Computation of invariant zeros of linear,time-invariant,multivariable systems

In this paper a numerical method is presented for computing the invariant zeros of a controllable linear, time-invariant, multivariable system described by the 4-tuplo (A, B, C, D) or the triple (A, B, C). The method is based on the fact. that a controllable system can be made maximally unobservable by means of state variable feedback, thereby causing the cancellation of the invariant zeros by an equal number of the system poles. The invariant zeros are obtained as the eigenvalues of a matrix of the same dimension as the number of invariant zeros. The method is applicable to both multivariable as well as single-input, single-output systems. Examples are given to illustrate the use of the method.     

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.     

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.     

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.     

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     

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.     

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.     

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     

Modeling of linear time-varying systems by linear time-invariant systems of lower order

A technique for modeling linear time-varying differential systems by linear time-invariant systems of lower order is discussed and criteria for determining stability, uniqueness, and the choice of the model of a given dimension are developed. In addition, a lower bound for the minimized performance index is derived and the effect on it of decreasing the dimension of the model is demonstrated.     

Simultaneous controller design for linear time-invariant systems

The use of generalized sampled-data hold functions (GSHF) in the problem of simultaneous controller design for linear time-invariant plants is discussed. This problem can be stated as follows: given plants P₁, P₂, . . ., P_N , find a controller C which achieves not only simultaneous stability, but also simultaneous optimal performance in the N given systems. By this, it is meant that C must optimize an overall cost function reflecting the closed-loop performance of each plant when it is regulated by C. The problem is solved in three aspects: simultaneous stabilization, simultaneous optimal quadratic performance, and simultaneous pole assignment in combination with simultaneous intersampling performance     

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.     

Nonlinear observers for perspective time-invariant linear systems

Perspective dynamical systems arise in machine vision, in which only perspective observation is available. This paper proposes and studies a Luenberger-type nonlinear observer for perspective time-invariant linear systems. Assuming a given perspective time-invariant linear system to be Lyapunov stable and to satisfy some sort of detectability condition, it is shown that the estimation error converges exponentially to zero. Finally, some simple numerical examples are presented to illustrate the result obtained.     

Stable controller synthesis for linear time-invariant systems

This paper is concerned with the problem of suboptimal stable mixed H 2 / H X control for linear time-invariant systems. The designed controller with the same order as that of the considered system is required to satisfy a prescribed H X performance bound or a prescribed degree of stability. By reducing the stable controller synthesis problem to a multiobjective state feedback control problem for two different state models, sufficient conditions for the solvability of the considered problem are given in terms of solutions to algebraic Riccati equations and matrix inequalities. An LMI-based iterative algorithm is developed to solve the stable controller synthesis problem. The proposed algorithm is shown to be convergent. Examples are given to illustrate the proposed methods.     

Semidefinite programming duality and linear time-invariant systems

Several important problems in control theory can be reformulated as semidefinite programming problems, i.e., minimization of a linear objective subject to linear matrix inequality (LMI) constraints. From convex optimization duality theory, conditions for infeasibility of the LMIs, as well as dual optimization problems, can be formulated. These can in turn be reinterpreted in control or system theoretic terms, often yielding new results or new proofs for existing results from control theory. We explore such connections for a few problems associated with linear time-invariant systems.     

The feedforward control of linear multivariable time-invariant systems

Necessary and sufficient conditions are derived for a stable feedforward control system to exist for a multivariable linear time-invariant system so that: (1) any measurable disturbances occurring in the system are asymptotically rejected, and (2) the outputs of the system asymptotically become equal to preassigned functions of a given class of outputs. The result may be interpretated as being a generalization of the single-input, single-output feedforward servo problem to multivariable systems or as being a solution to the asymptotic decoupling problem. Some numerical examples varying from 2nd to 9th order are included to illustrate the result.     

Eigenvalue-eigenvector sensitivity analysis of linear time-invariant singular systems

This note refers to the eigenvalue-eigenvector sensitivity analysis of linear time-invariant multivariable singular systems. Procedures that can be easily programmed on a digital computer are presented for the determination of the deviation of the eigenvalues and the corresponding right and left eigenvectors which are caused by the system's parameter changes about their nominal values. The method is illustrated by two examples.     

More on the controllability of linear time-invariant systems

This study shows that a controllable system [xdot] = Ax + Bu, where x is an n-vector, can be driven from an arbitrary initial condition x(0) = x₀ to an arbitrary final condition x(t_f = x_f by a polynomial control function of degree M = 2r + 1, where r = n ? rank (B), through a polynomial trajectory of degree M. A simple algorithm for finding u by solving a set of linear equations, the solution of which yields the polynomial coefficients, is given.