Output controllability and eigenvalue assignability in single-output linear systems

In this paper, a controller called dynamic matrix constrained variable structure controller (DM-CVSC) is proposed. The controller takes advantages of both dynamic matrix (DM) and constrained variable structure controllers. As a result, DM-CVSC is a robust trajectory tracking controller dealing with the constraints on control inputs and also makes decision based on the future behavior of the vehicle. The controller is applied to a linearized model of half-car suspension systems which are subject to different types of road disturbances and measurement noises. In this paper, it is shown that there is a simple formulation for calculating the range of sliding gains for single-input single-output (SISO) linear control systems. As for the multi-input multi-output (MIMO) linear control systems, the calculation of upper sliding gain profile for controller leads to a search problem. To show the efficiency of the proposed controller, it is applied to four different cases involving specific road disturbances and measurement noises. The performance of the proposed controller is compared to various control techniques.

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.

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.

We give a parameterization of all causal stabilizing repetitive learning controllers for single-input and single-output linear minimum phase systems. The past studies on the parameterization of a repetitive learning control system were to obtain the parameterization of all internally stabilizing controllers for the control system such that the plant is connected 1/(1-e/sup -sT/) in series. Since 1/(1-e/sup -sT/) has no zeros in the closed right half plane, also the serial connections of 1/(1-e/sup -sT/) and a minimum phase system have no zeros in the closed right half plane. That is, the parameterization of all stabilizing controllers for a repetitive learning control system may be obtained by using the parameterization for a minimum phase system in Glaria and Goodwin (1994). Using the above mentioned idea, at first, the parameterization of all stabilizing repetitive learning controllers for the biproper minimum phase system is carried out. Next, since the exact repetitive learning controllers to stabilize the strictly stable proper system do not exist, the parameterization of all modified repetitive learning controllers using a low-pass filter in the repetitive learning controller is given.

Adaptive regulation for minimum phase systems with unknown relative degree and uncertain exosystems

This article presents the central finite-dimensional H ∞ controller for linear time-varying systems with unknown parameters, that is suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, this article reduces the original H ∞ controller problem to the corresponding H 2 controller problem, using the technique proposed in Doyle et al. [Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989), ‘State-space Solutions to Standard H 2 and H Infinity Control Problems’, IEEE Transactions Automatic Control, 34, 831–847]. This article yields the central suboptimal H ∞ controller for linear systems with unknown parameters in a closed finite-dimensional form, based on the corresponding H 2 controller obtained in Basin and Calderon-Alvarez [Basin, M.V., and Calderon-Alvarez, D. (2008), ‘Optimal LQG Controller for Linear Systems with Unknown Parameters’, Journal of The Franklin Institute, 345, 293–302]. Numerical simulations are conducted to verify performance of the designed central suboptimal controller for uncertain linear systems with unknown parameters against the conventional central suboptimal H ∞ controller for linear systems with exactly known parameter values.

Necessary and sufficient conditions for output reachability and null output controllability of positive linear discrete systems with delays in state, input, and output are established. It is also shown that output reachability and null output controllability together imply output controllability.

A suboptimal method of reduced variance pole-assignment control is developed for single-input single-output linear discrete-time systems. The closed-loop system regulation may be made arbitrarily close to that obtainable by a linear quadratic Gaussian regulator and the computation of mathematical expectations is not required. A self-tuning control algorithm is developed and simulations indicate system and parameter identifiability occurs.

State space is one of the key concepts of system theory. In general, a state space is introduced into a system description without examining its specific physical meaning. It is known, however, that if we select a suitable state space representation, it becomes easier for us to understand or to manipulate the property of a system. Thus, although it may not be meaningful to try to arrive at a physical meaning of the state space, it is important to find a way where a choice of a state space can be made in accordance with the system of interest, taking into account the theoretical meaning of that choice. In this paper, the choice of a state space is discussed in terms of the concepts that general systems theory have provided and the canonical forms of linear control theory are interpreted using the state space representation. Four realizations of a single-input and single-output discrete basic linear system are constructed, whose state spaces are derived from system properties. such as reachability, past-det...

It is shown in this paper that any state space realization (A, b, c) of a given transfer function T(s) = β(s/α(s with α(s) monic and dim(A) = deg(α(s)), satisfies the identity β(A) = Qc(A, b)SαQo(A, c) where Qc(A, b) and Qo(A, c) are the controllability matrix and observability matrix of the matrix triple (A, b, c), respectively, and Sα is a nonsingular symmetric matrix. Such an identity gives a deep relationship between the state space description and the transfer function description of single-input single-output (SISO) linear systems. As a direct conclusion, we arrive at the well-known result that a realization of any transfer function is minimal if and only if the numerator and the denominator of the transfer function is coprime. Such a result is also extended to the SISO descriptor linear system case. As an applications, a complete solution to the commuting matrix equation AX = XA is proposed and the minimal realization of multi-input multi-output (MIMO) linear system is considered.

Algorithms are currently available for the solution of certain types of optimal-eigenvalue-assignment problems in which the eigenvalues of a given system are required to be shifted into preassigned locations or region while also minimising an appropriate quadratic-performance criterion. All the known methods for a solution of the above problem are based on manipulation of the original nth-order system matrices even if only r eigenvalues (r ≥ n) of the original system are to be reassigned. On the contrary, the method proposed in this paper, for a solution of the above problem, employs an rth-order equivalent model, which leads to a solution via manipulation of rth-order matrices only. The method also ensures that the remaining n — r eigenvalues of the original system are not disturbed and are carried over to the resultant feedback system. It has been shown in the paper that the suggested procedure brings about a considerable saving in computation time, and also requires less computer storage. Two numerical examples have been included to illustrate the suggested technique.

This article addresses the problem of simultaneous distributed state estimation, and control of linear systems with linear state feedback, subjected to process, and measurement noise, under the constraints of quantized, and rate-limited network data transmission. In the set-up adopted, sensors and actuators communicate through a network with a strongly connected topology. Unlike the case of centralized linear systems, for which the separation principle holds, the above practical assumption prevents the separate design of observers, and controller because each of the nodes does not necessarily have access to the control inputs generated at all the other nodes. We derive a linear distributed Luenberger observer, and a set of sufficient conditions that guarantee ultimate boundedness of the estimation error, and system state vectors, with bounds that depend on the L-infinity norm of the noise signals, and the number of bits used in the transmissions. A numerical example illustrates the performance and effectiveness of the proposed algorithm in controlling a network of open-loop unstable systems.

Using a Taylor series expansion, nonlinear control systems are approximated about desired equilibrium points in state space as linear control systems. The techniques of the linear regulator problem are then used to calculate the linear state variable feedback gains needed to keep the system at the desired equilibrium point. Liapunov's second method is the basis for determining stable regions of operation about the equilibrium points. When driving the system from one equilibrium point to another, the piecewise linearization method is also used but with intermediate equilibrium points successively utilized. Enough equilibrium points are used so that the system is stable throughout the region of interest. Thus, a nonlinear system is controlled using linear state variable feedback. A small digital computer is required to implement this strategy.

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.

This paper discusses the new optimal digital filter for the single-output linear time-invariant discrete-time system without the prior stochastic information about the initial state. The Kalman filter is obtained on the assumption that the prior information about the initial state is known. Since this assumption is not satisfied practically, the new optimal digital filter which is named the deadbeat Kalman filter is derived. The deadbeat Kalman filter is independent of the prior information about the initial state, and is unbiased and optimal after the minimal finite step. Finally, the duality between the deadbeat Kalman filter and the deadbeat LQ optimal control is shown.