2022 (vol. 32) - Number 1

*J. Cvejn:*

The magnitude optimum design of the PI controller for plants with complex roots and dead time

*S.F. Al-Azzawi, M.A. Hayali:*

Coexisting of self-excited and hidden attractors in a new 4D hyperchaotic Sprott-S system with a single equilibrium point

*M.A. Hammami, N. El Houda Rettab, F. Delmotte:*

On the state estimation for nonlinear continuous-time fuzzy systems

*M. Ilyas, M.A. Khan, A. Khan, Wei Xie, Y. Khan:*

Observer design estimating the propofol concentration in PKPD model with feedback control of anesthesia administration

*L. Moysis, M. Tripathi, M. Marwan:*

Adaptive observer design for systems with incremental quadratic constraints and nonlinear outputs – application to chaos synchronization

*S. Vaidyanathan, K. Benkouider, A. Sambas:*

A new multistable jerk chaotic system, its bifurcation analysis, backstepping control-based synchronization design and circuit simulation

*T.T. Tuan, H. Zabiri, M.I.A. Mutalib, Dai-Viet N. Vo:*

Disturbance-Kalman state for linear offset free MPC

*Yuan Xu, Jun Wang:*

A novel multiple attribute decision-making method based on Schweizer-Sklar *t*-norm and *t*-conorm with *q*-rung dual hesitant fuzzy information

*T. Kaczorek:*

Observers of fractional linear continuous-time systems

ACS Abstract:

**2021 (Volume 31)**

Number 4

Different linear control laws for fractional chaotic maps using Lyapunov functional

A. Othman Almatroud, M. Al-Sawalha Mossa(University of Ha'il, Saudi Arabia) | A. Ouannas(University of Larbi Ben M'hidi, Oum El Bouaghi, Algeria) | G. Grassi(Universita del Salento, Lecce, Italy) | I.M. Batiha(Ajman University, UAE) | A. Gasri(University of Larbi Tebessi, Algeria) |

Dynamics and control of discrete chaotic systems of fractional-order have received considerable attention over the last few years. So far, nonlinear control laws have been mainly used for stabilizing at zero the chaotic dynamics of fractional maps. This article provides a further contribution to such research field by presenting simple linear control laws for stabilizing three fractional chaotic maps in regard to their dynamics. Specifically, a one-dimensional linear control law and a scalar control law are proposed for stabilizing at the origin the chaotic dynamics of the Zeraoulia-Sprott rational map and the Ikeda map, respectively. Additionally, a two-dimensional linear control law is developed to stabilize the chaotic fractional flow map. All the results have been achieved by exploiting new theorems based on the Lyapunov method as well as on the properties of the Caputo h-difference operator. The relevant simulation findings are implemented to confirm the validity of the established linear control scheme.

**keywords:** discrete fractional calculus, chaotic maps, linear control, Lyapunov method

Efficient choice of parameters on delta-reachability bounded hybrid systems

K. Murillo, E. Rocha(University of Aveiro, Portugal) |

Hybrid systems (HS) are roughly described as a set of discrete state transitions and continuous dynamics modeled by differential equations. Parametric HS may be constructed by having parameters on the differential equations, initial conditions, jump conditions, or a combination of the previous ones. In real applications, the best solution is obtained by a set of metrics functional over the set of solutions generated from a finite set of parameters. This paper examines the choice of parameters on delta-reachability bounded hybrid systems.We present an efficient model based on the tool pHL-MT to benchmark the HS solutions (based on dReach), and a non-parametric frontier analysis approach, relying on multidirectional efficiency analysis (MEA). Three numerical examples of epidemic models with variable growth infectivity are presented, namely: when the variable of infected individuals oscillates around some endemic (non-autonomous) equilibrium; when there is an asymptotically stable non-trivial attractor; and in the presence of bump functions.

**keywords:** bounded hybrid system safety property delta-reachability multidirectional efficiency analysis

Effectiveness of Dynamic Matrix Control algorithm with Laguerre functions

P. Tatjewski(Warsaw University of Technology, Poland) |

The paper is concerned with the presentation and analysis of the Dynamic Matrix Control (DMC) model predictive control algorithm with the representation of the process input trajectories by parametrised sums of Laguerre functions. First the formulation of the DMCL (DMC with Laguerre functions) algorithm is presented. The algorithm differs from the standard DMC one in the formulation of the decision variables of the optimization problem – coefficients of approximations by the Laguerre functions instead of control input values are these variables. Then the DMCL algorithm is applied to two multivariable benchmark problems to investigate properties of the algorithm and to provide a concise comparison with the standard DMC one. The problems with difficult dynamics are selected, which usually leads to longer prediction and control horizons. Benefits from using Laguerre functions were shown, especially evident for smaller sampling intervals.

**keywords:** process control, model predictive control, DMC algorithm, Laguerre functions

On theoretical and practical aspects of Duhamel’s integral

M. Rozanski, B. Sikora, A. Smuda, R. Witula(Silesian University of Technology, Poland) |

The paper is a new approach to the Duhamel integral. It contains an overview of formulas and applications of Duhamel’s integral as well as a number of new results on the Duhamel integral and principle. Basic definitions are recalled and formulas for Duhamel’s integral are derived via Laplace transformation and Leibniz integral rule. Applications of Duhamel’s integral for solving certain types of differential and integral equations are presented. Moreover, an interpretation of Duhamel’s formula in the theory of operator semigroups is given. Some applications of Duhamel’s formula in control systems analysis are discussed. The work is also devoted to the usage of Duhamel’s integral for differential equations with fractional order derivative.

**keywords:** Duhamel?s integral, Duhamel?s principle, Duhamel?s formula, Laplace transformation, semigroup of operators, Leibniz integral rule, Volterra integral equation, Caputo fractional derivative

On optimal control problem subject to fractional order discrete time singular systems

T. Chiranjeevi(Rajkiya Engineering College Sonbhadra, U. P., India) | R.K. Biswas, N.R. Babu(National Institute of Technology, Silchar, India) | R. Devarapalli(BIT Sindri, Jharkhand, India) | F.P. García Márquez(University of Castilla-La Mancha, Spain) |

In this work, we present optimal control formulation and numerical algorithm for fractional order discrete time singular system (DTSS) for fixed terminal state and fixed terminal time endpoint condition. The performance index (PI) is in quadratic form, and the system dynamics is in the sense of Riemann-Liouville fractional derivative (RLFD). A coordinate transformation is used to convert the fractional-order DTSS into its equivalent non-singular form, and then the optimal control problem (OCP) is formulated. The Hamiltonian technique is used to derive the necessary conditions. A solution algorithm is presented for solving the OCP. To validate the formulation and the solution algorithm, an example for fixed terminal state and fixed terminal time case is presented.

**keywords:** fractional optimal control problem, discrete time singular system, fractional derivative, Hamiltonian technique

Application of maximum principle to optimization of production and storage costs

L. Popescu, R. Dimitrov(University of Craiova, Romania) |

A problem of optimization for production and storge costs is studied. The problem consists in manufacture of *n* types of products, with some given restrictions, so that the total production and storage costs are minimal. The mathematical model is built using the framework of driftless control affine systems. Controllability is studied using Lie geometric methods and the optimal solution is obtained with Pontryagin Maximum Principle. It is proved that the economical system is not controllable, in the sense that we can only produce a certain quantity of products. Finally, some numerical examples are given with graphical representation.

**keywords:** optimal control, Pontryagin Maximum Principle, controllability, production and storage

Explicit modeling of multi-period setup times in proportional lot-sizing and scheduling problem with variable capacity

W. Kaczmarczyk(AGH University of Science and Technology, Krakow, Poland) |

Small bucket models with many short fictitious micro-periods ensure high-quality schedules in multi-level systems, i.e., with multiple stages or dependent demand. In such models, setup times longer than a single period are, however, more likely. This paper presents new mixedinteger programming models for the *proportional lot-sizing and scheduling problem* (PLSP) with setup operations overlapping multiple periods with variable capacity.

A new model is proposed that explicitly determines periods overlapped by each setup operation and the time spent on setup execution during each period. The model assumes that most periods have the same length; however, a few of them are shorter, and the time interval determined by two consecutive shorter periods is always longer than a single setup operation. The computational experiments showthat the newmodel requires a significantly smaller computation effort than known models.

**keywords:** production; lot-sizing and scheduling; mixed-integer programming

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