Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: the discrete-time case

In this paper, a new methodology is developed for the closed-form solution of a generalized version of the finite-horizon linear-quadratic regulator problem for LTI discrete-time systems. The problem considered herein encompasses the classical version of the LQ problem with assigned initial state and weighted terminal state, as well as the so-called fixed-end point version, in which both the initial and the terminal states are sharply assigned. The present approach is based on a parametrization of all the solutions of the extended symplectic system. In this way, closed-form expressions for the optimal state trajectory and control law may be determined in terms of the boundary conditions. By taking advantage of standard software routines for the solution of the algebraic Riccati and Stein equations, our results lead to a simple and computationally attractive approach for the solution of the considered optimal control problem without the need of iterating the Riccati difference equation

Stabiity for Systems with Unknown Time Delays

Time delays are of long-standing interest in the study of control systems since they appear in many practical control problems and tend to degrade overall system performance. In this thesis, we consider two distinct problems involving uncertain time delays. The first problem that we consider is the achievable delay margin problem, which is determining the longest delay for which stability can be maintained when using a linear time invariant (LTI) controller. This problem has been considered in continuous-time, where bounds (often tight) have been found for plants with non-zero right half plane poles. In this work, we consider the discrete-time case, where we prove that an LTI controller exists which stabilizes the plant and the plant with a one step delay if and only if the plant has no negative, real unstable poles. The second problem that we consider is stabilizing any continuous-time single-input single-output LTI plant with an arbitrarily large time delay and gain. To solve this problem, we propose a simple generalized hold whose resulting discretized system is amenable to adaptive control. Furthermore, by exploiting the structure of the resulting discretized system, we propose purpose built estimators for the unknown gain and delay, which allows us to not only provide bounded-input bounded-output (BIBO) closed-loop stability, but also guarantees the exponential decay of any plant initial conditions, robustness to un-modelled dynamics, and tolerance to occasional, possibly persistent, jumps in the gain and delay. Furthermore, for the case of a first order plant, a similar, but suitably modified controller is shown to tolerate continuous variation of the unknown delay while still providing BIBO closed-loop stability

Pole Assignment With Improved Control Performance by Means of Periodic Feedback

This technical note is concerned with the pole placement of continuous-time linear time-invariant (LTI) systems by means of LQ suboptimal periodic feedback. It is well-known that there exist infinitely many generalized sampled-data hold functions (GSHF) for any controllable LTI system to place the modes of its discrete-time equivalent model at prescribed locations. Among all such GSHFs, this technical note aims to find the one which also minimizes a given LQ performance index. To this end, the GSHF being sought is written as the sum of a particular GSHF and a homogeneous one. The particular GSHF can be readily obtained using the conventional pole-placement techniques. The homogeneous GSHF, on the other hand, is expressed as a linear combination of a finite number of functions such as polynomials, sinusoidals, etc. The problem of finding the optimal coefficients of this linear combination is then formulated as a linear matrix inequality (LMI) optimization. The procedure is illustrated by a numerical example

Control-theoretic Approach to Communication with Feedback: Fundamental Limits and Code Design

Feedback communication is studied from a control-theoretic perspective, mapping the communication problem to a control problem in which the control signal is received through the same noisy channel as in the communication problem, and the (nonlinear and time-varying) dynamics of the system determine a subclass of encoders available at the transmitter. The MMSE capacity is defined to be the supremum exponential decay rate of the mean square decoding error. This is upper bounded by the information-theoretic feedback capacity, which is the supremum of the achievable rates. A sufficient condition is provided under which the upper bound holds with equality. For the special class of stationary Gaussian channels, a simple application of Bode's integral formula shows that the feedback capacity, recently characterized by Kim, is equal to the maximum instability that can be tolerated by the controller under a given power constraint. Finally, the control mapping is generalized to the N-sender AWGN multiple access channel. It is shown that Kramer's code for this channel, which is known to be sum rate optimal in the class of generalized linear feedback codes, can be obtained by solving a linear quadratic Gaussian control problem.Comment: Submitted to IEEE Transactions on Automatic Contro