DETERMINATION OF PERIODIC MODES IN TWO-DIMENSIONAL RELAY-PULSE SYSTEMS

V. N. Chudinov

(Presented by the scientific-technical seminar of the Department of Automation and Telemechanics)

In [1] questions of determining periodic modes in multichannel relay control were considered for the case when the controlled objects are characterized by only one controlled quantity. However, in industrial practice there exist many objects characterized by two or more controlled parameters. When multichannel regulators and various kinds of control machines are used for the purpose of controlling objects with several controlled parameters, information about the processes has a discrete character. Multichannel regulators implementing the simplest control law, for example positional control, and multidimensional control objects form multidimensional nonlinear discrete controlled systems. The difficulties of investigating such systems increase sharply in comparison with linear multidimensional systems. However, sometimes, taking into account the specificity of the connections between individual channels, multidimensional and, in particular, two-dimensional nonlinear discrete systems can be investigated by the usual methods adopted in the theory of one-dimensional systems.

We shall show the possibility of extending the harmonic-balance method to two-dimensional relay-pulse systems by the example of determining periodic modes in the control channel of a system for centralized monitoring and regulation of two-dimensional objects. In this case the equivalent structural diagram of the control channel can be reduced to the two-dimensional relay-pulse system of Fig. 1.

Fig. 1.

For the sake of simplifying the investigation it is expedient to divide the two-dimensional system of Fig. 1 into two elementary subsystems of Fig. 2. Such a division is all the more necessary because it makes it possible to reveal the mutual influence of one subsystem on the other. In this case the measure of the introduction of a separate

elementary subsystem into an interconnected two-dimensional system is determined by inverse, natural cross-couplings having transfer functions \(\kappa_{1,2}\) and \(\kappa_{2,1}\). Thus, the subsequent investigation reduces to the analysis of two one-dimensional relay-pulse systems and the subsequent comparison of the results. Taking into account that all the applicability requirements of the harmonic-balance method are satisfied, each elementary system in Fig. 2a can be analyzed as a one-dimensional one, at whose input a sinusoidal signal \(Z'[n]\) acts, in the general case of arbitrary amplitude and phase, Fig. 3.

Fig. 2.

Questions concerning the investigation of the properties of certain types of forced oscillations in one-dimensional relay-pulse systems were considered earlier in [2]. In [1] a definition of self-oscillations is given graphically by constructing critical regions of periodic solutions in the complex plane. Let us show how the geometry of the critical regions changes if the structural diagram has the form presented in Fig. 3. In this case the system of equations determining the complex amplitudes of the desired periodic regime can be represented in the following form:

Fig. 3.

\[ \dot a_{k,s}=\dot c_{k,3-s}-\dot b_{k,s}W_s^*,\qquad \left( \begin{array}{c} k=0,1,2\ldots L\\ s=1,2. \end{array} \right), \tag{1} \]

where

\[ \dot a_{k,s}=a_{k,s}E^{j\varphi_{k,s}} = \frac{2}{M_s}\sum_{\nu=0}^{L_s}\chi_s[\nu]E^{-jk\frac{2\pi}{M_s}\nu}, \tag{2} \]

\[ \dot c_{k,s}=c_{k,s}E^{j\varphi_{k,s}} = \frac{2}{M_s}\sum_{\nu=0}^{L_s}Z'_s[\nu]E^{-jk\frac{2\pi}{M_s}\nu}, \tag{3} \]

\[ \dot b_{k,s}=b_{k,s}E^{j\varphi_{k,s}} = \frac{2}{M_s}\sum_{\nu=0}^{L_s}F_s\{\chi_s[\nu]\}E^{-jk\frac{2\pi}{M_s}\nu}. \tag{4} \]

According to the harmonic-balance method, putting \(f_{0,1}=f_{0,2}=0.5\), the system of equations (1) can be reduced to the form:

\[ W_s^*=N_s^*-\frac{\dot c_{1,\,3-s}}{\dot a_{1,s}}\,N_s^*. \tag{5} \]

Neglecting the mutual influence of the channels, i.e., assuming \(Z'_s\equiv 0\), we obtain the result previously obtained in [1].

The difference consists in the fact that in the present case the region of existence of periodic regimes is determined by the vector sum of the function \(N_s^*\) and \(|\dot c_{1,\,3-s}/\dot a_{1,s}|\cdot N_s^*\).

Thus, it has been shown that taking channel interaction into account is equivalent to adding the vector quantity \(-[c_{1,3-s}/\dot a_{1,s}]\cdot N_s^*\) to the critical region \(N_s^*\), corresponding to the case of autonomous operation of the channels \([Z'_s \equiv 0]\).

From the computational point of view, it is simpler to reduce equation (5) to the form

\[ N_s^*=\frac{\dot a_{1,s}}{\dot a_{1,s}-c_{1,3-s}}\,W_s^*. \tag{6} \]

In this case, the geometry of the regions of existence of periodic modes, determined in [1], does not change. The characteristic vector \(W_s^*\), however, depending on the interaction of the channels, determined by the quantities \(\kappa_{1,2}\) and \(\kappa_{2,1}\), is in each particular case corrected by the displacement vector

\[ \eta_s=\frac{\dot a_{1,s}}{\dot a_{1,s}-c_{1,3-s}}. \]

The procedure for determining a periodic mode in a two-dimensional relay-pulse system consists, in general outline, of the following.

1. It is assumed that the interaction of the channels can be neglected, i.e., \(\kappa_{1,2}=\kappa_{2,1}=0\). By the method described earlier [1], the periodic mode \(Z_s[n]\) is determined in each elementary subsystem.

2. One of the cross-couplings is closed. The displacement vector \(\eta_g\) is determined, and the mutual position of the critical region, for example, \(N_2^*\), and of the vector \(\eta_{1,2}\cdot W_2^*\) is checked. If the end of the vector \(\eta_{1,2}\cdot W_2^*\) remains in the former subregion (item 1), then a conclusion is drawn about the autonomy of the second channel for the given \(\kappa_{1,2}\). If, however, the vector \(\eta_{1,2}\) transfers the vector \(W_2^*\) into another subregion, then the periodic mode \(Z_2[n]\) corresponding to the given subregion \(N_2^*\) is determined.

3. The remaining cross-coupling is closed. The displacement vector \(\eta'_{2,1}\) is determined. The mutual position of the critical region \(N_1^*\) and of the vector \(\eta_{2,1}\cdot W_1^*\) is checked. If it then turns out that the mutual position of the region \(N_1^*\) and of the vector \(\eta_{2,1}\cdot W_1^*\) is preserved (item 1), then a conclusion is drawn about the possibility of establishing in the two-dimensional system the periodic mode determined by items 1 and 2. Otherwise, a periodic mode of the given type cannot exist in the two-dimensional system. After this, the process of establishing a periodic mode is continued for new \(Z_s[n]\).

REFERENCES

1. V. N. Chudinov, V. M. Rikkonen. On the determination of an equivalent complex gain coefficient of an asymmetric relay-pulse element. Reports of the III Scientific-Technical Conference, Novokuznetsk, 1971.

2. Ya. Z. Tsypkin. Periodic regimes in nonlinear pulse automatic systems. Proceedings of the Tashkent Polytechnic Institute, issue 20, 1962.

3. V. S. Kuo. Forced oscillations and suppression of oscillations in nonlinear pulse systems. Mechanics, No. 2, 1968.