Direct Determination of Controller Parameters in Adaptive Pole Placement

Direct adaptive control of a single-input single-output linear time-invariant n-dimensional discrete-time system is approached in terms of a 2n-dimensional state representation constructed directly from the input/output difference equation. The central result of the paper de the derivation of a simple procedure for directly determining the controller parameters in adaptive pole placement. The only assumption on the system model is that the minimal order of the system is known a priori; in particular, the given system is not required to be minimum phase. The approach taken in the paper is based on the expression of the system input as a linear function of an expanded (4n-dimensional) regression vector with the coefficient vector of the linear relationship containing the desired controller parameters. In this way the determination of the controller parameters is reduced to a linear-in-the-parameters estimation problem which can be solved using existing parameter estimation schemes. The particular parametrization constructed in this paper is a “state-space” version of the parametrization derived by Elliott [1]. The estimation of controller parameters is based on the standard recursive least squares algorithm. If the expanded regression vector is sufficiently exciting (as opposed to persistently exciting), the closed-loop system is exponentially stable with the closed-loop poles located close to the desired poles for sufficiently large values of the time index.

Optimal performance of discrete-time control systems based on network-induced delay

There is a well-developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller. This theory describes limitations on achievable performance present when the plant has non-minimum phase zeros, unstable poles, and/or time delays. The parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants. This paper develops a theory of design limitations for sampled-data feedback systems wherein the response of the analogue system output is considered. This is done using the fact that the steady-state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows fundamental sensitivity and complementary sensitivity functions that re...

Simultaneous global external and internal stabilization of linear time-invariant discrete-time systems subject to actuator saturation

Block-form control of linear time-invariant discrete-time systems defined over commutative rings

In this paper we analyze proportional-derivative output feedback control for single-input single-output linear time-invariant dynamical systems with discrete state delays. The main focus of the presented work is the analysis of the effect of arbitrarily small implementation errors, such as a feedback delay or a finite-difference approximation of the derivative, on the stability of the closed loop. More specifically, it will be shown that the classical notion of stability does not suffice to adequately describe the asymptotic behavior of the closed loop system in the presence of these implementation errors, as stability might be lost even when these errors are made arbitrarily small. This paper therefore considers strong stability, which eliminates this well-posedness problem by not only requiring stability but by also enforcing that stability is preserved for sufficiently small perturbations. To conclude this paper, a controller design methodology for proportional-derivative output feedback controllers that strongly stabilize the system, is proposed.

Model Reference Adaptive Control (MRAC) of a wide repertoire of stable Linear Time Invariant (LTI) systems is addressed here. Even an upper bound on the order of the finite-dimensional system is unavailable. Further, the unknown plant is permitted to have both minimum phase and nonminimum phase zeros. Model following with reference to a completely specified reference model excited by a class of piecewise continuous bounded signals is the goal. The problem is approached by taking recourse to the time moments representation of an LTI system. The treatment here is confined to Single-Input Single-Output (SISO) systems. The adaptive controller is built upon an on-line scheme for time moment estimation of a system given no more than its input and output. As a first step, a cascade compensator is devised. The primary contribution lies in developing a unified framework to eventually address with more finesse the problem of adaptive control of a large family of plants allowed to be minimum or nonminimum phase. Thus, the scheme presented in this paper is confined to lay the basis for more refined compensators--cascade, feedback and both--initially for SISO systems and progressively for Multi-Input Multi-Output (MIMO) systems. Simulations are presented.

This paper concerns the order reduction of single-input single-output (SISO) linear time-invariant continuous systems based on the impulse response Gramian. From the recursive relationship among the Gramians, a new formula is derived for computing the system matrix in controllability canonical form. The result is applied to the model reduction problem. Reduced models obtained approximate the reduced-order Gramians while preserving some initial time moments and Markov parameters of the original system.

The main idea of the theory of discrete-time single-input single-output linear time-invariant systems (LTI-system) is that every such system has an input-output map that can be represented by the familiar convolution. Such a system is called system of convolution type. It was recently discovered by Boyd and Sandberg, that not all LTI-systems are systems of convolution type. Since not all LTI-systems are of convolution type it would be interesting to characterize the systems of convolution type. In this paper such a characterization is given with the help of the approximation behaviour of the system with respect to FIR-systems. A necessary and sufficient condition is given, such that the system is a system of convolution type. Some applications for the filter design are also described.

The Transmission Zero at s Radius and the Minimum Phase Radius of LTI Systems

Many different types of adaptive or universal controllers", capable of dealing with a very broad range of linear time-invariant systems, have been proposed. One class of such controllers uses switching between a finite, or at least countable, number of fixed controllers until stability is detected. Such controllers are very attractive from a theoretical viewpoint, in providing stabilization and asymptotic performance for a broad class of plants. However, such controllers are also known to have very poor transient properties, due to the long time required to search for a stabilizing feedback. The key contribution of this paper is to introduce the "method of localization" which can greatly improve the speed of the search. The method is described for linear time-invariant discrete time systems of known nominal order, with disturbances and noise present. Analysis and simulations demonstrate the potential for greatly improved transient performance.

Interval observer design for unknown input estimation of linear time-invariant discrete-time systems

Another algorithm for chaotification of any given linear time-invariant discrete-time systems is presented. The new chaotification algorithm uses the decentralized control and the continuous sawtooth function, which can generate discrete chaos with an arbitrarily desired amplitude bound. Based on the Marotto theorem, we mathematically prove that the controlled system is chaotic in the sense of Li and Yorke. Finally, a simple example is used to illustrate the effectiveness of the proposed theory and method.

: This report considers a feedback control law for linear time-varying and time invariant discrete systems based on a receding horizon concept applied to a minimum energy problem with fixed terminal constraints. The control law is shown to be asymptotically stable and to result in a new method for stabilizing linear time-varying systems as well as extending some well known methods for stabilizing time invariant systems. In particular, the stabilizing gains of the feedback control law are obtained from the solution to a discrete Riccati equation over an arbitrary finite time interval, which is relatively easy to compute. The gain matrix reduces to a constant matrix for linear time invariant systems. Some stability results will turn out to be special cases of these results. The results parallel those for linear continuous time systems, although the technical details are tedious and more involved.

This communication is concerned with stabilization via output static feedback of discrete-time linear time-invariant finite-dimensional systems. A new concept is introduced: output static stabilization in the relaxed sense. This (conservative) stabilization concept is defined so that it is characterized in terms of the discrete-time LQR problem. Properties, and alternative equivalent definitions of this new concept are analyzed. It is shown that for a particular class of plants, the stabilization problem in the above new sense can be cast as a convex programming problem (which moreover can be used as a tool for synthesis). A full characterization for the class of plants that are stabilizable in the above sense is presented. Important consequences of this characterization are somehow surprising. It is shown that the introduced new (conservative) stabilization concept is indeed much more important than it seems. The general problem of output static stabilization can be transformed into the stabilization problem in this new sense. A relationship between output static stabilizability and the discrete-time LQR problem is proved. Convex necessary and sufficient conditions, for joint stabilization (of multiple plants) in the above sense and for decentralized stabilization in the above sense, are presented.