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Nonlinear Systems
Published in Jitendra R. Raol, Ramakalyan Ayyagari, Control Systems, 2020
Jitendra R. Raol, Ramakalyan Ayyagari
Linear systems can be classified as either time-varying or time-invariant, depending on whether the coefficients of the differential equation vary with time or not; equivalently, whether the elements of the system matrix A vary with time or not. However, in the more general context of nonlinear systems, these adjectives are traditionally replaced by autonomous and non-autonomous. As we have seen in the previous part, a nonlinear system is said to be autonomous if f(x,t) does not depend explicitly on time; otherwise, the system is called non-autonomous. Hence, for obvious reasons, linear time-invariant (LTI) systems are autonomous and linear time-varying (LTV) systems are non-autonomous.
Introduction to Feedback Control Systems
Published in Ramin S. Esfandiari, Bei Lu, Modeling and Analysis of Dynamic Systems, 2018
Assume that the transfer function of the closed-loop system is Y(s)/R(s), where Y(s) is the system output and R(s) is the reference input. The bandwidth is defined as the frequency at which the magnitude of the closed-loop transfer function crosses −3 dB or 0.707. Recall that the steady-state response of a linear system to sinusoidal excitations is called the system’s frequency response. As shown in Figure 10.11, if the excitation frequency is lower than ωBW, the magnitude |Y(s)/R(s)| is close to 0 dB (or 1). This indicates that the system output follows the reference input. If the excitation frequency is higher than ωBW, the magnitude |Y(s)/R(s)| is reduced to a small value, and the system output no longer follows the reference input. The higher the bandwidth, the faster the reference signal the system can follow. Thus, the bandwidth is a measure of the speed of the response.
Transient Heat Flow through Building Elements
Published in T. Agami Reddy, Jan F. Kreider, Peter S. Curtiss, Ari Rabl, Heating and Cooling of Buildings, 2016
T. Agami Reddy, Jan F. Kreider, Peter S. Curtiss, Ari Rabl
Quite generally, if the input of a linear system is sinusoidal, the response is sinusoidal with the same frequency; the amplitude and phase of the response depend on the system parameters. In the case of transient heat flow through a wall, the time lag is the phase shift between the heat flux variations on either side of the wall, and the decrement factor is the ratio of the amplitude of the output wave to that of the input. This fact can be seen from the solution of the 1R1C network (Equation 8.48) by inserting sinusoidal driving functions on the right-hand side: the contribution of the initial state, whatever it was, vanishes exponentially with time, and purely sinusoidal response remains, as the reader can verify with some algebra and a table of integrals.
In-situ estimate of coating by equivalent circuit for PEO of AZ31B
Published in Surface Engineering, 2023
S. Wang, Y. C. Liu, H. M. Liu, Y. F. Lan
In stages I–II, D1 and D2 can be equated to wires due to the voltage being higher than withstand voltage value of Diodes, and the parallel resistance of R1 and R2 is approximately equal to R1 due to R1 << R2. Therefore, Figure 6 can be used to discuss the equivalent circuit model for stages I-II and analyse the transients as a linear time-invariant system. The transfer function Equation (7) is derived for the linear part of the model based on the transfer function calculation method of the principles of automatic control [18]. For a linear system with zero initial conditions, the transfer function is the ratio of the Laplace transform of the output to the Laplace transform of the input. It can be used to determine the output response of the system and to analyse the effect of changes in the system parameters on the output response. K = R1/R1 + R2; s is a complex variable in the Laplace transform; Equation (7) is compared with the transfer function form of a typical system to derive the damping ratio ξ of the second-order system and the damped oscillation period T of the second-order system as follows. To calculate Eqs. (8, 9), the raw oscilloscope data is imported into the script and the parameters are shown in Figure 10 are extracted.
Probabilistic assessment of vehicle derailment based on optimal ground motion intensity measure
Published in Vehicle System Dynamics, 2021
Zhibin Jin, Weizhan Liu, Shiling Pei, Jinzhe He
For a linear system, the frequency-response function is independent of the amplitude of the excitation. However, the vehicle exhibits significant nonlinearity at the wheel-rail interface and the suspension lateral stopper, and its frequency response is dependent on the amplitude of the excitation. To illustrate this point, the input acceleration was set at 0.5 and 3 m/s2, respectively, to produce different levels of nonlinearity. Figure 3 shows the amplitudes of lateral displacement and the roll angle of the car-body as a function of the excitation frequency (frequency-response curve). The left vertical axis represents the amplitude of the lateral displacement, and the right vertical axis represents the amplitude of the roll angle. When the amplitude of the input acceleration was 0.5 m/s2, the frequency at the maximum value of the frequency-response curve is 0.54 Hz. When the amplitude of the input acceleration was 3 m/s2, the corresponding frequency was 0.66 Hz. The move towards higher frequencies corresponding to the peak response for larger amplitude excitation is expected: the lateral stopper between the car-body and the bogies was not activated for a PGA of 0.5 m/s2, but it was activated for a PGA of 3.0 m/s2. Larger ground motion triggers the lateral stopper, which is much stiffer than the secondary springs.
Volterra series identification and its applications in structural identification of nonlinear block-oriented systems
Published in International Journal of Systems Science, 2020
If a discrete system is linear and time-invariant, then the first order Volterra series represents the linear input-output relationship of system as (Rugh, 1980; Schetzen, 1980) where x(k) is the input, y(k) the output, v(k) the noise of iid zero mean and finite variance, and L − 1 is the memory length of the impulse response function h(k). A linear system is determined uniquely by its impulse response function h(k). For nonlinear systems, the input-output relation is nonlinear. One class of nonlinear systems can be represented by the higher order Volterra series. Higher order Volterra systems are an extension of the familiar convolution from linear systems to nonlinear systems. For a discrete nonlinear time invariant system, the system often can be approximated by a P-order truncated Volterra series with sampling memory L − 1. In this case, it can be represented as (Rugh, 1980; Schetzen, 1980) where , and denotes the n-order Volterra output presented as where x(k), y(k)∈R are the system input and output, respectively, and hn (m1, … , mn) is the nth order discrete Volterra kernel function, and v(k) is an iid noise sequence of zero mean and finite variance.