### Automatics

**Chapter 17 Instability, or How Oscillations Are Created **

**Chapter 17.1 Introduction**

The

**oscillations**simply result from the solution of the

**differential equations**describing the feedback dynamic

**uni**t. You will get a time charts that, in response to a unit step

**x(t)**, the output

**y(t)**comes to a steady state

**y**:

**– without**oscillations

**– with fading**oscillations

With certain parameters, including

**K**gain, an unstable state with

**increasing oscillations**may also arise.

**Fig. 17-1**Steady state

**y**in a closed system when

**G(s)**

**static**or

**astatic**.

In

**steady**state,

**y(t)=y**is

**constan**t. We call such systems

**stable**. But that’s not always the case. With certain transmittance parameters

**G(s)**, especially with large

**K**and large delays, y(t)

**oscillations**of constant or

**increasing**amplitude will occur. Then we deal with unstable systems.

Of course, you can stop at the fact that for some

**G(s)**parameters the system is

**stable**and for others

**unstable**. These are the solutions of

**differential equations**and that’s it. This approach, however, causes some unsatisfactoriness. Therefore, let’s try to analyze the problem

**common sense**. We suspect that the

**oscillations**result from the

**inertia**of the system, from the fact that the response appears with a certain

**delay**.

**Example**

You sail on a sailboat. It’s night, the wind is blowing and you’re heading for the

**lighthouse**. For the sake of simplicity, let’s assume you only use the

**steer**. The control is simple. A small lighthouse to the

**right**of your course, you counter the steer to the

**left**and vice versa. If there are no disturbances such as wind, waves, etc., you are sailing almost in a straight line towards the lighthouse. By the way. There is an

**integral**relationship between the

**rudder**and the course angle, which can reduce the error to

**zero**. An experienced sailor sails in a

**straight**line along an almost

**invisible hose**. Now imagine that between the

**tiller**(i.e. what you hold in your hand) and the

**rudder blade**itself there is a

**delay**element. How this device is built – it doesn’t matter. The important thing is that your hand movements are delayed, e.g. by

**To=10 seconds**. A small gust of wind will change the course. You immediately counter. The effect will be visible only after

**To=10 sec**. The boat start sailing no longer on a hose but on a large boa snake. With a

**large**

**To**, the course may even go completely apart. You will not the goal. We will draw an important and quite obvious conclusion from the voyage. Objects with large

**inertias**and

**delays**are more difficult to control. They can cause a

**hos**e, pardon the

**oscillations**.

But let’s stop sailing and consider the operation of a pure delay term with

**K**gain and feedback. In such a system, the generation of

**oscillations**is the easiest to understand.

**Chapter 17.2 Instability of the delay element with feedback****Chapter 17.2.1 Introduction**

It is possible to analyze the system without using **differential calculus**. Calculate almost on fingers. We will come to an interesting conclusion, somewhat reminiscent of the **Nyquist Criterion**, which will be discussed in the next chapter.

An open system with a delay after closing with a negative feedback loop will be:**-stable** when **K<1****-unstable** when **K>1****-on the verge of stability** when **K=1****First** we will examine the **delay** term itself. We’ve done it before, but it doesn’t hurt to do it again. Repetitio est mater studiorum.**Chap. 17.2.2 Study of the delay unit in an open system**

Fig. 17-2**Delay unit transmittance** whose type clearly differs from typical **G(s) **transmittance.

**Fig. 17-3**The response

**y(t)**of the above

**delay unit**

**G(s)=exp(-1s)**is an exact copy of the input

**x(t)**with a delay

**To=1 sec**

**Chapter 17.2.3 Delay unit in a closed system K=0.75 and To=1sec**

Let’s close the

**delay unit**with a negative feedback loop.

**In front of the**

Fig.17-4

Fig.17-4

**delay unit**, an additional

**proportional unit**was placed, whose gain

**K=0.75**.

**Fig.17-5**In

**5**seconds, a rectangle pulse appears

**x(t)**=1.

At this time,

**y(t)=0**, therefore, at the

**G(s)**input after the comparison node, there will also be the same rectangular pulse

**0.75*x(t)=+0.75**.

In

**6**seconds

**y(1) = 0.75*x(0) = +0.75**

From

**6**seconds, the comparison node only reverses the phase, because

**x(n)=0**and

**e(n)=x(n)-y(n)=-y(n)**.

The next signal after the delay element will be counted according to of the formula

**y(n+1)= – 0.75*y(n)**

**y(2) = – 0.75*y(1)= – 0.75*0.75= – 0.5625**

**y(3) = – 0.75*y(2)= + 0.422**

**y(4) = – 0.75*y(3) = – 0.316**

**y(5) = – 0.75*y(4)= + 0.237**

**y(6) = – 0.75*y(5) = – 0.180**

**y(7) = – 0.75*y(6)= + 0.133**

**y(8) = – 0.75*y(7) = – 0.100**

**y(9) = – 0.75*y(8)= + 0.075**

e.t.c…

I emphasize that:

**y(2)**we calculated based on the known

**y(1)**,

**y(3)**we calculated based on the known

**y(2)**,

…e.t.c

This is the so-called

**recursive**method.

Subsequent pulses

**y(n)**are getting smaller and tend to zero. The impulse-type excitation threw the system out of balance and vanishing vibrations with a period of

**To=1 sec**.

From a mathematical point of view, the sequence y

**(1), y(2), y(3), …, y(n)**is a geometric progression in which:

**y(1) = 0.75**

The quotient of this progress

**q=- 0.75**

Since

**|q|<1**, the sequence tends to

**0**.

The tested transmittance was

**K=0.75<1.**What if

**K=1**?

**Chapter 17.2.4 Delay unit in a closed system K=1 and To=1sec**

**In**

Fig. 17-6

Fig. 17-6

**5**seconds, a single pulse

**x(t)**appears

**Fig. 17-7**

By analogy to Fig. 17-5:**y(1) = x(t) = +1y(2) = -1*y(1) = – 1y(3) = -1*y(2) = +1y(4) = -1*y(3) = – 1y(5) = -1*y(4) = +1y(6) = -1*y(5) = – 1**…etc

The impulse also knocked the system out of balance, but this time it caused

**undying**oscillations with a

**constant**amplitude. The system has become a generator!

So for systems with a delay and gain

**K=1**, the system is on the verge of stability.

What if

**K>**1, e.g.

**K=1.25**?

**Chapter 17.2.5 Delay unit in a closed system K=1.25**

**and To=1sec**

**In**

Fig. 17-8

Fig. 17-8

**5**seconds, a single pulse

**x(t)**appears.

**Fig. 17-9**The resulting oscillations whose amplitude tends to

**+/- infinity**!

They had to be created because:

**y(1) = 1.25*x(t) = + 1.250**

y(2) = -1.25*y(1) = – 1.562

y(3) = -1.25*y(2) = + 1.953

y(4) = -1.25*y(3) = – 2.441

y(5) = -1.25*y(4) = + 3.051

y(6) = -1.25*y(5) = – 3.815

y(7) = -1.25*y(6) = + 4.768

y(8) = -1.25*y(7) = – 5.960

y(9) = -1.25*y(8) = + 7.451

…itd

y(2) = -1.25*y(1) = – 1.562

y(3) = -1.25*y(2) = + 1.953

y(4) = -1.25*y(3) = – 2.441

y(5) = -1.25*y(4) = + 3.051

y(6) = -1.25*y(5) = – 3.815

y(7) = -1.25*y(6) = + 4.768

y(8) = -1.25*y(7) = – 5.960

y(9) = -1.25*y(8) = + 7.451

…itd

**Chapter 17.2.6 Conclusions**

**If we enclose the**

Fig.17-10

Fig.17-10

**delaying unit To**in a negative feedback loop and give a signal to the input, e.g. a

**single pulse**, the response will be oscillations about the period

**To**and the

**amplitude**:

**-decreasing**to 0..

**when k<1**–>

**stable system**

**-constant…………..when k=1**–>

**system on the verge of stability**

**-increasing**………

**when k>1**—

**> unstable system**

Note that based on the knowledge of the

**transmittance**of the

**open**system, we determined the

**stability**of the

**closed**system.

This is a bit like the

**Nyquist Criterion**–>

**Chapter 19**.

**Chap. 17.3 Instability of the Three-inertial Unit with feedback****Chapter 17.3.1 Introduction**Previously, we studied the

**delay unit**with

**negative feedback**. It was so simple that we only used elementary mathematics and intuition. By increasing the gain

**K**, the system became unstable. The source of this was the

**delay**. This is also the case in

**continuous**systems, but the concept of

**delay**should be replaced with

**inertia**. When there are many inertias

**T1, T2, T3,**such as in the

**three-inertial**unit, there will be something like a

**delay**

**To.**Then

**G(s)**can be approximated by the equivalent transfer

**transmittance**–> see

**Chapter 10.2**.

Fig. 17-11

Fig. 17-11

Substitute Transmittance

For small

**T**(not

**To**!), the equivalent transmittance becomes similar to the ideal

**delay unit**with

**k**gain in

**Fig. 17-3**. Let’s agree that when examining instability, the

**three-inertial**unit is a representative of all continuous

**G(s)**transmittances. A question may arise here. That’s why it’s not an even simpler

**two-inertial**,

**oscillating**or alone

**inertial**unit. The analysis will be easier.

Well, as it turns out later,

**units**with a

**monomial**, a

**binomial**in the denominator will always be

**stable**! Even with a very large

**K**. Yes, there will be

**oscillations**, but they will disappear!

*****assuming that the

**roots**of the

**monomial**or

**binomial**are positive. But that’s usually the case in open systems.

**Chapter 17.3.2 Test of the three-inertial unit in an open system**

**Fig. 17-12**Unstable systems can be stationary when input

**x(t)=0**, like theoretically a pencil placed vertically on a table. Therefore, we will unbalance this and the next (especially) systems with the x(t)-“flick” signal. It is a short pulse with a duration of

**t=0.02 sec**and an amplitude of

**50**. Its area is

**1**and therefore we treat it as a approximation of the Dirac pulse. Remember about its large amplitude, especially since the oscilloscope “cuts”

**y(t)**at

**+6**and

**-6**levels. The

**x(t)**input acts on the

**trhree-inertial**only for a short time from

**3 sec**to

**3.02**

**sec**. During this time, the energy from

**x(t)**is stored in the unit, which then “discharges” giving such a

**y(t)**waveform with a maximum for about

**t=5sec**.

Compare with the response of the

**ideal delay**unit in

**Fig. 17-3**. You can see a certain analogy. Only that

**y(t)**has been “blurred”. After some time

**t**, the output

**y(t)**reaches its maximum. This time is such a “pseudo delay”.

The analogy to the delay isn’t ideal, but the mechanisms causing instability in the

**ideal delay**unit and the

**trhree-inertial**unit are similar in

**feedback**. For example, that increasing the

**K**gain causes

**instability**.

**Rozdz. 17.3.3**.

**Three-inertial unit in a closed system system K=3****Fig.17-13**The input of the three-inertial unit is acted on by the signal

**e(t)=x(t)-y(t)**. At the beginning

**x(t)**has a very large amplitude

**x(t)=50**but it works for a very short time

**0.02 sec**. Then all the time

**x(t)=0**and further increase of

**y(t)**until reaching the maximum for

**t=4.3 sec**is mainly due to the energy received from

**x(t)**within

**0.02 sec**. Therefore, the initial time chart

**y(t**) is similar to the

**open**system in

**Fig. 17-13**. Similar, but not quite. The maximum in the

**open**system is slightly

**later**, i.e.

**for t=5 sec**. Why?

In the open system in

**Fig. 17-13**, after

**3.02 sec**, the signal

**x(t)=0**. The input signal on

**G(s)**is

**zero**and the time chart y(t) results only from the discharge of energy. However, in Fig.

**17-13**, the “braking force”

**e(t)=x(t)-y(t)=-y(t)**is active all the time! This was not in

**Fig. 17-12**. Therefore, the maximum occurred earlier, i.e. for

**t=4.35 sec**. Nay! “Braking” –

**y(t)**causes

**y(t)**to reach the state

**y(t)=0**in

**5.8 seconds**and continue to run towards

**negative**values. Now “braking” tries to turn

**y(t)**back to

**y(t)=0**. And so, with a few

**oscillations**, the system returns to a steady state

**y(t)=0**.

The study of the

**delay unit**shows that by increasing

**K**we can cause instability.

We expect that it is similar for the

**three-inertial**unit as a representative of continuous objects (read “normal”). So let’s increase the gain from

**K=3**to

**K=7**.

**Chapter 17.3.4**

**Three-inertial unit in a closed system**K=7

**Fig. 17-14**The same rectangular pulse (“almost dirac”)

**x(t)**is applied to the input.

The system

**swings**longer, but is still

**stable**. Greater rocking caused the larger

**K**, which pushes

**y(t)**more strongly in the direction of

**y(t)=0**. That means more oscillations. Then let’s increase the gain to

**K=10.035**. It will soon become clear why exactly

**10.035**.

**Chapter 17.3.5**

**Three-inertial unit in a closed**K=10.035**F****ig.17-15**The system is on the

**verge**of stability. For the delaying units, it was

**K=**1, and here

**K=10.035**, and there will be another value for the

**other dynamic units**. The

**stability criterion**is more complicated than for

**delay units**. I guess it’s not surprising. After all,

**three-inertial**is “more difficult” than

**delay unit**. Now

**K=10.035**caused the

**next**amplitude to have the same value as the

**previous**one. If

**K**were a little smaller, there would be no energy. The

**next**amplitude will be a

**little smaller**. When

**K**is greater, the amplitudes will increase. Let’s check for, for example,

**K=12**.

Chapter 17.3.6

Chapter 17.3.6

**Three-inertial unit in a closed**K=12**Fig.17-16**

The gain of **K=12** is such that each subsequent amplitude gets a bigger kick. Amplitudes grow to infinity. The system is **unstable**.

**Chapter 17.4 Three-inertial unit in a closed system – unit step x(t) as an input signal**

**Chapter 17.4.1 Introduction**

When examining the instability, it was enough to “click” the dirac impulse on the three-inertial term to possibly bring it out of equilibrium

**y(t)=0**. When the system was stable->

**Fig. 17-13**,

**Fig. 17.14**, it returned to the state of equilibrium

**y(t)=0**. Otherwise,

**oscillations**with a constant amplitude occurred–>

**Fig. 17.15**or increasing ->

**Fig. 17-16**.

And how will a stable and unstable system behave when the input signal

**x(t)**is a

**unit step**of

**1(t)**?*

*****more precisely, with a unit

**step x(t)=1(t-3)**because it is delayed with respect to

**t=0**by 3

**sec**.

**Chapter 17.4.2 Stable system unit step response**

**F****ig. 17-17**Classic. The

**y(t)**signal reaches a steady state

**y=0.75**acc. a well-known formula.

**Chapter 17.4.3 Unstable system unit step response**

It would seem that the steady-state

**Kz**gain makes no sense for an

**unstable**system.

It has a bit of that though.

**Fig. 17-18**The

**oscillation amplitude**increases to

**infinity**around the constant component

**y=0.923**determined according to

**known formula**.