STUDY OF A COUPLED CONTROL SYSTEM FOR A CLASS OF DISTRIBUTED-PARAMETER OBJECTS

V. I. Karnachuk, V. Ya. Durnovtsev

(Presented by the scientific seminar of the Department of PTF)

Multivariable control systems (MCS) are at present finding ever wider application in the automation of complex objects. This is due to the fact that comprehensive automation of production processes requires a transition from the control of a single parameter to the coupled control of several quantities that influence one another. Among such systems, an important place is occupied by single-type MCS, consisting of several identical, identically tuned controllers operating from a common source of raw material or on a common load. Multichannel automatic control systems (ACS) of distributed-parameter objects may be classed as single-type MCS; their task is the automatic optimization of parameter distribution. This problem cannot be correctly solved unless the mutual influence of the controlled parameters is taken into account. Taking mutual influence into account considerably complicates the analysis of the system, since in a coupled system the dynamics of each parameter is described by a differential equation of high order.

The founder of the theory of control of several parameters is I. N. Voznesenskii. He showed that, in order to eliminate the influence of parameters on one another, it is necessary to introduce artificial couplings into the system that compensate for the effect of the natural couplings. In this case the coupled system is transformed into an uncoupled, i.e., autonomous, one. The problem of autonomy is a specific problem that is absent in the theory of one-dimensional ACS. I. N. Voznesenskii solved this problem for a first-order object controlled by an ideal controller. Later, physically and technically realizable autonomy conditions were found for more complex systems [1–3]. In these works, the range of objects considered is, as a rule, limited to first-order objects. In practice, however, in studies in the field of control of distributed-parameter objects such as a rectification column, an oil-and-gas-bearing stratum, vulcanization chambers, various types of reactors, etc., a more complex approximation is often required.

In the present work, certain questions of the synthesis of a two-dimensional MCS for an astatic object with phase lead are considered.

when the object, for each controlled variable, is described by a second-order differential equation:

\[ T_2 \frac{d^2 x}{dt^2} + \frac{dx}{dt} = \kappa_0 T_1 \frac{dy}{dt} + \kappa_0 y . \tag{1} \]

The structural diagram of the coupled control system is shown in Fig. 1. The system is intended to maintain a specified value of the parameter \(X\) in two different regions of a large object.

Fig. 1. Structural diagram of a two-dimensional CCS

The controlled object is a multivariable system with a \(V\)-structure according to the accepted classification [3]. The transfer functions of the objects in each direct channel are equal to:

\[ W_{01}(p) = W_{02}(p) = \frac{\kappa_0 (T_1 p + 1)}{p (T_2 p + 1)} . \tag{2} \]

The interconnection of the controlled parameters is represented in the structural diagram through constant coefficients \(L_{12} = L_{21}\), although in the general case it is not time-invariant. Integral controllers with the transfer function are considered:

\[ W_{m1}(p) = W_{m2}(p) = \frac{\kappa_y \kappa_m}{p(T_m p + 1)} . \tag{3} \]

The controllers receive control signals from inertial sensors (thermocouples) located near the corresponding regulating elements. The transfer functions of the sensors are:

\[ W_{T1}(p) = W_{T2}(p) = \frac{\kappa_T}{T_T p + 1} . \tag{4} \]

Analysis of the coupled system by means of equations of motion, written even in operator form, is inconvenient because of the high order of the equations. The matrix method of writing equations offers much greater convenience, especially for structural synthesis.

In matrix form, the equation for an object with a \(V\)-structure has the form:

\[ X = W_0 \mu , \tag{5} \]

where

\[ W_0(p)= \begin{bmatrix} \dfrac{W_{01}}{1-W_{c1}W_{c2}L_{12}L_{21}} & \dfrac{W_{01}W_{c2}L_{12}}{1-W_{01}W_{c2}L_{12}L_{21}} \\[1.2em] \dfrac{W_{c2}W_{01}L_{21}}{1-W_{01}W_{c2}L_{12}L_{21}} & \dfrac{W_{c2}}{1-W_{01}W_{c2}L_{12}L_{21}} \end{bmatrix}, \]

and \(X\) and \(\mu\) are column matrices of the controlled and controlling variables, respectively.

For the controller one may write

\[ \mu = W_p(Sy-X), \tag{6} \]

where

\[ W_p(p)= \begin{bmatrix} W_{p1} & 0\\ 0 & W_{p2} \end{bmatrix}; \]

\(S\) is the transforming matrix of the control actions;

\(y\) is the column matrix of control actions.

The elements of the matrices \(W_p\) and \(S\) can be obtained after simple structural transformations:

\[ W_{p1}(p)=W_{p2}(p)= \dfrac{K_yK_mK_T}{p(T_m p+1)(T_T p+1)}. \]

Then the equation of the closed-loop CMS can be written in the following form (here and below we shall assume that the disturbances acting on the system are \(f=0\)):

\[ X=(I+W_0W_p)^{-1}\cdot W_0W_pSY, \tag{7} \]

where \(I\) is the identity matrix.

From (7) one can obtain the characteristic equation of the closed-loop CMS by setting equal to zero the determinant of the matrix \((I+W_0W_p)\):

\[ |I+W_0W_p|=0. \tag{8} \]

For a CMS, sufficiently general criteria for checking stability have not yet been found. Determination of the roots of the characteristic equation (8) is also a rather cumbersome task, since it can be shown that even in the two-dimensional case one has to solve an equation of the tenth order. Under such conditions, the use of computing facilities for calculating a CMS is not only desirable, but also necessary. The significance of analog models is especially great for solving problems of synthesis of CMSs possessing certain prescribed properties, and above all autonomous CMSs. It is known that realization of the autonomy conditions is often impossible; in any case, for each specific system the finding of autonomy conditions that could be realized by sufficiently simple elements is an independent problem. From expression (7) it is seen that the autonomy conditions reduce to diagonalization of the matrix

\[ \Phi_y=(I+W_0W_p)^{-1}\cdot W_0W_pS. \]

In this case the CMS equations split into \(n\) independent equations. It is evident that the matrix \(\Phi_y\) will be diagonal only in the case where the matrix \(W_0W_p\), which is the transfer matrix of the open-loop CMS, is diagonal. To realize these conditions, artificial compensating couplings are introduced into the CMS, whose transfer ...

Fig. 2. Electronic model of an autonomous SMR.

functions can be determined from a matrix equation of the MCS written in a form more convenient for these purposes [2]:

\[ \Phi_y = W_0 W_p (S-\Phi_y). \tag{9} \]

There are many possible ways of implementing compensating couplings. However, calculations performed according to equation (9) show that the most convenient variant for realization is the structural scheme in which the cross couplings are introduced between the inputs of the controller amplifiers. For this case the transfer functions of the compensating couplings have the form:

\[ K_{12}(p) = -W_{c1}L_{12}; \]

\[ K_{21}(p) = -W_{c2}L_{21}. \]

Taking expression (2) into account, we have:

\[ K_{12}(p) = K_{21}(p) = -\frac{K_0 L (T_1 p + 1)}{p (T_2 p + 1)}. \tag{10} \]

To study the two-dimensional MCS, an electronic model of the system was used, assembled on the basis of the EMU-8 analog installation. The circuit of the electronic model of the MCS is presented in Fig. 2. The numerical values of the parameters were taken as follows: \(k_0=10\); \(K_yK_mK_T=0.1\); \(T_1=10\) sec; \(T_2=0.1\) sec; \(T_m=0.3\) sec; \(T_T=0.5\) sec; \(L=0.1 \div 0.9\).

Fig. 3. Curves of transient processes in the channels of the nonautonomous (a) and autonomous (b) MCS

Studies of the model showed that the system without compensating couplings remains stable up to the interaction magnitude \(L=0.5\). A further increase in \(L\) leads to divergent oscillations of the controlled variable. However, even at \(L<0.5\) the character of the transient process in the system is unsatisfactory. The total settling time is \(25 \div 30\) sec with a maximum overshoot of 50%. The introduction of cross couplings satisfying the autonomy conditions makes it possible to sharply improve the quality of control.

As can be seen from the plots (Fig. 3), the sensitivity of each channel to a change in the setpoint in the neighboring channel is noticeably reduced. The duration of the transient process and the magnitude of the maximum overshoot can be reduced by lowering the gain coefficient of the amplifiers of both channels by a factor of 2 in comparison with the gain coefficient adopted for an uncoupled separate system.

Conclusions

1. Conditions of autonomy have been found, realizable by simple active \(RC\)-circuits for controlled systems of objects of the second order with phase lead.

2. Analysis of complex controlled systems by means of analog computers makes it possible to choose optimal values of the parameters of the controlled system.

An electronic model of a two-dimensional autonomous controlled system has been proposed. The influence of the magnitude of the interconnection on the stability of the system has been shown.

Literature

1. M. V. Meerov. Systems of Multivariable Control. Nauka Publishing House, 1965.
2. V. T. Morozovskii, “Avtomatika i telemekhanika,” 1962, No. 9.
3. M. D. Mezarovich. Multivariable control systems. Proceedings of the 1st IFAC Congress, Publishing House of the Academy of Sciences of the USSR, 1961.