2020 (vol. 30) - Number 4

*A.B. Malinowska, R. Kamocki, R. Almeida, T. Odzijewicz:*

On the existence of optimal consensus control for the fractional Cucker--Smale model

*R. Grymin, W. Bożejko, J. Pempera, M. Wodecki, Z. Chaczko:*

Algorithm for solving the Discrete-Continuous Inspection Problem

*T. Kaczorek, A. Ruszewski:*

Global stability of discrete-time nonlinear systems with descriptor standard and fractional positive linear parts and scalar feedbacks

*R. Arora, K. Gupta:*

Fuzzy goal programming technique for multi-objective indefinite quadratic bilevel programming problem

*P. Nowak, J. Czeczot, P. Grelewicz:*

Tuning rules for industrial use of the second-order Reduced Active Disturbance Rejection Controller

*M.I. Gomoyunov:*

Optimal control problems with a fixed terminal time in linear fractional-order systems

*Muhafzan, A. Nazra, L. Yulianti, Zulakmal, R. Revina:*

On LQ optimization problem subject to fractional order irregular singular systems

*M. Blizorukova, V. Maksimov:*

On one algorithm for reconstruction of an disturbance in a linear system of ordinary differential equations

ACS Abstract:

**2016 (Volume 26)**

Number 4

Practical and asymptotic stability of fractional discrete-time scalar systems described by a new model

Andrzej Ruszewski(Bialystok University of Technology, Poland) |

The stability problems of fractional discrete-time linear scalar systems described by the new model are considered. Using the classical D-partition method, the necessary and sufficient conditions for practical stability and asymptotic stability are given. The considerations are illustrated by numerical examples.

**keywords:** asymptotic stability, practical stability, fractional order, discrete-time linear system

On fault tolerant control structures incorporating fault estimation

D. Krokavec, A. Filasova, P. Liscinsky(Technical University of Kosice, Slovakia) |

The paper provides the minimal necessary modifications of linear matrix inequality conditions for the mixed *H*_{2}/*H*_{∞} control design as well as for the augmented observer-based fault estimation to be mutually compatible in joint design of integrated fault estimation and fault tolerant control. To be possible, within this integration, to design the controller which guarantees a pre-specified *H*_{∞} norm disturbance attenuation level, the design conditions has to be regularized using the *H*_{2} performance index and, moreover, augmented fault observer must be of enforced dynamics. Analyzing the ambit of performances given on the mixed *H*_{2}/*H*_{∞} design, the joint design conditions are formulated as a minimization problem subject to convex constraints expressed by a system of LMIs. The feasibility of the conditions is demonstrated by a numerical example.

**keywords:** linear systems, fault tolerant control, fault estimation, linear matrix inequalities, H2 norm, H2/H∞ control strategy

Hyperchaos, adaptive control and synchronization of a novel 4-D hyperchaotic system with two quadratic nonlinearities

Sundarapandian Vaidyanathan(Vel Tech University, India) |

This research work announces an eleven-term novel 4-D hyperchaotic system with two quadratic nonlinearities. We describe the qualitative properties of the novel 4-D hyperchaotic system and illustrate their phase portraits. We show that the novel 4-D hyperchaotic system has two unstable equilibrium points. The novel 4-D hyperchaotic system has the Lyapunov exponents *L*_{1} = 3.1575, *L*_{2} = 0.3035, *L*_{3} = 0 and *L*_{4} = -33.4180. The Kaplan-Yorke dimension of this novel hyperchaotic system is found as *D _{KY}* = 3.1026. Since the sum of the Lyapunov exponents of the novel hyperchaotic system is negative, we deduce that the novel hyperchaotic system is dissipative. Next, an adaptive controller is designed to stabilize the novel 4-D hyperchaotic system with unknown system parameters. Moreover, an adaptive controller is designed to achieve global hyperchaos synchronization of the identical novel 4-D hyperchaotic systems with unknown system parameters. The adaptive control results are established using Lyapunov stability theory. MATLAB simulations are depicted to illustrate all the main results derived in this research work.

**keywords:** chaos, hyperchaos, control, synchronization, Lyapunov exponents

Analytic solutions of transcendental equations with application to automatics

H. Górecki, M. Zaczyk(AGH University of Science and Technology, Poland) |

In the paper the extremal dynamic error *x*(τ) and the moment of time τ are considered. The extremal value of dynamic error gives information about accuracy of the system. The time τ gives information about velocity of transient. The analytical formulae enable design of the system with prescribed properties. These formulae are calculated due to the assumption that *x*(τ) is a function of the roots *s*_{1}, ..., *s _{n}* of the characteristic equation.

**keywords:** extremal problems, characteristic equation, transmittance

Predictor-based stabilization for chained form systems with input time delay

F. Mnif(Sultan Qaboos University, Oman) |

This note addresses the stabilization problem of nonlinear chained-form systems with input time delay. We first employ the so-called σ-process transformation that renders the feedback system under a linear form. We introduce a particular transformation to convert the original system into a delay-free system. Finally, we apply a state feedback control, which guarantees a quasi-exponential stabilization to all the system states, which in turn converge exponentially to zero. Then we employ the so-called -type control to achieve a quasi-exponential stabilization of the subsequent system. A simulation example illustrated on the model of a wheeled mobile robot is provided to demonstrate the effectiveness of the proposed approach.

**keywords:** σ-process transformation, chained form systems, mobile robots, exponential stabilization, time delay system

Multivariable predictive control considering time delay for load-frequency control in multi-area power systems

S. Daniar, R. Aazami(Ilam University, Iran) | M. Shiroei(Islamic Azad University, Iran) |

In this paper, a multivariable model based predictive control (MPC) is proposed for the solution of load frequency control (LFC) in a multi-area interconnected power system. The proposed controller is designed to consider time delay, generation rate constraint and multivariable nature of the LFC system, simultaneously. A new formulation of the MPC is presented to compensate time delay. The generation rate constraint is considered by employing a constrained MPC and economic allocation of the generation is further guaranteed by an innovative modification in the predictive control objective function. The effectiveness of proposed scheme is verified through time-based simulations on the standard 39-bus test system and the responses are then compared with the proportional-integral controller. The evaluation of the results reveals that the proposed control scheme offers satisfactory performance with fast responses.

**keywords:** load frequency control, model predictive control, time delay, generation rate constraint

Analysis and comparison of the stability of discrete-time and continuous-time linear systems

T. Kaczorek(Bialystok University of Technology, Poland) |

The asymptotic stability of discrete-time and continuous-time linear systems described by the equations *x _{i}*

_{+1}=

*Ā*and

^{k}x_{i}*ẋ*(

*t*) =

*A*(

^{k}x*t*) for

*k*being integers and rational numbers is addressed. Necessary and sufficient conditions for the asymptotic stability of the systems are established. It is shown that: 1) the asymptotic stability of discrete-time systems depends only on the modules of the eigenvalues of matrix

*Ā*and of the continuous-time systems depends only on phases of the eigenvalues of the matrix

^{k}*A*, 2) the discrete-time systems are asymptotically stable for all admissible values of the discretization step if and only if the continuous-time systems are asymptotically stable, 3) the upper bound of the discretization step depends on the eigenvalues of the matrix

^{k}*A*.

**keywords:** analysis, comparison, stability, discrete-time, continuous-time, linear system

An observer-based control of linear systems with uncertain parameters

M. Rachik, M. Lhous(Hassan II University of Casablanca, Morocco) |

In this paper, the observer-based control for a class of uncertain linear systems is considered. Exponential stabilizability for the system is studied and reduced-order observer is discussed. Numerical examples are given to illustrate obtained results.

**keywords:** observer-based control, stability, uncertain linear systems

Identification of stable elementary bilinear time-series model

L. Malinski(Silesian University of Technology, Poland) |

The paper presents new approach to estimation of the coefficients of an elementary bilinear time series model (EB). Until now, a lot of authors have considered different identifiability conditions for EB models which implicated different identifiability ranges for the model coefficient. However, all of these ranges have a common feature namely they are significantly narrower than the stability range of the EB model. This paper proposes a simple but efficient solution which makes an estimation of the EB model coefficient possible within its entire stability range.

**keywords:** bilinear model, time-series, identification

New observer-based control design for mismatched uncertain systems with time-delay

V.V. Huynh(Ton Duc Thang University, Vietnam) |

In this paper, the state estimation problem for a class of mismatched uncertain time-delay systems is addressed. The estimation uses observer-based control techniques. The mismatched uncertain time-delay systems investigated in this study include mismatched parameter uncertainties in the state matrix and in the delayed state matrix. First, based on a new lemma with appropriately choosing Lyapunov functional, new results for stabilization of mismatched uncertain time-delay systems are provided on the basis of a linear matrix inequality (LMI) framework and the asymptotic convergence properties for the estimation error is ensured. Second, the control and observer gains are given from single LMI feasible solution which can overcome the drawback of the bilinear matrix inequalities approach often reported in the literature. Finally, a numerical example is used to demonstrate the efficacy of the proposed method.

**keywords:** observer-based control, time-delay systems, linear matrix inequality (LMI), mismatched uncertain systems

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