## INTRODUCTION

**PID** (**proportional integral derivative**) control is one of the earlier control strategies. Its early implementation was in pneumatic devices, followed by vacuum and solid state analog electronics, before arriving at today’s digital implementation of **microprocessors**. It has a simple control structure which was understood by plant operators and which they found relatively easy to tune. Since many **control systems using PID control** have proved satisfactory, it still has a wide range of applications in industrial control. According to a survey for process control systems conducted in 1989, more than 90 of the control loops were of the **PID** type. PID control has been an active research topic for many years; see the monographs. Since many process plants controlled by **PID controllers** have similar dynamics it has been found possible to set satisfactory controller parameters from less plant information than a complete mathematical model. These techniques came about because of the desire to adjust controller parameters in situ with a minimum of effort, and also because of the possible difficulty and poor cost benefit of obtaining mathematical models. The two most popular **PID techniques** were the step reaction curve experiment, and a closed-loop “cycling” experiment under proportional control around the nominal operating point. In this chapter, several useful **PID**-type controller design techniques will be presented, and implementation issues for the algorithms will also be discussed in Sec. The proportional, integral, and derivative actions are explained in detail, and some variations of the typical PID structure are also introduced.

## ABOUT OUR TRAINER

The following figure shows the pictorial & Front panel

## FRONT PANEL VIEW

## THE SIMULATED CONTROLLER:

The controller comprises the following elements:

One proportional band control scaled in % proportional band.

One integral action control scaled in integral action time

One derivative action control scaled in derivative action time

The Error detector generates the error signal, the difference between the feedback signal (PV) and the set point and passes it to the three term controller comprising the proportional, integral and derivative controls. The figure shows the diagram of PID controllers.

**3 PID CONTROLLERS**

## THE SIMULATED PROCESS:

The process comprises the following elements, any of which may be inserted or omitted from a complete process by the use of patch chords.

- One simple lag with switched selected time constants of 0.5msec or 20msec.
- Two lag / Integrators with switched selected time constant of 0.5msec or 200msec
- One distance - velocity lag of delay 20msec.
- One inverter.

Every element produces a polarity inversion but the inverter may be included or not as necessary to produce the desired overall polarity. The process section consists of a fast function. When it is fast position, The time constant of the process is 0.5 to 20 msec.

The **PCS unit** has an inbuilt simulated process. The **simulated process** consists of following components.

## ERROR DETECTOR:

The error detector compares the set point signal and feedback signal and produces an error signal which is fed to controller for control action.

**ERROR DETECTOR**

The set point for the error detector as shown in Fig is given from two sources Set Value, and Set Value Disturbance. The Set Value is a DC voltage and Set Value Disturbances are test signals viz. Square, Sinusoidal, impulse, Ramp etc., are added in a summer and then gives as set point to error detector in PCS. The user can change the Set Value from -12V to +12V using the Set value knob. The Set value disturbance can be given from the external source / inbuilt waveform generator in PCS. The Feedback signal for the error detector is the process variable. The error detector compares the set point and the feedback and generates an error signal which can be measured in terminal T1. The error signal is branched out into three terminals E1, E2, E3 for Proportional, Integral and Derivative controllers.

## THEORETICAL DESCRIPTION:

## Introduction

Computer Architecture course is the part of the bachelor studies at our faculty for many years. Few years ago we adapt them significantly in order to make them appropriate for the majority of the students. Lectures are conceived to get students acquaint with the basic properties of computers in general with emphasis to basic building components and their properties. This is closely associated with possibilities of its practical use. In the higher level studies students are pointed more to the theoretical topics of the control systems analysis and synthesis, often with missing connection between the theory and practice: how to implement the control algorithms in the embedded microcomputer. Therefore we created a set of closely interconnected lessons and exercises on the embedded systems. The idea of lectures is to show an implementation of the digital PS controller based on single chip micro-controller Atmel AVR with RISC architecture [Turley 1997]. Besides to the controller algorithm itself we implemented also the algorithms for manual and automatic control,

Local and remote parameter adjustments, event logging and control system upgrades. As the hardware base it was used the school evaluation board called MiniMexle [Pospiech et al. 2006]. Its use was mentioned also in [Balogh 2008]. See also simmilar topics in [Lodge 2002].

Presented problem should also answer the following questions for students:

In this article we will show details of the interconnection between the real A/D converter and its sampling rate. Another focus will be on the replacement of the D/A converter with the internal **PWM** generator. The overall concept of the lecture is to bring the students from the theoretical design and simulation of the controller to the real (and operating) application. Also we try to break the idea that digital control system is not so good as corresponded analogue circuit. Requirements on the _nal application take into the account the implementation on the 8-bit single chip micro-controller:

….Range of the controlled voltage 0 { 5 V.

….Required precision better than 1%.

….Controlled voltage ripple less than 5 mV.

….Elimination of disturbances.

The last requirement will be mentioned just marginally. Major part of the theoretical works assumes simply addition of the disturbance to the system output and it does not change the system dynamics. In our system the disturbance not only that changes the output voltage, but also influences the overall characteristics of the system, including its dynamic behavior.

## Controlled System

The controlled system is the simplest order system created by the resistor and capacitor (see Fig). Our goal is to control the capacitor voltage. At this moment we control the system without any additional load on its output.

Measured step response of the system. Scope CH1 (blue) 5 V/div, CH2 (red) 1 V/div, Time base 200 ms/div. Measured T63 = 492:0 ms

The values of R = 10 k and C = 50 _F were selected. For these values, corresponding time constant T = RC is 0:5 s. Transfer function of such system is

We do not describe here the selection of the R and C values, but their choice is limited e.g. by the maximum output current available, maximum frequency on the inputs of the A/D converter of the micro-controller, input impedance of the A/D filter and other factors.

## Parameter Identification

Implemented manual control mode was used also for verification and identification of the system. Measured data were captured using an oscilloscope (see Fig.2) and the time constant and gain were measured.

Real parameters of the system measured from the step response of the system were very close to the theoretical values.

## CONTROLLER DESIGN

From the classical control theory point of view, our goal corresponds with the continuous control system with **PI controller**. This figure (and also the following ones) are displayed with, physical" quantities { figure contains also the

ranges for inputs and outputs. The transfer function of the **PI controller** is

Parameters for the controller were designed using the inverse dynamics method as following:

KR = 1[{] and TI = 0:5[s].

This corresponds to the following continuous version of the controller:

## First order system:

These are characterized by one pole and (or a zero. A pure integrator and a single time constant having transfer function of the form K/s and K/(TS+1). Unit step responses of the system are computed as follows and are shown in fig.

c(S) / R(s)=G(S)=k/s then

for a unit step input,

The time constant of the system is defined from the above equation at t=T which gives

c(t) = k(1 - e^{-1}) = 0.632k

This is an important characteristics of the system which is also defined in terms of the slope of the response curve at t=0. For a proper viewing on a CRO.The step input needs to be replaced by a square wave of sufficiently high frequency.

## Second order system:

These systems are characterized by two poles and up to two zeros. For the purpose of transient response studies, zeros are usually not considered primarily because of simplicity in calculations and also because the zeros do not affect the internal modes of the system.

A second order system is represented in the standard form as

Where

r = damping ratio

ò = undammed natural frequency

Depending upon the value of ò, the poles of the system may be real, repeated or complex conjugate

a)Under damped case (0 < ò < 1)

## Closed loop systems:

Closed loop or feedback systems involves the 'measurement' of the output of the system and generation of control signals, which is based upon the 'error' under the influence of a 'command' or 'reference' and the measured value of output. The Block Diagram of Closed Loop System is shown in Fig

**Block diagram of a closed loop system**

The closed loop transfer function for different open loop functions are shown below.

which gives the response of the second order system depending upon the value of K.

**Step response of an under damped system**

Thus the response of a closed loop system as shown above in Figure - 22 can be altered by varying the open loop gain and hence it should be possible to choose K to obtain a `suitable' performance. This leads to the concept of performance characteristics as defined on the step response of an under damped second order system.

The transient response of a system to a unit-step input depends on the initial conditions. For convenience in comparing transient responses of various subsystems, it is a common practice to use the standard initial condition that the system is at initially with output and all time derivatives are zero.

The transient response of a practical control system often exhibits damped oscillations before reaching steady state. In specifying the transient response characteristics of a control system to unit step input, it is common to specify the following.

- Delay time td
- Rise time tr
- Peak time tp
- Maximum overshoot Mp
- Settling time ts

## Delay time

The delay time is the time required for the response to reach half the final value

## Rise time

Rise time is the time required for the response to rise from 10% to 90%, 5% to 95% or 0% to 100% of its final value. For under damped second order systems, the 0% to 100% rise time is normally used. For over damped systems, the 10% to 90% rise time is commonly used.

## Peak time tp

The peak time is the time required for the response to reach the first peak of the overshoot.

t_{p}=∏ / ω_{n √ 1 - ζ2}

## Maximum Overshoot (Mp)

The maximum overshoot is the maximum peak value of the response curve measured from unity. If the final steady state response differs from unity, then it is common to use the maximum percent overshoot. The amount of maximum overshoot directly indicates the relative stability of the system.

maximum percent overshoot = c(t_{p}) - c(∞) / c(∞) X 100 % %M_{p} = e ^{-(ζ / √ 1 - ζ2)∏ }