**Chapter 22.1 Introduction
**You know the stable and instables systems now. Additionally-The lack of the motion doesn’t mean that the system is stable or instable. An example–>

**Fig. 20-4**

**Chapter 22**. The system is self-evidently instable but there is no motion up to

**5 sec**! The small pulse

**x(t)**unbalanced the system! Automatics should predict the closed looop system stability/instability when he knows open looop system parameters. The opened loop system is easier than closed and it’s

**99,999…%**stable. I intend the real opened physical object, not mathematically transformed as “Open System” under.

**Fig. 22-1**

The

**Nyquist**

**stability benchmark**is frequency type in contrast to

**Hurwitz benchmark**which is algebraic type.

The input

**x(t)**of the open loop

**G(s)**is a sinusoid x

**(t)=1*sin(ωt)**. You test response for all frequencies in the range

**0…infinity**. Theoretically of course. Practically there are some (

**30**for eample) frequnencies in the range

**low…high**. We measure the

**amplitude**of the output sinusoid

**y(t)-**mostly lower than input amplitude, and its

**phase**-mostly delayed. The output sinusoid

**y(t)=Ym*sin(ωt-φ)**parameters–>amplitude

**and phase**

**Ym****of the open loop**

**φ****G(s)**has easy to read information about closed loop

**G(s)**stability.

**Chapter 22.2 Vector as a comfortable presentation of the harmonic motion
**

**Fig. 22-2
**Motion as a formula

**y(t)=Ym*sin(ωt-φ)**is easier to analyze as o rotating point.

**The vector**

**A**is anticlockwise rotating. The projection of a vector

**A**for

**y**axis is changed on formula

**y(t)=Ym*sin(ωt-φ**

**)**where

**Ym**is a amplitude and

**φ**

**=0**is a phase.

Analogously:

Rotating vector

**B**formula

**y(t)=Ym*sin(ωt-φ**

**)**where

**Ym**is a amplitude and

**φ**

**=**is a phase.

**36°**Rotating vector

**C**formula

**y(t)=Ym*sin(ωt-φ**

**)**where

**Ym**is a amplitude and

**φ**

**=**is a phase. Other words

**180°****A**and

**C**are in antiphase.

Click

**harmonic motion**and observe the animation. Balls

**A,B**and

**C**

**are in motion according to their formulas.**

The author of this animation is profesor Miyazaki – Japanese Disney. By the way. It was wave animation and the wave moved right horizontally but all ball moves are vertical!

**Chapter 22.2.2 Sinusoids with the same amplitudes as vectors **

Call Desktor/PID/08_kryterium Nyquista/01_3_sinusoidy.zcos

**Fig. 22-3
**Generator

**1**represents ball

**A**–>

**T=2.2**–>angular velocity

**ω=2.86 1/sec ,**amplitude

**Ym=1,**phase

**φ=0°**

Generator

**2**represents ball

**B**–>

**T=2.2**–>angular velocity

**ω=2.86 1/sec ,**amplitude

**Ym=1,**phase

**φ=-36°**

Generator

**3**represents ball

**B**–>

**T=2.2**–>angular velocity

**ω=2.86 1/sec ,**amplitude

**Ym=1,**phase

**φ=-180°**

Click “Start”

**Fig**

**. 22-4**

The motions analyze isn’t comfortable as time functions. The easier are

**3**vectors.

**Fig. 22-5**

3balls harmonic motion as

3

**3**vectors. Vectors are anticlockwise rotating with angular velocity

**ω=2.86 1/sec**–>

**T=2.2sec**. The motion is a projection of a vector

**A**,

**B**and

**C**for

**y**axis.

**Chapter 22.2.3 Sinusoids with different amplitudes as vectors**

Call Desktop/PID/08_kryterium Nyquista/02_3_sinusoidy.zcos

**Fig. 22-6**

Click “Start”

** **

**Fig. 22-7
**Compare

**1**

***sin(ωt**

**)**and

**0.5**

***sin(ωt-36**

**)**time function

**Fig. 22-7**

**a**and their vectors

**Fig. 22-7**

**b**.

**Chapter 22.3 Amplitude phase characteristic
**We used characteristics time for dynamic units analyze by now. The step, ramp or dirac type pulse was given as input

**x(t)**signals. The output

**y(t)**was an important information about the object dynamic. Is the

**G(s)**lazy? Are the oscillations? My goal was response

**y(t)**connotation with the object

**G(s)**parameters. But there is a different approach to test dynamic objects:

Amplitude phase characteristic

We will build **A****mplitude Phase Characteristic** for the **inertial unit**:

**K=1**

**T=1 sek
**We will start at very small

**ω=0.31 1/sec**(T=20 sec!) and finish at

**ω=10.06 1/sec**(T=0.63 sek). The input signal

**x(t)=**

**1*sin(**.

**ω**t)**Chapter 22.3.2 Amplitude phase characteristic of the interial unit **

**Chapter 22.3.2.1 ω=0.31 1/sec (T=20 sek)**

Call Desktop/ PID/08_kryterium Nyquista/03_20.zcos

**Fig. 22-8
**Input signal

**x(t)=**

**1**

***sin(ωt**

**) ω=0.31/sec**–>

**T=20 sec**

**Fig. 22-9**

Ym=0.95 φ=-17.5°. The

Ym=0.95 φ=-17.5°

**φ**accuracy is a little suspect. Phase

**φ**especially. What is a measure method? Ruler? Simple matter.

I changed the oscilloscope time base in the separate experiment. The

**y(t)**amplitude and phase should be measured in steady state. Not at begining! The vectors are drawed in the

**Fig.22-9b**then.

**Chapter 22.3.2.2 ω=0.63 1/sek (T=10 sek)**

Wywołaj PID/08_kryterium Nyquista/04_10.zcos

The block diagram is the same but the generator frequency is bigger now.

Click “Start”

**Fig. 22-10
**The

**doubled but we don’t see it at the figure! But note that experiment time is shortened up**

**ω****t=30 sec**. This time will be the same at the next experiments.

The delay

**φ**was extended and the amplitude

**Ym**was minimized.

**Chapter 22.3.2.3 ω=1.26 1/sek (T=5 sek)**

Call Desktop/PID/08_kryterium Nyquista/05_5.zcos

Click “Start”

**Fig. 22-11**

The **red vector **is delayed and minimized further.

**Chapter 22.3.2.4 ω=2.51 1/sek (T=2.5 sek)**

Call Desktop/PID/08_kryterium Nyquista/06_2.5.zcos

Click “Start”

**Fig. 22-12**

The **red vector **is delayed and minimized further.

**Chapter 22.3.2.5 ω=5.03 1/sek (T=1.25 sek)**

Call Desktop/PID/08_kryterium Nyquista/07_1.25.zcos

Click “Start”

**Fig. 22-13**

The **red vector **is delayed and minimized further. Note that transition sinusoid state at the beginning is more distinct by bigger frequencies!

**Chapter 22.3.2.6 ω=10.06 1/sek (T=0.63 sek)**

Call Desktop/PID/08_kryterium Nyquista/08_0.625.zcos

Click “Start”

**Fig. 22-14
**The

**red vector**is delayed and minimized further. I expect that amplitude

**Ym**is aiming to

**0**and delay to

**φ=-90°**when

**ω****=infinity**.

Let’s combine all

**red vector**

**s**to one commone figure.

**Fig. 22-15**

The **Fig. 22-15a **is a common figure. **Green vektor **is a symbol of the **6** input **x(t) **sinusoids. The remaining **6 red vectors **are appropriate output

**y(t)**sinusoids.There were

**6**

only

**+**

**red vectors****1**

**green vektor**. But if were 100 or 1000000 vectors? We will have the

**Fig. 22-15b**then. The vector ends are drawing

**semicircle**here.

**Fig. 22-16**

A

A

**ends are seen as**

**ll red vectors****semicircle**. The

**x,y**axes are added yet. There is one question. How does treat the

**for start**

**red vector****ω0=0**? First-we haven’t done such a experiment. But more important. How to treat the

**sinusoid**with

**ω0=0**?

**Imagine**(ah Beatles…) that we have very slow frequency. For example

**T=1 year**. The sinusoid starts at 01.01.2016. The output will be

**+1**(almost!) and delay

**φ**

**=0**(almost!) after

**t=3 months**!

**Amplitude Phase Characteristic**is a generalisation of the

**Transmitted Frequency Band**concept. This concept is a

**gain**

**K**for all frequencies in range

**0…infinity**. The concept

**Amplitude Phase Characteristic**has more information about object dynamics. Not only

**gain**

**K**but

**phase**

**φ**(mainly

**delay**

**φ**) for all frequencies too!

**Chapter 22.4 Amplitude Phase Characteristic of other dynamic units**

**Chapter 22.4.1 Introduction
**We know the

**inertial unit**amplitude phase characteristic. It’s semicircle. But what about other dynamic units?

**Chapter 22.4.2 Proportional Unit**

**Fig. 22-17**

**Amplitude Phase Characteristic** of the Proportional Unit

There is state** x(t)=y(t)** always. So the output sinusoid is as input sinusoid for every **ω**. It’s one point (**+1,0**). It’s so easy that “until difficult”.

**Chapter 22.4.3 Static units
**They haven’t

**integral elements**. I.e. the aren’t single

**s**in the

**G(s)**denominator. Their

**Amplitude Phase Characteristic**isn’t a

**semicircle**as for

**inertial unit**but is a little similar.

**Fig. 22-18**

Triple inertial unitAmplitude Phase Characteristic for example. It wanders through

Triple inertial unit

**3 x/y**quarters. Analogously:

**Double inertial unit**–>2 quarters

**Inertial unit**–>1 quarter.

See

**Fig. 22-18**

Note that there is one crtical angular frequency

**ω=ωkr**when output sinusoid is in antiphase to input sinusoid

**. Double inertial unit**and

**Inertial unit**haven’t this attribute.

**Chapter 22.4.4 Integration unit as an astatic unit example
Chapter 22.4.4.1 Introduction
**

**Fig 22-16…18**shows

**static units**. You see the beginning for

**ω=0**and end for

**ω=∞**. But what about the

**astatic units**?

**Chapter 22.4.4.2 ω=0.31 1/sec (T=20 sec)**

Call Desktop/PID/08_kryterium Nyquista/09_calkujacy_20.zcos

**Fig. 22-19**

Click”Start”

**Fig. 22-20
**Never mind that the constant component was added. The interesting is only sinusoid output component! The sinusoid

**amplitude=3.2**and phase

**.**

**φ=-90°****The vectors are obvious.**

**Chapter 22.4.4.3 ω=0.63 1/sec (T=10 sec)**

Call Desktop/PID/08_kryterium Nyquista/10_calkujacy_10.zcos

Wciśnij “Start”

**Fig. 22-21**

The amplitude is double minimized, but the phase is the same **φ=-90°**.

**Chapter 22.4.4.4 ω=2.51 1/sek (T=5 sek)**

Call Desktop/PID/08_kryterium Nyquista/11_calkujacy_5.zcos

Click “Start”

**Fig 22-22**

The amplitude is double minimized, but the phase is the same **φ=-90°**.

**Chapter 22.4.4.5 ω=2.51 1/sec (T=2.5 sec)**

Call Desktop/PID/08_kryterium Nyquista/12_calkujacy_2.5.zcos

Click “Start”

**Fig. 22-23**

The amplitude is double minimized, but the phase is the same **φ=-90°**.

**Chapter. 22.4.4.6 ω=2.51 1/sec (T=1.25 sec)**

Call Desktop/PID/08_kryterium Nyquista/13_calkujacy_1.25.zcos

Click “Start”

**Fig. 22-24**

The amplitude is double minimized, but the phase is the same **φ=-90°**.

**Chapter 22.4.4.7 Amplitude Phase Characteristic of the integration unit**

Let’s combine **Fig. 22-20…Fig. 22-24**.

**Fig. 22-25**

**Fig. 22-25a **is a combined figure. **Fig. 22-25b** is made for all **0****<ω<∞**. Dla **ω=0** moduł wzmocnienie członu całkującego to **K=-∞** a dla **ω=∞** to **K=0**.

**Chapter 22.5 Nyquist stability benchmark**

We will test **3 opened loop **objects **G(s)**. Opened-it meanas that they are 99.99% stable–>see chapter beginning.

They have yellow, **green** and **red** ** Amplitude Phase Characteristic** see **Fig. 22-26.** Axis **x **points** (+7,0), (+10.035,0) **and** (+12,0) **are the **G(s) **numerators. They are “start points” of the **Amplitude Phase Characteristic**.

These characteristics are crossing axis **x **in the points **(-1.2,0) , (-1,0) i (-0.7,0****)**. All the **G(s) **are phase delayed **φ=-180°** here.

The scale doesn’t rule here.

**Fig. 22-26**

**Note
**All above

**3**charachterics are for

**opened loop G(s)**! The texts “stable” “on the border” “Instable” are for the states when they are

**closed loop**! T

**1. If**the

**opened loop**charachteristic doesn’t involve point

**(-1,0) then**the

**closed loo**

**p**will be

**stable**–>

**yellow characteristics**

**2. If**the

**opened loop**charachteristic crosses point

**(-1,0) then**the

**closed loo**

**p**will be

**on the border**–>

**green characteristics**

**3. If**the

**opened loop**charachteristic involve point

**(-1,0) then**the

**closed loo**

**p**will be

**instable**–>

**red charachterisctics**

**Chapter 22.6 Three experiments to check Nyquist benchmark**

**Chapter 22.6.1 Wstęp
**The dirac type pulse will be given to

**3 closed loop triple inertial units.**Their

**opened loop**charachteristics are shown in the

**Fig. 22-26**.

**Chapter 22.6.2 The “yellow” G(s) which should be stable when closed
**

**Amplitude Phase Characteristic**

We should to determine the characteristics theoretically for all

**ω****0**…

**infinity**. Practically there is a finite frequencies number. We will simplify more! There is interesting for us only the crosspoint

**(-0.7,0)**for

**. –>**

**ω3****Fig. 22-26**. Let’s assume that we don’t know this poin yet.

**Two**remaining points are obvious :

**x=7, y=0**for

**ω=0**

**x=0, y=0**for

**ω=∞**

Call Desktop/PID/08_kryterium Nyquista/14_K7_3_inercyjny_sinus.zcos

**Fig. 22-27**

Click “Start”

**Fig. 22-28**

I made some experiments for different

**ω**before and I found

**ω3=2*1/sek**where

**red sinusoid y(t)**has

**φ=-180°**to

**green sinusoid x(t)**.

By the way. Period

**T=3.14=Π**is a clear fortune here.

The input

**x(t)**amplitude is

**1.**So the

**y(t)**amplitude is a gain

**K**parameter for

**ω3=3.14**. So

**K(ω3)=-0.7**. Sign minus

**“-”**means that phase

**φ=-180°**.

So the opened loop characteristic doesn’t involve the point

**(-1,0)**inside as in

**FIg. 22-26**.

**Conclusion**. The closed loop will be stable. Let’s test it!

Call Desktop/PID/08_kryterium Nyquista/15__K7_sprzezenie_zwrotne.zcos

**Fig. 22-29**

The

**triple inertial unit K=7**closed loop. Input

**x(t)**is a Dirac type.

Click “Start”

**Fig. 22-30**

The input

**x(t)**tried to unbalance the system but without success. The system is stable. It confirms thesis

**1**of the

**Nyquist benchmark**–>

**Fig. 22-26**

**Chapter 22.6.3 The “green” G(s) which should be on the stability border when closed**

**Amplitude Phase Characteristic
**I made some experiments for different

**ω**before and I found

**ω2=2*1/sec**where

**.**

**red sinusoid y(t)**has φ=-180° to**green sinusoid x(t)**Call Desktop/PID/08_kryterium Nyquista/16_K10.035_3_inercyjny_sinus.zcos

**Fig. 22-31**

Click “Start”

**Fig. 22-32**

The

**for**

**red sinusoid y(t)****ω2=2*1/sec**

**has**

**φ=-180° to**and the amplitude=

**green sinusoid x(t)****1**! It means that

**K(**and the

**ω2**)=-1**G(s)**opened loop characteristic crosses

**axis x**in the point

**(-1,0)**as in

**Fig. 22-26**. It means that the

**G(s)**closed loop will be on the

**stability border**. Let’s test it!

Call Desktop/PID/08_kryterium Nyquista/17_K10.035_sprzezenie_zwrotne.zcos

**Fig. 22-33**

Click “Start”

**Fig. 22-34**

The input

**unbalanced the system and there are output**

**y(t)**steady amplitude oscillations.

**So the**

**grenn opened loop characteristic**crosses the point

**(-1,0)**as in

**FIg. 22-26**. The system is

**on stability border**. It confirms thesis

**the**

**2**of**Nyquist benchmark**–>

**Fig. 22-26****Chapter 22.6.4 The “red” G(s) which should be instable when closed
Amplitude Phase Characteristic
**I made some experiments for different

**ω**before and I found

**ω1=2*1/sec**where

**and the output amplitude**

**red sinusoid y(t)**has φ=-180° to**green sinusoid x(t)****y(t)**is more than

**1**.

Call Desktop/PID/08_kryterium Nyquista/18_K12_3_inercyjny_sinus.zcos

**Fig. 22-35**

Click”Start”

**Fig. 22-36**

The

**has**

**red sinusoid y(t)**for ω2=2*1/sec**φ=-180° to**and the

**green sinusoid x(t)****amplitude=**It means that

**1.2**!**and the**

**K(ω2)=-1.2****crosses axis**

**red****G(s)**opened loop characteristic**in the point**

**x****as**

**(-1.2,0)****in Fig. 22-26**and involve point

**(-1,0).**It means that

**will be**

**G(s)**closed loop**in**. Let’s test it!

**stable**Call Desktop/PID/08_kryterium Nyquista/19_K12_sprzezenie_zwrotne.zcos

**Fig. 22-37**

Wciśnij “Start”

**Fig. 22-38
**So the opened loop

**red characteristic**involve the point

**(-1,0)**inside as in

**FIg. 22-26**and system is

**instable**. It confirms thesis

**3**of

**the**

**Nyquist benchmark**–>

**Fig. 22-26****Chapter 22.7 Nyquist intuitively**

**22.7.1 Introduction
**

**Nyquist benchmark**has a very representative point

**(-1,0)**. The location of this point against

**Amplitude Phase Characteristic**says about

**closed G(s)**stability.

There was something similar in

**chapter 21 w p. 21.2 The delay unit with the feedback**

We close the

**delayed unit**–> dealy unit with the feedback

**system will be**

The

The

**:**

-stablewhen

-stable

**K<1**

-instablewhen

-instable

**K>1**

-on the borderwhen

-on the border

**K=1**

I will try to generalize this

**delay unit**for

**triple inertia unit**as a representative of all continuos objects

**Chapter 22.7.2 Why”yellow”** **G(s) from Fig.22-26 is **** stable?
**The gain

**K**

**(**see

**ω3=2*1/s**)=-0.7**Fig. 22-28**. It means that

**K=-0.7**for

**. That**

**ω3=2*1/s****y(t)**sinusoid is in

**antiphase**to input

**sinusoid**.

There is a

**negative feedback**in

**Fig. 22-29**. There is only

**-y(t)**after

**t=3sec**on the input here, because

**e(t)=x(t)-y(t)=-y(t)**and

**x(t)=0**after

**t>3sec**. The

**ω=1.76*1/s**–>

**Fig. 22-30**and it’s almost

**. But transfer function**

**ω3=2*1/s****G(s)**reverse

**y(t)**too. So this signal is

**doubled reversed**and returns with sign

**“+”**. It’s something similar to

**positive feedback**for

**. The**

**ω3=2*1/s****K=+0.7<1**and the system will be

**stable**.

**Chapter 22.7.3 Why”green”** **G(s) from Fig.22-26 is **** stable?
**The gain

**K**

**(**see

**ω3=2*1/s**)=-1**Fig. 22-32**.

Our conlusion is that it’s something similar to

**positive feedback**for

**. The**

**ω3=2*1/s****K=+1**and the closed system will be

**on the stability border**. The oscillations are self-maintained and the oscillations are constant.

**Chapter 22.7.4 Why”red”** **G(s) from Fig.22-26 is **** instable?
**The gain

**K**

**(**see

**ω3=2*1/s**)=-1 .2**Fig. 22-36**.

Our conlusion is that it’s something similar to

**positive feedback**for

**. The**

**ω3=2*1/s****K=+1.2>1**and the closed system will be

**instable**.

**Chapter 22.8 The Amplitude Phase Characteristic designating when G(s) is known
**We constructed the

**Amplitude Phase Characteristic**in the

**Chapters 22.3 and 22.4**by experimental method. We measured the output sinusoid amplitude and phase. The

**complex numbers**knowledge wasn’t necessary here. But the

**vectors**and

**complex numbers**are very near mutually.

**Vector**has

**x**and

**y**components on the

**xy**plane.

**Complex number**has real

**R**and imaginary

**Q**components on the

**PQ**plane too.

The

**vectors**may be added, subtracted, multiplied and divided as in the “normal” math.

**Complex numbers**too. There are special formulas.

The conclusion is that you calculate

**G(jω)**for many

**ω**. The calculated

**Amplitude Phase Characteristic**will be the same as made by experiments.

**Fig. 22-39**

**Amplitude Phase Characteristic**as a

**Spectral-Response Characteristic**

**G(jω)**.

Hey there just wanted to give you a brief heads up and let you know a few of the images aren’t loading properly.

I’m not sure why but I think its a linking issue.

I’ve tried it in two different web browsers and both show the same results.

All the chapter 22 figures are seen in my computer . I don’t know why you have problems. Please try from another computer.

Has anybody these same problems?

I read this article fully on the topic of the difference of latest and previous

technologies, it’s remarkable article.

Someone essentially help to make critically posts I’d state.

That is the first time I frequented your website page and thus far?

I surprised with the analysis you made to make this actual put up incredible.

Great activity!

I adore it when people come together and share views, great blog,

keep it up.