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The paper presents a basic description and examples of the use of so called descriptive functions, allowing analysing the influence of inherent and indispensable components of all mechatronic systems mechanical subsystems - so called hard nonlinearities. These parts “causing” - in addition to the centrifugal and Coriolis generalized forces- the non...
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Context 1
... is shown in Fig. 7. Nonlinear element output is often a periodic but generally non-sinusoidal function. Note that this occurs whenever the non-linearity f(x) is a uniquely invertible function because the output ...
Context 2
... describing function which describes a non-linear element, this element ˗ for the sinusoidal input -can be presented as a linear element with frequency transmission. This is shown in Fig.7.. ...
Context 3
... For linear dynamic system the frequency transition function is independent of the amplitude of the input signal. But describing function of the non-linear element differs from the frequency transition function of the linear element by being dependent on the amplitude of the input signal. Thus, the representation of the non- linear element of Fig. 7 ...
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Citations
... Using the time constants Ti and Td in real-time applications [27]. ...
The nonlinear effect in the control system is so important and it may have a hard or soft effect on the electrical, mechanical, biological, and many other systems. This paper analyzes the describing function (DF) which is the transfer function of the nonlinear (NL) control systems of many NL elements found such as saturation, and backlash. The effect of the NL on the third-order delayed system is considered. The PID controller is considered the heart of the control system and continuously finds the error between input and output, and formulates the desired signal for the actuator to control the plant. Experimental tanning of PID controller with the saturation NL as a case study with buffer Operation Amplifier (Op-Amp) to maintain the gain and phase shift. In addition, a low pass filter (LPF) is used in the feedback to minimize and attenuate the effect of the NL in the closed-loop control system. The Fourier series is used to analyze the DF. The results show the effectiveness of the proposed algorithm.
... The describing function method and its inversion serves nonlinear system analysis since many decades. References of [6,7,8,9,10], and [11] deal with nonlinear control system analysis using describing functions. ...
... To model the static nonlinearity, and to find conditions for any limit cycle of the nonlinear closed loop control system describing functions are used widely [5,6,7,8,9,10,11,12,13,15], and the interested reader should refer to. ...
The linear or the nonlinear feature of systems being modelled or designed is a core property we should be familiar with before taking up any system design or analysis-related task. The dynamic model-based closed loop control design is widely used in control engineering when dynamic or static nonlinearities are considered to be present in the closed loop control system. This paper addresses both dynamical and static nonlinearities modelled and handled in the closed loop control system
... Using the time constants Ti and Td in real-time applications [27]. ...
The nonlinear effect in the control system is so important and it may have a hard or soft effect on the electrical, mechanical, biological, and many other systems. This paper analyzes the describing function (DF) which is the transfer function of the nonlinear (NL) control systems of many NL elements found such as saturation, and backlash. The effect of the NL on the third-order delayed system is considered. The PID controller is considered the heart of the control system and continuously finds the error between input and output, and formulates the desired signal for the actuator to control the plant. Experimental tanning of PID controller with the saturation NL as a case study with buffer Operation Amplifier (Op-Amp) to maintain the gain and phase shift. In addition, a low pass filter (LPF) is used in the feedback to minimize and attenuate the effect of the NL in the closed-loop control system. The Fourier series is used to analyze the DF. The results show the effectiveness of the proposed algorithm.
... The concept of DF utilizes an approximation of the nonlinear relay behavior, usually via truncation of the Fourier series expansion [15]. The DF of the relay specified by Fig. 4 has the form [16]: ...
... ± 0.96169j, s 5,6 = −0.82529 ± 1.0529j (15) As can be seen, especially the real part of s 3,4 and the imaginary part of s 5,6 are pretty far from the required roots (13); however, the HPM 2 provides interesting time and frequency responses (see Subsect. 4.4). ...
The objective of this contribution is twofold. First, it demonstrates a case study on applying the standard single-run relay-feedback parameter identification test to a representative of infinite-dimensional systems. Namely, a delayed mathematical model of a circuit heating laboratory appliance process is used. Second, an initial estimation of the model parameters is done via the parameter identification of another – simpler – model. The transition between these two models adopts the idea of dominant spectrum assignment that is solved by using the well-established Levenberg-Marquardt algorithm. Finally, the remaining model parameters are estimated by solving another nonlinear optimization problem in the frequency domains. As transfer function denominator parameters are set independently to the numerator ones, the proposed technique significantly reduces the number of additional relay experiments. Numerical results indicate that the method needs improvements regarding time-response as well as frequency-response accuracy.
... In some engineering applications that deal with the study of dynamical control systems, the control input enters the differential equations in a nonlinear fashion [5]. It is of interest to identify the hidden nonlinearity while at the same time reduction is needed for robust simulations and control design [1]. ...
... , Y n th (njω i ) from the power spectrum.; 2 Fix k 1 to an arbitrary value and determine the scaled coefficient vector k = (k 1 , k 2 , . . . , kn) by solving the system in Eq. (5).; 3 Estimate the measurements of the linear transfer function from H(j ω) = Y ω, / n ≤i =0 k i φ i, . ; 4 Apply the linear Loewner framework for identification and reduction of the LTI. ...
We present an algorithm for data-driven identification and reduction of nonlinear cascaded systems with Hammerstein structure. The proposed algorithm relies on the Loewner framework (LF) which constitutes a non-intrusive algorithm for identification and reduction of dynamical systems based on interpolation. We address the following problem: the actuator (control input) enters a static nonlinear block. Then, this processed signal is used as an input for a linear time-invariant system (LTI). Additionally, it is considered that the orders of the linear transfer function and of the static nonlinearity are not a priori known.
... The condition under which the limit cycle exists is that the left side of (7) is 0 [15]. Therefore, there is the possibility that the limit cycle exists when the following equation holds. ...
... A strategy for the linearized analysis is the describing function method, which is a frequency domain approach that allows the limit cycle prediction and stability analysis. Authors propose a systematic way of multiple limit cycle determination, as well as the stability analysis of each one [13]. A novel analysis and design tool for nonlinear control system is presented [14]. ...
Dynamical technical systems are famous for large scale of time invariant nonlinearities applied inside. Some kinds of nonlinearities describe physical properties of devices used in the technical system. Some special kind of time invariant static nonlinearities are for ensure stability limiting and truncating signals inside the dynamical systems. Moreover, nonlinearity is a property of materials used both in static or dynamical induction machines. One of the widely spread and applied method to handle static nonlinearities is dynamical technical systems is the describing function method (DFM). The purpose of the author is to introduce and apply this technique to evaluate stability conditions of the automatic flight control system of the unmanned aerial vehicles (UAVs).
