Preliminary Automatics Course

Chapter 22. Nyquist stability criterion
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 Ym and phase φ of the open loop 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=36° is a phase.
Rotating vector C formula y(t)=Ym*sin(ωt-φ) where Ym is a amplitude and φ=180° is a phase. Other words 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
3
balls harmonic motion as 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. 22-7a and their vectors   Fig. 22-7b.

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 Amplitude 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(ωt).

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 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 ω doubled but we don’t see it at the figure! But note that experiment time is shortened up 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 and delay to  φ=-90° when  ω=infinity.
Let’s combine all red vectors 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
ll red vectors ends are seen as semicircle. The x,y axes are added yet. There is one question. How does treat the red vector for start  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 unit
Amplitude Phase Characteristic for example. It wanders through 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 loowill be stable–>yellow characteristics
2. If the opened loop  charachteristic crosses point (-1,0) then the  closed loowill be on the border–>green characteristics
3. If the opened loop  charachteristic  involve point (-1,0) then the  closed loowill 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  0infinity.  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 red sinusoid y(t) for ω2=2*1/sec  has  φ=-180° to green sinusoid x(t) and the amplitude=1! It means that K(ω2)=-1 and the 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 2 of the 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 red sinusoid y(t) has φ=-180° to green sinusoid x(t) and the output amplitude 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 red sinusoid y(t) for ω2=2*1/sec  has  φ=-180° to green sinusoid x(t) and the amplitude=1.2! It means that K(ω2)=-1.2 and the red G(s) opened loop characteristic crosses axis x in the point (-1.2,0) as in Fig. 22-26 and involve point  (-1,0). It means that G(s) closed loop will be instable. Let’s test it!
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
The
system will be :
-stable
when K<1
-instable
when K>1
-on the border
when 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(ω3=2*1/s)=-0.7 see Fig. 22-28. It means that K=-0.7 for ω3=2*1/s. That 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 ω3=2*1/s.  But transfer function G(s) reverse y(t) too. So this signal is doubled reversed and returns with sign “+”.  It’s something similar to positive feedback for ω3=2*1/s. The 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(ω3=2*1/s)=-1 see Fig. 22-32.
Our conlusion is that it’s something similar to positive feedback for ω3=2*1/s. The 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(ω3=2*1/s)=-1 .2 see Fig. 22-36.
Our conlusion is that it’s something similar to positive feedback for ω3=2*1/s. The 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 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).