This chapter was devoted to the analysis, electronic implementation, synchronization and chaos control of the MH oscillator. The equilibrium points of the mathematical model describing the proposed oscillator were determined and their stabilities analyzed taking into account the Routh-Hurwitz criteria. For specific parameters, the proposed oscillator showed two different forms of chaotic one-roller attractors and coexistence between the period 3 limit cycle and the chaotic one-roller attractor. The electronic implementation of the proposed oscillator was carried out on the ORCAD-PSpice software. The Orcard-PSpice results confirmed the results of the numerical simulations. In addition, the synchronization of identical chaotic memristor-Helmholtz oscillators with unidirectional coupling with simple controllers was achieved. Finally, based on the Routh-Hurwitz criteria, a simple and unique controller was developed to suppress the chaotic behavior of the MH oscillator. Future work will focus on integrated cryptographic applications with chaotic memristor-Helmholtz oscillators and explore the possibility of using memristors as synapses for neuron imitation. Helmholtz oscillator-based memristors could also be considered for neuromorphic implementations. The Routh–Hurwitz criteria discussed in Section 7.3.5.2 can be applied to the transfer function z, but the characteristic polynomial is written as a function of z. By applying the Routh-Hurwitz criteria, a stable system can be achieved. Similarly, the solution of the eigenvalue problem can be performed to determine the position of the poles in the complex plane of the discrete system. This eigenvalue problem can be performed to determine the position of the pole in the complex plane.
For discrete systems, the eigenvalue problem is written as follows: The interesting fact about the transfer function z is the use of stability conditions in the discrete region, which are the same techniques as shown in section 7.3. Second-order answers, the Routh–Hurwitz criteria, and the eigenvalue problem can be applied in both the discrete and continuous domains. (2) Second element: Multiply a0 by the diagonally opposite element of the next column (i.e. a5), then subtract it from the product of a1 and a4 (where a4 is a diagonally opposite element of the next column) and finally divide the result so that you get by a1. Mathematically, we write as a second similar element, we can calculate all the elements of the third line. (d) The elements in the fourth row can be calculated according to the following procedure:(1) First element: Multiply b1 by the diagonally opposite element of the next column (i.e. a3) then subtract this from the product of a1 and b2 (where b2 is a diagonally opposite element of the next column) and finally divide the result, you get by b1. Mathematically, as the first element, we write (2) Second element: Multiply b1 by the diagonally opposite element of the next column (i.e. a5), then subtract this from the product of a1 and b3 (where b3 is a diagonally opposite element from next to the next column) and finally divide the result so that you get by a1.
Mathematically, we write as a second similar element, we can calculate all the elements of the fourth line. Similarly, we can calculate all the elements of all the rows. Stability criteria If all elements in the first column are positive, the system is stable. However, if one of them is negative, the system will be unstable. Now, there are some special cases related to the Routh stability criteria, which are discussed below: In this chapter, we have proposed an adaptive observer to estimate the rotor speed of an excited DC motor separately without a speed sensor. We analysed the stability of the estimated speed using the Routh-Hurwitz criterion. DC motor speed is controlled by a hybrid controller that combines fuzzy logic and fractional PI controllers. In the proposed control method, fuzzy rules are adjusted in-line to optimize the gain of the fractional PI controller based on the error and its variation between the reference rotational speed and its measurement. The results obtained show that the overtaking, peak rise and stabilization time of the controller proposed for integration into the speed observer are lower than those of the conventional PI controller. In future research, we want to use this smart controller in the field of renewable energies and compare this controller with other conventional controllers (blur, neurons, blurred PSO, …) for electrical machines and in the design of an algorithm for adjusting the freedom factor of the fraction controller with fuzzy rules. After reading the theory of lattice synthesis, one can easily say that each pole of the system is on the right side of the origin of the plane s, this makes the system unstable.
Based on this condition, A. Hurwitz and E.J. Routh began to study the necessary and sufficient stability conditions of a system. We will discuss two criteria for the stability of the system. A first criterion is given by A. Hurwitz and this criterion is also known as the Hurwitz criterion for stability or Routh Hurwitz stability criterion (R-H). In the Routh–Hurwitz stability criterion, we can tell if the closed-loop poles are in the left half of the plane “s” or on the right half of the plane “s”, or on an imaginary axis.