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1
Content available Digraph-building method for finding a set of minimal realizations of 2-D dynamic systems
Markowski K. A., Hryniów K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 5
589--597
EN
This paper presents a digraph-building method designed to find the determination of realization of two-dimensional dynamic system. The main differences between the method proposed and other state-of-the-art solutions used include finding a set of realizations (belonging to a defined class) instead of only one realization, and the fact that obtained realizations have minimal size of state matrices. In the article, the proposed method is described, compared to state-of-the-art methods and illustrated with numerical examples. To the best of authors’ knowledge, the method shown in the paper is superior to all other state-of-the-art solutions both in terms of number of solutions and their matrix size. Additionally, MATLAB function for determination of realization based on the set of state matrices is included.
2
Content available remote Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2010
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Vol. 20, no 3
507-512
EN
The notion of a common canonical form for a sequence of square matrices is introduced. Necessary and sufficient conditions for the existence of a similarity transformation reducing the sequence of matrices to the common canonical form are established. It is shown that (i) using a suitable state vector linear transformation it is possible to decompose a linear 2D system into two linear 2D subsystems such that the dynamics of the second subsystem are independent of those of the first one, (ii) the reduced 2D system is positive if and only if the linear transformation matrix is monomial. Necessary and sufficient conditions are established for the existence of a gain matrix such that the matrices of the closed-loop system can be reduced to the common canonical form.
3
Content available remote Positivity and stabilization of fractional 2D linear systems described by the Roesser model
Kaczorek T., Rogowski K.
International Journal of Applied Mathematics and Computer Science
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2010
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Vol. 20, no 1
85-92
EN
A new class of fractional 2D linear discrete-time systems is introduced. The fractional difference definition is applied to each dimension of a 2D Roesser model. Solutions of these systems are derived using a 2D Z-transform. The classical Cayley-Hamilton theorem is extended to 2D fractional systems described by the Roesser model. Necessary and sufficient conditions for the positivity and stabilization by the state-feedback of fractional 2D linear systems are established. A procedure for the computation of a gain matrix is proposed and illustrated by a numerical example.
4
Content available remote Independence of asymptotic stability of positive 2D linear systems with delays of their delays
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2009
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Vol. 19, no 2
255-261
EN
It is shown that the asymptotic stability of positive 2D linear systems with delays is independent of the number and values of the delays and it depends only on the sum of the system matrices, and that the checking of the asymptotic stability of positive 2D linear systems with delays can be reduced to testing that of the corresponding positive 1D systems without delays. The effectiveness of the proposed approaches is demonstrated on numerical examples.
5
Content available remote Robust stabilization of discrete linear repetitive processes with switched dynamics
Bochniak J., Gałkowski K., Rogers E., Kummert A.
International Journal of Applied Mathematics and Computer Science
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2006
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Vol. 16, no 4
441-462
EN
Repetitive processes constitute a distinct class of 2D systems, i.e., systems characterized by information propagation in two independent directions, which are interesting in both theory and applications. They cannot be controlled by a direct extension of the existing techniques from either standard (termed 1D here) or 2D systems theories. Here we give new results on the design of physically based control laws. These results are for a sub-class of discrete linear repetitive processes with switched dynamics in both independent directions of information propagation.
6
Content available remote On the Interaction Between Theory, Experiments, and Simulation in Developing Practical Learning Control Algorithms
Longman R. W.
International Journal of Applied Mathematics and Computer Science
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2003
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Vol. 13, no 1
101-111
EN
Iterative learning control (ILC) develops controllers that iteratively adjust the command to a feedback control system in order to converge to zero tracking error following a specific desired trajectory. Unlike optimal control and other control methods, the iterations are made using the real world in place of a computer model. If desired, the learning process can be conducted both in the time domain during each iteration and in repetitions, making ILC a 2D system. Because ILC iterates with the real world, and aims for zero error, the field pushes the limits of theory, modeling, and simulation, to predict the behavior when applied in the real world. It is the thesis of this paper that in order to make significant progress in this field it is essential that the research effort employ a coordinated simultaneous synergistic effort involving theory, experiments, and serious simulations. Otherwise, one very easily expends effort on something that seems fundamental from the theoretical perspective, but in fact has very little relevance to the performance in real world applications.
7
Content available remote An Application of the Fourier Transform to Optimization of Continuous 2-D Systems
Dymkou V., Dymkov M.
International Journal of Applied Mathematics and Computer Science
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2003
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Vol. 13, no 1
45-54
EN
This paper uses the theory of entire functions to study the linear quadratic optimization problem for a class of continuous 2D systems. We show that in some cases optimal control can be given by an analytical formula. A simple method is also proposed to find an approximate solution with preassigned accuracy. Some application to the 1D optimization problem is presented, too. The obtained results form a theoretical background for the design problem of optimal controllers for relevant processes.
8
Content available Minimal order deadbeat functional observers for singular 2D linear systems
Kaczorek T.
Control and Cybernetics
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2003
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Vol. 32, no 2
301-311
EN
Sufficient conditions for the existence of minimal order deadbeat functional observers for singular 2D linear systems described by the general singular 2D model are established. A procedure for computing matrices of the functional observers is given and illustrated by numerical example.
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