Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1970. Vol. 191, No. 6
UDC 62.501.12
CYBERNETICS AND CONTROL THEORY
A. G. BUTKOVSKII
FINITE CONTROL AND CONTROLLABILITY IN DISTRIBUTED SYSTEMS
(Presented by Academician B. N. Petrov on 24 IX 1969)
An interesting problem of finite control (i.e., control on a finite time interval ([0,T])) ((^{1,2})) is the problem of damping oscillations in a branched and interconnected network of one-dimensional oscillatory systems (in particular, an electric power network with several power stations and consumers connected to one another by long transmission lines), for example, the network shown in Fig. 1. Here (C_1) and (C_2) denote power stations, and (P_1, P_2, P_3) are consumers.
Fig. 1 Fig. 2
Let us consider the simplest problem of this type (Fig. 2), in which there is only one control at the point (C) and two lines of lengths (S_1) and (S_2) with “fixed” ends (P_1) and (P_2). The precise formulation of such a problem in this simplest case is as follows. Suppose that the oscillations in the lines (S_1) and (S_2) are described respectively by the functions
[
Q_1(x_1,t), \qquad 0 \le x_1 \le S_1,
]
[
Q_2(x_2,t), \qquad 0 \le x_2 \le S_2,
]
for (t \ge 0).
The equations for these functions have the form:
[
\ddot Q_1 = Q_1'', \qquad 0 < x_1 < S_1,\qquad t>0,
\tag{1}
]
[
\ddot Q_2 = Q_2'', \qquad 0 < x_2 < S_2,\qquad t>0,
\tag{2}
]
where dots denote partial derivatives with respect to time (t), and primes denote partial derivatives with respect to the spatial variables (x_1) and (x_2).
The boundary and initial conditions have the form:
[
Q_1(0,t)=Q_2(0,t)=u(t),
\tag{3}
]
[
Q_1(S_1,t)=Q_2(S_2,t)=0,
\tag{4}
]
[
Q_1(x_1,0)=Q_{01}(x_1), \qquad \dot Q_1(x_1,0)=Q_{11}(x_1),
\tag{5}
]
[
Q_2(x_2,0)=Q_{02}(x_2), \qquad \dot Q_2(x_2,0)=Q_{12}(x_2),
\tag{6}
]
[
0 \le x_1 \le S_1,\qquad 0 \le x_2 \le S_2,\qquad t \ge 0.
]
The problem of finite control is to bring these systems to rest by means of the control (u(t)), (0 \leq t \leq T), i.e., to obtain the conditions
[
Q_1(x_1,T)=\dot Q_1(x_1,T)=0,\qquad 0\leq x_1\leq S_1,
\tag{7}
]
[
Q_2(x_2,T)=\dot Q_2(x_2,T)=0,\qquad 0\leq x_2\leq S_2.
\tag{8}
]
The first essential question that arises in solving this problem is: is this system controllable? In other words: does there exist a finite control (u(t)), (0\leq t\leq T), for finite (T) and arbitrary initial conditions (5), (6)? It turns out that, by the method of finite control, this problem is solved very simply, and the answer is negative. Indeed, the interpolation problem for this problem, as follows from formulas (13) in [1], has the form
[
\tilde u(m\pi/S_1)=\beta_{1m},\qquad
\tilde u(n\pi/S_2)=\beta_{2n},\qquad
m,n=\pm1,\pm2,\ldots,
\tag{9}
]
where the numbers (\beta_{1m}) and (\beta_{2n}) are computed by formula (14) in paper [1] with the corresponding changes of variables, in which, to compute (\beta_{1m}), one must put (Q_{01}(x_1)) instead of (Q_0(x)), (Q_{11}(x_1)) instead of (Q_1(x)), while to compute (\beta_{2n}) one must take (Q_{02}(x_2)) instead of (Q_0(x)), and (Q_{12}(x_2)) instead of (Q_1(x)). From (9) it is immediately clear that if the numbers (S_1) and (S_2) are commensurable, then for any (n=\pm1,\pm2,\ldots) there will be an (m) from the sequence (\pm1,\pm2,\ldots) such that
[
m/S_1=n/S_2.
\tag{10}
]
Since, by virtue of the arbitrariness of the initial distributions (Q_{01}(x_1)), (Q_{11}(x_1)), (Q_{02}(x_2)), and (Q_{12}(x_2)), the numbers (\beta_{1m}) and (\beta_{2n}), (m,n=\pm1,\pm2,\ldots), are different, the interpolation problem (9), in the case where (S_1) and (S_2) are commensurable, is unsolvable in entire functions (\tilde u(z)) of given finite degree, since the equality
[
\tilde u(m\pi/S_1)=\tilde u(n\pi/S_2)
]
must hold, whereas, generally speaking, at the same time (\beta_{1m}\ne\beta_{2n}).
It can also be shown that this interpolation problem (9) is unsolvable in the case of incommensurable (S_1) and (S_2). This can easily be shown by contradiction, using the fact that an entire function of finite degree belonging on the real axis to (L_2(-\infty,\infty)) has a bounded derivative [3].
Similarly, it can be shown that the system will remain uncontrollable if a new control (v(t)) is added at the right-hand ends of the intervals ([0,S_1]) and ([0,S_2]), i.e., if condition (4) is replaced by the condition
[
Q_1(S_1,t)=Q_2(S_2,t)=v(t),\qquad t\geq 0.
\tag{11}
]
In this case the interpolation problem has the form
[
\tilde u(m\pi/S_1)-(-1)^m\tilde v(m\pi/S_1)=\beta_{1m},
\tag{12}
]
[
\tilde u(n\pi/S_2)-(-1)^n\tilde v(n\pi/S_2)=\beta_{2n},
\tag{13}
]
[
m,n=\pm1,\pm2,\ldots,n,
]
whence it is seen that, in the case of commensurable (S_1) and (S_2), one can find such (m) and (n) that the left-hand sides of equations (12) and (13) will be identical, whereas the right-hand sides (\beta_{1m}) and (\beta_{2n}), generally speaking, are different. The unsolvability of the interpolation problem (12) and (13) can also be shown in the case of incommensurable (S_1) and (S_2).
The author expresses his gratitude to Prof. S. Rolewicz for a fruitful discussion of the controllability problems considered here.
Institute of Automation and Telemechanics
(Technical Cybernetics)
Moscow
Received
20 VII 1969
References Cited
- A. G. Butkovskii, DAN, 188, No. 3 (1969).
- A. G. Butkovskii, L. N. Postavskii, Automation and Telemechanics, No. 4 (1969).
- B. Ya. Levin, Distribution of Zeros of Entire Functions, Moscow, 1956.