A class of well-known canonical forms for single-input or single-output controllable and observable systems are extended to multivariable systems. It is shown that, unlike the single-variable case, the canonical forms are generally not unique, but that the structure of the canonical form can be controlled to some extent by the designer. A major result of the paper is that a multi-input system can be transformed to a set of coupled single-input subsystems.

Canonical forms for linear multivariable systems

An optimal control problem for a linear regulator with constant external disturbance is formulated. It is shown that, for a suitably selected quadratic-type performance index, the optimal control is not an explicit function of the external disturbance. Moreover, the optimal control can be synthesized as a time-invariant linear function of the state plus the first time integral of a certain other time-invariant linear function of the state.

Optimal control of the linear regulator with constant disturbances

The problem of transforming a single-input, single-output, time-variable linear system to phase-variable form is considered in this paper. Specific criteria for the existence of an equivalent canonical form are derived together with a method for obtaining the form when it exists. This method, when specialized to fixed systems, is particularly simple, and results in an easilyconstructed explicit form for the transforming matrix. Furthermore, the calculation of eigenvalues and eigenvectors necessary in previously published techniques is avoided.

Transformation of time-variable systems to canonical (phase-variable) form

Observer theory for constant-coefficient, linear systems is extended to systems with unknown, inaccessible inputs. Two types of observers for such systems are defined, and simplified criteria for their existence are developed. A theorem regarding the structure of these observers is given which allows the application of results for systems with single unknown inputs to systems with multiple unknown inputs.

Observers for systems with unknown and inaccessible inputs

In a previous paper [1], the conventional optimal linear regulator theory was extended to accommodate the case of external input disturbances \omega(t) which are not directly measurable but which can be assumed to satisfy d^{m+1}\omega(t)/dt^{m+1} = 0 , i.e., represented as m th-degree polynomials in time t with unknown coefficients. In this way, the optimal controller u^{0}(t) was obtained as the sum of: 1) a linear combination of the state variables x_{i}, i = 1,2,...,n , plus 2) a linear combination of the first (m + 1) time integrals of certain other linear combinations of the state variables. In the present paper, the results obtained in [1] are generalized to accommodate the case of unmeasurable disturbances \omega(t) which are known only to satisfy a given \rho th-degree linear differential equation D: d^{\rho}\omega(t)/dt^{\rho} + \beta_{\rho}d^{\rho-1}\omega(t)/dt^{\rho-1}+...+\beta_{2}d\omega/dt + \beta_{1}\omega=0 where the coefficients \beta_{i}, i = 1,...,\rho , are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulation x(t) \approx 0 in the face of any and every external disturbance function \omega(t) which satisfies the given differential equation D -even steady-state periodic or unstable functions \omega(t) . An essentially different method of deriving this result, based on stabilization theory, is also described, In each cases the results are extended to the case of vector control and vector disturbance.

Further study of the linear regulator with disturbances--The case of vector disturbances satisfying a linear differential equation

Linear transformation of arbitrary, completely controllable, single-input, time-variant linear dynamical system to canonical form

Another note on the transformation to canonical (phase-variable) form

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Another note on the transformation to canonical (phase-variable) form