CYBERNETICS AND CONTROL THEORY

K. A. ABGARYAN

ASYMPTOTIC SPLITTING OF THE EQUATIONS OF A CONTROLLED PROCESS UNDER SLOW VARIATION OF THE PARAMETERS OF THE CONTROLLED OBJECT AND OF THE CONTROL SYSTEM

(Presented by Academician A. Yu. Ishlinskii, 15 IV 1964)

1. In the present note we consider the following system of equations, whose study is connected with the solution of the problem of stability of a controlled process in the linear formulation:

[
\frac{dx}{dt}=u(t)x+a(t)z,\qquad v(t)=b(t)x,
]

[
z(t)=\int_{-\infty}^{t} w(t-t',t')\,v(t')\,dt',
\tag{1}
]

where (x) is a column matrix composed of the discrepancies of the controlled quantities; (z) is the control function; (v(t)) is the input signal of the automatic-control system; (b(t)) is a row matrix determining the law of formation of the input signal of the control system; (w(t-t',t')) is the impulse transition function of the automatic-control system; (u(t)) is a square matrix of order (n); (a(t)) is a column matrix.

In [1], by asymptotic integration of system (1), under the assumption that the eigenvalues of the matrix (u) are distinct and sufficiently separated from one another, a stability criterion was obtained that takes into account the variability of the parameters of the controlled object and of the parameters of the automatic-control system. Below we consider the general case, when the eigenvalues of the matrix (u) may be close and, at individual instants of time, multiple. We reduce the solution of system (1) to the solution of a certain number of mutually independent systems of first-order differential equations. In doing so we use the method of introducing “slow” time proposed by N. M. Krylov and N. N. Bogolyubov [2]. As applied to the system of homogeneous differential equations (dx/dt=u(t)x), which in our case represents the equations of the “open-loop” controlled system, this method was used for the asymptotic splitting of the equations into independent subsystems in [3–6].

In what follows, instead of system (1) we shall consider the equivalent system of integro-differential equations

[
\frac{dx}{dt}=u(t)x+a(t)\int_{-\infty}^{t} w(t-t',t')\,b(t')x(t')\,dt'.
\tag{2}
]

2. Let the eigenvalues of the matrix (u(t)) be divided into (p) groups (\lambda_{\sigma_1},\lambda_{\sigma_2},\ldots,\lambda_{\sigma k_\sigma}) ((\sigma=1,2,\ldots,p)), under the condition that on the segment (0\le t\le L)

[
|\lambda_{\sigma i}-\lambda_{sj}|>0
\qquad
(\sigma\ne s;\ i=1,2,\ldots,k_\sigma;\ j=1,2,\ldots,k_s).
\tag{3}
]

Using Sylvester’s formula for the decomposition of matrices into components (see, for example, [7]), in accordance with the indicated partition the matrix (u) …

can be represented in the form

[
u=\sum_{\sigma=1}^{p} K_{\sigma}\Lambda_{\sigma}M_{\sigma},
\tag{4}
]

where (K_{\sigma}), (\Lambda_{\sigma}), (M_{\sigma}) are matrices of type (n\times k_{\sigma}), (k_{\sigma}\times k_{\sigma}), (k_{\sigma}\times n), respectively, and

[
M_{\sigma}K_s=
\begin{cases}
E_{k_{\sigma}}, & s=\sigma,\
0, & s\ne\sigma
\end{cases}
\tag{5}
]

((E_{k_{\sigma}}) is the identity matrix of order (k_{\sigma})),

[
K_{\sigma}\Lambda_{\sigma}M_{\sigma}
=
\Delta_{\sigma_1\ldots\sigma_{k_{\sigma}}}(u)
\sum_{i=1}^{k_{\sigma}}
\frac{\lambda_{\sigma_i}}{\Delta_{\sigma_i}(\lambda_{\sigma_i})}
\prod_{j\ne i}(u-\lambda_{\sigma_j}E_n),
]

where

[
\Delta_{\alpha_1\ldots\alpha_r}(\lambda)
=
\frac{\displaystyle\prod_{j=1}^{n}(\lambda-\lambda_j)}
{\displaystyle\prod_{i=1}^{r}(\lambda-\lambda_{\alpha_i})}.
]

The matrix

[
\sum_{i=1}^{k_{\sigma}}
\frac{\lambda_{\sigma_i}}{\Delta_{\sigma_i}(\lambda_{\sigma_i})}
\prod_{j\ne i}(u-\lambda_{\sigma_j}E_n)
\equiv
\Phi_{\sigma}(u)
]

is a polynomial of degree (k_{\sigma}-1) in the matrix (u). On the segment ([0,L]) the coefficients of this polynomial are continuous functions of (t), including those points at which the eigenvalues belonging to the group (\sigma) ((\sigma=1,2,\ldots,p)) become equal to one another. The indeterminacies that arise at the indicated points are easily resolved.

The rank of the matrix (\Delta_{\sigma_1\ldots\sigma_{k_{\sigma}}}) is equal to (k_{\sigma}). Therefore this matrix can be represented as the product of a matrix (K_{\sigma}) of type (n\times k_{\sigma}) and a matrix (M_{0\sigma}) of type (k_{\sigma}\times n) ((\Delta_{\sigma_1\ldots\sigma_{k_{\sigma}}}(u)=K_{\sigma}M_{0\sigma})). After this the matrices (M_{\sigma}) and (\Lambda_{\sigma}) are determined by the formulas

[
M_{\sigma}=(M_{0\sigma}K_{\sigma})^{-1}M_{0\sigma},
\qquad
\Lambda_{\sigma}=M_{0\sigma}\Phi_{\sigma}K_{\sigma}
]

(the matrix (M_{0\sigma}K_{\sigma}) is nonsingular).

Let us note that expression (4) can also be represented as follows:

[
u=K\Lambda M,
\tag{6}
]

where

[
K=(K_1,\ldots,K_p),
\qquad
\Lambda=
\begin{pmatrix}
\Lambda_1 & 0\
\cdot & \cdot\
0 & \Lambda_p
\end{pmatrix},
\qquad
M=
\begin{pmatrix}
M_1\
\cdot\
\cdot\
M_p
\end{pmatrix}.
]

1. Consider the system of integro-differential equations

[
\frac{dx}{dt}
=
u(\tau)x
+
\varepsilon a(\tau)
\int_{-\infty}^{t}
w(t-t',\tau')\,b(\tau')x(t')\,dt'
\qquad
(\tau=\varepsilon t),
\tag{7}
]

which for (\varepsilon=1) coincides with system (2).

We shall seek the solution of equations (7) in the form

[
x=\sum_{\sigma=1}^{p}\widetilde K_{\sigma}(\tau)\,\widetilde y_{\sigma}(t),
\tag{8}
]

where the vector functions (\widetilde y_\sigma) are solutions of the equations

[
\frac{d\widetilde y_\sigma}{dt}=\widetilde\Lambda_\sigma(\tau)\widetilde y_\sigma
\qquad (\sigma=1,2,\ldots,p).
\tag{9}
]

We represent the matrices (\widetilde K_\sigma(\tau)) and (\widetilde\Lambda_\sigma(\tau)) in the form of formal expansions

[
\widetilde K_\sigma=\sum_{k=0}^{\infty}\varepsilon^k K_\sigma^{[k]},\qquad
\widetilde\Lambda_\sigma=\sum_{k=0}^{\infty}\varepsilon^k \Lambda_\sigma^{[k]} .
\tag{10}
]

In turn, we seek the solution of equations (9) in the form

[
\widetilde y_\sigma=\sum_{k=0}^{\infty}\varepsilon^k y_\sigma^{[k]} .
\tag{11}
]

Substitute equalities (8) and (9) into equation (7) and equate to zero the sum of all terms containing (\widetilde y_\sigma). We obtain

[
\left(\varepsilon\frac{d\widetilde K_\sigma}{d\tau}
+\widetilde K_\sigma\widetilde\Lambda_\sigma-u\widetilde K_\sigma\right)\widetilde y_\sigma
-\varepsilon a\int_{-\infty}^{t} w(t-t',\tau')\,b(\tau')\,\widetilde K_\sigma(\tau')\widetilde y_\sigma(t')\,dt'=0 .
\tag{12}
]

In the last equality substitute the expansions (9) and (10) and equate to zero the sum of all coefficients at equal powers of (\varepsilon). We shall have

[
L_\sigma^{[0]}y_\sigma^{[0]}=0,
]

[
L_\sigma^{[k]}y_\sigma^{[0]}+
\sum_{\alpha=1}^{k}\left[L_\sigma^{[k-\alpha]}y_\sigma^{[\alpha]}
+I_\sigma^{[k-\alpha]}\left(y_\sigma^{[\alpha-1]}\right)\right]=0
\qquad (k=1,2,\ldots),
\tag{13}
]

where

[
L_\sigma^{[0]}=K_\sigma^{[0]}\Lambda_\sigma^{[0]}-uK_\sigma^{[0]},
]

[
L_\sigma^{[i]}=\frac{dK_\sigma^{[i-1]}}{d\tau}
+K_\sigma^{[i]}\Lambda_\sigma^{[0]}+\cdots+K_\sigma^{[0]}\Lambda_\sigma^{[i]}
-uK_\sigma^{[i]}
\qquad (i=1,2,\ldots),
\tag{14}
]

[
I_\sigma^{[i]}(y)=-a(\tau)\int_{-\infty}^{t}
w(t-t',\tau')\,b(\tau')\,K_\sigma^{[i]}(\tau')\,y(t')\,dt'
\qquad (i=0,1,2,\ldots).
]

Put (K_\sigma^{[0]}\equiv K_\sigma,\ \Lambda_\sigma^{[0]}\equiv\Lambda_\sigma) ((K_\sigma,\Lambda_\sigma) are the matrices appearing in expansion (4)). Then, as is not difficult to verify, (L_\sigma^{[0]}\equiv0), and in each (k)-th equality (13) the term containing (y_\sigma^{[k]}) vanishes.

From equalities (9), (10), (11) there follow the following recurrence formulas for determining (y_\sigma^{[i]}) ((i=1,2,\ldots)):

[
y_\sigma^{[i]}=
Y_\sigma(t)\int_{t_0}^{t}Y_\sigma^{-1}(t')
\left[\Lambda_\sigma^{[1]}(\tau')Y_\sigma^{[i-1]}(t')+\cdots+
\Lambda_\sigma^{[i]}(\tau')Y_\sigma(t')\right]dt'\,C
=
Y_\sigma^{[i]}(t)C
\qquad (i=1,2,\ldots),
\tag{15}
]

where (Y_\sigma(t)) is the fundamental matrix of solutions of the equation

[
\frac{dy_\sigma}{dt}=\Lambda_\sigma y_\sigma,
\tag{16}
]

and (C) is a matrix of arbitrary constants.

Taking into account equalities (15), equalities (13) may be brought to the form

[
uK_\sigma^{[k]}=K_\sigma^{[k]}\Lambda_\sigma+K_\sigma\Lambda_\sigma^{[k]}+D_\sigma^{[k-1]}
\qquad (k=1,2,\ldots),
\tag{17}
]

where

[
D_\sigma^{[k-1]}=
\sum_{\alpha=1}^{k-1}K_\sigma^{[k-\alpha]}\Lambda_\sigma^{[\alpha]}
+\sum_{\alpha=1}^{k}\left[L_\sigma^{[k-\alpha]}Y_\sigma^{[\alpha]}
+I_\sigma^{[k-\alpha]}\left(Y_\sigma^{[\alpha-1]}\right)\right]Y_\sigma^{-1}
+\frac{dK_\sigma^{[k-1]}}{d\tau}.
\tag{18}
]

If the matrices (Y_\sigma, \Lambda_\sigma^{[1]}, K_\sigma^{[1]}, Y_\sigma^{[1]}, \ldots, \Lambda_\sigma^{[k-1]}, K_\sigma^{[k-1]}, Y_\sigma^{[k-1]}) have already been found, then the matrix (D_\sigma^{[k-1]}) is a known matrix, and from the (k)-th equality (17) one can determine (\Lambda_\sigma^{[k]}) and (K_\sigma^{[k]}).

Indeed, for compatibility of this equality it is sufficient to require that

[
M_\sigma\left(K_\sigma \Lambda_\sigma^{[k]} + D_\sigma^{[k-1]}\right) = 0.
\tag{19}
]

Hence we immediately find

[
\Lambda_\sigma^{[k]} = - M_\sigma D_\sigma^{[k-1]}.
\tag{20}
]

With the aid of equality (20), the (k)-th equality (17) can be represented as

[
\Lambda Q_\sigma^{[k]} = Q_\sigma^{[k]}\Lambda_\sigma + M G_\sigma D_\sigma^{[k-1]},
\tag{21}
]

where

[
G_\sigma = E_n - K_\sigma M_\sigma, \qquad Q_\sigma^{[k]} = M K_\sigma^{[k]}.
\tag{22}
]

We split the matrix (Q_\sigma^{[k]}), consisting of (n) rows and (k_\sigma) columns, into (p) blocks so that the first block contains (k_1) rows, the second block (k_2) rows, and so on:

[
Q_\sigma^{[k]} =
\begin{pmatrix}
Q_{\sigma 1}^{[k]}\
\vdots\
Q_{\sigma p}^{[k]}
\end{pmatrix}.
\tag{23}
]

Then equality (21) decomposes into (p) mutually independent matrix equations

[
\Lambda_\sigma Q_{\sigma\sigma}^{[k]} = Q_{\sigma\sigma}^{[k]}\Lambda_\sigma,
\tag{24}
]

[
\Lambda_s Q_{\sigma s}^{[k]} = Q_{\sigma s}^{[k]}\Lambda_\sigma + M_s D_\sigma^{[k-1]} \qquad (s \ne \sigma).
\tag{25}
]

On the basis of equality (24) we may take (Q_{\sigma\sigma}^{[k]}=0). Since the matrices (\Lambda_s) and (\Lambda_\sigma) for (s \ne \sigma) have no common eigenvalues, the remaining equalities (25) uniquely determine the matrices (Q_{\sigma s}^{[k]}) ((s \ne \sigma)). Having thus determined the matrix (Q_\sigma^{[k]}), it is easy to compute the matrix (K_\sigma^{[k]}) by the formula

[
K_\sigma^{[k]} = K Q_\sigma^{[k]}.
\tag{26}
]

The relations given above make it possible successively to determine the matrices (\Lambda_\sigma^{[1]}, K_\sigma^{[1]}, \Lambda_\sigma^{[2]}, K_\sigma^{[2]}, \ldots), which occur in the formal expansions (10), by means of which the system of integro-differential equations of order (n) is split into (p) independent systems of differential equations (9) of orders (k_1, k_2, \ldots, k_p) ((k_1 + k_2 + \ldots + k_p = n)). Taking the matrices (\widetilde K_\sigma) and (\widetilde\Lambda_\sigma) to be partial sums of the expansions (10), we obtain an approximate splitting of the integro-differential equations. In essence, the integration of equations (7), and hence also of the original system (1), is reduced to the integration of (p) independent systems of differential equations (16), for, having the matrix (Y_\sigma(t)) of fundamental solutions of equations (16), one can determine (\widetilde y_\sigma(t)) by means of formulas (15) and (11).

Moscow Aviation Institute
named after S. Ordzhonikidze

Received
8 IV 1964

CITED LITERATURE

(^{1}) I. M. Rapoport, Dokl. Akad. Nauk SSSR, 158, No. 2 (1964).
(^{2}) N. M. Krylov, N. N. Bogolyubov, Introduction to Nonlinear Mechanics, Kiev, 1937.
(^{4}) S. F. Feshchenko, Dop. AN UkrSSR, 1 (1959).
(^{4}) S. F. Feshchenko, Ukr. Mat. Zhurn., 7, No. 2, No. 4 (1955).
(^{5}) Yu. L. Daletskii, S. G. Krein, Ukr. Mat. Zhurn., 2, No. 4, 71 (1950).
(^{6}) Yu. L. Daletskii, DAN, 92, 881 (1953).
(^{7}) R. F. Feshchenko, Dunkan, A. Kodlar, Matrix Theory and Its Applications to Differential Equations and Dynamics, IL, 1950.