UDC 62-53.001

APPLICATION OF THE DECOMPOSITION METHOD TO THE SYNTHESIS OF AUTONOMOUS UNIFORM MULTIVARIABLE CONTROL SYSTEMS

V. I. Karnachuk, V. A. Voiloshnikov, G. P. Nagorny, A. V. Logutov

(Presented by the scientific seminar of the Faculty of Physics and Engineering)

Among multivariable control systems (MCS) for complex production processes, an important place is occupied by the class of uniform MCS, consisting of several identical control channels operating from a common source of raw material or on a common load.

As examples of uniform MCS, we may point to electric-power systems consisting of a large number of identical generators operating in parallel [1], systems for synchronizing the propellers of turboprop aircraft engines [2], automatic control systems for nuclear reactors [3], systems for automatic distribution of blast by tuyeres of a blast furnace [4], etc.

A distinctive feature of these systems is that the coupling between the individual control channels can be represented by dynamic elements whose transfer functions differ only in gain coefficients.

The identity of a number of elements of the transfer matrix of uniform MCS makes it possible to use decomposition techniques extensively for the purposes of their analysis. The decomposition method consists in the fact that, instead of the original interconnected MCS, certain equivalent systems are considered, consisting of simpler uncoupled subsystems. The transition to equivalent systems is carried out in such a way that the results of the investigations can be recalculated fairly simply for the original MCS [5].

Let, in the general case, the matrix equation of motion of a uniform MCS, shown in Fig. 1, have the form

\[ [E + (H + A)R]\varphi = (H + A)\cdot(\bar q + R\lambda), \tag{1} \]

where \(H(p)\) and \(R(p)\) are diagonal matrices of the plant and the controller, respectively; \(E\) is the identity matrix; \(\varphi\), \(\bar q\), \(\lambda\) are vectors of the controlled coordinates, disturbing actions, and control actions, respectively; \(A\) is the matrix of cross-couplings.

The matrix \(A\) can be represented as

\[ A = m(p)\cdot N, \tag{2} \]

where \(m(p)\) is a scalar transfer function,

\[ N \]

is a numerical matrix.

Let us pass to a new coordinate basis, connected with the old one by the nonsingular transformation matrix \(C\),

\[ \overline{\Phi}=C^{-1}\cdot \varphi;\qquad \overline{Q}=C^{-1}\cdot q;\qquad \overline{\Lambda}=C^{-1}\cdot \Lambda), \tag{3} \]

The equation of motion of the system in the new basis takes the form

\[ \left[E+(H' + A')R'\right]\overline{\Phi}=(H' + A')\cdot (\overline{Q}+R\overline{\Lambda}), \tag{4} \]

where

\[ H'=C^{-1}\cdot H\cdot C;\qquad A'=C^{-1}\cdot A\cdot C;\qquad R'=C^{-1}\cdot R\cdot C. \]

Since the scalar matrices \(H\) and \(R\) commute with \(C\), the similarity transformation does not change their form,

\[ H'=C^{-1}\cdot C\cdot H=H, \]

and so on.

If the matrix \(C\) is now chosen so that the matrix \(A\) becomes diagonal, then equation (4) decomposes into \(n\) independent equations, and the original MVS can be represented by equivalent uncoupled systems. As the matrix \(C\), the canonical basis of the matrix \(N\) is usually taken.

In the special case when the matrix of cross-connections is symmetric, a uniform MVS can be represented by an equivalent system consisting of simple uncoupled subsystems: an averaged motion, defined as the mean value of like-named coordinates of all separate control channels, and \((n-1)\) relative motions, defined as the difference of like-named coordinates of separate channels

Fig. 1. Matrix structural diagram of an MVS

\[ x_{i\,\mathrm{avg}}=\frac{1}{n}\sum_{\kappa=1}^{n}x_{i\kappa}, \tag{5} \]

\[ x_{ij\,\mathrm{rel}}=x_i-x_j. \tag{6} \]

In what follows we shall consider only symmetric uniform MVSs, which are the most widely used in engineering applications. A characteristic feature of such systems is the presence in them of a single controller that performs correction according to the mean value of a parameter of all separate channels. Thus, in a system of synchronously operating parallel generators there is a separate voltage (or frequency) controller on the common bus. In the MVS of a large nuclear reactor there is a controller of the overall power level. In the MVS for blast distribution among the tuyeres of a blast furnace there is an independent controller of the total flow rate, and so on.

The controller of the mean value of the distributed parameter is, in this case, the controller of the averaged motion. Therefore, for symmetric uniform MVSs one can propose a method for synthesizing autonomous MVSs that differs from the classical methods and is based on the use of the decomposition method.

The most widespread method for constructing autonomous MVSs is the method of cross compensating connections imposed between the separate channels. In this case the controller matrix in equation (1) is assumed to be nondiagonal, and by selecting the nondiagonal elements \(r_{ij}\) one achieves diagonality of the matrix \((H+A)R\). In this case equation (1) decomposes into \(n\) uncoupled equations [5].

It is not difficult to see that the same result is obtained under decomposition of the system; however, to achieve autonomy here, not a change of the controller matrix \(R\) is used, but a change of the basis of the controlled coordinates.

In other words, to achieve autonomy in a uniform symmetric MCS it is necessary to pass to such a basis of controlled coordinates in which the separate control channels will be fully or partially independent of one another.

Let the transfer matrix of the plant be symmetric. For simplicity of notation, we restrict ourselves to three channels:

\[ (H+A)= \begin{bmatrix} W(p) & m(p) & m(p)\\ m(p) & W(p) & m(p)\\ m(p) & m(p) & W(p) \end{bmatrix}. \tag{7} \]

We represent the vector of controlled coordinates in the form

\[ \bar{\varphi}= \begin{bmatrix} x^{*}\\ x_1\\ x_2 \end{bmatrix}, \tag{8} \]

where \(x^{*}\) is the mean value of the controlled parameter, determined by formula (5).

Let us pass to the new coordinate basis (3). Using the ambiguity in the choice of the vectors of the canonical basis for the symmetric matrix \((H+A)\), we require that in the equivalent system the quantity \(x^{*}\) remain unchanged and experience a one-sided action from the other coordinates.

For example, let us specify the vector \(\bar{\Phi}\) in the form

\[ \bar{\Phi}= \begin{bmatrix} x^{*}\\ x_1-x^{*}\\ x_2-x^{*} \end{bmatrix}. \tag{9} \]

Then, on the basis of (3), we determine the transformation matrix

\[ C^{-1}= \begin{bmatrix} 1&0&0\\ -1&1&0\\ -1&0&1 \end{bmatrix}, \qquad C= \begin{bmatrix} 1&0&0\\ 1&1&0\\ 1&0&1 \end{bmatrix}. \tag{10} \]

The transfer matrix of the plant in the new basis takes the form

\[ (H'+A')= \begin{bmatrix} (W+2m) & m & m\\ 0 & (W-m) & 0\\ 0 & 0 & (W-m) \end{bmatrix}. \tag{11} \]

Expression (11) shows that, along with the autonomy of the separate channels from one another, a one-sided autonomy of the entire subsystem of separate channels from the regulator of the mean value of the parameter \(x^{*}\) is also achieved here.

Depending on the conditions of technical implementation, the vector of controlled coordinates may be specified in a somewhat different form, for example,

\[ \bar{\Phi}= \begin{bmatrix} x^{*}\\ x_1-\bigl[(x^{*}-\kappa x_0^{*})+\kappa x_0^{*}\bigr]\\ x_2-\bigl[(x^{*}-\kappa x_0^{*})+\kappa x_0^{*}\bigr] \end{bmatrix} \quad \text{or} \quad \bar{\Phi}= \begin{bmatrix} x^{*}\\ x_1-\kappa x^{*}\\ x_2-\kappa x^{*} \end{bmatrix}, \tag{12} \]

where \(\kappa=\mathrm{const}\).

Then the transfer matrix of the plant, respectively, takes the form

\[ (H' + A') = \begin{bmatrix} (W + 2m) & \kappa m & \kappa m \\ 0 & (W - m) & 0 \\ 0 & 0 & (W - m) \end{bmatrix}, \tag{13а} \]

\[ (H' + A') = \begin{bmatrix} (W + 2\kappa m) & m & m \\ m(1+\kappa-2\kappa^2) & (W-\kappa m) & (1-\kappa)m \\ m(1+\kappa-2\kappa^2) & (1-\kappa)m & (W-\kappa m) \end{bmatrix}. \tag{13б} \]

Figure 2 presents the structures of three synthesized autonomous multivariable control systems constructed on the basis of expressions (9) and (12). Obviously,

Fig. 2. Structural diagrams of synthesized autonomous multivariable control systems

that the problem of synthesizing autonomous uniform multivariable control systems by the decomposition method has a set of solutions. Thus, the system possesses the indicated properties when passing to a new basis of the form

\[ \bar{\Phi}= \begin{bmatrix} x^{*}\\ x_{1}-x^{*}\\ x_{2}-x^{*} \end{bmatrix}. \tag{14} \]

The transfer matrix of the plant for this case coincides completely with expression (11).

For purposes of illustration, Fig. 3 gives curves of transient processes in the AMS, constructed according to the type of structures b and c (Fig. 2).

Fig. 3. Transient processes in the channels of the AMS caused by a change in the setpoint of the regulator of the mean value of the parameter

The curves were obtained on the analog computer EMU-8 for three-channel uniform AMS of the indicated types, with the following matrices of the plant, the regulator, and the sensors, respectively:

\[ (H+A)= \begin{bmatrix} \dfrac{k_{11}(T_{1}p+1)}{p(T_{2}p+1)} & \dfrac{k_{12}(\tau p+1)}{p(T_{2}p+1)} & \dfrac{k_{13}(\tau p+1)}{p(T_{2}p+1)} \\[1.2em] \dfrac{k_{21}(\tau p+1)}{p(T_{2}p+1)} & \dfrac{k_{22}(T_{1}p+1)}{p(T_{2}p+1)} & \dfrac{k_{23}(\tau p+1)}{p(T_{2}p+1)} \\[1.2em] \dfrac{k_{31}(\tau p+1)}{p(T_{2}p+1)} & \dfrac{k_{32}(\tau p+1)}{p(T_{2}p+1)} & \dfrac{k_{33}(\tau p+1)}{p(T_{2}p+1)} \end{bmatrix}, \tag{15} \]

\[ R=\left[ \begin{array}{ccc} \dfrac{\kappa_{1}}{p\left(T_{3}p+1\right)} & 0 & 0\\[6pt] 0 & \dfrac{\kappa_{1}}{p\left(T_{3}p+1\right)} & 0\\[6pt] 0 & 0 & \dfrac{\kappa_{1}}{p\left(T_{3}p+1\right)} \end{array} \right], \tag{16} \]

\[ D=\left[ \begin{array}{ccc} \dfrac{\kappa_{2}}{T_{4}p+1} & 0 & 0\\[6pt] 0 & \dfrac{\kappa_{2}}{T_{4}p+1} & 0\\[6pt] 0 & 0 & \dfrac{\kappa_{2}}{T_{4}p+1} \end{array} \right]. \tag{17} \]

REFERENCES

1. V. S. Kulebakin, V. T. Morozovskii, I. M. Sindeev. Aircraft Electrical Power Supply. Moscow, Oborongiz, 1956.

2. V. A. Bodner. Automation of Aircraft Engines. Moscow, Oborongiz, 1965.

3. A. Dzh. D. Truxal. Stability of interconnected control systems. Proceedings of the First International Congress on Automatic Control, Moscow, IFAC, vol. 1, 1961.

4. O. S. Sobolev. Analysis of the structure of a multivariable automatic system with nonlinear feedback. Avtomatika i telemekhanika, 1964, no. 3.

5. V. T. Morozovskii. Multivariable Automatic Control Systems. Moscow, Energiya, 1970.