Introduction

The quality of the synthesized control system depends on utilization of information about input signals. Therefore, the so-called absorption principle [7] in the automatic control theory is widespread. The absorption principle is built on class description of the input signals by homogeneous differential or difference equations with arbitrary initial conditions [1, 3–5, 8–11]. In this paper, the larger class of input signals of a continuous-time linear control system is specified by the stationary heterogeneous differential equation with arbitrary initial conditions and a restricted right member [2, 6]. This approach can be easily extended to discrete systems.

1. Statement of the problem

Let

, ,                                                                                        (1)

be transfer functions corresponding to a desirable control system and a synthesized control system, where  is a stable polynomial.

In this paper, we shall determine the signals class  such that

                                                                                (2)

here

, .                                                                           (3)

2. Main results

Consider the error E(s) determined by (3). Using (1), we get

                                                                                      (4)

or

,                                                                                                 (5)

where ,  , and

.                                                                                        (6)

We show that the input signals class  is defined by the set of solutions of the linear differential equation

, ,                                                                                            (7)

with arbitrary initial conditions and a piecewise continuous restricted right member, i.e.,

, .                                                                                   (8)

Here  and  is piecewise continuous. Next we define . The application of the Laplace transformation to equation (7) yields

,                                                                                             (9)

where L₀ is a polynomial of degree m-1. Note that the Laplace transformation of the functions ,  and  is existed. Combining (5) and (9), we obtain

.                                                                                            (10)

Since D(s) is the Hurwitz polynomial, it follows that

.                                                                                                     (11)

Here

                                                                                                     (12)

is a transient components of the error ε(t). We can therefore write

.                                                                                 (13)

Let us consider now a steady-state components

                                                                                                     (14)

— of the error ε(t). Let

,                                                                                            (16)

where

.                                                               (18)

Taking into account (8), we get

,                                                         (21)

Combining (21) and (13), we get the following proposition:

.                                                          (23)

Let  and

.                                                                                      (24)

Thus proposition (2) is executed for the input signals satisfying conditions (7), (8), and (24).

Notice that relationships (17) and (22) indicate the practical methods for the decrease of the steasy-state error. If , then the steady-state error is equal to 0.

3. Example

Let us consider

,   ,  .                                                       (25)

Therefore,

,   .                                                                          (26)

From (22), we obtain . Thus the signals class  is defined by the following conditions:

,                                                                                   (27)

.                                                                  (28)

This implies that

,  .                                                             (29)

Clearly, algebraic polynomials of degree 1, trigonometric polynomials, decreasing exponents, and logarithmic functions belong to the selected class . Note also that the class  are not exhausted the listed functions.

Let us consider now

, .                                                                        (30)

Hence,

,   .                                                       (31)

Using (22), we get . Consequently the signals class  is determined by the equation

,   ,                                                                         (32)

where

.                                                                             (33)

If , then the element with transfer function (30) realizes the asymptotically fine noise-proof differentiating of amplitude modulated signals