resides in a quotient space of an auxiliary extended state space system. The dimension of that auxiliary space is equal to the state space dimension of the original system (ignoring the delay).

Control Issues in Systems with Loop Delays

This chapter discusses properties of feedback control systems containing loop delays (dead-time systems) and approaches to controller design for such systems. Consider the feedback system depicted in Fig. 1, where P is a plant, C is a controller, r is a reference signal, d is a disturbance, u is a control signal, and y is an output (measurement) signal. It is assumed throughout that both the measured signal y and the control signal u are delayed by hy and hu units of time, respectively. This is reflected in Fig. 1 by the two delay blocks containing the delay element Dh defined by

Control of systems with I/O delay via reduction to a one-block problem

In this paper, the standard (four-block) H/sup /spl infin// control problem for systems with a single delay in the feedback loop is studied. A simple procedure of the reduction of the problem to an equivalent one-block problem having particularly simple structure is proposed. The one-block problem is then solved by the J-spectral factorization approach, resulting in the so-called dead-time compensator (DTC) form of the controller. The advantages of the proposed procedure are its simplicity, intuitively clear derivation of the DTC form of the H/sup /spl infin// controller, and extensibility to the multiple delay case.

Technical communique: On stability of second-order quasi-polynomials with a single delay

In this note the stability of a second-order quasi-polynomial with a single delay is studied. Although there is a vast literature on this problem, most available solutions are limited to some particular cases. Moreover, some published results on this subject appear to contain imprecise, or even wrong, conditions. The purpose of this note is to show that by accurate application of known theories, a complete explicit characterization of stability regions can be derived in a most general case. As a byproduct of the proposed analysis, we show that in the high-order case the quasi-polynomial is delay-independent unstable whenever its delay-free version has an odd number of unstable roots (or, equivalently, a negative static gain).