Classification Of A Control System Computer Science Essay

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Engineering is applying technical, mathematical & scientific knowledge to design numerous applications which help us in day-to-day life. Engineers are concerned with understanding the forces of nature and controlling the materials for the benefit of mankind. Control systems engineers are concerned with controlling and understanding the segments of their environment called as system. Control systems engineering consists of various branches such as Classical Controls, Modern Controls, Robust Controls, Optimal Controls, Adaptive Controls, Non-linear Controls, & Game Theory. Control systems are an integral part of modern society. A control system consists of subsystems and processes assembled for the purpose of controlling the outputs of the processes. They are four components for a control system. They are Process, Measuring element, the Controller and the Final control element. A control system is build for four primary reasons. They are Power Amplification, Remote Control, Convenience of Input form and Compensation for Disturbances. It also enables us to learn about the mathematical model and use it effectively. The mathematical model can be controlled using control system tools based on their complexity of the model. These tools will also enable us to control it in time domain and also in frequency domain. Control systems are also useful in dangerous or remote locations and also used to provide convenience by changing the form of the input.

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This report mainly discusses about the characteristics of Dead-Time compensation, its effect and rectification.

CHAPTER 1

CONTROL SYSTEM

A control system can be defined as a set of device or interconnection of components to manage, command, direct or regulate the system configuration that will provide a desired system response.

Figure 1 Basic flow diagram of a control system

1.1Classification of a Control system

Control systems are classified into two classes based on combinations and variations.

Open-Loop Control System

Closed-Loop Control System

1.1.1Open-Loop Control System:

An Open Loop Control System is controlled only by input signal. The system does not include feedback. Such a system can be represented by a block diagram which is shown in Figure 1.1b.

Figure 1.1a Process to be Controlled

An Open Loop Control System uses an actuating device to control the process without using feedback.

Figure 1.1b Block Diagram of Open-Loop System

The output remains constant for a constant input signal provided the external conditions remain unaltered. The response of an open-loop system is dependent on the characteristics of the system. The output may be changed to any desired value by appropriately changing the input signal but variations in external conditions or internal parameters of the system may cause the output to vary from the desired value in an uncontrolled fashion. The open loop system is satisfactory only if such fluctuations can be tolerated or system components are designed and constructed so as to limit parameter variations and environmental conditions are well-controlled.

1.1.2Closed-Loop Control System:

A Closed-Loop System is the instantaneous state of the output is feedback to the input and is used to modify it in such a manner so as to achieve the desired output i.e. in a closed loop system control input is affected by the system output. Such a system can represent the block diagram shown in Figure 1.2.

Figure 1.2 Block Diagram of Closed-Loop System

By means of the negative feedback loop the accuracy of the system output is improved when compared with open-loop system and the system becomes more stable. The closed loop system can achieve greater accuracy than open-loop systems, but it is acceptable only if fluctuations (noises) can be tolerated or system components are designed and constructed so as to limit parameter variations and environmental conditions are well-controlled. An error detector compares the signal received through the feedback elements, which is a function of the output response, with the reference input. Any dissimilarity between these two signals constitutes an error or actuating signal, which actuates the control elements. The control elements (Controller) in turn alter the conditions in the system in such a manner so as to reduce the original error.

1.2 Control System Design

The control system is majorly used when it satisfies these conditions

Suppressing the influence of external disturbances

Ensuring the stability of the process

Optimizing the performance of the system

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The following table shows the control system design process.

Table 1.2 Design Process

Based on the criteria mentioned above a system can be designed based on certain objective.

CHAPTER 2

CONTROLLERS

A controller monitors and affects the output variables of the system, which is affected by adjusting the input variables. The controllers are classified into six types.

Proportional Controller (P)

Integral Controller (I)

Derivative Controller (D)

Proportional Integral Controller (PI)

Proportional Derivative Controller (PD)

Proportional Integral Derivative Controller (PID)

Since all the controllers have several drawbacks and PID has fewer drawbacks when compared with other controllers, we mostly use PID controllers.

Proportional Integral Derivative Controller (PID):

A PID controller is the most widely used controller in industries. It is used to control devices and to maintain control of a varying system. A PID controller corrects error among a measured process variable and a desired set-point. The integral term allows the elimination of a step disturbance and the derivative term is used to provide damping or shaping of the response. The response of PID controller can be explained in terms of responsiveness of the controller to an error, i.e. the amount to which the controller overshoots the set-point and the amount of system oscillation. Tuning of the PID controller is important, which is nothing but tuning of three constants in PID, which provide us control action for a specified process requirements. The output of PID is defined by a combination of P+I+D. The mathematical form is given below.

U (t) = KP e (t) + KI d + KD ......................................................... (1)

Where,

U (t): output of PID

KP: Proportional Gain

KI: Integral Gain

KD: Derivative Gain

E: Error signal = SP - PV

T: Instantaneous Time

Figure 2 Block Diagram of PID controller

Limitations of PID controller:

PID controller loop gains must be condensed so that the control system does not overshoot or oscillate. It can be used effectively among Set-Point (SP) and the Process Variable (PV). Another limitation is they are linear. So, it is often enhanced through methods like PID gain scheduling or Fuzzy Logic. High sampling rate, measurement precision and measurement accuracy are necessary to achieve complete control performance. The overall performance can be improved by combining the feedback control of a PID controller with Feed-Forward control. Another limitation with controller using PID gains alone is that the proportional term requires an error to generate an output and integrator term requires an error and time.

CHAPTER 3

DEAD-TIME

Whenever an input variable of a system changes, there is a time interval (short or long) during which no effect is observed on the outputs of the system. This time interval is called dead time or transportation lag or pure delay or distance-velocity lag. By overall Dead-time is the delay when a controller output signal is received until when the measured process variable first begins to respond.

Figure 3 (A) Feedback with Dead-time

Figure 3 (B) Feedback with Dead-time compensation

Figure 3 (C) Feedback with Dead-time compensated

3.1 Dead-time Compensation:

A time lag (time delay or dead time) limiting the permissible process gain (PG) reduces the ability to control the process. So, a controller mechanism is necessary to reduce this limitation. This mechanism is called the 'Dead-time compensator'. Smith Predictor is extensively used to reduce Dead-time. Smith Predictor gives a new controlled variable that is the reaction of the process variable to its controller output without Dead-time. It needs three parameters such as process gain, Dead-time and the time constant. Smith Predictor uses these parameters to construct models of the process from the controller output. The maximum permissible controller gain is inversely proportional to Dead-time. The controller gain can be hypothetically increased without any limit provided ignoring justifying circumstances such as loop interaction, measurement noise, resolution and the final element dead band.

3.2 Effects of Dead-time:

Dead-time is practised when using fast timing discriminators for counting or timing measurements. It is also practised when linear amplifiers combined with a discriminator for a pulse-height selection and also due to photon counting distribution. Quantum cryptography speed is limited by Dead-time. Dead-time effects can be seen in phase of a system. It reduces phase margin (PM) of a system. PM is positive without Dead-time and negative with Dead-time.

3.3 Need for a Dead-time compensator:

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A small adjustment made in the input variable will affect the whole process. A Dead-time compensator is used to reduce the controller gain (CG), the control action, time delay, and to stabilise the process. For the effective use of the controller there arises a need for Dead-Time Compensator.

CHAPTER 4

NYQUIST CRITERION

The frequency response of a system can be represented either as a polar plot or as separate magnitude and phase diagrams. The Nyquist criterion relates the stability of a closed loop system to the open loop frequency response and the open loop pole location. Thus, knowledge of the open-loop system's frequency response yields information about the stability of the closed-loop system.

Distinct advantages for using Nyquist criterion

Modeling transfer function from physical data

Designing lead compensators to meet a steady-state error requirement and a transient response requirement

Finding the stability of non-linear systems and

Settling ambiguities when sketching a root locus

4.1 Derivation of the Nyquist Criterion:

Consider a system shown in figure. The Nyquist criterion can tell us how many closed-loop poles are in the right half-plane. We learn four major concepts based on this derivation. They are,

Relationship between the poles of 1+G(s)H(s) and the poles of G(s)H(s)

Relationship between the zeroes of 1+G(s)H(s) and the poles of closed-loop transfer function T(s)

Concept of mapping points

Concept of mapping contours

Figure 4 System used for Nyquist Criterion

From these equations we can conclude that the poles of 1+G(s)H(s) are same as the poles of G(s)H(s), open-loop system and the zeroes of 1+G(s)H(s) are same as the poles of T(s), the closed-loop system.

4.2 Sketching the Nyquist Diagram:

The contour that encloses the right half-plane can be mapped through the function G(s)H(s) by substituting points along the contour into G(s)H(s).

The points along the positive extension of the imaginary axis yield the polar frequency response of G(s)H(s).

Approximations can be made to G(s)H(s) for points around the infinite semi-circle by assuming that the vectors originate at the origin.

4.3 Stability using the Nyquist Diagram:

The Nyquist Diagram can be used to determine a system's stability, using the simple equation Z = P - N. The values of P, the number of open-loop poles of G(s)H(s) enclosed by the contour, and N, the number of encirclements of the Nyquist diagram makes about -1, are used to determine Z, the number of right-half plane poles of the closed-loop systems. The approach is to set the loop gain equal to unity and draw the Nyquist diagram. Since the gain is simply a multiplying factor, the effect of the gain is to multiply the resultant by a constant anywhere along the Nyquist diagram.

CHAPTER 5

CONTROLLER TUNING

The adjustment of the controller parameters to achieve satisfactory control is called controller tuning. Tuning a control loop is adjustment of control parameters so that the controller will be able to reduce an error quickly without causing the process variable to oscillate. For PID controller three control parameters have to be tuned, namely, gain or proportional band, integral gain or reset, and derivative gain or rate. There are three approaches for tuning a controller.

Use one-quarter decay ratio

Use time integral performance such as ISE, IAE or ITAE

Use semi-empirical rules

Other than these approaches and the two process of controller tuning there is also Manual Tuning method. Manual tuning method is trial-error method. It is setting KI, KD values to Zero and increasing the value of KP so that the system oscillates. Mostly, manual tuning process is used for only studying about the controller. A tabular column will show how increasing the parameters affect the system process.

Effect of increasing parameters

Serial No.

Factor

Rise Time

Overshoot

Settling Time

Inaccuracy at equilibrium

1.

KP

Decrease

Increase

Increase

Decrease

2.

KI

Decrease

Increase

Increase

Eliminated

3.

KD

Indefinite

Decrease

Decrease

None

Table 5.1 Manual Tuning Method

5.1 Controller tuning techniques

They are two process of controller tuning based on classification of control system.

Open loop tuning process

Closed loop tuning process

5.1.1 Open loop tuning process

John G. Zeigler and Nathaniel B. Nichols proposed a method for tuning a control loop effectively. Their method is a way of relating the process parameters - delay time, process gain and time constant with the control parameters - controller gain and reset time. Zeigler-Nichols has proposed three ways for open loop tuning.

Zeigler-Nichols - Open Loop Reaction Curve Method

Zeigler-Nichols - Open Loop Point Of Inflection Method

Zeigler-Nichols - Open Loop Process Gain Method

Based on the methods suggested above the open loop tuning can be done. There is always the same procedure for open loop response but the calculation varies in each method. The common procedure is

Look at the Open Loop response

Analyse the Open Loop response

Evaluate the Controller parameters

Use any one of the tuning method

Run the process again with the corrective action

The essential drawbacks are it is difficult to determine whether the system has reached the steady state and the disturbances influence the result significantly.

5.1.1.1 Zeigler-Nichols-Open Loop Reaction Curve Method

The process reaction curve is an approximate model of the process assuming the process behaves as a first-order system with Dead-time.

Figure 5.1.1.1 Process Reaction curve of an Open Loop System

Zeigler-Nichols - Open Loop Reaction Curve Method

Serial No.

Controller

Gain

Reset

Derivative

1

P

X/DR

None

None

2

PI

0.9 X/DR

0.3/D

None

3

PID

1.2 X/DR

0.5/D

0.5/D

Table 5.1.1.1 Open Loop System Reaction Curve Method

Where,

X % Change of output

R % Change at the Point of Inflection (POI)

D min Time till the intercept of tangent line and the process value

5.1.1.2 Zeigler-Nichols-Open Loop Point of Inflection Method

Figure 5.1.1.2 Point of Inflection of an Open Loop System

Zeigler-Nichols - Point of Inflection Method

Serial No.

Controller

Gain

Reset

Derivative

1

P

X/DR

None

None

2

PI

0.9 X/DR

0.3/D

None

3

PID

1.2 X/DR

0.5/D

0.5/D

Table 5.1.1.2 Open Loop System Point of Inflection Method

Where,

X % Change of output

R % Change at the Point of Inflection (POI)

Ti min Time from output change to POI

P % Process value change at POI

D = Ti - (P/R)

5.1.1.3 Zeigler-Nichols-Open Loop Process Gain Method

Figure 5.1.1.3 Process Gain of an Open Loop System

Zeigler-Nichols - Open Loop Process Gain Method

Serial No.

Controller

Gain

Reset

Derivative

1

P

L/GPD

None

None

2

PI

0.9 L/GPD

0.3/D

None

3

PID

1.2 L/GPD

0.5/D

0.5/D

Table 5.1.1.3 Open Loop System Process Gain Method

Where,

L m Predominate Lag

GP % Process Gain

D min Time till the intercept of tangent line and the process value

5.1.2 Closed loop tuning process

The main weakness of the transient response method is that it is sensitive to disturbances because it depends on open-loop environments, but closed loop method avoids this difficulty because it is performed in closed loop. John G. Zeigler and Nathaniel B. Nichols proposed a method for tuning a control loop effectively. Zeigler-Nichols closed tuning is based on stability margins. Their method is based on critical gain and critical period. This method has more advantage over others, i.e. it is easy to control the amplitude of the limit cycle by an appropriate choice of the relay amplitude. The procedure for closed loop tuning is discussed below

Place the controller with low gain, i.e. only proportional control

Gradually raise the gain

Make small change to the Set-point until the oscillations start

The value of gain and the period of oscillations give the GU, PU

Figure 5.1.2 Ultimate Gain of a Closed Loop System

Zeigler-Nichols - Closed Loop Method

Serial No.

Controller

Gain

Reset

Derivative

1

P

0.5 GU

None

None

2

PI

0.45 GU

1.2/PU

None

3

PID

0.6 GU

2/ PU

PU /8

Table 5.1.2 Calculation for a Closed Loop System

Where,

PU Ultimate Period of oscillations

GU Ultimate Gain

Different tuning methods are based on different performance criteria. But the best way of tuning a controller is Zeigler-Nichols tuning method. Derivative control is very sensitive to noise but best suited for Dead-time and lag. Higher the gain is

The greater the rejection of disturbance

The greater the response the set-point changes

Based on the gain we can arrive into two results

Reducing the Dead-time increases the maximum gain and the controllability

Rising the ratio of the lag increases the controllability

Smaller the gain gives more marginal stability as the ultimate gain.

Advantages of Zeigler-Nichols method

No need to operate at a stability limit for a process

Controller settings are easy to calculate

Determining Dead-time and process time lag is easy

Less chance of saturating control loop components

Disadvantages of Zeigler-Nichols method

The relations stated holds good for some limit only and there is restrictions also

If the system is so noisy it is difficult to determine the reaction curve

CHAPTER 6

Classical Smith-Predictor Sensitivity & Stability Criteria

Proposition to illustrate the sensitivity and system criteria with relation to system gain and time-delay variation

Smith Predictor is a good way to compensate dead-time in a system. Figure 6.1 shows a classical Smith-Predictor structure which has the transfer function as,

Figure 6.1 Classical Smith Predictor Structure

Here GO denotes the real system & Gm describes the real system, whereas, τM and τO are the real delay and model delay. Let ω1 and ω2 be the frequency values.

Let's assume main controller as Gc(s) = and also assume GO is a stable system.

Proposition

The system's characteristic equation can be expressed as follows.

We assume =

So we can design and obtain,

The system characteristic equation can be divided into two parts:

Therefore,

Then the dual-locus diagram is obtained when s traverses the Nyquist contour. The argument of is the angle between the vector joining the corresponding points on loci and and the real axis. Therefore the following is derived.

Due to and being a periodic function of ω and symmetric to real axis, it is obvious that the Nyquist curves have numerous intersection between and . Let ω1 and ω2 be the frequency value, respectively, at the intersection point. The system will be unstable, if the point on curve crosses the Nyquist line before the point on the Nyquist line reaches the intersection. Then the system will be stable if ω1 < ω2. When for ω1, ω2 ϵ R, the proof of the proposition is achieved.

So,

Inference made from proposition:

The right hand sides of (1) and (2) are periodic and are bounded by a real number 2. So, KP is delay-dependent with τO and τO-τM. Since ω1 < ω2, KP has to be decreased. KP decreases if τO becomes larger and KP decreases if τO-τM becomes larger at some fixed τO.

This can be clearly understood with an example. Let us apply this proposition to a first-order stable system. Assume that where, KP 0 and by a simple analysis, we reduce . Based on the proposition 1 and equation (1)-(3), we get,

Therefore, the solution to (4)-(6) is,

Where, ωS describes the proper solution of ω2. From these equations we come into conclusions like KP decreases if τO becomes larger and KP decreases if τO-τM becomes larger at some fixed τO.

From the equations (4)-(6) (7), if τO = 1 and τM = 1.1 the solution ωS = 12.9030 and KP = 19.3125 can be obtained. For the values of KP greater than 19.4 the system becomes unstable. For different values of KP is simulated using Matlab.

CHAPTER 7

New Modified Smith-Predictor

New Modified Smith-Predictor is proposed based on the proposition derived earlier.

Figure 7.1New Modified Smith-Predictor

The symbol r denotes the set-point input. GC is employed to take care of the set-point servo-tracking. GC1 makes the system output error. If GO = GM together with Ï„O = Ï„M is satisfied, and GC1 works nothing to be zero and the standard Smith-Predictor structure is obtained. The Output error feedback is.

The closed-loop response to set-point input is given as,

If, then (11) becomes,

Control Scheme:

There are two controllers GC and GC1 in our MSP structure to be designed. The controller GC1 has the self-adaptive mechanism, which can make the simulated model output change adaptive to real system output. Assume E1 and Y1 denote the main controller output and simulated output. Therefore,

Then apply steady-state final value theorem,

If then,

Then the characteristic equation is stable. Now we can achieve the design of the controller GC1. It is designed, for simplicity, to be P type for the plant assumed as an integrator, tracking this form and to be a PI type for the plant assumed as a form

Controller tuning:

A. For the plant assumed as an integrator with dead-time:

The model assumed as and respectively. The delay free part of the above equation becomes,

where λ is the closed-loop parameter. So the parameter can be obtained by a suitable choice of λ. Based on the analysis above, we obtain,

where . The threshold value of KC1 can be obtained easily by dual-diagram Nyquist curve as follow:

Introducing the parameter φPM, the phase margin of the closed-loop system with the loop transfer function, we obtain:

The choice of φPM = 600 gives the system satisfactory performance, correspondingly

B. For the plant assumed as first-order stable plant with dead-time:

Consider with and . Correspondingly and choice of φPM = 600 gives

CHAPTER 8

PROPOSED MODIFIED

SMITH PREDICTOR STRUCTURE

Consider the plant with transfer function

(8.1)

where represents the time delay of L seconds. Assume that the time delay L is known and process is .

Here (8.2)

and L (process) = L (model). The modification in Figure 8.1 consists of the additional feedback path (0) from the difference of the plant output Y and the model output to the control input u. When = 0 the Smith predictor is used. When 0, then modified Smith predictor is obtained. The controller used in this Smith predictor is an I-PD controller where the integrator is in the forward path and the proportional and derivative control are in the feedback acting on the feedback signal. The controller was tuned by Ziegler -Nichols method and by relay method, since relay method gives the immediate value of ultimate gain and Ultimate period, when compared to ZN method it is also preferred for auto-tuning.

Y

-

-

+

R(s)

+

+

+

D(s)

+

+

+

-

-

+

+

-

d

K0

ym

Figure 8.1 Proposed modified smith predictor

The set point and the load disturbance response is given by

Where

(8.3)

(8.4)

where, (8.5)

and represents the estimated transfer function of the plant. From (8.3) and (8.4) we have

(8.6)

(8.7)

for the perfect model of the plant and perfect dead time of the process. If the closed loop system is stable, the set point and the disturbance response for step inputs are as follows

lim Lim (8.8)

sï‚®0 sï‚®0

lim Lim (8.9) sï‚®0 sï‚®0

for Ko ≠0.

This shows that there will be no steady state error for a constant load disturbance.

If we have the perfect model representation of the plant it follows from (8.6) and (8.7) that the stability of the modified smith predictor depends on the roots of the characteristics equation

(8.10)

Detailed analysis of the roots of the equation

(8.11)

can be performed by root locus technique. To derive the tuning formulae for the modified smith predictor, only the ultimate gain should be determined.

Equation (8.11) can be transformed to the following equation

(8.12)

where (8.13)

Then the Nyquist criterion can be applied to find the ultimate gain, guaranteeing that for. is obtained setting the phase margin to zero in the relations

(8.14)

=1 (8.15)

Thus the ultimate gain is given by

(8.16)

The parameter is the phase margin of the closed loop system with the loop transfer function (8.13) and appears just in consequence of the procedure used to analyze the roots of (8.11)

When the gain is given by, the closed loop system shown in figure (8.1) is stable. Even for higher order systems without resonant poles parameter , corresponding to ,provides satisfactory closed loop performances both in set point response and load disturbance rejection. So the next tuning formula is proposed for the gain .

(8.17)

From figure (8.1) and (8.17), it follows that tuning the modified smith predictor requires the adjustment of only three parameters and . The disturbance gain parameter can be found once we know the values of and which is obtained from the model.

Here tuning of the P,I and D values are performed by the relay feedback test. There are two parameters that result from this auto-tune test. One is the period (time between successive peaks),P, and the other is the amplitude of the process output, a. The period has units of time; the ultimate frequency can be found from

(8.18)

And the ultimate gain can be found from the amplitude

(8.19)

Conclusion

The traditional Smith-Predictor's limitations are analyzed and a proposition is derived for stability range with relation to system gain and dead-time

A New Modified Smith-Predictor for integrative and stable system proposed by Wang and Liu is given a detailed study for set point tracking

Based on the above study a New Modified Smith-Predictor approach is done using an I-PD controller for varying time-delay with adaptive tuning technique using least square algorithm

The New Modified Smith-Predictor design is completed and the adaptive tuning technique is under process