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UDC 517.949.22
APPROXIMATE SOLUTION OF THE PROBLEM OF ANALYTIC CONSTRUCTION OF A REGULATOR FOR AN EQUATION WITH RETARDATION
E. M. MARKUSHIN, S. N. SHIMANOV
The article considers the problem of the analytic construction of a regulator for an equation with retardation [1].
A justification is given for an approximate method of solving the problem by decomposing the motion described by an equation with retardation into canonical variables [4, 5].
§ 1. PRELIMINARY REMARKS
Consider an automatic-control system whose perturbed motion is described by the equation
\[ \frac{dx(t)}{dt}=ax(t-\tau)+m\xi, \tag{1.1} \]
where \(a, m\) are constants; \(\tau\) is the retardation \((\tau=\mathrm{const}>0)\); \(\xi\) is the unknown action of the regulator.
In accordance with [1], we shall assume that the admissible controls \(\xi\) belong to the set of functionals defined on continuous curves \(x(t)\) \((t\geq 0)\) and satisfying the Cauchy–Lipschitz conditions
\[ \|\xi[x'(t)]-\xi[x(t)]\|^{(h)}\leq L_{\xi}\|x'(t)-x(t)\|^{(h)}, \]
where the norm is defined by the equality
\[ \|x(\vartheta)\|^{(h)}=\sup [x(\vartheta)] \quad [-\tau\leq \vartheta\leq 0]. \]
The problem of the analytic construction of a regulator for systems with retardation was formulated in [1]. In the present case, for equation (1.1), it is required to find such a functional \(\xi[x(\vartheta)]\) that every solution with any initial function \(\varphi(\vartheta)\)
\[ x_0(\vartheta)=\varphi(\vartheta) \quad [-\tau\leq \vartheta\leq 0] \]
tends to zero as \(t\to\infty\). In addition, it is required that the value of the integral
\[ I(\xi)=\int_{0}^{\infty} V\,dt \tag{1.2} \]
of the positive-definite quadratic form
\[ V=\alpha x^2(t)+c\xi^2, \tag{1.3} \]
where \(\alpha, c\) are constants, be minimal.
The problem of the analytic construction of a regulator for systems of ordinary equations was considered in [2–3].
Since equation (1.1), for any \(a\) and \(m \ne 0\), is stabilizable, it follows, according to [1], that there exists an optimal control \(\xi^*[x(\vartheta)]\) minimizing the integral (1.2) and having the form
\[ \xi^*=\beta_1 x(0)+\int_{-\tau}^{0}\sigma(\vartheta)x(\vartheta)\,d\vartheta, \tag{1.4} \]
where \(\beta_1=\mathrm{const}\), and \(\sigma(\vartheta)\) is some function \([-\tau \leq \vartheta \leq 0]\). In the practical computation of the coefficient \(\beta_1\) and the function \(\sigma(\vartheta)\) of the optimal control \(\xi^*[x(\vartheta)]\), certain technical difficulties arise [1]; therefore it is expedient to consider approximate methods for determining it. This article presents a method for the approximate determination of the optimal action, based on selecting in the functional space \(C[-\tau,0]\) a subspace with a finite-dimensional basis [4, 5], on which the motion of the system with retardation can be represented by a system of ordinary differential equations.
It is necessary to note that questions of approximation of problems of the analytic construction of a regulator for systems with retardation were considered in [12–15].
An approximate method for solving this problem, based on selecting systems of ordinary differential equations, was given in the works of M. E. Salukvadze [14], Yu. M. Repin and V. E. Tret’yakov [13, 15]. A justification of these approximate methods was given in the works [12, 13] by N. N. Krasovskii. In addition, N. N. Krasovskii in [1] proposed a method for the approximate determination of the coefficients of the optimal controlling action by expanding them in Fourier series.
§ 2. SPLITTING OF EQUATION (1.1). APPROXIMATE SOLUTION OF THE PROBLEM OF ANALYTIC CONSTRUCTION OF A REGULATOR FOR AN EQUATION WITH RETARDATION
It is known [4–6] that equation (1.1) in the space of continuous functions \(C[-\tau,0]\) corresponds to the operator equation
\[ \frac{dx_t(\vartheta)}{dt}=Ax_t(\vartheta)+R\xi, \tag{2.1} \]
where \(x_t(\vartheta)=x(t+\vartheta)\) \([-\tau \leq \vartheta \leq 0]\), and the operators \(A\) and \(R\) are defined as follows:
\[ Ax(\vartheta)= \begin{cases} \dfrac{dx(\vartheta)}{dt}, & -\tau \leq \vartheta < 0,\\[6pt] ax(-\tau), & \vartheta=0; \end{cases} \tag{2.2} \]
\[ R= \begin{cases} 0, & -\tau \leq \vartheta < 0,\\ m, & \vartheta=0. \end{cases} \tag{2.3} \]
To equation (1.1), or, what is the same, to equation (2.1) with operator right-hand side, there corresponds the characteristic equation
\[ \lambda=ae^{-\lambda\tau}. \tag{2.4} \]
Suppose \(a \ne -\dfrac{1}{\tau e}\). Then the roots of the characteristic equation (2.4) will be simple. Arrange them in order of decreasing real parts
\[ \operatorname{Re}\lambda_1 \geqslant \operatorname{Re}\lambda_2 \geqslant \ldots \geqslant \operatorname{Re}\lambda_N \geqslant \ldots . \tag{2.5} \]
To equation (2.1) there corresponds the following equivalent system of equations [4, 5]:
\[ \frac{dy_1}{dt}=\lambda_1 y_1+m\xi, \]
\[ \cdots \]
\[ \frac{dy_N}{dt}=\lambda_N y_N+m\xi, \tag{2.6} \]
\[ \frac{dz_t(\vartheta)}{dt} = Az_t(\vartheta) - \left[ \sum_{i=1}^{N}\frac{e^{\lambda_i\vartheta}}{\Delta'(\lambda_i)} \right]m\xi + R\xi, \]
where \(y_i=f_i[x]\) \((i=1,2,\ldots,N)\) are linear functionals,
\[ f_i[x]=x_t(0)+a\int_{-\tau}^{0}x_t(\vartheta)e^{-\lambda_i(\tau+\vartheta)}\,d\vartheta, \tag{2.7} \]
\[ x_t(\vartheta)=x(t+\vartheta)\qquad [-\tau\leqslant \vartheta\leqslant 0], \]
\[ f_j[z_t(\vartheta)]=0\qquad (j=1,2,\ldots,N) \]
and
\[ \Delta'(\lambda_i)=1+\lambda_i\tau. \tag{2.8} \]
The coordinate \(z(\vartheta)\) is the remainder in the expansion of the element \(x(\vartheta)\in C[-\tau,0]\) into a series with respect to the eigenvalues of the operator \(A\), i.e.
\[ z_t(\vartheta)=x(t+\vartheta)-\sum_{i=1}^{N}\frac{e^{\lambda_i\vartheta}}{\Delta'(\lambda_i)}\,y_i(t). \tag{2.9} \]
It is natural to assume that, in the approximate determination of the optimal functional \(\xi^{*}[x(\vartheta)]\) for equation (1.1), it is sufficient to solve the problem of the analytic construction of the optimal control \(\xi_N^{*}(y_1,\ldots,y_N)\) for the truncated system of ordinary equations
\[ \frac{dy_i}{dt}=\lambda_i y_i+m\xi_N\qquad (i=1,2,\ldots,N), \tag{2.10} \]
obtained from (2.6) for \(z\equiv0\). That is, for system (2.10) it is necessary to find an action \(\xi_N^{*}\) such that, when inserted into (2.10), it forms an asymptotically stable system: stabilizes system (2.10) and minimizes the integral
\[ I_N(\xi_N)=\int_{0}^{\infty}V_N\,dt, \tag{2.11} \]
where \(V_N\) is a form of the type
\[ V_N=\alpha\left(\sum_{j=1}^{N}\frac{y_j}{\Delta'(\lambda_j)}\right)^2+c\,\xi_N^2. \tag{2.12} \]
The indicated problem for systems of ordinary equations was solved by A. M. Letov [2]. We shall make use of his solution. Note that for \(m \ne 0\) system (2.10) is controllable; therefore problem (2.10), (2.11) has a solution. Suppose that the optimal control \(\xi_N^*(y_1,\ldots,y_N)\) has been found; it has the form \(\xi_N^*=\sum_{i=1}^{N} p_i^{(N)} y_i^{(N)}\), where \(p_i^{(N)}\) are constants. The functional \(\xi_N^*\) can be represented in the form
\[ \xi_N^*=\sum_{i=1}^{N} p_i^{(N)} y_i^{(N)} = \left(\sum_{i=1}^{N} p_i^{(N)}\right)x(t) + a \int_{-\tau}^{0} \left(\sum_{i=1}^{N} p_i^{(N)} e^{-\lambda_i(\tau+\vartheta)}\right) x(t+\vartheta)\,d\vartheta . \tag{2.13} \]
This functional is naturally regarded as approximate with respect to the optimal functional (1.4) of problem (1.1), (1.2). Consequently, in the present paper the following approximate method for solving the problem of analytic construction of a regulator for an equation with retardation is considered. In the approximate solution of the problem we choose the first \(N\) roots of the characteristic equation (2.4) and at the same time consider the system of ordinary differential equations of order \(N\), (2.10). For system (2.10) we solve the problem of analytic construction of a regulator with the quality criterion (2.11), (2.12), which is obtained from (1.2), (1.3), setting the quantity \(z(t)\) in (2.9) equal to zero. The solution of this problem is described in [2, 3]. Suppose that we have solved this problem. The optimal control of problem (2.10), (2.11) has the form (2.13).
The control (2.13) is regarded as an approximate optimal functional of problem (1.1), (1.2), since, for sufficiently large \(N\), it will minimize the integral (1.2) with the form (1.3) as accurately as desired, i.e., the value of the integral (1.2) with the form (1.3) will be arbitrarily close to the minimum. More precisely, it will differ from the optimal value of the integral by the quantity
\(\varepsilon_N\{\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}+\|\ddot{\varphi}(\vartheta)\|^{(h)}\}^2\), where \(\varepsilon_N\) is arbitrarily small. In the present paper a justification of this method is given. The approximate method described for solving the problem of analytic construction of a regulator for an equation with retardation naturally extends to the general case of systems with retardation.
§ 3. On the convergence of optimal controls
Let, for system (2.10), for some sufficiently large \(N\), an optimal control \(\xi_N^*(y_1,\ldots,y_N)\) be found, minimizing the integral (2.11) of the form (2.12). The following proposition holds.
Theorem 3.1. If the functional (2.13) is substituted into equation (1.1), then the value of the integral (1.2), for sufficiently large number \(N\), will be arbitrarily close to the minimum; i.e., the estimate holds
\[ \left| I(\xi_N^*)-I(\xi^*) \right| \leq \delta \left\{ \|\varphi(\vartheta)\|^{(h)} + \|\dot{\varphi}(\vartheta)\|^{(h)} + \|\ddot{\varphi}(\vartheta)\|^{(h)} \right\}^{2}, \]
where \(\delta\) is an arbitrarily small positive quantity for sufficiently large number \(N\).
Consider the integrals (1.2) and (2.11) and show that
\[ \left| I(\xi^*)-I_N(\xi_N^*) \right| \leq \eta \left\{ \|\varphi(\vartheta)\|^{(h)} + \|\dot{\varphi}(\vartheta)\|^{(h)} + \|\ddot{\varphi}(\vartheta)\|^{(h)} \right\}^{2} \tag{3.1} \]
\[ (\eta=\text{const}>0), \]
where \(\eta \to 0\) as \(N \to \infty\), and \(\|\varphi(\vartheta)\|^{(h)}\) is the norm of the initial perturbation.
In inequality (3.1), on the left, \(I(\xi^*)\) is the minimum value of the integral (1.2), and \(I_N(\xi_N^*)\) is the minimum value of the integral (2.11). For this purpose we first show that the difference \(\left| I(\xi_N^*) - I_N(\xi_N^*) \right|\) has an estimate of the form (3.1), i.e.,
\[
\left| I(\xi_N^*) - I_N(\xi_N^*) \right|
\leq
a_1 \left\{ \|\varphi(\vartheta)\|^{(h)}
+ \|\dot{\varphi}(\vartheta)\|^{(h)}
+ \|\ddot{\varphi}(\vartheta)\|^{(h)} \right\}^2
\]
\[
(a_1=\mathrm{const}>0),
\tag{3.2}
\]
where \(a_1 \to 0\) as \(N \to \infty\), and \(I(\xi_N^*)\) is the value of the integral (1.2) when the optimal control of system (2.10) is used as the control of equation (1.1).
Indeed, for sufficiently large \(N\), let us find the optimal control \(\xi_N^*(y_1,\ldots,y_N)\) of system (2.10), where \(y_1,\ldots,y_N\) are defined by formulas (2.7), and substitute it into (1.1):
\[
\frac{dx(t)}{dt}=a x(t-\tau)+m\xi_N^*[x_t(\vartheta)]
\tag{3.3}
\]
\[
(\xi_N^*[x_t(\vartheta)]=\xi_N^*[y_1(x_t(\vartheta)),\ldots,y_N(x_t(\vartheta))]).
\]
Splitting equation (3.3), we obtain, on the basis of (2.6),
\[
\frac{dy_1}{dt}=\lambda_1 y_1+m\xi_N^*(y_1,\ldots,y_N),
\]
\[
\cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot\quad \cdot
\]
\[
\frac{dy_N}{dt}=\lambda_N y_N+m\xi_N^*(y_1,\ldots,y_N),
\tag{3.4}
\]
\[
\frac{dz_t(\vartheta)}{dt}
=Az_t(\vartheta)-
\left[\sum_{i=1}^{N}\frac{e_i^{\lambda_i\vartheta}}{\Delta'(\lambda_i)}\right]m\xi_N^*
+R\xi_N^*.
\]
Therefore
\[
I(\xi_N^*)=I_N(\xi_N^*)+
2\int_{0}^{\infty}\left[
a z(t)\sum_{i=1}^{N}\frac{y_i(t)}{\Delta'(\lambda_i)}
\right]dt+
\]
\[
+\int_{0}^{\infty} a z^2(t)\,dt.
\tag{3.5}
\]
Let us estimate the quantity \(z(t)\) appearing in the last two integrals in equality (3.5). According to [7, 8, 13], the general solution of the nonhomogeneous equation (3.3) with initial perturbation \(\varphi(\vartheta)\) \([-\tau\leq\vartheta\leq 0]\) can be written in the form
\[
x(t,\varphi,(\vartheta))=\varphi(0)u(t)+
\]
\[
+a\int_{-\tau}^{0}\varphi(\vartheta)u(t-\tau-\vartheta)\,d\vartheta
+\int_{0}^{t}u(t-\vartheta)m\xi_N^*\,d\vartheta,
\tag{3.6}
\]
where \(u(t)\) is the solution of the homogeneous equation with initial perturbation
\[
\varphi^*(\vartheta)=
\begin{cases}
1, & \vartheta=0,\\
0, & -\tau\leq \vartheta<0.
\end{cases}
\tag{3.7}
\]
The solutions of system (2.10) can be written in the form
\[ y_i=y_i(0)e^{\lambda_i t}+\int_0^t e^{\lambda_i(t-\vartheta)}m\xi_N^*\,d\vartheta, \tag{3.8} \]
where, in accordance with (2.7),
\[ y_i(0)=\varphi(0)+a\int_{-\tau}^0 \varphi(\vartheta)e^{-\lambda_i(\tau+\vartheta)}\,d\vartheta. \tag{3.9} \]
For the remainder \(z_N(t)\) of the nonhomogeneous equation, on the basis of (3.6), (3.8), (3.9), we obtain
\[ \begin{aligned} z_N(t)&=x(t)-\sum_{i=1}^N \frac{y_i(t)}{\Delta'(\lambda_i)}=\\ &=\mu_N(t)\varphi(0)+\int_{-\tau}^0 \mu_N(t-\tau-\vartheta)\varphi(\vartheta)a\,d\vartheta+\\ &\quad+\int_0^t \mu_N(t-\vartheta)m\xi_N^*\,d\vartheta, \end{aligned} \tag{3.10} \]
where
\[ \mu_N(t)=u(t)-\sum_{i=1}^N \frac{e^{\lambda_i t}}{\Delta'(\lambda_i)}. \]
Let the initial curve \(\varphi(\vartheta)\) be continuous and bounded together with its first two derivatives; then its expansion converges uniformly on the interval \([-\tau,0]\) [8, 16]. In this case the remainder of the expansion of the function \(\varphi(\vartheta)\) into the series
\[ z_N(\vartheta)=\varphi(0)-\sum_{i=1}^N \frac{e^{\lambda_i\vartheta}}{\Delta'(\lambda_i)} f_i[\varphi(\vartheta)] \]
has a uniform estimate of the form
\[ |z_N(\vartheta)|\leqslant \varepsilon_N\{\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}+\|\ddot{\varphi}(\vartheta)\|^{(h)}\}, \tag{3.11} \]
where the constant \(\varepsilon_N\to 0\) as \(N\to\infty\). Further, it follows from [11] that the optimal control \(\xi_N^*(y_1,\ldots,y_N)\) for each truncated system (2.10) has the uniform estimate
\[ |\xi_N^*(y_1,\ldots,y_N)|\leqslant A\{\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}\}e^{-\gamma t}, \tag{3.12} \]
where \(A\) is a positive constant independent of the choice of \(N\); \(\varphi(\vartheta)\) is the initial function \([-\tau\leq \vartheta\leq 0]\), and \(\gamma\) is a positive number greater than \(k\); \(2k\) is the modulus of the least real part of the roots of the characteristic equation of system (1.1), in which, instead of \(\xi\), there stands the optimal functional \(\xi^*[x(\vartheta)]\). This equation has the form
\[ \Delta_1=\Delta(\mu)\Delta(-\mu)+\frac{mc}{\alpha}=0;\qquad \Delta(\mu)=\mu-a\exp(-\mu\tau). \]
We note that the last equation was obtained by Yu. M. Repin in considering the problem of the analytic construction of a regulator by another method. The estimate of \(\gamma\) follows from the circumstance that, as the number \(N\) increases, the roots with negative real parts for the truncated system approach the roots of the quasipolynomial \(\Delta_1(\mu)\).
From estimates (3.11) and (3.12), on the basis of (3.10), it follows that \(z_N(t)\) is an arbitrarily small decreasing function, i.e.,
\[ |z_N(t)| \leq B_N \{ \|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}+\|\ddot{\varphi}(\vartheta)\|^{(h)} \} e^{-\gamma t}. \tag{3.13} \]
Using inequality (3.13), we obtain estimates of the last two integrals in (3.5), from which inequality (3.2) follows.
Suppose that the optimal control of problem (1.1), (1.2) is known. Then substitute it into equation (1.1) and consider the integral (1.2):
\[ I(\xi^*)=\int_0^\infty [\alpha x^2(t)+c\xi^{*2}]\,dt = \tag{3.14} \]
\[ = \int_0^\infty \left[ \alpha\left(\sum_{i=1}^{N}\frac{y_i(t)}{\Delta'(\lambda_i)}+z(t)\right)^2 +c\bigl(\xi_N+\xi^*[z_t(\vartheta)]\bigr)^2 \right]\,dt. \]
Here the notation has been introduced
\[ \xi_N=\xi^*\left[\sum_{i=1}^{N}\frac{e^{\lambda_i\vartheta}}{\Delta'(\lambda_i)}\right] \tag{3.15} \]
and the splitting formula (2.9) has been used,
\[ x_t(\vartheta)=\sum_{i=1}^{N}\frac{e^{\lambda_i\vartheta}}{\Delta'(\lambda_i)}\,y_i(t)+z_t(\vartheta). \tag{3.16} \]
Then, putting
\[ I_N(\xi_N)=\int_0^\infty \alpha\left[ \left(\sum_{i=1}^{N}\frac{y_i(t)}{\Delta'(\lambda_i)}\right)^2+c\xi_N^2 \right]\,dt, \tag{3.17} \]
we obtain the following expression for \(I(\xi^*)\):
\[ I(\xi^*)=I_N(\xi_N)+\Delta^*(z_t(\vartheta)), \tag{3.18} \]
where
\[ \Delta^*(z_t(\vartheta)) = 2\int_0^\infty \alpha\left(\sum_{i=1}^{N}\frac{y_i(t)}{\Delta'(\lambda_i)}\right)z(t)\,dt + \]
\[ +\int_0^\infty \alpha z^2(t)\,dt + c\int_0^\infty \left[2\xi_N\xi^*[z_t(\vartheta)]+\xi^*[z_t(\vartheta)]\right]\,dt. \tag{3.19} \]
But any function \(x_t(\vartheta)\) that is continuous and bounded together with its two derivatives can be represented in the form of a series
\[ x_t(\vartheta)=\sum_{i=1}^{N}\frac{e^{\lambda_i\vartheta}}{\Delta'(\lambda_i)}\,y_i(t)+z_t(\vartheta), \]
where for \(z_t(\vartheta)\) the estimate holds
\[ |z_t(\vartheta)| \leq \theta_N\left\{\|x_t(\vartheta)\|^{(h)}+\|\dot{x}_t(\vartheta)\|^{(h)}+\|\ddot{x}_t(\vartheta)\|^{(h)}\right\}. \tag{3.20} \]
and \(\theta_N \to 0\) as \(N \to \infty\).
Since the solution of equation (1.1) with the optimal control \(\xi^*\) satisfies the estimate
\[ |x_t(\vartheta)| \leq B_1 \|\varphi(\vartheta)\|^{(h)} e^{-qt}, \]
\[ q=\mathrm{const}>0,\qquad B_1=\mathrm{const}<\infty, \]
then, using (3.20), for the quantity \(\Delta^*(z_t(\vartheta))\) on the basis of (3.19) we obtain
\[ |\Delta^*(z_t(\vartheta))|\leq \zeta\left\{\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}+\|\ddot{\varphi}(\vartheta)\|^{(h)}\right\}^2 \tag{3.21} \]
\[ (\zeta=\mathrm{const}>0), \]
where \(\zeta \to 0\) as \(N \to \infty\).
On the basis of (3.21) and (3.18) we conclude that
\[ \left|I(\xi^*)-I_N(\xi_N)\right|\leq \zeta\left\{\|\varphi(\vartheta)\|^{(h)}+\|\dot{\varphi}(\vartheta)\|^{(h)}+\|\ddot{\varphi}(\vartheta)\|^{(h)}\right\}^2 . \tag{3.22} \]
Since either the inequality \(I_N(\xi_N^*)\leq I(\xi^*)\) or the inequality \(I_N(\xi_N^*)\geq I(\xi^*)\) holds, in the first case we have
\[ I_N(\xi_N^*)\leq I(\xi^*)\leq I(\xi_N^*), \tag{3.23} \]
and in the second
\[ I(\xi^*)\leq I_N(\xi_N^*)\leq I_N(\xi_N). \tag{3.24} \]
Taking into account the estimates of the differences (3.2) and (3.22), from the inequalities (3.23) and (3.24) we obtain inequality (3.1).
From inequalities (3.1) and (3.2) follows the assertion of Theorem 3.1.
It can be proved that the following assertion holds.
Theorem 3.2. From the sequence of functionals \(\xi_N^*\) (2.13) one can always choose a convergent subsequence whose limit will be the functional \(\xi^*\) (1.4) minimizing, by virtue of equation (1.1), the integral (1.2).
The proof of this theorem can be carried out using the continuity of the integral \(I(\xi)\) on all \(\xi\) that ensure the asymptotic stability of the solution \(x=0\) of equation (1.1), and the uniform boundedness of the functionals \(\xi_N^*\).
In conclusion it should be noted that this paper was discussed at the citywide seminar on the theory of controlled processes. The authors thank the head of the seminar, Corresponding Member of the Academy of Sciences of the USSR N. N. Krasovskii, for useful comments.
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Received by the editors
July 30, 1965
Ural State University
named after A. M. Gorky