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1
Zeros in linear systems with time delay in state
EN
The concept of invariant zeros in a linear time-invariant system with state delay is considered. In the state-space framework, invariant zeros are treated as triples: complex number, nonzero state-zero direction, input-zero direction. Such a treatment is strictly related to the output-zeroing problem and in that spirit the zeros can be easily interpreted. The problem of zeroing the system output is discussed. For systems of uniform rank, the first nonzero Markov parameter comprises a certain amount of information concerning invariant zeros, output-zeroing inputs and zero dynamics. General formulas for output-zeroing inputs and zero dynamics are provided.
EN
The notion of zeros in linear time-invariant multi input multi output systems with delay in state or/and control input is not extensively discussed in the relevant literature. The concept of invariant zeros in a linear time-invariant MIMO system with delay in control vector is considered. In the state-space framework the invariant zeros are treated as the triples: complex number, nonzero state-zero direction, input-zero direction. Such treatment is strictly related to the output-zeroing problem and in that spirit the zeros can be easily interpreted. The problem of zeroing the system output is also addressed. For systems of uniform rank the first nonzero Markov parameter comprises a certain amount of information concerning invariant zeros, output-zeroing inputs and the zero dynamics. General formulas for output-zeroing inputs, the corresponding solutions and the zero dynamics are provided. The obtained results are illustrated by simple numerical examples.
3
Zeros and the output-zeroing problem in linear fractional-order systems
EN
The concept of invariant zeros in fractional order LTI systems with the Caputo derivative is introduced in the paper. The problem of zeroing the system output is discussed. For systems of uniform rank explicit formulas for output-zeroing inputs and the corresponding solutions to the state equation are provided. The zero dynamics and invariant zeros for such systems are also characterized.
4
An equivalent matrix pencil for bivariate polynomial matrices
EN
In this paper, we present a simple algorithm for the reduction of a given bivariate polynomial matrix to a pencil form which is encountered in Fornasini-Marchesini’s type of singular systems. It is shown that the resulting matrix pencil is related to the original polynomial matrix by the transformation of zero coprime equivalence. The exact form of both the matrix pencil and the transformation connecting it to the original matrix are established.
5
A general solution to the output-zeroing problem for MIMO LTI systems
EN
The problem of zeroing the output in an arbitrary linear continuous-time system S(A,B,C,D) with a nonvanishing transfer function is discussed and necessary conditions for output-zeroing inputs are formulated. All possible real-valued inputs and real initial conditions which produce the identically zero system response are characterized. Strictly proper and proper systems are discussed separately.
6
Zeros in Discrete-Time Mimo Lti Systems and the Output-Zeroing Problem
EN
A geometric interpretation of invariant zeros of MIMO LTI discrete-time systems is provided. The zeros are treated as the triples: complex number, state zero direction, input zero direction. Such a treatment is strictly connected with the output zeroing problem and in that spirit the zeros can be easily interpreted even in the degenerate case (i.e. when each complex number constitutes an invariant zero). Simply, in the degenerate case, to each complex number we can assign an appropriate real initial condition and an appropriate real input sequence which produce a non-trivial solution to the state equation and a zero system response. Clearly, when zeros are treated merely as complex numbers, such an interpretation is impossible. The proposed definition of invariant zeros is compared with other commonly known definitions. It is shown that each Smith zero of the system matrix is also an invariant zero in the sense of the definition adopted in the paper. On the other hand, simple numerical examples show that the considered definition of invariant zeros and the Davison-Wang definition are not comparable. The output-zeroing problem for systems decouplable by state feedback is also described.
7
On some characterization of invariant and decoupling zeros of singular systems
EN
In this paper a geometric characterization of invariant and decoupling zeros of a generalized state-space model described by the matrix 5-tuple (E,A,B,C,D), where E is singular but the pencil sE-A is regular, is presented. These zeros are characterized as invariant zeros of an appropriate linear system S(A,B,C,D). Several numerical examples are included to illustrate the proposed results as well as to discuss different definitions of zeros.
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