**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 criterion**is frequency type in contrast to

**Hurwitz criterion**which is an 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<∞**. The **K=- ∞ **for

**ω=0**and

**K=0**for

**ω=∞**.

*** K**is a amplification coefficient.

**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**!

**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)**.

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