2018 (vol. 28) - Number 4

*T. Kaczorek:*

A new method for computation of positive realizations of fractional linear continuous-time systems

*A. Biswas, A. Sarkar:*

Development of dual hesitant fuzzy prioritized operators based on Einstein operations with their application to multi-criteria group decision making

*L. Li, R. Zhang, J. Wang, X. Shang:*

Some q-orthopair linguistic Heronian mean operators with their application to multi-attribute group decision making

*J. Duda:*

Lyapunov matrices approach to the parametric optimization of a system with two delays and a PD controller

*M. Lhous, M. Rachik, El M. Magri:*

Ideal observability for bilinear discrete-time systems with and without delays in observation

*P.T. Anh, A. Babiarz, A. Czornik, M. Niezabitowski:*

Variation of constant formulas for fractional difference equations

*K. Bingi, R. Ibrahim, M. Noh Karsiti, S. Miya Hassan, V. Rajah Harindran:*

A comparative study of 2DOF PID and 2DOF fractional order PID controllers on a class of unstable systems

ACS Abstract:

**2004 (Volume 14)**

Number 2

**Using control theory to make cancer chemotherapy beneficial from phase dependence and resistant to drug resistance**

Marek Kimmel(Rice University, USA) | Andrzej ¦wierniak(Silesian University of Technology, Poland) |

**keywords:** optimal control, biomathematical models, cancer chemotherapy, bilinear systems, cell cycle dynamics.

**- norm computation for stable linear continuous-time periodic systems**

B.P. Lampe(University of Rostock, Germany) | M. Obraszov and E. Rosenwasser(St. Petersburg State University of Ocean Technology, Russia) |

*T*is the period of the system, and it operates only with matrices of finite dimension. An example is given, where the -norm is determined exactly, and the proposed method turns out to be superior to other known methods.

**keywords:** time-varying systems, periodic motion, Laplace transforms, Green functions, transfer functions, norms.

**A controllability problem for a class of uncertain - parameters discrete-time linear dynamic systems**

Krzysztof Oprzedkiewicz(University of Mining and Metallurgy, Poland) |

**keywords:** linear discrete-time systems with uncertain parameters, controllability.

**A note on zeros, output-nulling subspaces and zero-dynamics in MIMO LTI systems**

Jerzy Tokarzewski(Military University of Technology, Poland) |

*S(A,B,C)*the classical notion of the Smith zeros does not characterize fully the output-zeroing problem nor the zero dynamics. The question how this notion can be extended and related to the state-space methods is discussed. Nothing is assumed about the relationship of the number of inputs to the number of outputs nor about the normal rank of the underlying system matrix. The proposed extension treats multivariable zeros (called further the invariant zeros) as the triples (complex number, nonzero 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 any complex number is such a zero).

A simple sufficient and necessary condition of nondegeneracy is presented. The condition decomposes the class of all systems

*S(A,B,C)*such that

*B*

*0*and

*C*

*0*into two disjoint subclasses: of nondegenerate and degenerate systems. In nondegenerate systems the Smith zeros and the invariant zeros are exactly the same objects which are determined as the roots of the so-called zero polynomial. The degree of this polynomial equals the dimension of the maximal

*(A,B)*-invariant subspace contained in ker

*C*, while the zero dynamics are independent upon control vector. In degenerate systems the zero polynomial determines merely the Smith zeros, while the set of the invariant zeros equals the whole complex plane. The dimension of the maximal

*(A,B)*-invariant subspace contained in ker

*C*is strictly larger than the degree of the zero polynomial, whereas the zero dynamics essentially depend upon control vector.

**keywords:** linear systems, zeros, degeneracy, zero dynamics, Markov parameters, singular value decomposition.

**A time-varying switching plane design for variable structure control of the third order system with input constraint**

Andrzej Bartoszewicz and Aleksandra Nowacka(Technical University of £ód¼, Poland) |

**keywords:** sliding mode control, variable structure systems, time-varying switching planes, switching surface design.

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