Modern Control Engineering

32 Chapter 2 / Mathematical Modeling of Control Systems

m

k

b

u(t)

y(t)

Figure 2–15

Mechanical system.

u(t)

D(t)

B(t)

A(t)

C(t)

x•(t) y(t)

Údt

x(t)

++ ++

Figure 2–14

Block diagram of the

linear, continuous-

time control system

represented in state

space.

the operating state are discussed in Section 2–7.) A block diagram representation of

Equations (2–12) and (2–13) is shown in Figure 2–14.

If vector functions fandgdo not involve time texplicitly then the system is called a

time-invariant system. In this case, Equations (2–12) and (2–13) can be simplified to

(2–14)

(2–15)

Equation (2–14) is the state equation of the linear, time-invariant system and Equation

(2–15) is the output equation for the same system. In this book we shall be concerned

mostly with systems described by Equations (2–14) and (2–15).

In what follows we shall present an example for deriving a state equation and output

equation.

EXAMPLE 2–2 Consider the mechanical system shown in Figure 2–15. We assume that the system is linear. The

external force u(t)is the input to the system, and the displacement y(t)of the mass is the output.

The displacement y(t)is measured from the equilibrium position in the absence of the external

force. This system is a single-input, single-output system.

From the diagram, the system equation is

(2–16)

This system is of second order. This means that the system involves two integrators. Let us define

state variables x 1 (t)andx 2 (t)as

Then we obtain

or

(2–17)

(2–18)

The output equation is

y=x 1 (2–19)

x# 2 =-

k

m

x 1 -

b

m

x 2 +

1

m

u

x

1 =x 2

x

2 =

1

m

A-ky-by

B+

1

m

u

x# 1 =x 2

x 2 (t)=y#(t)

x 1 (t)=y(t)

my$+by# +ky=u

y

(t)=Cx(t)+Du(t)

x

(t)=Ax(t)+Bu(t)

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