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Dynamic Systems and Control Theory
Published in Mohammad Monir Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, 2019
When subjected to the step input, the system will initially have an undesirable output. This is called the transient response. transient response The transient response occurs because a system is approaching its final output value. If the time goes to infinity, system response is called steady-state response. steady-state response The steady-state response of the system occurs when the transient response has ended. The amount of time to take place the transient response is known as the rise time. The amount of time it takes for the transient response to end and the steady-state response to begin is known as the settling time. It is common practice for systems engineers to try and improve the step response of a system. In general, it is desirable to have the transient response reduced, the rise and settling times be shorter, and the steady-state to approach a desired “reference” output. Using the MATLAB command step, we can compute the step response of the step response system (2.4). However, for a large-scale system, we can apply the implicit Euler method that is implicit Euler method summarized in Algorithm 6 for computing the step response of the system (2.4).
Continuous-Time Systems
Published in Harold Klee, Randal Allen, Simulation of Dynamic Systems with MATLAB® and Simulink®, 2018
Except for the case when ζ = 0, the unit step response approaches the limiting or steady-state value y(∞) = K, which means that K is the DC or steady-state gain of the second-order system in Equation 2.16. The parameter ζ, which determines the existence and extent of the oscillations as well as the duration of the transient response, is called the damping ratio of the system. The last two parameters ωn and ωd are the natural frequency and damped natural frequency of the second-order system, respectively. The first, ωn, is the frequency of the sustained oscillations (ζ = 0) in Equation 2.27, and the second, ωd, is the frequency of the decaying oscillations (0 < ζ < 1) in Equation 2.24. It follows from Equation 2.22 that ωd < ωn The natural frequency ωn is an indication of the speed of the step response (and the system in general) since the oscillatory natural modes are damped by the exponential term with time constant 1/ζωn in Equation 2.23.
Transverse Vibration of Rotor Systems Integrated with Active Magnetic Bearings
Published in Rajiv Tiwari, Rotor Systems: Analysis and Identification, 2017
An iterative procedure is used for selecting the values of kD, which is called the tuning of the controller. The procedure includes finding the step response of the whole system for different values of kD, and examining the obtained responses. For plot that has better tr (rise time) and ts (settling time), the corresponding kD value is taken for further numerical simulation. For linear control systems, the characterization of the transient response is often done by the use of a step function as an input, and it is called the step response. For the step response obtained for the system, the rise time tr and settling time ts should be small so that the steady-state response is reached in a short time (see Figure 18.15).
Critical damping design method of vibration isolation system with both fractional-order inerter and damper
Published in Mechanics of Advanced Materials and Structures, 2022
Yandong Chen, Jun Xu, Yongpeng Tai, Xiaomei Xu, Ning Chen
The main aim of including an inerter in a VIS is to utilize its inertia effect, i.e. to increase the mass effect. Therefore, in general, the isolation system is equipped with independent dampers [12–16]. In textbooks of classical vibration theory, the concept of critical damping in Refs. [41, 42] is analyzed in detail in the study of single-degree-of-freedom damped vibration systems, and its value can be analytically solved using a characteristic equation (CE). Similarly, the calculation of the critical damping of a second-order system in control system engineering [43, 44] has a similar introduction. Åström [43] indicated that in industrial control applications, the overshoot is frequently 8–10%. Ogata [44] mentioned that, in practice, if the resonant peak is between 1.1 and 1.5, the system is considered to have a good performance, and the corresponding damping ratio is approximately 0.3–0.8, and the overshoot is approximately 5–40%. One of the basic characteristics of the transient response of a second-order system is overshoot, which primarily depends on the damping [43, 44]. In other words, by selecting the desired overshoot, the damping ratio can be obtained, and vice versa. The calculation of critical damping is key to the design of damper parameters.
A novel hybrid many optimizing liaisons gravitational search algorithm approach for AGC of power systems
Published in Automatika, 2020
Prangya Mohanty, Rabindra Kumar Sahu, Sidhartha Panda
The PID controller is well known and most accepted feedback controllers in industrial applications because of its usefulness, simple design, cost-effective and effectiveness for linear plants. The controller with only proportional action has the ability to reduce rise time, but steady-state error cannot be removed. By using integral action this steady-state error can be eradicated but the transient response of the system becomes poorer. This transient response can be improved by using derivative action and it also reduces overshoot and stability of the system. But, the conventional PID controllers may be ineffectual because of its linear structure, particularly, for complex systems associated with delay time and uncertainties. Alternatively, the fuzzy logic controller (FLC) can handle nonlinearity and uncertainties and can be designed to get the desired system performance. Fuzzy PID structure has been proposed in literature to get overall improved performance [22,23]. Therefore, fuzzy PID are chosen in this study for AGC. The configuration of fuzzy PID structure is displayed in Figure 2 which is a mixture of fuzzy PD and PID structures from [24], with input scaling parameters (K1 and K2) of FLC and gains of PID (KP, KI, KD). The controllers take individual ACEs (e1 (t) and e2 (t)) as inputs as expressed by:
Dynamic analysis and design of a semiconductor supply chain: a control engineering approach
Published in International Journal of Production Research, 2018
Junyi Lin, Virginia L.M. Spiegler, M.M. Naim
The transfer function represents the dynamic properties of the system. In particular, the characteristic equation, defined by equating the denominator of overall transfer function to zero, can be used to find poles (roots), which give an initial understanding of the underlying dynamic mechanism of the semiconductor hybrid MTS-MTO system including system stability and unforced system dynamic property (i.e. natural frequency and damping ratio). Stability is a fundamental property of a supply chain system. From the linear system perspective, the system is stable if the trajectory will eventually return to an equilibrium point irrelevant to the initial condition, while an infinity trajectory is presented if the system is unstable (Wang, Disney, and Wang 2012).Thus, the system response to any change in an input (demand) will result in uncontrollably increasing oscillations in the supply chain (Disney and Towill 2002). A system also has critical stability when it is located at the edge of the stability boundary, and system oscillations are regular and infinite for such situation. More details of supply chain stability can be found in Riddalls and Bennett (2002), Warburton et al. (2004), Sipahi and Delice (2010). Regarding the unforced system dynamic property, natural frequency (ωn) determines how fast the system oscillates during the transient response and can be used to indicate the system’s speed to reach the steady state condition for responding external demand signal, e.g. the inventory recovery speed. Damping ratio (ζ), on the other hand, describes how the system’s oscillatory behaviour (i.e. variability) decays with time, and can be perceived as initial insight of the system’s unforced dynamic performance; for instance, the extent to which the order rate and inventory will oscillate with time.