On the Determination of Periodic Regimes in Discrete Systems Containing a Relay Element with a Multivalued Asymmetric Characteristic
Abstract Generated abstract
The paper addresses the determination of periodic modes in discrete relay-pulse systems whose nonlinear element has a multivalued asymmetric relay characteristic with hysteresis. It extends the equivalent complex gain approach by accounting for dependence of the relay output on both the input samples and the sign of their increments, deriving relations for switching instants, sample counts in a half-period, and the corresponding inverse equivalent gain mapping. The work describes how to partition the input-signal plane into subregions associated with different periodic modes and transform these regions into the complex-gain plane. It concludes that hysteresis shifts the resulting regions toward lower relative frequencies, providing a graphical criterion for identifying or avoiding undesirable periodic regimes.
Full Text
ON THE DETERMINATION OF PERIODIC MODES IN DISCRETE SYSTEMS CONTAINING A RELAY ELEMENT WITH A MULTIVALUED ASYMMETRIC CHARACTERISTIC
V. N. Chudinov
(Presented by the scientific-technical seminar of the Department of Automation and Telemechanics)
In [1] questions were considered concerning the construction of the equivalent complex gain for the case when the NE was characterized by a single-valued asymmetric relay characteristic. With such a type of nonlinearity, the output signal of the nonlinear element \(\Phi\{\chi[n]\}\) depends only on the value of the input signal at the instant of its application.
If the nonlinearity is multivalued, for example, possesses hysteresis, then the output signal of the nonlinear element is determined not only by the values of the input signal at discrete instants of time, but also by the character of their variation, i.e., by the sign of the difference \(\Delta \chi = \chi_n - \chi_{n-1}\). The signal at the output of the NE in this case is determined by the dependence \(\Phi\{\chi[n]; \operatorname{sign}\Delta \chi\}\).
The influence of hysteresis on the parameters of periodic modes of relay-pulse systems was considered in [2, 3]. However, in the indicated works the authors were interested only in symmetric periodic modes.
In the present case, questions are considered concerning the construction of the equivalent complex gain of the NE for the case of a multivalued asymmetric relay characteristic.
From consideration of the character of variation of the signal at the input of the NE [1], the following relations may be written:
\[ a_1 \cdot \cos \left[\pi - \left(2\frac{\pi}{M}\alpha + \varphi\right)\right] = \Delta (m - 1), \tag{1} \]
\[ a_1 \cdot \cos \left[\pi + \left(2\frac{\pi}{M}\beta - \varphi\right)\right] = \Delta (m + 1) \quad \text{for mode } A \text{ and even } M, \tag{2} \]
\[ a_1 \cdot \cos \left[\pi - \left(2\frac{\pi}{M}\alpha + \varphi\right)\right] = -\Delta (m + 1), \tag{3} \]
\[ a_1 \cdot \cos \left[\pi + \left(2\frac{\pi}{M}\beta - \varphi\right)\right] = -\Delta (m - 1) \quad \text{for mode } B \text{ and even } M, \tag{4} \]
\[ a_1 \cdot \cos \left[\pi - \left(2\frac{\pi}{M}\alpha + \frac{\pi}{M} + \varphi\right)\right] = \Delta (m - 1), \tag{5} \]
\[ a_1\cdot \cos\left[\pi-\left(2\frac{\pi}{M}\beta-\frac{\pi}{M}-\varphi\right)\right] =\Delta(m+1)\quad \text{for mode } A \tag{6} \]
and odd \(M\),
\[ a_1\cdot \cos\left[\pi-\left(2\frac{\pi}{M}\alpha+\frac{\pi}{M}+\varphi\right)\right] =-\Delta(m+1), \tag{7} \]
\[ a_1\cdot \cos\left[\pi+\left(2\frac{\pi}{M}\beta-\frac{\pi}{M}-\varphi\right)\right] =-\Delta(m-1)\quad \text{for mode } B \tag{8} \]
and odd \(M\).
For the case in which the nonlinear element has an ambiguous relay characteristic, we shall determine the displacement of the center of oscillations in the following way:
\[ \pm \chi_0=m\Delta, \]
where \(2\Delta\) is the width of the loop of the relay characteristic,
\(m\) is the magnitude of the relative static offset [4].
Let us note that, for the case under consideration, \(m=0,1,\ldots E\left\{\dfrac{M}{2}\right\}\).
From the relations given above, the following expressions can be obtained for determining the number of samples of the function \(\Phi\{\chi[n];\operatorname{sign}\Delta\chi[n]\}\) in a half-period, respectively to the left and to the right of \(\pi\):
\[ n_{1,2}=E\left\{\frac{M}{2\pi}\left[\arccos\left(\pm\frac{m\pm1}{a_1/\Delta}\right)\pm\varphi\pm\frac{\pi}{M}\right]\right\}. \tag{9} \]
The signs \(+\) and \(-\) are determined by the modes \(A, B\) and by the parity of \(M\); moreover, for even \(M\) the term \(\pi/M\) is absent.
The general expression for determining the equivalent complex gain \(S^*\) is in this case defined as follows:
\[ S^*=\frac{2}{a_1M}\sum^{M}\Phi\{\chi[\nu];\operatorname{sign}\Delta\chi[\nu]\} \exp\left[-j\left(2\frac{\pi}{M}\nu+\varphi\right)\right]. \tag{10} \]
By analogy with the case considered earlier [1], we shall define the inverse equivalent complex gain \(N^*\) as a conformal mapping of the regions \(G\) in the input-signal plane onto the regions \(G^*\) in the plane of the complex gain. The transformation vector \(\dot Z\) in each specific case is determined by means of the summation limits \(n_1\) and \(n_2\), defined earlier.
The general procedure for constructing the regions of the input signal and for separating the subregions corresponding to different types of possible periodic modes, for a nonlinear element with an ambiguous symmetric relay characteristic, is given in [3]. However, the use of the method of [3] for the case of asymmetric oscillations becomes extremely complicated. In this case it is expedient to separate the subregions by writing the boundary relations for the amplitude and phase of the input signal.
It should be noted that, for the case of a nonlinear element with an ambiguous characteristic, it is necessary separately to distinguish subregions in the input-signal plane \(\dot a\) for the range of variation of \(\varphi\) from \(-\pi/M\) to \(0\) and from \(0\) to \(+\pi/M\) for even \(M\), and from \(-\pi/M\) to \(-\pi/2M\) and from \(+\pi/2M\) to \(+\pi/M\) for mode \(B\) and odd \(M\), and from \(-\pi/2M\) to \(0\) and from \(0\) to \(+\pi/2M\) for mode \(A\) and odd \(M\).
The region \(G\) of the input signal in this case is determined by the following relations:
\[ |\operatorname{Re}\{\dot a\}|\geq \Delta(m-1)\quad \text{for } \Delta\chi<0, \tag{11} \]
\[ |\operatorname{Re}\{\dot a\}| \geq \Delta(m+1) \quad \text{for } \Delta\chi > 0 \text{ for mode } A, \tag{12} \]
\[ |\operatorname{Re}\{\dot a\}| \geq \Delta(m+1) \quad \text{for } \Delta\chi < 0, \tag{13} \]
\[ |\operatorname{Re}\{\dot a\}| \geq \Delta(m-1) \quad \text{for } \Delta\chi > 0 \text{ for mode } B. \tag{14} \]
In this case, for the range of variation of \(\varphi\) from \(-\pi/M\) to \(0\) (the case of even \(M\)), the switching instant corresponding to the minimum amplitude of the input signal will correspond to the boundary value \(\Delta(m+1)\), and for the range of variation of \(\varphi\) from \(0\) to \(+\pi/M\), to the boundary value \(\Delta(m-1)\). Further, as the amplitude of the input signal increases, the boundary values of the switching lines alternate.
Fig. 1.
Visible diagram labels: \(M=4\), \(M=5\), \(M=6\); subfigures \(a)\), \(b)\), \(c)\); \(\tau=1;3\), \(\tau=2\), \(\tau=2;3\), \(\tau=1;4\), \(\tau=3;2\), \(\tau=1;5\), \(\tau=2;4\), \(\tau=3\); axis label \(j\).
Fig. 2.
Visible diagram labels: Mode \(A\), Mode \(B\); \(a)\ M=4\), \(b)\ M=5\), \(c)\ M=6\); labels \([0;1]\), \([1;1]\), \([1;0]\), \([0;0]\), \([1;2]\), \([2;1]\), \([2;2]\); axis label \(j\).
The division of the region \(G\) into subregions corresponding to different \(\tau\) is effected by a series of straight lines:
\[ \operatorname{Re}\left\{\dot a \exp\left(\pm j\frac{2\pi}{M}r\right)\right\} = (r+1)a_n \quad \text{for even } M \text{ and} \tag{15} \]
\[ \operatorname{Re}\left\{\dot a \exp\left[\pm j\frac{\pi}{2M}(2r+1)\right]\right\} = (r+1)a_n \quad \text{for odd } M. \tag{16} \]
The boundary value \(a_n\) corresponds to one of the relay actuation threshold values, which, owing to the ambiguity of the characteristic of the relay element, depends on the direction of variation of the input signal, i.e., on the sign of the difference \(\Delta x\).
The transformation of the regions of the input signal \(G\) into the regions \(G^*\) of the equivalent complex gain is carried out, as before [1], by means of the transformation vector \(Z\), by rotating the latter through the angle \(\gamma=\arg \dot Z\) and changing the scale by a factor of \(Z\).
The procedure for solving the equation
\[ W^*\left[\exp\left(j\frac{2\pi}{M}\right)\right]=N^* \]
remains the same [1].
Figures 1 and 2 show the partition into subregions in the plane of the input signal and plots of the regions \(N^*\) for a fixed hysteresis magnitude and various \(M\), making it possible to judge the possible types of periodic modes for a given \(W^*\). In addition, one can determine how the frequency characteristic \(W^*\) must be shaped in order to avoid undesirable modes. In conclusion, we note that the influence of hysteresis is manifested in a shift of the subregions \(N^*\) toward lower relative frequencies (larger \(M\)).
REFERENCES
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V. N. Chudinov, V. M. Rikkonen. On the determination of the equivalent complex gain of an asymmetric relay-pulse element. Proceedings of the III Scientific-Technical Conference, Novokuznetsk, 1971.
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Yu. M. Korshunov. On the construction of the equivalent complex gain of a nonlinear pulse element. Avtomatika i telemekhanika, vol. XXIII, 1962, No. 5.
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