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1

An output sensitivity problem for a class of fractional order discrete-time linear systems

Benfatah Youssef, El-Bhih Amine, Rachik Mostafa, Lafif Marouane

Acta Mechanica et Automatica

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2021

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Vol. 15, no. 4

227--235

EN

Consider the linear discrete-time fractional order systems with uncertainty on the initial state {Δαxi+1=Axi+Bui, i≥0x0=τ0+τ̂0∈Rn, τ̂0∈Ωyi=Cxi, i≥0}, where A,B and C are appropriate matrices, x0 is the initial state, yi is the signal output, α the order of the derivative, τ0 and τ̂0 are the known and unknown part of x0, respectively, ui=Kxi is feedback control and Ω⊂Rn is a polytope convex of vertices w1,w2,...,wp. According to the Krein–Milman theorem, we suppose that τ̂0=Σ pj=1αjwj for some unknown coefficients α1≥0,...,αp≥0 such that Σ pj=1αj=1. In this paper, the fractional derivative is defined in the Grünwald–Letnikov sense. We investigate the charac-terisation of the set χ(τ̂0,ϵ) of all possible gain matrix K that makes the system insensitive to the unknown part τ̂0, which means χ(τ̂0,ϵ)={K∈Rm×n / ∥∂yi∂αj∥≤ϵ, ∀j=1,...,p,∀i≥0}, where the inequality ∥∂yi∂αj∥≤ϵ showing the sensitivity of yi relative-ly to uncertainties {αj}j=1p will not achieve the specified threshold ϵ>0. We establish, under certain hypothesis, the finite determination of χ(τ̂0,ϵ) and we propose an algorithmic approach to made explicit characterisation of such set.

2

Novel fault detection criteria based on linear quadratic control performances

Krokavec D., Filasová A.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 4

929-938

EN

This paper proposes a new approach to designing a relatively simple algorithmic fault detection system that is potentially applicable in embedded diagnostic structures. The method blends the LQ control principle with checking and evaluating unavoidable degradation in the sequence of discrete-time LQ control performance index values due to faults in actuators, sensors or system dynamics. Design conditions are derived, and direct computational forms of the algorithms are given. A simulation example subject to different types of failures is used to illustrate the design process and to demonstrate the effectiveness of the method.

3

H-infinity control of discrete-time linear systems constrained in state by equality constraints

Filasová A., Krokavec D.

International Journal of Applied Mathematics and Computer Science

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2012

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Vol. 22, no. 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 discrete-time 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 outper forms 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.

4

Asymptotics of solution for singularly perturbed nonlinear discrete periodic optimal control problems

Kurina G., Nekrasova N.

Systems Science

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2008

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Vol. 34, no 1

23-37

EN

The asymptotic expansion of the solution of a singularly perturbed nonlinear discrete time periodic optimal control problem is constructed as series with respect to non-negative integer powers of a small parameter. The terms of asymptotic expansion are the solutions of optimal control problems which are essentially simpler than the original perturbed problem. The solvability of the perturbed problem is established in the neighborhood of a solution of the simpler non-perturbed problem of the lower dimension. The estimates are obtained for the proximity of the approximate solutions to the exact one. The nice property is proved, namely, the values of the minimized functional do not increase when higher-order approximations to the optimal control are used. Numerical examples are given in order to illustrate the method proposed.

5

Optimal flow control strategies for a single virtual circuit in a connection-oriented communication network

Ignaciuk P., Bartoszewicz A.

Systems Science

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2007

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Vol. 33, no 4

93-101

EN

In this paper, the congestion control problem in a single virtual circuit of a connection-oriented network is addressed. The network is modeled as a discrete time, n-th order system. On the basis of the state space description of the plant a feedback control law is derived by minimizing linear quadratic cost functional. Two controllers are proposed. The first one is obtained when the whole state vector is considered in the optimization procedure, while in the latter case the emphasis is placed on the output variable, with its influenceon the cost functional moderated by a weighting coefficient. Closed loop system stability is demonstrated, and conditions for no data loss and full bottleneck link bandwidth utilization in the network are presented, and strictly proved. Since the rates generated by the controller are always nonnegative and bounded, the proposed scheme can be implemented in real systems.

6

Discrete time sliding mode control using multirate output feedback with chattering elimination

Bandyopadhyay B., Thakar V. K.

Systems Science

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2005

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Vol. 31, no 4

49-61

EN

This paper presents a inultirate output feedback based sliding mode control algorithm which eliminates the chattering. The proposed algorithm uses the output samples for the switching function evaluation and sliding mode control computation. The control is based on the reaching law approach and drives the system trajectory to sliding manifold in finite time. The control law obtained has two parts, nonlinear and linear. The nonlinear part brings the trajectory in the vicinity of the sliding manifold and linear control guarantees that the sliding manifold is readied in one step and thereafter the motion of the system confines to the manifold. The proposed algorithm does not require knowledge of the complete state vector. As in many practical situations, all the states are not available for mcasurement: the proposed algorithm gives better choice for obtaining similar result. The algorithm is applied to a dc motor system with position measurement only.