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EN
The notion of finite zeros of discrete-time positive linear systems is introduced. It is shown that such zeros are real nonnegative numbers. It is also shown that a square positive strictly proper or proper system of uniform rank with the observability matrix of full column rank has no finite zeros. The problem of zeroing the system output for positive systems is defined. It is shown that a square positive strictly proper or proper system of uniform rank with the observability matrix of full column rank has no nontrivial output-zeroing inputs. The obtained results remain valid for non-square positive systems with the first nonzero Markov parameter of full column rank.
2  Finite zeros of positive linear continuous-time systems
EN
The notion of finite zeros of continuous-time positive linear systems is introduced. It is shown that such zeros are real numbers. It is also shown that a square positive strictly proper or proper system of uniform rank with observability matrix of full column rank has no finite zeros. The problem of zeroing the system output for positive systems is defined. It is shown that a square positive strictly proper or proper system of uniform rank with observability matrix of full column rank has no nontrivial output-zeroing inputs. The obtained results remain valid for non-square positive systems with the first nonzero Markov parameter of full column rank.
3  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.
4  Zeros and the output-zeroing problem in linear systems with time delay in control vector
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.
5  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.
6  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.
7  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.   Strona / 1    JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.