2.1 Linear input-output models

Lecture

The mathematical model of a dynamic system is called the set of mathematical symbols that uniquely determine the development of processes in the system, i.e. her movement. At the same time, depending on the symbols used, analytical and graphic-analytical models are distinguished. Analytical models are constructed with the help of alphabetic characters, while grapho-analytical ones allow the use of graphic symbols (see clause 2.1.2).

Depending on the type of signals, continuous and discrete models of systems differ. Depending on the operators used - linear and nonlinear , as well as time and frequency models. Temporary are models in which the argument is (continuous or discrete) time. These are differential and difference equations written explicitly or in operator form. Frequency models involve the use of operators whose argument is the frequency of the corresponding signal, i.e. Laplace, Fourier, etc. operators

This section discusses continuous linear temporal models of dynamic systems.

2.1 Linear input-output models

An input-output (BB) model is a description of the relationship between the input and output signals of a dynamic system. The need for such a description appears when considering the behavior of individual units and, in particular, the control object (OU), and the control system as a whole. Differences in the mathematical description of the blocks and the control system are not fundamental, but they require the use of different designations (see Clause 1.5). So, the input signal of the ACS is the specifying effect y * ( t ), and the output is the variable y ( t ) . When describing blocks, the notation x ₂ ( t ) and x ₁ ( t ) are often used , respectively. In the following, we use the notation typical of the control object, where the input signal is the control action u ( t ) and the output is the variable variable y ( t ).

2.1.1. Analytical models. The linear input-output model of a single-channel dynamic system (here, the control object) can be represented by an ordinary differential equation of the form:

[M1] 2.1 Linear input-output models ,

where a _i , b _{i are} coefficients ( model parameters ), a ₀ 2.1 Linear input-output models 0 , b ₀ 0, n - model order, 0 m <n . Equation [M1] connects the input signals and their derivatives with output signals y ( t ) and their derivatives on some time interval, i.e. at . Meanings , , ..., are called initial values (conditions ), and the number r = n - m 2.1 Linear input-output models 1 - relative degree of the model.

There are stationary systems for which the parameter values are unchanged: 2.1 Linear input-output models , and you can put , and non-stationary models , where the parameters are functions of time, i.e. , . In the case when , the equation is called reduced.

System for which 2.1 Linear input-output models , called autonomous . The description of an autonomous system is given by a homogeneous differential equation of the form

[M1a] 2.1 Linear input-output models .

Model [M1] can be rewritten in operator form. To do this, we introduce into consideration the differentiation operators

2.1 Linear input-output models

and suppose that

2.1 Linear input-output models .

Taking into account the introduced notation, the equation [M1] is easily converted to the operator form

[M2] 2.1 Linear input-output models ,

where differential operators are used

(2.1) 2.1 Linear input-output models ,

(2.2) 2.1 Linear input-output models .

The operator a ( p ) is the characteristic polynomial of the differential equation [M1], and the complex numbers 2.1 Linear input-output models , which are the roots of the characteristic equation

(2.3) 2.1 Linear input-output models ,

are called the poles of the system [M1] . The differential operator b (p) is the characteristic polynomial of the right side. Roots of the equation

(2.4) 2.1 Linear input-output models ,

those. complex numbers 2.1 Linear input-output models are called system zeros [M1].

From equation [M2] we find an explicit connection of the variables y ( t ) and u ( t ) in the form of an operator equation:

[M3] 2.1 Linear input-output models ,

where is integral differential operator

(2.5) 2.1 Linear input-output models

called the transfer function of the system [M1].

The advantage of using operator models like [M2] and [M3] is, firstly, the brevity of the corresponding equations, and secondly, the convenience of converting complex (composite) models (see Section 2.4).

Consider a special case of a dynamic system with coefficients b ₀ = b ₁ = ... = b _m-1 = 0 . When b _m = b 2.1 Linear input-output models 0 the system has a relative degree r = n-1 , and zeros are missing. The equation [M1] takes the form

(2.6) 2.1 Linear input-output models ,

equation [M2] -

(2.7) 2.1 Linear input-output models ,

and the equation [M3] is

(2.8) 2.1 Linear input-output models .

Example 2.1. Let be 2.1 Linear input-output models and . The differential equation of the system has the form

2.1 Linear input-output models

with initial conditions 2.1 Linear input-output models ; . Here - speed output variable. The operator form of the model is

2.1 Linear input-output models ,

and

2.1 Linear input-output models .

System characteristic equation

2.1 Linear input-output models

has two (real or complex) roots

2.1 Linear input-output models .

2.1.2. Structural schemes. The most common graph-analytical form of a dynamic system model is a structural diagram - a type of directional graph. Elements of such a scheme are (Fig. 2.1)

Fig. 2.1. Elements of the structural scheme

letter designations of signals ( x ( t ) , u ( t ) , y ( t ) and so on) and so on;
letter symbols of operators (for example, W ( p ));
graphic symbols - arrows indicating the direction of the signals, nodes (signal forks), blocks with indication of the input and output signals, as well as operators describing the connections between the signals.

The simplest blocks used in structural schemes are (Fig. 2.2):

unit of comparison;
adder;
proportional block;
integrator

(see also clause 2.3).

2.1 Linear input-output models

Fig.2.2. Simplest blocks

Example 2.2. The input-output heating furnace, RC-chains and motor acceleration (see Example 1 .1) are described by a first-order differential equation.

(2.9) 2.1 Linear input-output models

where T, K are constant coefficients (parameters). The operator form of the model is

(2.10) 2.1 Linear input-output models .

Here 2.1 Linear input-output models - the characteristic equation, which has one root (pole of the system) p ₁ = - 1 / T. From equation (2.10) we find the operator connection of the input and output

2.1 Linear input-output models

2.1 Linear input-output models .

Therefore, the transfer function of the block is the operator

2.1 Linear input-output models .

Note that equation (2.9) can be reduced to

(2.11) 2.1 Linear input-output models

2.1 Linear input-output models

where a = 1 / T, b = K / T. Then the operator form (2.10) takes the form

(2.12) 2.1 Linear input-output models

and form (2.12) is

2.1 Linear input-output models .

Example 2.3. Consider the motion of a material point of mass m under the action of a force (input action) u = F ( t ). This dynamic system is described by a second-order equation (Newton's 2nd law)

(2.13) 2.1 Linear input-output models

with initial conditions y ₀ = y ₀ (0), 2.1 Linear input-output models where y ( t ) is a linear displacement. The operator form of the model takes the form

(2.14) 2.1 Linear input-output models

and the characteristic equation of the system

(2.15) 2.1 Linear input-output models

2.1 Linear input-output models

has two roots (poles of the system) p _1,2 = 0 . From equation (2.14) we find the operator connection of the input and output

(2.16) 2.1 Linear input-output models

where b = 1 / m . Therefore, the transfer function of the block is the operator

2.1 Linear input-output models

(2.17) 2.1 Linear input-output models .

In structural diagrams of multidimensional and multichannel systems, vector signals 2.1 Linear input-output models , and sometimes emit double arrows.

2.1 Linear input-output models

2.1.3. Multichannel models. First, we consider a multichannel system with independent (autonomous) channels. The system is described by m operator equations.

[M2m] 2.1 Linear input-output models

each of which characterizes the behavior of one of its channels.

Let us consider the vectors of output variables y and control u :

2.1 Linear input-output models , ,

accordingly, and we write the system of equations in vector-matrix form:

2.1 Linear input-output models

or,

[M2m] A ( p ) y = B ( p ) u

If matrix A ( p ) - is reversible, i.e. there is an inverse matrix

2.1 Linear input-output models ,

then from the equation [М2m] we find

[M3m] y = W ( p ) u,

where W ( p ) = { W _ij } is the transfer matrix of the system (matrix integro-differential operator), calculated as

W ( p ) = A ^-1 ( p ) B ( p ) = 2.1 Linear input-output models .

It is easy to see that in the considered case the transfer matrix is diagonal, i.e.

W ( p) = diag { W _ii (p) } = { b _i (p) / a _i (p) }.

Now consider a multiply connected system , i.e. multichannel system with connected channels, described by a system of operator equations

a ₁₁ (p) y ₁ + a ₁₂ (p) y ₂ + ... + a _1m (p) y _m = b ₁₁ (p) u ₁ + b ₁₂ (p) u ₂ + ... + b _1m (p) u _m

a ₂₁ (p) y ₁ + a ₂₂ (p) y ₂ + ... + a _2m (p) y _m = b ₂₁ (p) u ₁ + b ₂₂ (p) u ₂ + ... + b _2m (p) u _m

[M2m] . . .

a _m1 (p) y ₁ + a _m2 (p) y ₂ + ... + a _mm (p) y _m = b _m1 (p) u ₁ + b _m2 (p) u ₂ + ... + b _mm (p) u _m

The system is reduced to the vector-matrix form [M2m], where

2.1 Linear input-output models ;

and the form [M3m], where the transfer matrix W ( p ) is determined by the expression

W ( p ) = A ^-1 ( p ) B ( p ) = 2.1 Linear input-output models

Model [M3m] can also be written in scalar form:

y ₁ = W ₁₁ (p) u ₁ + W ₁₂ (p) u ₂ + ... + W _1m (p) u _m

y ₂ = W ₂₁ (p) u ₁ + W ₂₂ (p) u ₂ + ... + W _2m (p) u _m

. . .

y _m = W _m1 (p) u ₁ + W _m2 (p) u ₂ + ... + W _mm (p) u _m

Note that the diagonal operators W _ii ( p ) belong to the main channels , and the remaining transfer functions W _ij ( p ) characterize the cross-links of the multichannel system.

For a two-channel multiply-connected system ( m = 2), we obtain:

y ₁ = W ₁₁ (p) u ₁ + W ₁₂ (p) u ₂ ,

y ₂ = W ₂₁ (p) u ₁ + W ₂₂ (p) u ₂ ,

where W ₁₁ ( p ) , W ₂₂ ( p ) are the transfer functions of the main channels of the system, and W ₁₂ ( p ) , W ₂₁ ( p ) - transfer functions of cross-links.

2.1.4. Perturbed model of the system. The disturbing effect f ( t ), which characterizes the effect on the control object of the external environment (see § 1.2), is considered as an additional input signal. 2.1 Linear input-output models Then the linear model of the single-channel dynamic system takes the form

[M1f] 2.1 Linear input-output models

where d _{i are the} coefficients that determine the influence on the processes in the perturbation system f ( t ) and its derivatives f ^{( i )} ( t ) , d ₀ 2.1 Linear input-output models 0 , 0 <n . After substituting the differentiation operators p ⁱ and the corresponding transformations, we obtain the operator form of the model [M1f]: