Tanel Mullari

Equivalence of Different Realization Methods for Higher Order Nonlinear Input–Output Differential Equations*

Several state space realizability conditions and realization algorithm for nonlinear single-input single-output higher order input–output differential equation are compared. Three different necessary and sufficient state space realizability conditions, respectively, given in terms of integrability of the subspaces of one-forms, involutivity of the conditionally invariant distributions, and commutativity of iterative Lie brackets of vector fields, are proved to be equivalent. Moreover, the algorithm-based solutions are demonstrated for computing the integrable basis of the subspaces of oneforms. Finally, explicit formulas are provided for calculation of the differentials of the state coordinates which, in case the necessary and sufficient realizability conditions are satisfied, can be integrated to obtain the state coordinates.

Extended observer form: simple existence conditions

The paper focuses on the problem of transforming the discrete-time single-input single-output nonlinear state equations into the extended observer form, which, besides the input and output, also depends on a finite number of their past values. The simple necessary and sufficient conditions for the existence of the extended coordinate change and the output transformation, allowing to solve the problem, are formulated in terms of certain partial derivatives, related to the input–output equation, corresponding to the state equations. Moreover, a certain algorithm for transforming the state equations into the observer form is proposed.

Discussion on: Unified Approach to the Problem of Full Decoupling via Output Feedback

The article under discussion studies the unmeasurable disturbance decoupling problem (DDP) via the dynamic output feedback from a unified viewpoint for continuousand discrete-time nonlinear control systems as well as for discrete event systems. This is a challenging problem, not having today a complete solution, that is the necessary and sufficient solvability conditions together with the constructive algorithm to compute the feedback law. In [1]1 the intrinsic coordinate-free necessary solvability conditions were obtained in terms of the maximal conditioned invariant subspace and the smallest controlled invariant subspace. The article under discussion provides the sufficient algorithm-based solvability conditions. Unfortunately it does not provide the explicit algorithm for problem solution; the steps of the algorithm are scattered implicitly in the different parts of the article, partly presented in the unified manner, partly treated separately for the continuous and discrete cases. Note that there exist certain differences in the problem statement as well as in the assumptions made on the control system, not to mention the mathematical tools used in [1] and in the article under discussion. We will comment these differences below. First, in [1] only analytic state affine systems are considered whereas the article under discussion allows the systems nonlinear in control, and the functions in system

State space realization of bilinear continuous-time input-output equations

This paper studies the realizability property of continuous-time bilinear input–output (i/o) equations in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The paper proves that the 2nd order bilinear i/o differential equation, unlike the discrete-time case, is always realizable in the classical state space form. The complete list of 3rd and 4th order realizable i/o bilinear models is given and two subclasses of realizable i/o bilinear systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for all realizable 2nd and 3rd order bilinear i/o equations, and for one realizable subclass of bilinear i/o equations of arbitrary order.

Transformation of nonlinear control systems into the observer form: necessary conditions

This paper gives the necessary conditions for existence of the output and state coordinate transformations, allowing to transform the nonlinear single-input single-output control system into the observer form. The suggested conditions strengthen the existing ones. The new conditions require (additionally) the certain n 1-forms to be closed.

On Classical State Space Realizability of Bilinear Input-Output Differential Equations

This paper studies the realizability property of continuous-time bilinear i/o equations in the classical state space form. Constraints on the parameters of the bilinear i/o model are suggested that lead to realizable models. The paper proves that the 2nd order bilinear i/o differential equation, unlike the discrete-time case, is always realizable in the classical state space form. The complete list of 3rd and 4th order realizable i/o bilinear models is given and two subclasses of realizable i/o bilinear systems are suggested. Our conditions rely basically upon the property that certain combinations of coefficients of the i/o equations are zero or not zero. We provide explicit state equations for all realizable 2nd and 3rd order bilinear i/o equations, and for one realizable subclass of bilinear i/o equations of arbitrary order

Forward and Backward Shifts of Vector Fields: Towards the Dual Algebraic Framework

An algebraic framework for discrete-time nonlinear control systems is introduced, based on the forward and backward shifts of the vector fields. It appears to be the dual of the framework based on the differential 1-forms. As applications, the characterization of accessibility and the solution to the static state feedback linearization problem are stated in terms of the given vector fields.

On accessibility conditions for state space nonlinear control systems on homogeneous time scales

Abstract An algebraic formalism for nonlinear control systems defined on homogeneous time scales is introduced. This formalism is based on the forward, backward shifts, delta and nabla operators of differential one-forms and vector fields. The accessibility conditions for considered control systems both in terms of vector fields and one-forms are presented.

Discrete-time Lie derivative with respect to system dynamics ?

The paper extends the concept of the Lie derivative of the vector eld for the discrete-time case, preserving its geometrical meaning. The new concept is illustrated on the problem of lowering the input shifts in the generalized state equations.

Weak reachability and controllability of discrete-time nonlinear systems: generic approach and singular points

ABSTRACT The paper finds the singular points from which (to which) the generically accessible system is not weakly reachable (controllable) in k steps. These points are found with the help of the space of vector fields, being the discrete-time analogue of the strong accessibility distribution. Unlike in the continuous-time case, a separate object is needed to find the singular points related to weak reachability.