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CYBERNETICS AND CONTROL THEORY
L. S. GNOENSKII
ON A PROBLEM IN THE SYNTHESIS OF CONTROLLED SYSTEMS
(Presented by Academician A. Yu. Ishlinskii, 26 XI 1963)
A system of automatic control is considered whose behavior under the influence of disturbing actions is described on the time interval \([0,T]\) by a linear differential equation of order \(n\),
\[ L(y)=f(t) \tag{1} \]
with zero initial conditions. The disturbance \(f(t)\) belongs to the class \(F\) of piecewise-continuous functions bounded in absolute value by the constant \(m_0\).
B. V. Bulgakov \((^1)\) solved the problem of determining a function \(f^*(t)\) from \(F\) for which the corresponding solution attains the greatest absolute value at the time \(T\) (the maximum accumulated error). This problem was also considered in \((^{2,3})\).
The present communication is devoted to the problem of synthesis of automatic control systems according to the principle of minimizing the maximum accumulated error. The case studied is when
\[ L(y)=L_1(y)+c(t)y. \]
Here \(L_1(y)\) is an operator of order \(n\) with constant or variable coefficients, corresponding to the unchangeable part of the system. The coefficient \(c(t)\), which is to be chosen, belongs to the class \(H\) of piecewise-continuous functions bounded in absolute value by the constant \(m_1\), and has the physical meaning of a variable gain coefficient.
Denote the solution of equation (1) by \(y(t,f(t),c(t))\). It is required to find such a \(c^0(t)\) from \(H\) at which
\[ I=\min_c \max_f |y(T,f(t),c(t))|,\qquad c\in H,\quad f\in F,\quad 0\leq t\leq T. \]
A method is presented that makes it possible to determine successively functions \(c_i^0(t)\) from \(H\) in such a way that
\[ R_{i+1}=\max_f |y(T,f(t),c_{i+1}^0(t))|<R_i=\max_f |y(T,f(t),c_i^0(t))|. \]
Sufficient conditions are indicated under which this process leads to the determination of \(I\); an estimate is given for the number of steps required for this, and an estimate of the number of switchings in the function \(c^0(t)\). The results obtained can also be extended to the case where it is necessary to choose the coefficients at the derivatives of \(y(t)\) of order less than \(n-1\).
Let \(y_1(t,f)\) be the solution of the equation \(L_1(y)=f\) with zero initial conditions; \(G_1(T,t)\) the Cauchy function of the operator \(L_1(y)\). Then \(f_1(t)=m_0\operatorname{sign}G_1(T,t)\) is the function on which the greatest possible value of the absolute value of the solution at the time \(T\) is attained. Choose an arbitrary point \(t_0\) from \([0,T]\) for which \(y_1(t_0,f_1)\ne0\). Put
\[ c(t)=m_1\operatorname{sign}\bigl(y_1(t_0,f_1)G_1(T,t)\bigr)\quad \text{for } t\in\gamma,\qquad c(t)=0\quad \text{for } t\in\bar{\gamma}; \]
\[ \gamma=[t_0-\mu,t_0+\nu];\quad \mu\geq0,\quad \nu\geq0,\quad \nu-\mu=\Delta,\quad \gamma\in[0,T]. \tag{2} \]
Let \(c(t)\) in (1) be defined by relations (2), and let \(y(t,f,\Delta)\) be the solution of equation (1) with zero initial conditions. By \(f_\alpha(t)\) denote such a function from \(F\) on which
\[ \max_f \bigl((y_1(t_0,f_1)-y_1(t_0,f))\operatorname{sign}y_1(t_0,f_1)\bigr) \quad \text{for } y_1(T,f_1)-y_1(T,f)=\alpha \]
\[ 0\leq \alpha<2y_1(T,f_1),\qquad f\in F. \]
This function, for \(y_1(t_0,f_1)>0\), has the form
\[ f_\alpha(t)=-m_0\operatorname{sign}G_1(T,t) \quad \text{for } t\in \sigma(x_\alpha), \]
\[ f_\alpha(t)=m_0\operatorname{sign}G_1(T,t) \quad \text{for } t\notin \sigma(x_\alpha). \]
Here \(\sigma(x)\) is the set in \([0,T]\) on which \(G^*(t)\ge x\);
\[ G^*(t)=\frac{G_1(t_0,t)}{G_1(T,t)} \quad \text{for } t\in[0,t_0]; \qquad G^*(t)=0 \quad \text{for } t\in(t_0,T]. \]
By \(x_\alpha\) we denote the unique root of the equation
\[ m_0\int_{\sigma(x)} |G_1(T,\tau)|\,d\tau=\frac{\alpha}{2}. \]
If
\[ m_0\int_{\sigma(+0)} |G_1(T,\tau)|\,d\tau<\frac{\alpha}{2}, \qquad m_0\int_{\sigma(-0)} |G_1(T,\tau)|\,d\tau>\frac{\alpha}{2}, \]
then
\[ f_\alpha(t)=-m_0\operatorname{sign}G_1(T,t) \quad \text{for } t\in \sigma(+0)\cup(t_0,u_0), \]
\[ f_\alpha(t)=m_0\operatorname{sign}G_1(T,t) \quad \text{for } t\notin \sigma(+0)\cup(t_0,u_0). \]
Here \(u_0\) is the unique root of the equation
\[ m_0\int_{\sigma(+0)} |G_1(T,\tau)|\,d\tau + m_0\int_{t_0}^{u} |G_1(T,\tau)|\,d\tau = \frac{\alpha}{2}. \]
If \(G^*(t)=\mathrm{const}\) for \(t\in[0,t_0)\), then
\[ f_\alpha(t)=-m_0\operatorname{sign}G_1(T,t) \quad \text{for } t\in[0,u_1], \]
\[ f_\alpha(t)=m_0\operatorname{sign}G_1(T,t) \quad \text{for } t\in(u_1,T]. \]
Here \(u_1\) is the root of the equation
\[ m_0\int_{0}^{u} |G_1(T,\tau)|\,d\tau=\frac{\alpha}{2}. \]
If \(y_1(t_0,f_1)<0\), then \(f_\alpha(t)\) is obtained by a slight modification of the formulas (4) given above. The expression \(y_1(t_0,f_\alpha)\) is a convex function of the argument \(\alpha\), vanishing at the point \(\alpha_0\). Let us also denote
\[ a_0 = T\sup_{\tau,t}\left|\frac{\partial G_1(t,\tau)}{\partial t}\right| + \sup_{\tau,t}|G_1(t,\tau)|, \qquad a_1=\sup_{\tau,t}|G_1(t,\tau)| \]
\[ (0\le \tau\le t\le T). \]
Theorem 1. Suppose that the quantity \(\Delta\) specified in (2) does not exceed \(\min\{T,\Delta^0\}\); then for any \(f\) from \(F\)
\[ |y(T,f,\Delta)| \le y_1(T,f_1) - \frac{\alpha_0 m_1|y_1(t_0,f_1)|}{4y_1(T,f_1)} \int_{t_0-\mu}^{t_0+\nu}|G_1(T,\tau)|\,d\tau; \]
\[ \Delta^0= \frac{4\alpha_0 y_1(T,f_1)} {\alpha_0 m_1 a_1 |y_1(t_0,f_1)|+4q y_1(T,f_1)}, \tag{4} \]
\[ q=\max\left\{ 4m_1a_1a_0 + 8\frac{m_0a_0y_1(T,f_1)}{|y_1(t_0,f_1)|}, \; 2m_1a_1y_1(T,f_1) \right\}, \]
We describe a multistage process of decreasing the maximum accumulated error. Define the differential operator \(L_{i+1}\) from the relations
\[ L_{i+1}=L_i+c_i y \quad (i=1,2,\ldots), \qquad c_i^{-}(t)\le c_i(t)\le c_i^{+}(t), \]
\[ c_i^{+}(t)=c_{i-1}^{+}(t)-c_{i-1}^{*}(t), \qquad c_i^{-}(t)=c_i^{+}(t)-2m_1, \qquad c_1^{+}(t)=m_1. \]
Here \(c_{i-1}^*(t)\) is the function chosen at the \((i-1)\)-st step. Denote by \(E_i\) the subset of \([0,T]\) on which
\[ \operatorname{sign} c_i^-(t) \leq \operatorname{sign}\bigl(y_i(t,f_i)G_i(T,t)\bigr) \leq \operatorname{sign} c_i^+(t) \quad (i=2,3,\ldots), \qquad E_1=[0,T]. \]
In these inequalities \(y_i(t,f_i)\) is the solution of the equation \(L_i(y)=f_i\) with zero initial conditions; \(G_i(T,t)\) is the Cauchy function of the operator \(L_i\),
\[ f_i(t)=m_0 \operatorname{sign} G_i(T,t). \]
Suppose that \(i-1\) steps have been carried out; \(\operatorname{mes} E_i=e_i\), and the Cauchy function \(G_i(T,t)\) has \(s_i\) zeros on \((0,T)\). The set \(E_i\) contains either an interval \(E_{i1}\) of length greater than \(e_i/2s_i\), or a subset \(E_{i1}\) of measure greater than \(e_i/2\), which consists only of intervals whose boundary points are zeros of \(y_i(t,f_i)\) and discontinuity points of the function \(c_i^+(t)\).
On \(E_{i1}\) we choose a point \(t_i\) at which \(y_i(t,f_i)\) attains its greatest value in absolute value. The subinterval of \(E_{i1}\) on which \(t_i\) lies is denoted by \(\beta_{i1}\). From relations (3), (4), applied to the operator \(L_i\) and the point \(t_i\), we determine \(\Delta_i^0\) (if \(m_1^*=\max\{|c_i^+(t_i)|, |c_i^-(t_i)|\}=2m_1\), then in (3), (4) \(2m_1\) must be substituted instead of \(m_1\)). Let \(\gamma_i\) denote the interval \([t_i-\mu_i,t_i+\nu_i]\) containing the point \(t_i\). If \(\operatorname{mes}\beta_{i1}\leq \Delta_i^0\), then \(\gamma_i\) coincides with \(\beta_{i1}\). If \(\operatorname{mes}\beta_{i1}>\Delta_i^0\), then \(\gamma_i\subset\beta_{i1}\), and \(\operatorname{mes}\gamma_i=\Delta_i^0\). We define the function \(c_i^*(t)\) by the relations:
\[ c_i^*(t)=m_1^* \operatorname{sign}\bigl(y_i(t_i,f_i)G_i(T,t)\bigr) \quad \text{for } t\in\gamma_i, \qquad c_i^*(t)=0 \quad \text{for } t\notin\gamma_i. \]
In order that \(I\) be realized at the \((i-1)\)-st step, it is necessary that
\[ \operatorname{mes} E_i=0. \tag{5} \]
Some sufficient conditions are given by
Theorem 2. Suppose that after the \((i-1)\)-st step condition (5) is satisfied. If, moreover: 1) \(y_i(\tau,f_i)>0\) for every \(\tau\) in \((0,T)\); 2) \(G_i(t,\tau)\), considered for fixed \(\tau\) as a function of the argument \(t\), is sign-constant for \(\tau\leq t\leq T\), then \(y_i(T,f_i)=I\).
An estimate of the degree of decrease of the maximal accumulated error \(R_i\) and of the measure \(e_i\) of the set \(E_i\) as functions of the number of steps is given by
Theorem 3. Let \(A,\varepsilon\) be arbitrary positive numbers satisfying the inequalities \(A<y_1(T,f_1)\), \(\varepsilon<T\). Then after the \(r\)-th step either \(y_{r+1}(T,f_{r+1})<A\), or \(\operatorname{mes} E_{r+1}<\varepsilon\). Here
\[ r \leq r^0 = 3\left[\frac{y_1(T,f_1)-A}{\lambda}+1\right], \qquad \lambda=kA^{s_1}\varepsilon^{s_2}. \]
The function \(c^0(t)\), on which the minimum of the maximal accumulated error \(I\) is realized, can take only two values: \(m_1\) and \(-m_1\). An estimate of the number of switching points of \(c^0(t)\) on the interval \((0,T)\) is given by
Theorem 4. If \(I>A>0\), then the function \(c^0(t)\) has on \((0,T)\) no more than \(\eta=[k_1/A]\) switching points.
In Theorems 3 and 4 the square brackets denote the integer part of a number. The coefficients \(k,k_1\) depend on the upper bound of the modulus of the coefficients of the operator \(L_1(y)\) and of the adjoint operator \(L_1^*(y)\), on the order of equation (1), and on the quantities \(m_1,m_0,T\). The coefficients \(s_1\) and \(s_2\) are determined by the order of equation (1).
All-Union Correspondence
Machine-Building Institute
Received
21 XI 1963
REFERENCES
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- Ya. N. Roitenberg, Applied Mathematics and Mechanics, 22, issue 4 (1958).
- L. S. Gnoenskii, Applied Mathematics and Mechanics, 26, issue 1 (1962).