Time delay estimation in continuous linear time-invariant systems

The use of a time delay in modeling LTI systems is quite common. However, attempts to estimate these time delays in continuous systems typically resorted to methods which increase the number of parameters in the system, in contradiction to the use of time delay in the model to begin with. We present here an attempt to estimate the time delays directly. The algorithm we present is supported both by analysis and simulations with very encouraging results     

Determination of zeros and zero directions of linear time-invariant systems by matrix generalized inverses

Consider the linear control system whose state-space equations are dx/dl = Ax + Bu, y = Cx + Du where x, u and y are, respectively, the state, input and output vectors, and A, B, C, D are constant matrices. Problem : find constant scalars r and constant vectors v, w such that an input of the form v exp (rt) 1(t) will yield a state of the form x = w exp (rt) and output y ≡ 0, t ≥ 0. The method developed is this: the output equation is solved for x and the solution is substituted into the state equation. The resulting equation must satisfy a consistency condition involving r, in order for a solution of the problem to exist. Matrix generalized inverses are employed both to derive consistency conditions and to obtain solutions. Several examples illustrate a variety of conditions.     

Assessing the stability of linear time-invariant continuous interval dynamic systems

It is known that stability analysis of linear time-invariant dynamic systems under parameter uncertainties can be equated to estimating the range of the eigenvalues of matrices whose elements are intervals. In this note, first the problem of finding tight outer bounds on the eigenvalue ranges is considered. A method for computing such bounds is suggested which consists, essentially, of setting up and solving a system of n mildly nonlinear algebraic equations, n being the size of the interval matrix investigated. The main result of the note, however, is a method for determining the right end-point of the exact eigenvalue ranges. The latter makes use of the outer bounds. It is applicable if certain computationally verifiable monotonicity conditions are fulfilled. The methods suggested can be applied for robust stability analysis of both continuous- and discrete-time systems. Numerical examples illustrating the applicability of the new methods are also provided.     

On the application of matrix generalized inverses to the design of observers for time-varying and time-invariant linear systems

A method for the design of Luenberger observers is developed utilizing the simplest generalized inverse of matrices. The method can be applied to time-invariant and to time-varying systems.     

On the design of optimal time-invariant compensators for linear stochastic time-invariant systems

Design equations are developed for an optimal limited state feedback controller problem. These equations are developed for the stochastic case, in which both plant noise and measurement noise may be present, and for the case of a dynamic compensator. Four possible approaches to the solution of the nonlinear design equations are described. A fourth-order example illustrates some of the difficulties associated with the solution of these equations and suggests additional areas for study.     

Controllability of linear time-invariant systems

In this paper it is shown that for any controllable linear time-invariant system, an arbitrary initial point can be driven to an arbitrary final point by means of an input, polynomial in t of degree 2k— I, with k the controllability index of the system. Also, the state trajectory is polynomial in time and of the same degree. The bound on the degree is an improvement with respect to similar results in the literature. The proof of this result is rather direct compared to the involved proof in the literature of a weaker theorem. An extension of this result considering sinusoidal inputs and state trajectories is provided.     

On strong stabilization of asymptotically time-invariant linear time-varying systems

This paper considers the strong stabilization problem: given a linear time-varying system which is stabilizable by dynamic feedback, when can the stabilizer be chosen to be itself stable? We consider here the case of algebras of discrete time, time-varying systems which are asymptotically time-invariant, in the sense that as time evolves the time-varying transfer operator converges to a time-invariant transfer operator. Convergence here is in the sense of uniform or strong convergence of sequences of operators on an appropriate Hilbert space of input–output signals.     

Output-feedback synchronizability of linear time-invariant systems

The paper studies the output-feedback synchronization problem for a network of identical, linear time-invariant systems. A criterion to test network synchronization is derived and the class of output-feedback synchronizable systems is introduced and characterized by sufficient and necessary conditions. In particular it is observed that output-feedback stabilizability is sufficient but not necessary for output-feedback synchronizability. In the special case of single-input single-output systems, conditions are derived in the frequency domain. The theory is illustrated with several examples.     

Controllability of generalized linear time-invariant systems

This note deals with the problem ofc-controllability of generalized systems. A new criterion for determiningc-controllability of generalized systems is established. With controllability established, it is shown that it is always possible to transfer the system from the initial position to any target by a control vector-valued function whose elements are polynomials of bounded degree. The polynomial coefficients are obtained as a solution to a set of linear equations.     

Robust stabilization of linear time-invariant systems

The problem of robustly stabilizing families of linear time-variant plants consisting of a nominal plant and its frequency domain neighborhood is discussed. Necessary and sufficient conditions are obtained for robust stabilizability of such families of plants. The set of robustly stabilizing controllers is parameterized and the largest achievable stability margin is characterized for a linear time-invariant plant     

On ultimate boundedness around non-assignable equilibria of linear time-invariant systems

In this note we investigate the following questions: given a (finite-dimensional) linear time-invariant (LTI) multivariable system and a constant desired value for its output, say y_?. Assume there is no assignable equilibrium point corresponding to y_?. How “close” to y_? can we ultimately keep the output using LTI static state-feedback stabilizing controllers? Can this neighborhood of y_? be reduced with dynamic, nonlinear, time-varying controllers? Our main contributions are the proof that the optimal ultimate boundedness neighborhood is achieved with LTI static state-feedback, the explicit computation of the neighborhood's size and the proof, under some reasonable rank assumptions, that the system has non-assignable values for the output if and only if it has a transmission zero at zero. Interestingly, there is no connection between this problem and the more familiar concepts of controllability and observability.     

Noncausal inverses for linear systems

Suppose we are to guide a plant with a square transfer matrix through a series of way points. This can be accomplished by output tracking of trajectories which are fitted together in a sufficiently smooth manner. Each segment of each output is prescribed in terms of a polynomial or sine function or the product thereof. Our technique is to compute: 1) a noncausal “particular solution” based on the desired trajectory over each segment and the movement from one segment to the next, and this “particular solution” consists of a “particular control” and a “particular state trajectory”; and 2) a regulator to handle modeling errors, disturbances, etc. We concentrate on the “particular solution”     

Comments on "On the application of matrix generalized inverses to the design of observer for time-varying and time-invariant linear systems"

It is shown by counterexample that the correction paper [1] of the paper is incorrect. Following a definition and a theorem, we present a corrected version of the authors' theorem to make the results of the paper precise.     

Comments on "On the application of matrix generalized inverses to the design of observers for time-varying and time-invariant linear systems"

In this discussion, several comments on the eigenvalue assignment algorithm in the above paper are made for the purpose of better illustration and clarity.     

Dynamic observers for linear time-invariant systems

An alternative state estimator structure for linear time-invariant systems named for dynamic observer is presented, which can be considered as an extension of the usual observer in its configuration. As a possible design method for the dynamic observer, an efficient and plausible design algorithm is also provided. The mechanism of the proposed dynamic observer design is the dual of the output feedback controller design. The essential characteristics of the dynamic observer to be qualified as an effective observer are addressed.     

Lyapunov stability robustness measures for multivariable,continuous, time-invariant,linear systems

In this paper general robustness measure bounds are introduced for any multivariable, continuous, time-invariant, linear systems. Bounds are obtained for allowable non-linear time-varying perturbations such that the resulting system remains stable. Bounds are also derived for linear perturbations. The robustness measures and the related theorems are applied to optimal LQ state feedback, direct output feedback, and to generalized dynamic output feedback designs.     

On the identification of multi-output linear time-invariant and periodic dynamic systems

In this paper we describe an efficient computational algorithm for estimating the coefficients of the characteristic polynomial of a linear time-invariant multi-output dynamic system, using only output observations, for qualitative analysis of the transition matrix or for evaluating its eigenvalues. We also give some computational results of the identification of those systems using the Ho-Kalman approach. Furthermore, an identification scheme for high-frequency periodic systems of unknown periods is described in detail.     

On impulsive algebraic multiplicities of linear time-invariant singular systems under feedback

Impulsive algebraic multiplicities of linear time-invariant singular systems under output feedback or decentralized output feedback are characterized in terms of system matrices directly. The multiplicities describe the number of impulsive modes that can be eliminated and indicate the number of finite poles that can be assigned by the respective feedback. These characterizations also lead to new regularizability conditions for singular systems.     

Stochastic robustness of linear time-invariant control systems

A simple numerical procedure for estimating the stochastic robustness of a linear time-invariant system is described. Monte Carlo evaluation of the system's eigenvalues allows the probability of instability and the related stochastic root locus to be estimated. This analysis approach treats not only Gaussian parameter uncertainties but also nonGaussian cases, including uncertain but bound variations. Confidence intervals for the scalar probability of instability address computational issues inherent in Monte Carlo simulation. Trivial extensions of the procedure admit consideration of alternate discriminants; thus, the probabilities that stipulated degrees of instability will be exceeded or that closed-loop roots will leave desirable regions can also be estimated. Results are particularly amenable to graphical presentation