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CYBERNETICS AND CONTROL THEORY
Academician B. N. PETROV, S. V. EMEL'YANOV, N. E. KOSTYLEVA
ON CONTROL OF LINEAR PLANTS WITH VARIABLE PARAMETERS
A system of automatic control of a linear plant with variable parameters is considered, the differential equation of motion of which has the form
\[ Q(p)x_1=P(p)z_{m-1}, \tag{1} \]
where \(x_1\) is the controlled coordinate; \(z_{m-1}\) is the control action;
\[ Q(p)=\sum_{i=0}^{n} a_{i+1}(t)p^i,\qquad a_{n+1}=1; \]
\[ P(p)=\sum_{i=0}^{m-1} b_{i+1}(t)p^i,\qquad b_m=1; \]
\[ p=\frac{d}{dt};\qquad n\geqslant m; \]
\(a_i(t), b_i(t)\) are certain analytic functions of time, with
\[ a_{i\min}\leqslant a_i\leqslant a_{i\max} \]
\[ b_{i\min}\leqslant b_i\leqslant b_{i\max}. \]
The problem arises of synthesizing the control law in such a way that, when \(a_i(t), b_i(t)\) vary within the indicated range, the dynamic properties of the system change only slightly.
In works \((^1,{}^2)\) it is shown that, in the case \(P(p)=1\), the use of automatic control systems with variable structure (\(z_{m-1}\) changes discontinuously) makes it possible, for a certain set of values of the coordinates \(x_1, \dot{x}_1,\ldots,x_1^{(n-1)}\), to ensure independence of the motion of the system from the parameters \(a_i(t), b_i(t)\). If the degree of the polynomial \(P(p)\) is not equal to zero \((m>1)\), then a discontinuous change of the control action \(z_{m-1}\) leads to the fact that, for arbitrary initial conditions, the solution of the differential equation of motion (1) depends on the variable parameters of the plant \((^3)\).
In the present article an attempt is made to form a control law in such a way that, in the space of the coordinates \(x_1,\dot{x}_1,\ldots,x_1^{(n-1)}\), there exists a region in which the motion does not depend on the coefficients \(a_i(t), b_i(t)\). This is achieved by connecting, in series with the principal correcting device 1 considered in \((^2)\), a passive filter 2 with local switched feedbacks (Fig. 1).
The motion of the dynamic system (Fig. 1) in the \((n+m-1)\)-dimensional space \((x_1,\ldots,x_n,z_1,\ldots,z_{m-1})\) is described by the system of differential equations
\[ \frac{d\mathbf{R}}{dt}=\mathbf{G}(\mathbf{R},\mathbf{T}), \tag{2} \]
where
\[ \begin{gathered} \mathbf{R}=(x_1,\ldots,x_n,\ z_1,\ldots,z_{m-1});\\ \mathbf{G}=(f_1,\ldots,f_n,\ h_1,\ldots,h_{m-1});\\ \mathbf{T}=(\psi_1,\ldots,\psi_{n-1},\ \Phi_1,\ldots,\Phi_{m-1});\\ f_i=x_{i+1}\quad (i=1,2,\ldots,n-1);\\ f_n=\sum_{i=1}^{m-1} b_{i+1} z_{m-1}^{(i)}+b_1 z_{m-1}-\sum_{i=1}^{n} a_i x_i; \tag{3}\\ h_i=\frac{1}{T_i}(z_{i-1}-z_i)\quad (i=1,2,\ldots,m-1); \tag{4}\\ z_0=-\sum_{i=1}^{n-1}\psi_i x_i-\sum_{i=1}^{m-1}\Phi_i z_i; \end{gathered} \]
\(x_2,\ldots,x_n\) are the coordinates of the plant; \(z_1,\ldots,z_{m-2}\) are the coordinates of the controller,
\[ \psi_i= \begin{cases} \omega_i, & \text{if } x_i\sigma>0,\\ \lambda_i, & \text{if } x_i\sigma<0 \end{cases} \quad (i=1,\ldots,n-1)^*; \tag{5} \]
\[ \Phi_i= \begin{cases} \mu_i, & \text{if } z_i\sigma>0,\\ \gamma_i, & \text{if } z_i\sigma<0 \end{cases} \quad (i=1,\ldots,m-1)^{**}; \tag{6} \]
\[ \sigma=\sum_{i=1}^{n} c_i x_i; \]
\(\omega_i,\lambda_i,\mu_i,\gamma_i,c_i\) are constant coefficients, \(c_n=1\).
Fig. 1
From (3) and (4), the function \(f_n\) can be expressed in terms of the coordinates of the plant and controller:
\[ \begin{aligned} f_n={}&\left(\sum_{i=1}^{m-1} b_{i+1} A_{0,i}+b_1-A_{m-1,m-1}\Phi_{m-1}\right)z_{m-1}+\\ &+\sum_{j=1}^{m-2}\left(\sum_{i=j}^{m-1} b_{i+1} A_{j,i}-A_{m-1,m-1}\Phi_{m-1-j}\right)z_{m-1-j}-\\ &-\sum_{i=1}^{n-1}\left(a_i+A_{m-1,m-1}\psi_i\right)x_i-a_n x_n. \end{aligned} \tag{7} \]
\[ \begin{aligned} &{}^*\ \text{In the case } x_i\sigma=0\\ &\qquad \psi_i=\omega_i \text{ when } x_i\sigma\to +0;\qquad \psi_i=\lambda_i \text{ when } x_i\sigma\to -0.\\ &{}^{**}\ \text{In the case } z_i\sigma=0\\ &\qquad \Phi_i=\mu_i \text{ when } z_i\sigma\to +0;\qquad \Phi_i=\gamma_i \text{ when } z_i\sigma\to -0. \end{aligned} \]
Here
\[ A_{0,1}=-\frac{1}{T_{m-1}};\qquad A_{1,1}=\frac{1}{T_{m-1}}; \]
\[ A_{j,i+1}=A_{j-1,i}\frac{1}{T_{m-j}}+A_{j,i}\frac{1}{T_{m-1-j}}; \]
\[ A_{-1,i}=0;\qquad A_{i+1,i}=0. \]
Consequently, if \(z_{m-1}\) belongs to the class \(C_{m-2}\), then, according to the equations of motion (2), (4), (7), in the space \((x_1,\ldots,x_n,z_1,\ldots,z_{m-1})\) the phase trajectories are continuous functions. It is not difficult to show that in this case in the subspace \((x_1,\ldots,x_n)\) there exists a region \(U\subset S\) (\(S\) is the hyperplane defined by the equation \(\sigma=0\)), the motion in which is determined only by the coefficients \(c_i\) (4). In order that, after the representative point reaches the hyperplane \(S\), the subsequent motion in the subspace \((x_1,\ldots,x_n)\) should not depend on the variable parameters of the object, the region \(U\) must coincide with the hyperplane \(S\).
The region \(U\) on the hyperplane \(S\), according to (2), is determined by the relations
\[ \mathbf{c}\,\frac{d\mathbf{R}_x}{dt}<0 \quad \text{for } \sigma>0, \]
\[ \mathbf{c}\,\frac{d\mathbf{R}_x}{dt}>0 \quad \text{for } \sigma<0, \tag{8} \]
where
\[ \mathbf{c}=(c_1,\ldots,c_n), \qquad \mathbf{R}_x=(x_1,\ldots,x_n). \]
From (2) and (8) we obtain
\[ \sum_{i=1}^{n-1}(c_{i-1}-a_i-A_{m-1,m-1}\psi_i)x_i+(c_{n-1}-a_n)x_n+ \]
\[ +\left(\sum_{i=1}^{m-1} b_{i+1}A_{0,i}+b_1-A_{m-1,m-1}\Phi_{m-1}\right)z_{m-1}+ \]
\[ +\sum_{j=1}^{m-2}\left(\sum_{i=j}^{m-1} b_{i+1}A_{j,i}-A_{m-1,m-1}\Phi_{m-1-j}\right)z_{m-1-j}<0 \quad \text{for } \sigma>0; \tag{9} \]
\[ \sum_{i=1}^{n-1}(c_{i-1}-a_i-A_{m-1,m-1}\psi_i)x_i+(c_{n-1}-a_n)x_n+ \]
\[ +\left(\sum_{i=1}^{m-1} b_{i+1}A_{0,i}+b_1-A_{m-1,m-1}\Phi_{m-1}\right)z_{m-1}+ \]
\[ +\sum_{j=1}^{m-2}\left(\sum_{i=j}^{m-1} b_{i+1}A_{j,i}-A_{m-1,m-1}\Phi_{m-1-j}\right)z_{m-1-j}>0 \quad \text{for } \sigma<0; \]
\[ c_0=0. \]
The control vector \(\mathbf{T}\) must ensure fulfillment of conditions (9) for any point
\[ \left(x_1,x_2,\ldots,x_{n-1},-\sum_{i=1}^{n-1}c_i x_i\right) \]
of the hyperplane \(S\), i.e.
\[ \sum_{i=1}^{n-1}(c_{i-1}-a_i-c_i c_{n-1}+c_i a_n-A_{m-1,m-1}\psi_i)x_i+ \]
\[ +\left(\sum_{i=1}^{m-1} b_{i+1}A_{0,i}+b_1-A_{m-1,m-1}\Phi_{m-1}\right)z_{m-1}+ \]
\[ +\sum_{j=1}^{m-2}\left(\sum_{i=j}^{m-1} b_{i+1}A_{j,i}-A_{m-1,m-1}\Phi_{m-1-j}\right)z_{m-1-j}<0 \quad \text{for } \sigma>0, \tag{10} \]
\[ \sum_{i=1}^{n-1} \left(c_{i-1}-a_i-c_i c_{n-1}+c_i a_n-A_{m-1,m-1}\psi_i\right)x_i+ \]
\[ +\left(\sum_{i=1}^{m-1} b_{i+1}A_{0,i}+b_1-A_{m-1,m-1}\Phi_{m-1}\right)z_{m-1}\xi+ \]
\[ +\sum_{j=1}^{m-2}\left(\sum_{i=j}^{m-1}b_{i+1}A_{j,i}-A_{m-1,m-1}\Phi_{m-1-j}\right)z_{m-1-j}>0 \quad \text{for } \sigma<0. \]
From expressions (5), (6), and (10) we obtain the necessary and sufficient conditions for the coincidence of the domain \(U\) with the hyperplane \(S\) under variation of the coefficients \(a_i(t)\), \(b_i(t)\) within the prescribed range:
\[ A_{m-1,m-1}\omega_i>\sup_{a_i,a_n}\left(c_{i-1}-a_i-c_i c_{n-1}+c_i a_n\right), \]
\[ A_{m-1,m-1}\lambda_i<\inf_{a_i,a_n}\left(c_{i-1}-a_i-c_i c_{n-1}+c_i a_n\right) \quad (i=1,\ldots,n-1); \]
\[ A_{m-1,m-1}\mu_j>\sup_{b_2,\ldots,b_m}\sum_{i=j}^{m-1} b_{i+1}A_{j,i}, \]
\[ A_{m-1,m-1}\gamma_j<\inf_{b_2,\ldots,b_m}\sum_{i=j}^{m-1} b_{i+1}A_{j,i} \quad (j=1,\ldots,m-2); \]
\[ A_{m-1,m-1}\mu_{m-1}>\sup_{b_1,\ldots,b_m} \left(\sum_{i=1}^{m-1} b_{i+1}A_{0,i}+b_1\right), \]
\[ A_{m-1,m-1}\gamma_{m-1}<\inf_{b_1,\ldots,b_m} \left(\sum_{i=1}^{m-1} b_{i+1}A_{0,i}+b_1\right). \tag{11} \]
If the representative point falls on the hyperplane \(S\) and \(\omega_i,\lambda_i,\mu_j,\gamma_j,\mu_{m-1},\gamma_{m-1}\) satisfy (11), then the subsequent change of the regulated coordinate \(x_1\) will not depend on \(a_i(t)\), \(b_i(t)\).
Thus, the use of systems with variable structure for controlling objects whose motion is described by an equation of the form (1) makes it possible to substantially weaken the dependence of the quality of the control process on the variable parameters of the object.
Institute of Automation
and Telemechanics
Received
29 XI 1963
CITED LITERATURE
- S. V. Emel’yanov, M. A. Bermant, DAN, 145, No. 4 (1962).
- S. V. Emel’yanov, V. I. Utkin, DAN, 152, No. 2 (1963).
- S. V. Emel’yanov, N. E. Kostyleva, DAN, 153, No. 4 (1963).
- S. V. Emel’yanov, Izv. AN SSSR, Energetika i avtomatika, No. 4 (1962).