The robustness of controllability and observability of linear time-varying systems

Fixed point methods from nonlinear analysis are used to establish conditions under which the uniform complete controllability, of linear time-varying systems is preserved under nonlinear perturbations in the state dynamics and the zero-input uniform complete observability of linear time-varying systems is preserved under nonlinear perturbation in the state dynamics and output read-out map. Robustness of partial controllability., observability, and a specific kind of nonzero input observability are also proven.     

On the controllability and observability of discrete-time linear time-delay systems

This article studies the controllability and observability of discrete-time linear time-delay systems, so that the two properties can play a more fundamental role in system analysis before controller and observer design is engaged. Complete definitions of controllability and observability, which imply the stabilisability and detectability, respectively, and determine the feasibility of eigenvalue assignment, are proposed for systems with delays in both state variables and input/output signals. Necessary and sufficient criteria are developed to check the controllability and observability efficiently. The proofs are based on the equivalent expanded system, but the criteria only involve the delays and matrices of the same dimension as the original system. Finally, the duality between the suggested controllability and observability is presented.     

On the concepts of controllability and observability of linear systems

The objectives of this paper are to give a self-contained presentation of the subject of "controllability" and "observability" (concepts due to Kalman) developed in a simple yet general and rigorous way; to introduce these concepts in terms of the output as well as in terms of the state; and to point out and illustrate some of the subtleties which exist in the time-varying case.     

Signal insensitivity, controllability, and observability in linear systems

Signal insensitivity, suitably defined in terms of Frèchet derivative, is proved to be equivalent to signal invariance in linear time-invariant systems. Then, relations between signal-insensitivity and controllability-observability properties of the system are briefly reviewed.     

Structural controllability and observability

In this paper new criteria for the analysis of structural controllability and observability are derived for systems where additional structure in the parameter variations is taken into consideration. Existing results are shown to be generalized in this way. Simple examples are included to illustrate the usefulness of the conditions.     

Nonlinear controllability and observability

The properties of controllability, observability, and the theory of minimal realization for linear systems are well-understood and have been very useful in analyzing such systems. This paper deals with analogous questions for nonlinear systems.     

Decomposition of linear sequential machines

Linear sequential machines can sometimes be decomposed into parallel and series connections of smaller linear sequential machines. Necessary and sufficient conditions are given for such decompositions to exist for finite linear sequential machines.Research sponsored by the Air Force Office of Scientific Research, Grant AF-AFOSR 639-67, and by the National Science Foundation, Grant GP-6945     

Analysis of observability and controllability of nonlinear dynamic systems by linear methods

The problem of finding sufficient unobservability and uncontrollability criteria for nonlinear dynamic systems (including systems with nondifferentiable nonlinearities) is considered on the basis of known unobservability and uncontrollability criteria for linear systems and additional linear conditions. Duality relations supplementing the well-known duality principle for linear systems are established.     

A necessary algebraic condition for controllability and observability of linear time-varying systems

In this note, we give an algebraic condition which is necessary for the system x'(t)=A(t)x(t)+B(t)u(t), y(t)=C(t)x(t), either to be totally controllable or to be totally observable, where x/spl isin//spl Ropf//sup d/, u/spl isin//spl Ropf//sup p/, y/spl isin//spl Ropf//sup q/, and the matrix functions A, B and C are (d-2), (d-1) and (d-1) times continuously differentiable, respectively. All conditions presented here are in terms of known quantities and therefore easily verified. Our conditions can be used to rule out large classes of time-varying systems which cannot be controlled and/or observed no matter what the nonzero time-varying coefficients are. This work is motivated by the deep result of Silverman and Meadows.     

Composite system controllability and observability

A differential operator approach to modeling the dynamical behavior of linear, time-invariant, multivariable systems is employed in order to develop some new techniques for determining the controllability and observability properties of series, parallel, and feedback composite systems. The tests outlined involve polynomial matrix manipulations and rely on some recent results which deal with the controllability and observability of systems in differential operator form.     

On stochastic observability and controllability

New definitions of stochastic observability, detectability, controllability and reachability are proposed. For both continuous and discrete time linear systems easily verifiable necessary and sufficient conditions are obtained, which are compatible with the usual ones for the corresponding (degenerate) deterministic systems. Furthermore, for stochastically observable, linear systems explicit expressions for the LUMCE of the state at time $\text{t≧0}$, with no information of the initial state, are obtained.     

Duality of the notions of controllability and observability for degenerate linear hybrid systems

Consideration was given to the linear system of equations in continuous and discrete time with constant matrices of coefficients which is not explicit in the derivative of the continuous component of the desired vector function. The notion of conjugate system was introduced, and an analog of the Kalman duality theorem relating the notions of total controllability and observability was proved.     

Band criteria and recursive tests of complete controllability and observability of linear algebraic-differentiable systems

New criteria are introduced of complex controllability and observability of controllable linear stationary algebraic-differentiable systems with many inputs and outputs, which reduce to the calculation of the rank of special band matrices. Recursive tests are set up, which permit reducing computational expenses in the verification for the fulfillment of the conditions of complete controllability and observability of algebraic-differentiable systems of a large dimension.     

Measures of controllability and observability and residues

The relationship between two measures of controllability and observability is derived, given a linear time-invariant system. Both of the measures are based on angles between various subspaces of the state space. It is shown that these measures can be used to bound the norm of the residues of the transfer function     

On the semigroups of linear sequential machines

A new and considerably shorter proof is given to the characterization of the groups of linear nonsingular sequential machines due to Ecker.