system prototypes

7/27/2019 System Prototypes

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SIMPLE SYSTEM PROTOTYPES


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In order to give an idea about the various types of behav-

ior of dynamical systems we will next discuss a numberof simple system prototypes.

Recall that linear dynamical systems can be described by

its transfer function G(p),

y(t) = G(p)u(t), G(p) =

B(p)

A(p)

The system properties are uniquely determined by its

transfer function:

Poles of G(p)The zeros pi of the denominator polynomial A(p),

i.e., the solutions to

A(pi) = 0

determine how the autonomous behavior of a system

(i.e., when u(t) = 0). The poles pi thus determine

system stability and whether the output oscillates or

not.

Remark. pi are real or (often) complex-valued num-bers. The term pole is used in the theory of analyt-

ical functions to denote a point at which a function

is infinite; in this case G(p) is infinite at p = pi.

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Zeros of G(p)

The zeros zi of the numerator polynomial B(p), i.e.,the solutions to

B(zi) = 0

determine the response to the input u(t), such as

whether the system has an inverse response or not.

Just as the poles, the zeros of a system may be real

or complex.

The system behavior will be demonstrated by considering

the step response, i.e., the systems output when the input

is a step,

u(t) =

0 , for t < 0

usteg , for t

0

and y(t) = 0, t < 0.

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The systems we will consider are:

First-order systems.Cf. above.

Systems with two time constants.These are second-order systems consisting of two first-

order systems in series.

Underdamped system.Systems with an oscillating response. Second-order

systems with complex-conjugate poles can be under-

damped.

Systems with time delay.

Systems with integration.

Systems with inverse response.The response starts in the wrong direction before

changing direction. These systems can be described

(and are often caused) by two subsystems operating

in parallel which act in opposite directions.

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First-order systems

A first-order system is described a first-order differential

equationdy(t)

dt+ ay(t) = bu(t)

or

Tdy(t)

dt+ y(t) = Ku(t)

where T = 1/a and K = b/a.Transfer function:

G(p) =b

p + a=

K

T p + 1

Step response:

ystep

(t) = K(1

et/T)ustep

, t

0

Static gain = K: y(t) Kustep as t .Time constant = T: y(T) = K

1 e1ustep = 0.632 y()

Stability:

|y(t)| as t if (and only if) T 0 (a 0) the system is unstable if a 0Or:System is unstable if (and only if) the pole p1 = a ofG(p) is positive (or zero)

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Step response of first-order system with

a = 1, b = 1(T = 1, K = 1)

0

1

u(t)

1 0 1 2 3 4 5

0

1

y(t)

time

Note:

Output derivative reacts instantaneously to a discontin-

uous change in the input.

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System with two time constants

Two first-order systems in series:

G(p) =b2

p + a2 b1p + a1

=b1b2

p2 + (a1 + a2)p + a1a2

This is a second-order system with the real poles

p1 = a1 and p2 = a2.Differential equation:

d2y(t)

dt2+ (a1 + a2)

dy(t)

dt+ a1a2 y(t) = b1b2 u(t)

Response:

IfT1 = T2 the system can be represented in terms of twofirst-order systems in parallel:

Define the time constants T1

= 1/a1, T

2= 1/a

2and

static gains K1 = b1/a1, K1 = b2/a2. Then

G(p) =K

(T2p + 1)(T1p + 1), K = K1K2

Partial fraction expansion:

G(p) =KT1/(T1 T2)

T1p + 1

+KT2/(T2 T1)

T2p + 1ifT1 = T2.

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Then

y(t) = G(p)u(t) = y1(t) + y2(t)

where

y1(t) =KT1/(T1 + T2)

T1p + 1u(t), y2(t) =

KT2/(T1 + T2)

T2p + 1u(t)

Step response:

ystep(t) = y1,step(t) + y2,step(t)

=KT1

T1 T2

1 et/T1

ustep +KT2

T2 T1

1 et/T2

ustep

= K

1 T1

T1 T2et/T1 T2

T2 T1et/T2

ustep

The case T1

= T2: taking the limit as T

1 T2

0 gives

ystep(t) = K

1 (1 + t

T)et/T

ustep

Stability: |y(t)| if (and only if) a1 = 1/T1 0and/or a2 = 1/T2 0= system is stable if and only if the poles ofG(p) are

all negative; p1 = a1 < 0 and p2 = a2 < 0.

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Step response of system with two time constants

0.2

0

0.2

0.4

0.6

0.8

1

1.2

u

2 0 2 4 6 8 10 12 14 16

0

0.2

0.4

0.6

0.8

1

1.2

y

tid

Note:

Response smoother than for first-order system,dy(t)dt is not affected immediately by discontinuous change

in input.

Instead second derivatived2y(t)dt2 reacts instantaneously to

the discontinuous change in the input.

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Underdamped systems

General second-order system:

d2y(t)

dt2+ a1

dy(t)

dt+ a2y(t) = b1

du(t)

dt+ b2u(t)

has transfer function:

G(p) =b1p + b2

p2

+ a1p + a2In the general case, G(p) may have complex conjugate

poles.

IfG(p) has poles p1 and p2, we have

p2 + a1p + a2 = (p p1) (p p2)= p2 (p1 + p2)p + p1p2

= a1 = (p1 + p2) , a2 = p1p2System properties can be conveniently characterized in

terms of the parameters n, (zeta) and K defined by

2n = a2 = p1p2.n is called the natural frequency of the system

2n = a1 =

(p1 + p2).

is called the relative damping of the system.

K2n = b2K is the static gain of the system.

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Then (assuming for simplicity b1 = 0):

d2y(t)

dt2+ 2n

dy(t)

dt+ 2ny(t) = K

2nu(t)

and

G(p) =K2n

p2 + 2np + 2nThe system poles are given by

p2 + a1p + a2 = p2 + 2np +

2n = 0

=p1,2 = n

2 1 n

We see that the determines the nature of the poles:

assuming that > 0,

If 1, the system has real poles If = 1, the system has a double pole at

p1 = p2 = n If < 1, the system has complex conjugate poles

p1,2 = j.In this case, n =

|p1,2

|=

2 + 2

and = /|p1,2|.

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Step response:

The case 1 corresponds to a system with two timeconstants, which we have already considered.

For < 1, the step response is given by

ystep(t)=K

1 1

ent [sin(nt) + cos(nt)]

ustep

=K

1

1

ent sin(nt + )

ustep

where

=

1 2, = arccos()We observe:

The response oscillates with frequency n.For = 0, the oscillation frequency equals the natu-

ral frequency n. The amplitude of the oscillations decreases exponen-

tially according to ent. Hence, the larger (smaller)the relative damping is, the faster (slower) the ex-

ponential decay of the oscillation.

The relative damping reduces the oscillation frequen-cy n: from n = n for = 0 to n = 0 for = 1.

Stability: If n < 0, the output diverges as t in-creases, and the system is unstable.

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As n =

1

2

(p1 + p2) (cf. above), we have that the

system is unstable if it has a pole with a positive real

component.

The relative damping determines the amount of oscil-

lation of the system. We have the following classification: > 1: the system is overdamped.

It has two real negative poles, corresponding to sys-

tem with two time constants.

= 1: the system is critically dampled.It has a real negative double pole.

< 1: the system is underdamped.It has two complex conjugate poles with negative real

components.

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0 1 2 3 4 5 6 7 8 9 100.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

y

tid

Step responses of second-order systems with relative damp-

ings = 2, 1, 0.5 and 0.1 (from below), natural frequency

n = 1 and static gain K = 1.

Maximum overshoot M (in % of static value):

M = exp

100%

Time tM at which the maximum is obtained:

tM =

n

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Example

Mechanical system composed of spring and damper.

u(t)

(t)

m

bk

y(t): deviation from equilibrium

u(t): external force acting on mass

Newtons law:

md2y(t)dt2

= u(t) + Ff(t) + Fd(t)

where:

- Ff(t): spring force, proportional and directed in the

opposite direction to deviation y(t),

Ff(t) = ky(t)- Fd(t): damping force, proportional and directed in theopposite direction to rate of change dy(t)/dt,

Fd(t) = b dy(t)dt

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=

m

d2y(t)

dt2= u(t) ky(t) b dy(t)

dt

ord2y(t)

dt2+

b

m

dy(t)

dt+

k

my(t) =

1

mu(t)

This is a second-order system with transfer function

y(t) = 1/mp2 + bmp +

km

u(t)

We see that the system has:

- natural frequency n =

km

- static gain K = 1/m2n

= 1k

- relative damping = b/m2n =12

bmk

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Example - Harmonic oscillator.

Case with zero relative damping; = 0:

d2y(t)

dt2+ 2ny(t) = K

2nu(t)

When left to itself (u(t) = 0), the output is sinusoidally

oscillating with the natural frequency frequency n.

Indeed, the signal

y(t) = A sin(nt)

has second derivative

d2y(t)

dt2= A2n sin(nt) = 2ny(t)

so that

d2y(t)

dt2+ 2ny(t) = 0

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Remark

Response of system with b1 = 0,

G(p) =b1p + b2

p2 + a1p + a2

We can write G(p) as a parallel coupling of two second-

order systems:

G(p) = b1pp2 + a1p + a2

+ b2p2 + a1p + a2

Hence

y(t) = G(p)u(t) = y1(t) + y2(t)

where

y1(t) = G1(p)u(t), G1(p) =b1p

p2

+ a1p + a2u(t)

and

y2(t) = G2(p)u(t), G2(p) =b2

p2 + a1p + a2u(t)

We already know how to compute step response y2(t) of

G2(p).

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The first system G1(p) can be written as a series connec-

tion

G1(p) = pb1

p2 + a1p + a2= p G1(p)

where

G1(p) =b1

p2 + a1p + a2

Hence we obtain the output by first computing the output

ofG1(p),y1(t) = G1(p)u(t)

followed by the differential operator p,

y1(t) = p y1(t) =dy1(t)

dt

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Systems with inverse response

Inverse response systems typically arise when two dynam-

ical processes act in parallel, but in opposite directions

and with different time constants. Examples of inverse

0.2

0

0.2

0.4

0.6

0.8

1

1.2

u

2 1 0 1 2 3 4 5 6 7 80.5

0

0.5

1

1.5

y

tid

response systems:

Response of water level in boiler to changes in waterflow to boiler.

Response of temperature to changes in fuel input ratein combustion of solid fuels.

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Altitude control of airplanes.

Economic systems: response of tax revenues to changesin tax level(?).

To see how inverse response behavior may arise, consider

two first-order system in parallel, so that

y(t) = y1(t) + y2(t)

where

y1(t) =b1

p + a1u(t)

and

y2(t) =b2

p + a2u(t)

Inverse response as shown in the figure is obtained if andonly if the following conditions hold:

Final condition:y(t) = y1(t) + y2(t) > 0, as t

Initial condition:dy(t)

dt

t=0

= dy1(t)dt

t=0

+ dy2(t)dt

t=0

< 0

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But

y1(t) b1a1

ustep as t and

y2(t) b2a2

ustep as t Final condition =

b1

a1+

b2

a2> 0

which implies (as a1, a2 > 0; stable systems assumed)

a1b2 + a2b1 > 0

Moreover,dy1(t)

dt

t=0

= b1 ustep

anddy2(t)

dt

t=0

= b2 ustep

Initial condition =b1 + b2 < 0

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Relation to transfer function of the parallel connected

system:

Gpar(p) =b1

p + a1+

b2p + a2

=(b1 + b2)p + a1b2 + a2b1

(p + a1)(p + a2)

Zero:

(b1 + b2)p + a1b2 + a2b1 = 0

= pzero = a1b2 + a2b1

b1 + b2Inverse response conditions

a1b2 + a2b1 > 0

and

b1 + b2 < 0

imply that pzero > 0.

General property of inverse response systems:

Transfer function of an inverse response system has at

least one zero which is positive (if real) or has a positive

real component (if complex valued)

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Systems with integration

These are systems consisting of pure integration:

y(t) = kt0

[u() u0] dHere the output keeps changing as long as u(t) u0 = 0,cf. figure.

0.2

0

0.2

0.4

0.6

0.8

1

1.2

u

2 0 2 4 6 8 100

1

2

3

4

5

6

y

tid

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Examples of integrators:

Inventory models, where y(t) is the amount of ma-terial in stock, and u(t) is inflow and u0 the outflow

of material. For example, y(t) could be the liquid

volume of a container and u(t) and u0 the in- and

outflows, respectively.

Servo motors used for position control. The posi-

tion is controlled by a voltage u(t), where the rate

of change of position is proportional to the voltage,

dy(t)/dt = ku(t).

Differential equation and transfer function

Differentiation gives

dy(t)dt

= k [u(t) u0]=

y(t) =k

p[u(t) u0]

Transfer function:

G(p) =k

pThe system has a pole atp = 0. Hence it is on the stability

boundary.

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Systems with time delay

A time delay means that it takes a time L before the

input affects the system. The figure shows the response

of a first-order system with time delay L = 1:

0.2

0

0.2

0.4

0.6

0.8

1

1.2

u

2 0 2 4 6 8 100.2

0

0.2

0.4

0.6

0.8

1

1.2

y

tid

The control of time delay systems is difficult as the effect

of a control action is seen only after a certain time. Forgood control, one should predict the future system output

y(t + L), which depends on the past control actions

u(s), s [t L, t].

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Examples of time-delay systems:

In the process industry time delays arise typicallydue to transport delays (for example, when material

is transported in pipes etc.).

In a paper machine it is the time it takes for the webto travel from the point where control actions are

taken to the point where measurements are made.

In web services, time delays may arise due to the timeit takes to start a new server.

Transfer function:

A pure time-delay system is described by

y(t) = u(t

L)

We can express this relation with the differential operator

p = ddt by introducing the Taylor-series expansion

u(t + h) = u(t) + hdu(t)

dt+

h2

2!

d2u(t)

dt2+ + h

k

k!

dku(t)

dtk+

=

1 + hp +

(hp)2

2! + +(hp)k

k! +

u(t)

= ehpu(t)

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With p =

L we have

y(t) = u(t L) = eLpu(t)or

y(t) = GL(p)u(t)

where the transfer function of the time delay is

GL(p) = eLp

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Higher-order systems

In general, dynamical systems are not restricted to first-

order or second-order systems.

However, by partial fraction expansion a transfer func-

tion G(p) with simple poles can be written as a parallel

connection of first-order and second-order systems:

G(p) =B(p)

A(p)

=k

KkTkp + 1

+

l

bl1p + bl2p2 + al1p + al2

When G(p) has multiple poles, there are also terms con-

sisting of series connections of first-order or second-order

systems.

The decomposition above implies:

The output of a higher-order system can be comput-ed as a sum of the outputs of first-order and second-

order systems.

The system G(p) is stable if and only if all the first-order and second-order systems in the parallel con-

nection are stable. Equivalently, the system is stableif and only if all the poles of G(p) (i.e., the zeros of

A(p)) have negative real parts.

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