On transmission zeros and zero directions of multivariable time-invariant linear systems with input-derivative control

The purpose of this paper is to investigate the problem of finding inputs which generate zero outputs in linear systems whose state equation contains the derivative of the input. The method developed makes use of the simplest matrix generalized inverse, the {1}-inverse.     

Finite transmission zeros of linear time-invariant generalized state space systems

In this paper, the problem of finding inputs which generate zero outputs in linear time-invariant control systems of the form E(dx/dt) = Ax + Bu is considered where E and A are not necessarily square matrices. This problem is solved using the theory of matrix generalized inverses.     

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 minimal-order inverses of linear continuous time-invariant systems

Minimal-order inverses of a linear continuous time-invariant system are defined and an algebraic algorithm for constructing them is presented. A necessary and sufficient condition for the applicability of the algorithm is shown to be the same as that for the invertibility of the given system. Some important properties of the minimal-order inverses relevant to the design of linear multivariable systems are derived. In particular the characteristic polynomials of all minimal-order inverses of a given system are shown to be identical and invariant under duality, coordinate transformations and proportional state feedback.     

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.     

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.     

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.     

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.     

Transfer-function matrix synthesis by matrix generalized inverses

The matrix generalized inverse provides a symbolism for establishing the consistency and finding the solution of a system of linear algebraic equations. This concept is employed to find a control law for a linear time-invariant control system which causes the transfer-function matrix of the resulting system to be a specified function of s.     

Decoupling of continuous-time linear time-invariant systems using generalized sampled-data hold functions

We investigate the possibility of input-output decoupling without stability a square continuous-time LTI plant (C, A, B) by designing discrete devices, the generalized sampled-data hold functions. By adopting sequential design procedures for the resulting diagonalizing problem, two cases are found to be solvable and yield nontrivial solutions: (1) CB nonsingular; (2) mn2m, rk(CB)=n-m where n, m are the number of states and inputs (outputs) respectively; C=I_n-C^T(CC^T)⁻¹C.     

Pole assignment using matrix generalized inverses

The problem of pole assignment in a completely controllable linear time-invariant system dx/t = Ax + Bu, y = Cx is considered. A method using matrix generalized inverses is developed for the computation of a matrix K such that the matrix A + BK has prescribed eigenvalues which need satisfy only the condition that a certain number of them are distinct and real; then a feedback law of the form u = r + Kx can be used to achieve the desired pole-placement. The method does not require solution of sets of non-linear equations or manipulation of polynomial matrices, and no knowledge of eigenvalues and/or eigenvectors of A is necessary. If the computed matrix K and the given matrix C satisfy a consistency condition, a matrix Kν such that KνC = K can be directly obtained from K and the desired pole-placement can be realized by an output feedback law u = r + Kνy.     

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.     

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.     

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     

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.     

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”     

On the application of matrix generalized inverses to the construction of inverse systems

A discrete-time fast regulator with fast observer is considered in this paper. The system is assumed as a linear time-invariant nth order r-input, m-output system. Complete controllability and complete observability are assumed. No assumptions are made on the non-singularity of the system matrix and integrality of n/r and n/m. It is shown that the fast regulator with fast observer transfers any initial state to the origin within the number of steps equal to the sum of observability and controllability indices. Simple design methods for fast regulator and fast observer are presented. Also, it is illustrated geometrically how the state point is transferred to a subspace with successively decreased dimension for each step until it arrives at the origin.