Robotics process control book Part 9 doc

Input variables may take different random values through time.. The variables that at any time are assigned to a real number by some statement from a space of possible values are called

The graph of G(jω)

G(jω) = |G(jω)|ej arg G(jω)= <[G(jω)] + j=[G(jω)] (4.3.36)

in the complex plane is called the Nyquist diagram The magnitude and phase angle can be expressed as follows:

tan ϕ = =[G(jω)]

ϕ = arctan=[G(jω)]

Essentially, the Nyquist diagram is a polar plot of G(jω) in which frequency ω appears as an implicit parameter

The function A = A(ω) is called magnitude frequency response and the function ϕ = ϕ(ω) phase angle frequency response Their plots are usually given with logarithmic axes for frequency and magnitude and are referred to as Bode plots

Let us investigate the logarithm of A(ω) exp[jϕ(ω)]

The function

defines the magnitude logarithmic amplitude frequency response and is shown in the graphs as

L is given in decibels (dB) which is the unit that comes from the acoustic theory and merely rescales the amplitude ratio portion of a Bode diagram

Logarithmic phase angle frequency response is defined as

Example 4.3.1: Nyquist and Bode diagrams for the heat exchanger as the first order system The process transfer function of the heat exchanger was given in (4.3.2) G(jω) is given as

T1jω + 1=

Z1(T1jω − 1) (T1jω)2+ 1

(T1ω)2+ 1 − j Z1T1ω

(T1ω)2+ 1

p(T1ω)2+ 1e

−j arctan T 1 ω

The magnitude and phase angle are of the form

A(ω) = Z1

p(T1ω)2+ 1 ϕ(ω) = − arctan T1ω

Nyquist and Bode diagrams of the heat exchanger for Z1= 0.4, T1= 5.2 min are shown in Figs 4.3.2,4.3.3, respectively

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

−0.2

−0.18

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

Re

Figure 4.3.2: The Nyquist diagram for the heat exchanger

−25

−20

−15

−10

−5

Frequency [rad/min]

−80

−60

−40

−20

0

Frequency [rad/min]

Figure 4.3.3: The Bode diagram for the heat exchanger

-20

L [dB]

ω[rad/min]

ω1

Figure 4.3.4: Asymptotes of the magnitude plot for a first order system

In general, the dependency =[G(jω)] on <[G(jω)] for a first order system described by (4.3.2) can easily be found from the equations

u = <[G(jω)] = Z1

v = =[G(jω)] = −(TZ1T1ω

Equating the terms T1ω in both equations yields

(v − 0)2−



u −Z21



= Z1 2

2

(4.3.47)

which is the equation of a circle

Let us denote ω1= 1/T1 The transfer function (4.3.2) can be written as

G(s) = ω1Z1

s + ω1

The magnitude is given as

A(ω) = Z1ω1

and its logarithm as

L = 20 log Z1+ 20 log ω1− 20 log

q

This curve can easily be sketched by finding its asymptotes If ω approaches zero, then

and if it approaches infinity, then

q

This is the equation of an asymptote that for ω = ω1 is equal to 20 log Z1 The slope of this asymptote is -20 dB/decade (Fig4.3.4)

Table 4.3.1: The errors of the magnitude plot resulting from the use of asymptotes.

ω 15ω1 14ω1 12ω1 ω1 2ω1 4ω1 5ω1

δ(dB) 0.17 -0.3 -1 -3 -1 -0.3 -0.17

0

-π/2

ϕ

ω 1

ω

-π/4

[rad/min]

Figure 4.3.5: Asymptotes of phase angle plot for a first order system

The asymptotes (4.3.51) and (4.3.53) introduce an error δ for ω < ω1:

δ = 20 log ω1− 20 log

q

and for ω > ω1:

δ = 20 log ω1− 20 logqω2

1+ ω2− [20 log ω1− 20 log ω] (4.3.55) The tabulation of errors for various ω is given in Table4.3.1

A phase angle plot can also be sketched using asymptotes and tangent in its inflex point (Fig4.3.5)

We can easily verify the following characteristics of the phase angle plot:

If ω = 0, then ϕ = 0,

If ω = ∞, then ϕ = −π/2,

If ω = 1/T1, then ϕ = −π/4,

and it can be shown that the curve has an inflex point at ω = ω1= 1/T1 This frequency is called the corner frequency The slope of the tangent can be calculated if we substitute for ω = 10z

(log ω = z) into ϕ = arctan(−T1ω):

˙

1 + x2, x = −T1ω

dϕ

dz =

−2.3

1 + (−T110z)2T110z dϕ

d log ω =

−2.3

1 + (−T1ω)2T1ω for ω = 1/T1

dϕ

d log ω = −1.15 rad/decade

-1.15 rad corresponds to −66o The tangent crosses the asymptotes ϕ = 0 and ϕ = −π/2 with error of 11o400

4.3.4 Frequency Characteristics of a Second Order System

Consider an underdamped system of the second order with the transfer function

Its frequency transfer function is given as

G(jω) =

1

T 2 k

r



1

T 2

k − ω22+T2ζ

k

2

ω2

exp j arctan −2ζT kω

1

T 2

k − ω2

!

(4.3.57)

A(ω) =

1

T 2 k

r



1

T 2

k − ω22+T2ζ

k

2

ω2

(4.3.58)

ϕ(ω) = arctan −T2ζkω

1

T 2

The magnitude plot has a maximum for ω = ωk where Tk = 1/ωk (resonant frequency) If

ω = ∞, A = 0 The expression

M = A(ωk)

A(0) =

Amax

is called the resonance coefficient

If the system gain is Z1, then

L(ω) = 20 log |G(jω)| = 20 log

Z1 (jω) 2

ω 2 k

+ 2ωζ

kjω + 1

(4.3.61)

L(ω) = 20 log

Z1

T2

k(jω)2+ 2ζTkjω + 1

L(ω) = 20 log Z1− 20 log

q (1 − T2ω2)2+ (2ζTkω)2 (4.3.63)

It follows from (4.3.63) that the curve L(ω) for Z1 6= 1 is given by summation of 20 log Z1 to normalised L for Z1= 1 Let us therefore investigate only the case Z1= 1 From (4.3.63) follows L(ω) = −20 log

q (1 − T2

In the range of low frequencies (ω  1/Tk) holds approximately

For high frequencies (ω  1/Tk) and T2

kω2 1 and (2ζTkω)2 (T2

kω2)2 holds L(ω) ≈ −20 log(Tkω)2= −2 20 log Tkω = −40 log Tkω (4.3.66) Thus, the magnitude frequency response can be approximated by the curve shown in Fig 4.3.6 Exact curves deviate with an error δ from this approximation For 0.38 ≤ ζ ≤ 0.7 the values of δ are maximally ±3dB

The phase angle plot is described by the equation

ϕ(ω) = − arctan 2ζTk

1 − T2

At the corner frequency ωk = 1/Tk this gives ϕ(ω) = −90o

Bode diagrams of the second order systems with Z = 1 and T = 1 min are shown in Fig.4.3.7

-40

L [dB]

ω

ω = 1/Τk k

[rad/min]

Figure 4.3.6: Asymptotes of magnitude plot for a second order system

−60

−40

−20

0

20

40

Frequency (rad/s)

−200

−150

−100

−50

0

Frequency (rad/s)

ζ =0.05

ζ =0.2

ζ =0.5

ζ =1.0

Figure 4.3.7: Bode diagrams of an underdamped second order system (Z1= 1, Tk= 1)

Im

- π /2

ω =

ω→ 0 0

Figure 4.3.8: The Nyquist diagram of an integrator

The transfer function of an integrator is

G(s) = Z1

where Z1= 1/TI and TI is the time constant of the integrator

We note, that integrator is an astatic system Substitution for s = jω yields

G(jω) = Z1

jω = −jZω1 = Z1

ω e

−j π

From this follows

A(ω) = Z1

The corresponding Nyquist diagram is shown in Fig 4.3.8 The curve coincides with the negative imaginary axis The magnitude is for increasing ω decreasing The phase angle is independent of frequency Thus, output variable is always delayed to input variable for 90o Magnitude curve is given by the expression

L(ω) = 20 log A(ω) = 20 logZ1

The phase angle is constant and given by (4.3.71)

If ω2= 10ω1, then

thus the slope of magnitude plot is -20dB/decade

Fig.4.3.9shows Bode diagram of the integrator with TI= 5 min The values of L(ω) are given

by the summation of two terms as given by (4.3.73)

Consider a system with the transfer function

Its frequency response is given as

G(jω) =

n

Y

i=1

G(jω) = exp j

n

X

ϕi(ω)

! n

Y

10−1 100 101 102

−60

−50

−40

−30

−20

−10

0

10

Frequency [rad/min]

−91

−90.5

−90

−89.5

−89

Frequency [rad/min]

Figure 4.3.9: Bode diagram of an integrator

and

20 log A(ω) =

n

X

i=1

ϕ(ω) =

n

X

i=1

It is clear from the last equations that magnitude and phase angle plots are obtained as the sum

of individual functions of systems in series

Example 4.3.2: Nyquist and Bode diagrams for a third order system

Consider a system with the transfer function

s(T1s + 1)(T2s + 1). The function G(jω) is then given as

jω(T1jω + 1)(T2jω + 1). Consequently, for magnitude follows

ωp(T1ω)2+ 1p(T2ω)2+ 1 L(ω) = 20 log Z3− 20 log ω − 20 logp(T1ω)2+ 1 − 20 logp(T2ω)2+ 1

and for phase angle:

ϕ(ω) = −π2 − arctan(T1ω) − arctan(T2ω)

Nyquist and Bode diagrams for the system with Z3= 0.5, T1= 2 min, and T2= 3 min are given in Figs.4.3.10and 4.3.11

−2.5 −2 −1.5 −1 −0.5 0

−50

−40

−30

−20

−10

0

10

Re

Figure 4.3.10: The Nyquist diagram for the third order system

−100

−50

0

50

100

Frequency [rad/min]

−300

−250

−200

−150

−100

−50

Frequency [rad/min]

Figure 4.3.11: Bode diagram for the third order system

4.4 Statistical Characteristics of Dynamic Systems

Dynamic systems are quite often subject to input variables that are not functions exactly spec-ified by time as opposed to step, impulse, harmonic or other standard functions A concrete (deterministic) time function has at any time a completely determined value

Input variables may take different random values through time In these cases, the only characteristics that can be determined is probability of its influence at certain time This does not imply from the fact that the input influence cannot be foreseen, but from the fact that a large number of variables and their changes influence the process simultaneously

The variables that at any time are assigned to a real number by some statement from a space

of possible values are called random

Before investigating the behaviour of dynamic systems with random inputs let us now recall some facts about random variables, stochastic processes, and their probability characteristics

Let us investigate an event that is characterised by some conditions of existence and it is known that this event may or may not be realised within these conditions This random event is char-acterised by its probability Let us assume that we make N experiments and that in m cases, the event A has been realised The fraction m/N is called the relative occurrence It is the experimen-tal characteristics of the event Performing different number of experiments, it may be observed, that different values are obtained However, with the increasing number of experiments, the ratio approaches some constant value This value is called probability of the random event A and is denoted by P (A)

There may exist events with probability equal to one (sure events) and to zero (impossible events) For all other, the following inequality holds

Events A and B are called disjoint if they are mutually exclusive within the same conditions Their probability is given as

An event A is independent from an event B if P (A) is not influenced when B has or has not occurred When this does not hold and A is dependent on B then P (A) changes if B occurred or not Such a probability is called conditional probability and is denoted by P (A|B)

When two events A, B are independent, then for the probability holds

Let us consider two events A and B where P (B) > 0 Then for the conditional probability

P (A|B) of the event A when B has occurred holds

For independent events we may also write

Let us consider discrete random variables Any random variable can be assigned to any real value from a given set of possible outcomes A discrete random variable ξ is assigned a real value from

0 1 2 3 4 5 6

b)

a)

F (x) P

0, 2

0, 4

0, 6

0, 8 1

0, 1

0, 2

0, 3

t t t

x

x

Figure 4.4.1: Graphical representation of the law of distribution of a random variable and of the

associated distribution function

a finite set of values x1, x2, , xn A discrete random variable is determined by the set of finite values and their corresponding probabilities Pi (i = 1, 2, , n) of their occurrences The table

ξ =



x1, x2, , xn

represents the law of distribution of a random variable An example of the graphical representation for some random variable is shown in Fig.4.4.1a Here, the values of probabilities are assigned to outcomes of some random variable with values xi The random variable can attain any value of

xi (i = 1, 2, , n) It is easily confirmed that

n

X

i=1

Aside from the law of distribution, we may use another variable that characterises the proba-bility of a random variable It is denoted by F (x) and defined as

F (x) = X

x i ≤x

and called cumulative distribution function, or simply distribution function of a variable ξ This function completely determines the distribution of all real values of x The symbol xi ≤ x takes into account all values of xi less or equal to x F (x) is a probability of event ξ ≤ x written as

Further, F (x) satisfies the inequality

When the set of all possible outcomes of a random variable ξ is reordered such that x1 ≤ x2 ≤

· · · ≤ xn, the from the probability definition follows that F (x) = 0 for any x < x1 Similarly,

F (x) = 1 for any x > xn Graphical representation of the distribution function for a random variable in Fig.4.4.1a is shown in Fig.4.4.1b

Although the distribution function characterises a completely random variable, for practical reasons there are also defined other characteristics given by some non-random values Among the possible, its expected value, variance, and standard deviation play an important role

The expected value of a discrete random variable is given as

µ = E[ξ] =

n

X

i=1

In the case of uniform distribution law the expected value (4.4.11) can be written as

µ = 1

n

n

X

i=1

For a play-cube tossing yields

µ =

6

X

i=1

xiPi = (11

6+ 2

1

6+ 3

1

6+ 4

1

6+ 5

1

6+ 6

1

6) = 3.5 The properties of the expected value are the following:

Constant Z

Multiplication by a constant Z

Summation

Multiplication of independent random variables

If ξ is a random variable and µ is its expected value then the variable (ξ − µ) that denotes the deviation of a random variable from its expected value is also a random variable

Variance of a random variable ξ is the expected value of the squared deviation (ξ − µ)

σ2= D[ξ] = E(ξ − µ)2 =

n

X

i=1

Whereas the expected value is a parameter in the neighbourhood of which all values of a random variable are located, variance characterises the distance of the values from µ If the variance is small, then the values far from the expected value are less probable

Variance can easily be computed from the properties of expected value:

σ2= Eξ2

− 2ξE[ξ] + (E[ξ])2 = E[ξ2

i.e variance is given as the difference between the expected value of the squared random variable and squared expected value of random variable Because the following holds always

a b

x f(x)

F(a) F(b) F(x)

a)

b) 0

x

a b

1

Figure 4.4.2: Distribution function and corresponding probability density function of a continuous

random variable

variance is always positive, i.e

The square root of the variance is called standard deviation of a random variable

A continuous random variable can be assigned to any real value within some interval if its distribution function F (x) is continuous on this interval The distribution function of a continuous random variable ξ

is the probability the random variable is less than x A typical plot of such a distribution function

is shown in Fig.4.4.2a The following hold for F (x):

lim

lim

The probability that a random variable attains some specified value is infinitesimally small On the contrary, the probability of random variable lying in a some interval (a, b) is finite and can be calculated as

The probability that a continuous random variable is between x and x + dx is given as

P (x ≤ ξ < x + dx) = dF (x)

where the variable

f (x) = dF (x)

is called probability density Figure 4.4.2b shows an example of f (x) Thus, the distribution function F (x) may be written as

F (x) =

Z x

−∞

Because F (x) is non-decreasing, the probability density function must be positive

The probability that a random variable is within an interval (a, b) calculated from its density function is given as the surface under curve f (x) within the given interval Thus, we can write

P (a ≤ ξ < b) =

Z b a

Correspondingly, when the interval comprises of all real values, yields

Z ∞

−∞

Expected value of a continuous random variable is determined as

µ = E[ξ] =

Z ∞

−∞

A random variable can be characterised by the equation

E[ξm] =

Z ∞

−∞

which determines m-th moment of a random variable ξ The first moment is the expected value The second moment is given as

E[ξ2] =

Z ∞

−∞

Central m-th moment is of the form

E[(ξ − µ)m] =

Z ∞

Variance of a continuous random variable ξ can be expressed as follows

σ2 = E[(ξ − µ)2] =

Z ∞

The standard deviation is its square root

Normal distribution for a continuous random variable is given by the following density function

f (x) = 1

σ√

2πe

−(x − µ)

2

Let us now consider two independent continuous random variables ξ1, ξ2 defined in the same probability space Their joint density function is given by the product

where f (x ), f (x ) are density functions of the variables ξ , ξ

? ?91

? ?90 .5

? ?90

− 89. 5

− 89< /small>

Frequency [rad/min]

Figure 4.3 .9: Bode... in Figs.4.3.10and 4.3.11