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A new methodology for solving inverse dynamics problem is developed. The methodology is based on using a mathematical model of a dynamical system and robust stabilization methods for a system under uncertainty. Most exhaustively the theory is described for linear finite-dimensional time-invariant scalar systems and multiple-input multiple-output systems. The study shows that with this approach, the zero dynamics of the original system is of crucial significance. This dynamics, if exists, is assumed to be exponentially stable. It is established that zero-dynamics, relative order, and the corresponding equations of motion cannot be defined correctly in multiple-input multiple-output systems. For correct inverse transformation of the solution of the problem, additional assumptions have to be introduced, which generally limits the inverse system category. Special attention is given to the synthesis of elementary (minimal) inverters, i.e., least-order dynamical systems that solve the transformation problem. It is also established that the inversion methods sustain the efficiency with finite parameter variations in the initial system as well as with uncontrolled exogenous impulses having no impact on the system's internal dynamics. [ABSTRACT FROM AUTHOR]

We consider the problem of determining the equations of zero dynamics of a nonlinear system that is affine with respect to the control. All known methods for solving this problem have a restricted scope. We obtain a new algorithm for solving this problem which permits determining equations of zero dynamics of some systems to which the previously known methods cannot be applied. [ABSTRACT FROM AUTHOR]

We consider the problem of inversion of a dynamical system, that is, the problem of reconstructing the unknown input of a system on the basis of the measured output. To form a continuous estimate of the unknown input, we suggest to use a controlled model of the system in which the control is constructed with the use of higher-order sliding modes. [ABSTRACT FROM AUTHOR]

We generalize the notion of relative degree for a controlled dynamical system (possibly, discrete). We show that the new notion of leading incomplete relative degree is uniquely determined for a broad class of 'square' dynamical systems (i.e., systems with as many inputs as outputs), including controllable systems. If r is the vector of relative degree of a system in the classical sense, then it is so in the generalized sense as well. We indicate a constructive algorithm for computing the leading incomplete relative degree. [ABSTRACT FROM AUTHOR]

The article discusses a study on the asymptotic reconstruction of the state vector of a dynamical system depicted by differential-difference equations with a measurable time delays. It mentions that asymptotic observability consists of two parts which are differential-difference and difference equations. It states that the observer uses the input and output of the system measured at current and previous times to solve the problems and create an asymptotically estimate of the unknown state vector. It adds that for time delay systems, the observability is defined in different ways that lead in differences in the solutions of the problem.

Sufficient conditions for the invertibility of linear stationary dynamical systems are formulated. It is shown that the a priori information that the input signal is bounded substantially expands the class of systems for which the inversion problem is solvable. [ABSTRACT FROM AUTHOR]