### Automatics

**Chapter 23 P Control**

**Chapter 23.1 Introduction**

The length of this and the following chapters on

**PID**is a bit

**scary**. But courage! Most experiments are repeatable with different parameters. The

**P**type

**Controller**is the simplest continuous

**Controller**. In the following chapters, you will learn about more sophisticated

**PI**and

**PD**controllers and the most sophisticated option – the

**PID**controller. They form a group of

**PID**type controllers. It can be said with some exaggeration that the

**controller**is by default a

**PID**controller.

We will examine the control of the

**P-type**regulator with the

**object**:

**One-inertial**K=1, T1=10 sec.

**Two-inertion**K=1, T1=5 sec. T2=3 sec.

**Three-inertial**K=1, T1=5 sec. T2=3 sec. T3=0.5 sec.

Such

**objects**are often found in

**industry**, especially in the

**chemical**industry. Their

**time**constants are of course larger. Each

**experiment**is a response to a step

**x(t)**, and then to a step

**x(t)**and a disturbance

**z(t)**. The duration is

**1 minute**for the step

**x(t)**and

**2 minutes**for the step

**x(t)**and disturbance

**z(t)**.

**Chapter 23.2 Transmittance Gz(s) and fixed gain Kz of a closed system**

This is a short reminder** Chapter 16 How does feedback work?**

**Fig. 23-1**

At the entrances of objects**– G(s)** in an **open system****– Gz(s) **or** G(s)** in a **closed system**

A unit step **x(t)=1(t-3)** is given, i.e. a **step x=1** with a delay of **To=3sec**

Open system **G(s)**

G(s) is a **three-inertia**l object with gain **K=10** and time constants **T1=1sec, T2=0.5sec and T3=0.1sec**. From the **G(s) numerator** is a gain **K=10** in steady state. This amplification as **K=y(t)/x(t)** from the response to step **x(t)** confirms the theory. Indeed, in **steady state**, i.e. after about **20 seconds**–>**y1(t)=10, x(t)=1. So K=10/1=10.****Closed system Gz(s)**

The formula for the **transmittance** of a **closed** system **Gz(s)** and the resulting formula for the **steady-state** gain** Kz** are also presented. You will see this in experiments as the steady state output **y(t)=y** when **x(t)=1** . For example, **y=0.91** or **y=0.99**.**Let me remind you that the formula for Gz(s) and for Kz is based on the assumption that the equation K*e(t)=y(t) is satisfied in the steady state.**

Then the output signal** y(t)** tries to follow (more or less exactly) the input signal setpoint **x(t)**, and is immune to **disturbances**. This is the basis of **automatics**. If this is not obvious to you, go back to **chapter 16**.

**Chapter 23.3 P controller with a one-inertial object****Chapter 23.3.1 Introduction**

The **P** controller has only one setting – **Kp**. In **chapter 22** (Fig. 22-2d) we have shown that** static objects** usually have gain **K=1**. Therefore, the **Kp** of the controller is also a **gain** of the entire open system containing the **controller** and the **object**.

This approach greatly facilitates the adjusting of the settings of the **controller**. We will analyze the **y(t)** response to the **x(t) step** at various **Kp** settings. **First**, let’s examine the **object itself**. This is where the adjusting of the settings of each **controller** often begins.**Chapter 23.3.2 An one-inertial object in an open system**

**Fig. 23-2**The animation confirms that it is an

**one-inertial unit**with

**K=1**and

**T=10sec**

**Chapter 23.3.3 P Control Kp=10**

The

**proportional**controller, otherwise the

**P**-type

**controller**, performs the simplest mathematical operation. Namely, it constantly observes the output

**y(t)**and the input

**x(t)**and calculates the control signal

**s(t)**:

**s(t) = Kp*e(t) = Kp*[x(t) – y(t)]**. Here

**Kp=10**.

**Fig. 23-3**Response to a

**x(t)**unit step of the controller

**P**when

**Kp=10**

Entire

**chapter 16**is to repeat ad nauseam that

**negative feedback**systems tend to a

**steady state**where

**K*e(t)=y(t)**. And we called the time-varying quantity

**K*e(t)**the goal pursued by

**y(t)**. This target is nothing more than the control signal

**s(t)**of the

**P-type**controller.

The calculated

**K=0.91**is confirmed with the

**time chart**. It can also be justified by simple intuition.

**The one-inertial**unit, which may be e.g. an

**RC**system, will be in equilibrium when

**s(t)=y(t)**and additionally nothing “moves”, i.e. when the time charts

**s(t)**and

**y(t)**are horizontal.

For

**t=3 sec**, the output

**y(t)=0**, because the

**one-inertial unit**(e.g. RC) is “loaded” from

**0**. Initially, i.e. in

**3 sec**, the control signal

**s(t)=10*(1-0)=10**. The oscilloscope clips the time charts at level

**2**and therefore you don’t see

**s(t)=10**. So

**RC**is loaded at maximum

**speed**. In a “moment” there will be a small positive

**y(t)**and the control signal

**s(t)=10*[s(t)-y(t)]**is a little smaller, and the rate of increase of the signal

**y(t)**will decrease a bit. When will

**y(t)**stop growing? Then when

**y(t)=s(t)**. This will happen after approx.

**15 seconds**.

One more thing. The process is similar to that of the previous

**chapter**when the

**Client**was steering manually. Similar, but not quite. This is due to human imperfection. With more experience with manual controls, the “human”

**time chart**would be more like that.

And if the

**Client**did it precisely by calculating:

**s(t)=10*[s(t)-y(t)]**every

**0.01 sec**?

This would get the

**exact time**chart in

**Fig. 22-3**. This is what the

**Scilab**program, which I use for

**all simulations**, did. The

**Client**wasn’t as good at steering as

**Scilab**, but he subconsciously steered as a

**proportional**controller algorithm. At the beginning there was a

**large**error, it gave a

**large**control

**s(t)**signal. The

**e(t)**error decreased, it reduced the control

**s(t)**, until it reached a steady state.

The gain

**Kp=10**of the proportional controller resulted in:

**1.**

**11-reduction**of the gain to

**Kz=0.91**

**2.**

**11-reduction**of the time constant to

**Tz=0.91 sec**

This is

**generally**the case for

**P**control. A large

**Kp**causes almost

**Kz=1**(always a little less than 1!) and increases the speed of the system.

I also propose a full view of the time chart, without the

**truncated**control signal

**s(t)**.

You will get exactly the same diagram as Fig.

**23-3**. What’s the difference? In the (invisible) oscilloscope

**settings**that provide a full view of all

**signals**, especially the control signal

**s(t)**.

**Fig. 23-4**

Unlike the previous animation, you can see how large the control signal **s(t)** is at the start of the **step s(t)**. The greater the gain **Kp** of the **P** controller. Thanks to this, the output **s(t)** signal at the beginning increases much **faster** than in the open system in **Fig. 23-2**. Let’s compare these **time charts** to appreciate the good job of the** P**-type controller.

**Fig. 23-5**The

**closed**system is

**11**times

**faster**. This is an obvious plus of

**P**control. Unfortunately, there was a steady-state error

**e=1-0.91=0.09**. This is typical for

**Proportional**control. The error can be reduced by increasing the

**Kp**gain .

I hope you don’t come to a brilliant conclusion.

An open (without controller) system is

**slower**than a

**closed**system, but its steady-state error is

**zero**!

Indeed, then

**y(t)=x(t)**. It’s a holy truth, but a system without controller is vulnerable to

**disturbances**!

**Chapter 23.3.4 P Control Kp=100**

Let’s also examine the influence of the

**Kp**setting of the

**P**controller on the quality of control

**Fig. 23-6**The system is clearly

**faster**and the

**e(t)**error is

**smaller**. In steady state, the

**red y(t)**almost coincides with the

**black x(t)**while the

**green e(t)**is almost

**zero**. This is also confirmed by the theory where approximately

**Kz=0.99**and

**Tz=0.1 sec**. The oscilloscope clips

**s(t)**at height

**2**. So let’s see the full time chart with other oscilloscope settings. The diagram will be identical, but with different oscilloscope settings.

**Fig. 23-7**

Now you see the whole **s(t)**, but **x(t), y(t) and e(t)** are almost invisible. At high gains **Kp**, the control signals reach **s(t)** very large **values**. For example – a furnace in a steady state requires only **10 kW** of power, and at the beginning of the stroke as much as **1 MW**! Only a mad designer would give such power. Practically, it will be, for example, **30 kW**. Of course, this will adversely affect the time chart, but not as much as **1 MW** differs from **30 kW**! We will return to the topic in** Chapter 31 Influence of Non-linearity on Control**

**Chapter 23.3.5 Summary of the inertial object controlled by the P controller**

It is easy to control. Increasing the

**Kp**parameter will not cause

**instability**or even

**oscillation**, but it will increase the control accuracy. Structural

**stability**at any

**Kp**is easily proved by

**Hurwitz**or

**Nyquist**, whose

**amplitude-phase**characteristic only passes through

**one quadrant**.

**Chapter 23.4 P controller with a two-inertial object****Chapter 23.4.1 Introduction**

We will repeat the same experiments, but with a **two-inertial** object. The responds may be more interesting.**Chapter 23.4.2 Two-inertial object in an open system**

**Fig. 23-8**

The object is a serial connection of **2** **single-inertial** objects with gains** K1=K2=1** and time constants **T1=3 sec** and **T2=5 sec**. The point of inflection characteristic for multi-inertial systems is visible. Determination of the time constants **T1**, **T2** is not as simple as for the **inertial** one.**C****hapter 23.4.3 P Control Kp=10**

**Fig. 23-9**

The diagram differs from **Fig. 23-5** only in another more complicated object.

Also steady state will occur when **s(t)=y(t)**, and “nothing is moving”. So all derivatives are **zero**. This is after **30 seconds**. Unlike the control of the inertial term, s(t) sometimes exceeds y(t). Then s(t) becomes less than **y(t)**, then exceeds **y(t)** again. And so after a few “pendulums”** y(t)** will arrive exactly at the calculated fixed value **y=0.91**. The control signal **s(t)** has large amplitudes** cutted** by the oscilloscope. To see the whole **s(t)** we will repeat the experiment with other oscilloscope settings.

**Fig. 23-10**But would you be satisfied with a

**controller**that you set the setpoint

**x(t)=100ºC**and it only gives

**91ºC**. Unfortunately, that’s the beauty of

**P-controller**. Notice that

**s(t)**is

**negative**at times. So it cools. In order to reach a steady state

**faster**.

There is no

**zero**error

**e(t)**. The only thing I can do is increase the

**Kp**of the controller, e.g. to

**Kp=100**.

**Chapter 23.4.4 P Control Kp=100**

**Fig. 23-11**The block diagram obscures the lower

**e(t)**and

**x(t)**signals a bit, but you will see them in the animation.

It is true that in steady state

**y(t)=0.99*x(t)**, i.e. y(t) is almost equal to the setpoint

**x(t)**, but at the expense of larger overshoots of

**y(t)**. In

**Fig. 23-6**, we studied P-control for an

**one-inertial**term and

**Kp=100**. Here we could arbitrarily increase

**Kp**and the system reached the

**steady state**quickly and without

**overshoots**. At large

**Kp**, the

**fixed**error

**e(t)**was almost

**zero**. With a

**two-inertial**term and

**Kp=100**it is not so nice. Control

**time**and

**oscillations**are

**unacceptable**.

**Chapter 23.4.5 Summary of a two-inertial object controlled by the P controller**

It is more difficult to control than

**one-inertial**. It can be proved, for example, from the

**Hurwitz**criterion that increasing the

**Kp**parameter will not cause instability. Also with

**Nyquist**which “walks” only in

**2**quarters of the open

**object**. Instead, there will be

**oscillations**and a longer

**calm down time**.

**Chapter 23.5 P controller with a three-inertial object****Chapter 23.5.1 A three-inertial object in an open system**

**Fig****. 23-12**An

**one-inertial unit**with a time constant

**T=0.5 sec**was added in series to the

**two-inertial**. You will see how such a small change (small

**To=0.5 sec!**) can mess up a

**feedback system**.

**Chapter 23.5.2 P Control Kp=10**

The system differs from Fig.

**23-9**only by an additional

**one-inertial unit**connected in series.

**Fig. 23-13**

If I proposed such a control to the Client, he would set off the dogs. Not only is there a large **error e(t)=0.09**, but it dangles after one **minute**. Increasing the gain to **Kp=30** probably won’t improve the situation!**Chapter 23.5.3 Control Kp=30**

**Fig****. 23-14**The system has become

**unstable**. Although the

**closed-loop gain**formula for instability is a bit of a no-brainer, it’s not entirely so. Here

**Kz=0.97**means that there is a

**DC**component of

**0.97**in the

**oscillation**. Let’s see what the oscilloscope cuts off. You will get a

**diagram**exactly the same as in

**Fig. 23-14**. Only the oscilloscope settings are such that the entire control signal

**s(t)**is visible

**Fig. 23-15**The scale of the oscilloscope is so small that the unit step

**x(t)**is almost invisible, close to the

**time axis**. However, you can see how

**y(t)**starts to swing, especially the much larger

**s(t)**.

**Chapter 23.5.4 Summary of the inertial object controlled by the P controller**

The hardest to control. The

**amplitude-phase characteristic**of the open system passes through

**3 quadrants**. Therefore, according to the

**Nyquist criterion**, a closed system may (but does not have to)

**be unstable**.

**Chapter 23.6 How does the P controller suppress disturbance?**

**Chapter 23.6.1 Introduction**

Suppression of

**disturbances**is the main job of the

**controller**. There would be no

**automatics**if he couldn’t do it. We previously studied

**the response**to a

**unit step x(t)**that lasted

**1 minute**. It’s still bearable. But the

**next experiments**will be

**2 minutes**. First there will be a

**unit step x(t)**and then at

**70 seconds**a

**disturbance z(t)**(also unit step)

**positive**or

**negative**. It’s like putting an extra

**heater**or

**cooler**into the liquid. The

**automatics**should compensate for this

**disturbance**. That is, with an additional

**heater**, the

**controller**should

**reduce**the supplied power, and with a

**cooler**,

**increase**it. In total, the

**experiment**will last

**2 minutes**. It’s not much, but some people may find it irritating.

We will examine the

**P**-type control with

**positive**or

**negative**disturbance for previously known

**objects**:

**– single-inertial**

**– two-inertial**

**– three-inertial**

**Chapter 23.6.2 Positive disturbance with a one-inertial object Kp=10**

We will start with a one-inertial object with a controller

**Kp=10.**

**Fig. 23-16**The disturbance

**z(t)**here is additional heating

**+0.2**. As if an additional

**heater**appeared, or the voltage on the heater increased. Up to

**70 sec**. as in

**Fig. 26-3**. Per disturbance

**z(t)=+0.2**in 7

**0 sec**. controller reacted correctly. He lowered the

**heating power**on the heater behind the controller. Although there is a slight disturbance effect, it has been attenuated

**11 times**. Complete

**damping**, as you will see later, will only be provided by the integral control

**e(t)**i.e.

**I, PI**or

**PID**.

Compare with the manual control in

**Fig. 22-16**in

**Chapter 22**. Similar time charts? Yes, only

**transients**are so much

**better**! But with

**manual control**, the

**steady-state**error seems a little

**less**. Weird a bit. Why

**?**Because you subconsciously turned on the

**I**component, which is

**integration**. You reacted to a constant error by

**gradually slowly**reducing the control signal

**s(t)**. We will deal with integration in control later.

**Chapter 23.6.3 Negative disturbance with a one-inertial object Kp=10**

**Fig. 23-17**Here the

**disturbance**is negative,

**z(t)=-0.2**. The

**controller**reacted

**correctly**, i.e. with

**additional**heating. But the

**error**is not

**zero**here

**e=0.09**. It can be reduced by increasing the

**controller**gain, e.g.

**Kp=100**. Let’s check it.

**Chapter 23.6.4 Positive disturbance with an one-inertial object Kp=100**

**Fig. 23-18**The

**oscilloscope**cuts the control signal

**s(t)**at the

**+2**level. But you probably realize that at the beginning of the

**unit step x(t)**the control signal

**s(t)**reaches

**+100**! The greater

**Kp**caused the

**e**error to be almost

**zero**and the reaction almost

**instantaneous**. The system hardly reacts to

**disturbances**.

**Chapter 23.6.5 Negative disturbance with an one-inertial object Kp=100**

**Fig. 23-19**The regulator reacted correctly, i.e. with additional heating.

**Chapter 23.6.6 Positive disturbance with a two-inertial object Kp=10**

**Fig. 23-20**The beginning i.e. the

**response**to the

**x(t)**step is of

**time chart**as in

**Fig. 23-9**. Note that the

**time scale**is different. A bit too much

**oscillation**and long

**calm down time**time. The parameter

**Kp=10**is not too large and therefore the fixed error is

**considerable**. The suppression of

**disturbance**looks better. Less

**oscillation**and shorter settling time. Often the response to disturbances is more important than reaching the set point

**x(t)**. Why? Because the disturbance suppression is

**continuous**, while changes of the setpoint

**x(t)**are less frequent. Nevertheless, it is difficult to present such an arrangement to the

**client**. A bit of a shame about the large fixed error. The experiment is an example that the response to a

**unit step**is different than to a

**disturbance**! This is because the disturbance transmittance

**Gzakl(s)=y(s)/z(s)**is different from the

**closed-loop**transmittance

**Gz(s)=y(s)/x(s)**. This is often forgotten when examining the response to a setpoint

**x(t)**and not to a disturbance

**z(t)**.

**Chapter 23.6.7 Negative disturbance with a two-inertial object Kp=10**

**Fig. 23-21**Correct reaction. The system tries to compensate for the cooling with additional heating.

**Chapter 23.6.8 Positive disturbance with a three-inertial object Kp=10**

**Fig. 23-22**

Even with such a small gain **Kp=10**, the oscillations and the signal calm down time **y(t)** are too **large**. Although it suppressed the “heating” disturbance **z(t)=+0.2 by **lower control **s(t)**. Fortunately, there is a **PD** control. We’ll talk about that in the next chapter.**Chapter 23.6.9 Negative disturbance with a three-inertial object Kp=10**

**Fig. 23-23**

No comments.**Fig. 23-14** and **23-15** show that for** Kp=30** the system is **unstable**. Therefore, we will not test the disturbance response **z(t)** for this **Kp**

**Chapter 23.7 Why is a closed system better than an open one?**

**The**

Fig. 23-24

Fig. 23-24

**open**-top and

**closed**-bottom systems, which are affected by the following signals:

– input

**x(t)**

– disturbance

**z(t)**.

The problem affects all controllers. Not only

**P-controllers**. For many it is a question like why is it better to be

**healthy**and

**rich**than

**poor**and

**sick**? I, for example,prefer

**healthy**and

**rich**.

But if you still have doubts, compare the

**simultaneous response**of the

**open**system

**y1(t)**with the response of the

**feedback system y2(t)**to the input signal

**x(t)**and the disturbance

**z(t)**. The

**controlled**and

**open object**will be the previously known

**two-inertial**unit. The regulator is of type

**P**, because we do not know another one yet.

**Fig. 23-25**Compare the response

**y1(t)**of the

**open**system and

**y2(t)**of the

**closed**system. Up to

**70 seconds**, some may still be wondering what they like more. The “open” signal

**y1(t)**takes a little longer to reach

**x1(t)=1**, but without

**oscillation**.

And the most

**important**

**y1(t)=x(t)=1**!

What cannot be said about

**y2(t)**, where the error is as much as

**9%**! It would seem to be

**1:0**for an

**open**system. But the spell disappearsat

**70 seconds**when

**negative**disyurbance z(t) (cooling) occurs. “Open”

**y1(t)**dropped by as much as

**20%**and “closed”

**y2(t)**by only

**2%**.

You can still complain about

**error**of the

**closed**system. Fortunately, automatics has the tools to bring it down to

**zero**. The

**PI**and

**PID**controllers in the following chapters will do this.

**Chapter 23.8 Conclusions of P-controller****P**-type **control**

– It is the **simplest** controllwe from the **PID** group

– **Accelerates** the output **time chart** and suppresses **disturbance** quickly.

– Does not provide **zero** error **e** in **steady state**. That is, the **steady** output signal **y(t)** is always **smaller** than the setpoint **x(t)**.

– The **e** error is smaller the **higher** the controller gain **Kp**

– For **one-inertial** units, the gain **Kp** can be very large. We can then assume that the **error** is almost** zero** and the response **y(t)** is immediate. This approach is also possible for **multi-inertial** units, when the remaining time constants **T1, T2** … **Tn** are much smaller than the basic time constant **T1**. Then we treat it “almost” as **one-inertial**. I**nstability** may occur, but only with very large **Kp**.

The disadvantage of the **P** type control is that it requires a larger **Kp** than **PI** or **PID**. And it’s not just **oscillations** and possible **instability**. A large **Kp** causes that at the very beginning the control signal **s(t)** is, for example, **10 times** greater than it is necessary for the output signal **y(t)** to come to a value close to the setpoint **x(t)**. In the case of water heating, this means that the heater power is also **10** times **greater** than in the** steady** state. Also the **cables** should be **bigger**.

So far we **have assumed** that there are no **power constraints**. In practice, there will always **be some**, even from the **supply voltage**. This will result in a slower **time charts** than **ideal**. We will come back to this topic in **chapter 30**.

Generally, in the** P control** (and not only) there is a basic **contradiction** of **goals**. Large **Kp** is a small control **error** – great, but also **oscillations** and even **instability**. In the next chapter, you will learn that adding a** derivative** component **D** to the controller **P** works like a **balm**. It “calms down” and allows you to give a larger **Kp**->smaller control **error**, although it still does not provide zero **error e(t)**. Then we are dealing with a **PD** type regulation.