### Automatics

**Chapter 19. Nyquist Stability Criterion****Chapter 19.1 Introduction**An open system is usually

**stable**. It can become

**unstable**after being closed with a

**feedback**loop. The

**Nyquist Criterion**can predict, based on the amplitude-phase characteristics of an

**open system**(which is “easier” than a closed one), the

**stability**of a

**closed system**. An open system is, firstly, simpler than a closed one, and secondly, it is mostly stable. I mean an open system, so to speak “by nature”, and not one that after applying the formula for

**Gz(s)**becomes “open” as below.

**The Nyquist criterion belongs to**

Fig. 19-1

Fig. 19-1

**frequency**, unlike the algebraic

**Hurwitz**criterion from the

**next chapter**. The

**frequency**response of an open system is relatively easy to determine as in

**chapter 18**. You can do it on the “living facility” without fear that you will lose control over the

**industrial installation**. After all, it is an

**open**system, and therefore

**stable**. And this is the advantage of the

**Nyquist Criterion**.

**Chapter 19.2 Nyquist Criterion**

In **chapter 18** we got to know **the amplitude-phase characteristics**. They are fundamental to understanding the **Nyquist Criterion**. We will examine **3** open **dynamical systems**, which in **Fig. 19-2** have **yellow**, **green**, and **red** **amplitude-phase characteristics**. They correspond to three successive **three-inertial units** from **Chapter 19.4, 19.5 and 19.6.****Fig. 19-2**All

**3**characteristics apply to

**open**systems.

The inscriptions

**“Stable”**,

**“On the border”**and

**“Instable”**refer to the

**state,**but only after closing them with a

**negative feedback**loop! The

**characteristic**is not to scale. The points on the

**x**axis, i.e.

**(7.0), (10,035.0)**and

**(12.0)**are the

**numerators**of these

**transmittances**, i.e. gains

**K**in steady state. From them “start” individual characteristics with initial pulsation

**ω=0**. In

**ω1, ω2**and

**ω3**pulsations, the

**characteristics**pass through the

**x**axis at points

**x= (-1.2.0), (-1.0)**and

**(-0.7.0)**. The

**sinusoids**here have a phase shift of

**φ=-180°**. Since these are

**three-inertial**units, they pass through

**3**quadrants and for

**ω=∞**they end at

**(0,0)**.

**And**now the most important, the

**Nyquist Criterion**!

**1. If**the

**characteristic**does not cover the point

**(-1.0)**, then the

**closed**system is

**stable**->

**yellow characteristic**.

**2. If**the

**characteristic**passes the point

**(-1.0)**, then the

**closed**system will be on the

**border of stability**->

**green characteristic**.

**3. If**the

**characteristic**covers the point

**(-1.0)**, then the

**closed**system will be

**unstable**->

**red characteristic**.

**Chapter 19.3 Checking the Nyquist Criterion for a stable, boundary and unstable system.**

Specific examples will be** 3 three-inertial** units corresponding to the **yellow, green, red** characteristics in **Fig. 19-2**. They will differ only in the **numerator** of the transmittance **G(s)**, i.e. the gain** K=7, 10.035** and **12**. First, we will determine in the most simplified way the **amplitude-phase characteristics** of a given transmittance **G(s)** – i.e. **yellow, green, red**, in an open system. Then we will close them with a **negative feedback** loop and with the Dirac impulse we will throw the systems out of equilibrium.

**Chap. 19.4 Checking the Nyquist Criterion for the transmittance with gain “K=7”.**

That is, a stable **yellow****–>Fig. 19-2****Chap. 19.4.1 Amplitude-phase characteristics of an open system**

We should determine the **amplitude-phase characteristics** for all pulsations in the range **ω=0…∞** by feeding **sinusoids x(t)=1sin(ωt)** of different pulsations to the input. In practice, it is enough for a finite number of **ω** pulsations. Similarly, we studied, for example, the **inertial unit** in **Chapter 18**. Let’s simplify the problem even more. We will limit ourselves only to the **ω3** pulsation, in which the characteristic will intersect the** x** axis at the point **(-0.7,0)**.

The remaining **2** points, i.e.:**(7.0) for ω=0****(0,0) for ω=∞**

are derived from **Fig. 19-2** and are obvious.

**Fig. 19-3**For the pulsation

**ω3=2/sec**(corresponding to the period

**T=3.14sec**), the output sine wave

**y(t)**in steady state is shifted in phase relative to

**x(t)**by

**φ=-180°**. I found the

**ω3**pulsation by trial and error. The output signal

**y(t)**as a sine wave stabilized after approx.

**20 seconds**. Only then can its parameters be measured, i.e.

**amplitude**,

**period**and

**phase**. The input sine wave

**x(t)**has an amplitude of

**1**. Therefore, the amplitude

**y(t)=0.7**is also a gain for

**ω3**. So

**K(ω3)=-0.7**. The

**minus sign**results from the phase

**φ=-180°**. This means that the

**yellow**characteristic intersects the

**x**axis for

**ω3**at the point

**(-0.7.0)**. So it does not include the point

**(-1.0)**and it is in

**Fig. 19-2**.

**Conclusion**

The

**closed**system should be

**stable**.

**Chap. 19.4.2 Will the closed system be stable?**

**Fig. 19-4Three-inertial** unit with

**K=7**in a

**closed**system. The input

**x(t)**is the Dirac pulse. The Dirac impulse knocked the system out of equilibrium, but after several oscillations

**ω=1.76/sec**, the system returned to the state

**y(t)=0**. So the system is

**stable**. The “

**yellow**” characteristic in

**Fig. 19-2**does not include the point

**x=(-1.0)**. This confirms thesis

**No. 1**of the

**Nyquist Criterion**.

Note:

Initially

**x(t)=dirac**and then

**x(t)=0**. Therefore, almost all the time

**e(t)=0-y(t)**, i.e.

**e(t)=-y(t)**in the

**Fig 19-4**.

**Chap. 19.5 Checking the Nyquist criterion for transmittance with gain “K=10.035”.**

That is, on the stability border-**green-->Fig. 19-2.****Chap. 19.5.1 Amplitude-phase characteristics of an open system**We will limit ourselves to the

**ω2**pulsation, in which the

**green**characteristic intersects the

**x**axis at the point

**(-1.0)**. Notice that

**ω2=ω3=2/sec**. This is obvious, because all

**3**transmittances differ only in the gain

**K**in the numerator. So for the next transmittance “

**K=12**” there is also

**ω1=2/sec**.

**Fig. 19-5**

For **ω2≈2/sec**, the **steady-state** output sine wave **y(t)=1** is shifted in phase by **φ=-180°** and its amplitude is equal to the input **x(t)**. So **K(ω2)=-1**. So we have determined **3** important points of the **green** characteristic for** ω=0, ω=ω2=2/sec and ω=∞** as in **Fig. 19-2**.**Conclusion**

The **closed** system should be on the **verge of stability**.

Let’s check.**Chap. 19.5.2 Will the closed system be at the limit of stability?**

**Fig. 19-6****Three-inertial** unit with **K=10.035** in a **closed** system.

Input **x(t)=Dirac** impulse knocked the system out of equilibrium and **oscillations** of** constant** amplitude and **pulsation ω=2/sec** were created. So the system is on the **verge of stability**. The **green** characteristic of the **open system** in **Fig. 19-2** intersects the point **(-1.0)**. This confirms thesis** No. 2** of the **Nyquist Criterion**.

**Chap. 19.6 Checking the Nyquist criterion for transmittance with gain “K=12”.**That is unstable

**red**

**–>Fig. 19-2.**

**Chap. 19.6.1 Amplitude-phase characteristics of an open system**

We will limit ourselves to the pulsation

**ω1**, in which the

**characteristic**will intersect the

**x**axis at the point

**x=-1.2**The other

**2**points for

**ω=0**and

**ω=∞**are obvious.

**Fig. 19-7**

“For the pulsation **ω1=2/sec**, the steady-state output sine wave **y(t)** is shifted in phase by** φ=-180°** and its amplitude is **1.2**. So **K(ω1)=-1.2**. So we have determined **3** important points of the **red** characteristic for **ω=0**, **ω1=2/sec** and **ω=∞** from **Fig. 19-2**. This means that the** red** characteristic intersects the** x-axis** for** ω1** at the point **(-1.2.0)**. That is, it covers the point** (-1.0)**. And so it is in **Fig. 19-2**.**Conclusion**

A **closed** system should be unstable.

Let’s check.**Chapter 19.6.2 Will a closed system be unstable?**

**Fig.19-8**

Three-inertial unit with **K=12** in a **closed** system.

Input** x(t)=Dirac** impulse knocked the system out of **equilibrium** and oscillations of increasing amplitude and pulsation **ω=2.14/sec** were created. So the system is **unstable**. The **red** characteristic of the open system **Fig. 19-2** covers the point **(-1.0)**. This confirms thesis **No. 3** of the **Nyquist Criterion**.

**Chap. 19.7 Nyquist intuitively****Chap. 19.7.1 Introduction**

I don’t know if it will work, but I’ll try. Well, in the **Nyquist criterion** there is a very characteristic point **(-1.0)**. The stability of a closed system depends on the **amplitude-phase characteristics** of an **open** system with respect to this point.

Go back to **Chapter 17 Instability, or How oscillations are created.**

There was something similar here:

When reinforcement:**K<1**, then the **closed** system is **stable****K=1** the** closed** system is **“at the limit”****K>1** the **closed** system is **unstable**

We found that the **responses** to the **Dirac** of the **Delay** and **Three-inertial** unitsare **similar**, only in the **latter** the answer is more “fuzzy”. Here again, some simplistic **Mr. Nyquist** came out, tipping his hat.

So I will try to explain how the true **Nyquist Criterion** works for the** three** previously studied** three-inertial** units:**–”Yellow“****–”Green“****– “Red“**

**Chapter 19.7.2 Why is “Yellow” stable?**

Why the

**yellow characteristic**of the open system in

**Fig. 19-2**not including the point

**(-1.0)**means the stability of the

**closed system**?

**Open system**for pulsations

**ω3=2/s**acc.

**Fig. 19-3**has a gain

**K=-0.7**. The

**negative**sign is the

**phase shift**.

Now look at

**Fig. 19-4**where

**negative feedback**was used.

The input of the

**three-nertial**is

**y(t)**with

**reversed phase**, because after

**3 seconds**it is

still

**x(t)=0**and

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

In turn, we know that for this frequency in a steady state

**ω=1.76*1/s**–>

**Fig. 19-4**, the

**three-inertial**unit almost reverses the

**phase**. Almost … because according to

**Fig. 19-3**reverses the phase for

**ω3=2*1/sec.**

So the

**phase**is

**reversed twice**, i.e. in a steady state, the sine wave

**y(t)**is in phase with the

**input signal**! Something

**similar**to

**positive feedback**with a gain

**K=+0.7**in

**Fig. 17-5**of

**chapter 17**! Since

**K=+0.7<1**, the signal will not sustain itself (as in positive feedback) and the

**oscillations**will disappear. The

**closed**system will be

**stable**.

**Chapter 19.7.3 Why is “green” on the verge of stability?**

For

**Yellow**, the open system for

**ω3=2*1/s**had

**K=-0.7**. Similarly, for

**Green**, the

**open**system for

**ω2=2*1/s**had

**K=-1**(Fig. 19-5).

By reasoning similarly to

**“Yellow”**, we will come to the conclusion that the “pseudo-positive” feedback” with the gain

**K=+1**will cause vibrations with a constant amplitude, as in

**Fig. 17-7**

**chapter 17**. The

**oscillations**will sustain themselves and the closed system will be on the

**border**of stability.

**Chapter 19.7.4 Why “Red” is unstable**

Open system gain

**K=-1.2**(Fig. 19-7). Therefore, a “pseudo-positive” feedback with a gain

**K=+1.2>1**will cause

**oscillations**with increasing amplitude similar to

**Fig.17-9**

**chapter 17**. The

**closed**system will be

**unstable**.

**Chap. 19.8 Determining the amplitude-phase response from a known transfer function G(s)**

In **chapter 19.3 **we determined the characteristics of the **G(s)** units with the experimental method by applying a sine wave** x(t)** with an **amplitude** of **1** to the **input** and measuring the **amplitude** and **phase** of the output** sine** wave **y(t)**. We did it for different **ω** pulsations, theoretically in the range **0…∞**. No knowledge of** complex numbers** was needed. It was enough to interpret the sine **wave** as a** vector** of the appropriate length and angle in the **x,y** plane.

From **vectors** to **complex numbers** is very **close**. Just as a **vector** has coordinates of the **x, y** plane, a **complex number** has coordinates of the** P, Q** plane, where** P** is the axis of the **real component** and **Q** of the **imaginary component**. Operations known in “ordinary” mathematics are also performed on** complex numbers**. Addition, subtraction, multiplication and division. There are appropriate **formulas** for this and that’s it. So, knowing them, you can determine the **amplitude-phase characteristics** of a given **G(s)**. Substitute **s** with the imaginary number **jω**, where **ω** is the appropriate pulsation and the complex number **G(jω)** will show us the appropriate value for the pulsation **ω**.

Let me remind you that **G(s)** is a fraction of** two polynomials** of the numerator **L(s)** and the denominator** M(s)**. For this, we only need monkey dexterity to perform **4 basic** mathematical operations on **complex numbers**. The amplitude-phase characteristic calculated in this way for various ω pulsations will look similar to the previous one:

**Fig.19-9**

The **amplitude-phase** characteristic **G(jω)** calculated from the transmittance **G(s)**.**G(jω)** is the so-called **spectral transmittance**. It contains the same information about the** static** and **dynamic** properties of the object as the operator transfer function **G(s)**.