Determination of the Mean Time of Reliable Operation of an Automatic Control System Within Specified Limits

F. F. Idrisov

(Presented by the scientific seminar of the Department of Automation and Telemechanics)

Consider an automatic control system (ACS) subjected to random disturbances. Suppose that the variation of the system’s output coordinate in time, characterizing its operability, is a stationary process with correlation function \(K_u(\tau)=\sigma^2 \exp\{-\alpha|\tau|\}\). Here \(\sigma^2\) is the variance, and \(\tau\) is the correlation time.

It is required to determine a quantitative estimate of the ability of the ACS to maintain its output parameter within the limits \((a,b)\), specified by technical requirements, under random conditions.

An adequate mathematical apparatus describing an ACS subjected to random disturbances and having random initial conditions is the apparatus of the Kolmogorov–Planck equations. The integral of these equations is the probability density of the process of interest. Unfortunately, however, methods for the exact solution of the Kolmogorov–Planck equations for stationary processes have not yet been obtained. Approximate methods, meanwhile, are very cumbersome in terms of the computational work required.

But in a whole range of practical cases encountered, it is sufficient to know not the probability density of the random process, but the moments of its distribution, say, the mathematical expectation. It is shown below how, by a comparatively simple reduction of the Kolmogorov–Planck equation to an ordinary differential equation and by applying the Bubnov–Galerkin method to the latter, one can obtain an analytic expression for the mathematical expectation of the process.

Thus, following the method set forth in [1], let us consider a stationary continuous process \(U(t)\), which at the initial instant of time \(t=0\) assumes a value equal to \(x\). Then the probability density \(w(y,T/x)\) of the transition of the process \(U(t)\) from state \(x\) to state \(y\) over the time \(T\) will satisfy the following Kolmogorov–Planck equation:

\[ \frac{\partial w(y,T/x)}{\partial T} = \frac{\partial [xw(y,T/x)]}{\partial x} + \frac{\partial^2 w(y,T/x)}{\partial x^2}, \tag{1} \]

with the boundary conditions:

\[ w(y,0/x)=\delta(y-x), \tag{2} \]

\[ w(a,T/x)=w(b,T/x)=0. \tag{3} \]

where \(\delta(y-x)\) is the Dirac delta function, \(a<x<b\). The function

\[ Q(T/x)=\int_a^b w(y,T/x)\,dy \tag{4} \]

is the probability that the values of the random process under consideration, over a time interval of duration \(T\), will lie within the segment of the real axis \((a,b)\), provided that at the initial instant its value belonged to the specified segment. This probability satisfies the obvious conditions:

\[ Q(0/x)=1,\qquad Q(\infty/x)=0. \tag{5} \]

The probability that the first attainment of the boundaries will occur in a time interval whose duration does not exceed \(T\) is equal to \(1-Q(T/x)\). The probability density \(q(T/x)\) of the instant of first attainment of a boundary is obtained by differentiating the indicated quantity with respect to \(T\), i.e.,

\[ q(T/x)=-\frac{\partial Q(T/x)}{\partial T}. \tag{6} \]

Using (6), we find the mean time to attainment of the boundaries:

\[ m_T(x)=\int_0^T Tq(T/x)\,dT =-\int_0^T T\frac{\partial Q(T/x)}{\partial T}\,dT. \tag{7} \]

Taking (5) into account and integrating by parts, we obtain an expression for the mean time to attainment of the boundaries:

\[ m_T(x)=\int_0^\infty Q(T/x)\,dT. \tag{8} \]

Let us integrate both sides of equation (1) with respect to \(y\) over the limits from \(a\) to \(b\), and then with respect to \(T\) over the limits from zero to infinity, taking (4)—(8) into account. We obtain an ordinary differential equation with variable coefficients:

\[ \frac{d^2m_T(x)}{dx^2} +x\frac{dm_T(x)}{dx} +m_T(x)=-1, \tag{9} \]

under the boundary conditions:

\[ m_T(a)=m_T(b)=0. \tag{10} \]

This equation can be reduced to an equation with initial conditions and solved by means of the Laplace transform. But this is a comparatively long procedure, leading to cumbersome expressions. It can be solved much more simply and rapidly by the Bubnov–Galerkin method.

Thus, we solve equation (9) by the Bubnov–Galerkin method. For simplicity of calculation, we shall first take: \(|a|=|b|=1\).

Let the unknown function \(E(x)\), according to [2], satisfy in a certain domain \(\Omega\) the inhomogeneous differential equation

\[ LE(x)=f(x), \tag{11} \]

where \(L\) is an operator generating a differential equation and satisfying certain homogeneous boundary conditions. An infinite sequence of coordinate functions \(\varphi_i(x)\), \((i=1,n)\), is chosen, which

which are a sufficient number of times (in accordance with the data of the problem) continuously differentiable in the closed domain \(\overline{\Omega}=\Omega+S\) and which satisfy all boundary conditions of our problem (by \(S\) is denoted the boundary of the domain \(\Omega\)). We shall assume that both equation (11) and the boundary conditions corresponding to it are linear. Then the function \(E_n(x)=\sum_{k=1}^{n} a_k \varphi_k(x)\), where \(a_k\) are arbitrarily chosen constants, satisfies all the boundary conditions of the problem. According to the Bubnov–Galerkin method, the coefficients \(a_k\) are determined from the requirement that the left-hand side of equation (11), after substituting \(U_n(x)\) into it instead of \(E(x)\), be orthogonal to the functions \(\varphi_i(x)\), \((i=1,n)\).

The Bubnov–Galerkin method thus leads to the following system of linear algebraic equations:

\[ \sum_{k=1}^{n} a_k (L\varphi_k,\varphi_m)=(f,\varphi_m)\quad m=\overline{1,n}; \tag{12} \]

(here the parentheses denote the scalar product).

For our problem, as coordinate functions one may take:

\[ \varphi_k(x)=\cos\frac{k\pi x}{2}, \quad \text{where } k=2n+1,\; n=0,1,2,\ldots \]

The proposed functions \(\varphi_k(x)\) satisfy the imposed boundary conditions (10). We restrict ourselves to the number of terms for which \(k\leq 3\). Put

\[ m_T(x)=a_1\cos\frac{\pi x}{2}+a_3\cos\frac{3\pi x}{2}. \tag{13} \]

Let us construct the system of algebraic equations relating the coefficients \(a_k\) by formula (12):

\[ \left. \begin{aligned} a_1\left(1-\frac{\pi^2}{4}\right)-a_3\frac{3}{2\pi}&=-\frac{4}{\pi}\\ -a_1\frac{1}{4}+a_3\left(\frac{3}{2}-\frac{9\pi^2}{4}\right)&=-\frac{4}{3\pi} \end{aligned} \right\}. \tag{14} \]

Whence

\[ a_1=0.86;\quad a_3=0.1. \]

Thus,

\[ m_T(x)=0.86\cos\frac{\pi x}{2}+0.1\cos\frac{3\pi x}{2}. \]

Conclusions. By measuring the value of the output parameter of an automatic control system at an arbitrary instant of time, one can comparatively simply determine the mean time for the parameter to reach the prescribed limits, i.e., in other words, determine the mean time of reliable operation of the system.

REFERENCES

1. B. R. Levin. Theoretical Foundations of Statistical Radio Engineering, vol. 1. Sovetskoe Radio Publishing House, Moscow, 1966.

2. S. G. Mikhlin. Variational Methods in Mathematical Physics. Gostekhizdat, Moscow, 1957.