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Similarity transformation of matrices to one common canonical form and its applications to 2D linear systems

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2010
|
tom 20
|
nr 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.
2
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Elimination of finite eigenvalues of the 2D Roesser model by state feedbacks

100%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2001
|
tom 11
|
nr 2
369-376
EN
A new problem of decreasing the degree of the closed-loop characteristic polynomial of the 2D Roesser model by a suitable choice of state feedbacks is formulated. Sufficient conditions are established under which it is possible to choose state feedbacks such that the non-zero closed-loop characteristic polynomial has degree zero. A procedure for computation of the feedback gain matrices is presented and illustrated by a numerical example.
3
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Hybrid stabilization of discrete-time LTI systems with two quantized signals

88%
Zhai G., Matsumoto Y., Chen X., Imae J., Kobayashi T.
International Journal of Applied Mathematics and Computer Science
|
2005
|
tom 15
|
nr 4
509-516
EN
We consider stabilizing a discrete-time LTI (linear time-invariant) system via state feedback where both the quantized state and control input signals are involved. The system under consideration is stabilizable and stabilizing state feedback has been designed without considering quantization, but the system's stability is not guaranteed due to the quantization effect. For this reason, we propose a hybrid quantized state feedback strategy asymptotically stabilizing the system, where the values of the quantizer parameters are updated at discrete time instants. We also extend the result to the case of static output feedback.
4
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Positivity and linearization of a class of nonlinear continuous-time systems by state feedbacks

88%
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
|
2015
|
tom 25
|
nr 4
827-831
EN
The positivity and linearization of a class of nonlinear continuous-time system by nonlinear state feedbacks are addressed. Necessary and sufficient conditions for the positivity of the class of nonlinear systems are established. A method for linearization of nonlinear systems by nonlinear state feedbacks is presented. It is shown that by a suitable choice of the state feedback it is possible to obtain an asymptotically stable and controllable linear system, and if the closed-loop system is positive then it is unstable.
5
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$H_∞$ control of discrete-time linear systems constrained in state by equality constraints

75%
Krokavec D.
International Journal of Applied Mathematics and Computer Science
|
2012
|
tom 22
|
nr 3
551-560
EN
In this paper, stabilizing problems in control design are addressed for linear discrete-time systems, reflecting equality constraints tying together some state variables. Based on an enhanced representation of the bounded real lemma for discretetime systems, the existence of a state feedback control for such conditioned stabilization is proven, and an LMI-based design procedure is provided. The control law gain computation method used circumvents generally an ill-conditioned singular design task. The principle, when compared with previously published results, indicates that the proposed method outperforms the existing approaches, guarantees feasibility, and improves the steady-state accuracy of the control. Furthermore, better performance is achieved with essentially reduced design effort. The approach is illustrated on simulation examples, where the validity of the proposed method is demonstrated using one state equality constraint.
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