Lab 2 Report control systems

7/24/2019 Lab 2 Report control systems

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Experiment 1:

1.1. Requirements of the Experiment:

This experiment focused on understanding the system errors within all control systems. Observing

how the steady state error is affected with different function inputs. Looking specifically at step,

ramp and parabolic inputs and their associated outcomes.

For the second order transfer functions we were to determine the system type, constant errors for

different inputs and the steady state errors of inputs of various types. Furthermore, to easily

represent these characteristics, MATLAB scripts were used to calculate all error types and values.

To observe how different values and inputs altered the system response and steady state error this

was done with two different control systems.


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1.2. Introduction:

In real life applications of control systems, because of non ideal configurations, input applied and

non-linear sources, errors in systems are likely to appear. The steady state error is an error for stable

systems is the difference between the input command and the actual response of the system. An

example is shown below.

Graph 1: Example of steady state error

To calculate the steady state error, first the static error constants must be determined. Position (Kp),

velocity (Kv) and acceleration (Ka) error constants depend on what input type the systems receives.

It will determine which of these values are finite or not. The three input types directly relate to

which type of error will be in the system. The three input types are step, ramp and parabolic.

With the static error constants calculated, it is possible to determine the steady state error (Ess) for

each type of input

Step:

Ramp:

Parabolic:


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The type of a system depends on the number of pure integrations in the forward path. This is the

coefficient of sof the denominator of the systems transfer function. The table below shows how all

systems types and their errors relate to the type of input of the system.

Table 1:Relationship of inputs, errors and system types.


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1.3. Solution Description:

For each of the systems shown below, We were to calculate the steady state error for each the step,

ramp and parabolic inputs. To calculate the errors within each system first we need to simplify the

system into a open loop transfer function.

System 1:

Where K = 10

() =

()

1 + ()()

After transferring the G(s) and H(s) functions of system one into matlab we computed the combined

transfer function, T(s). This can be easily achieved using the matlab function "feedback", the function

takes both G and H as inputs and returns the combined transfer function, Ge with unity feedback.

This line of code is seen in appendix 1, Code 1.

After calculating the unity transfer function, it is used to determine the position, velocity and

acceleration constant errors. This is done with the "dcgain" matlab function, it takes in Ge and

returns the constant error values. For the velocity and acceleration errors Kv and Ka, the transfer

function is first multiplied by a transfer function with value and 2respectively before using the

"dcgain" function to alter the input from a step to instead a ramp and parabolic input.

With the error constants calculated, we can now use the pre defined formulas of steady state error

for each input, giving us Ess for each type of input. All code working can be found in Appendix 1.

System 2:


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To convert system 2 into a single transfer function with unity feedback the system needed to be

broken up into sections and simplified. First G2(s) and 10 were combined to give G3(s) using the

same method as the previous system. The next step was to combine the newly formed G3(s) and

G1(s). This is done by simply multiplying the functions together to form Gt(s). Finally Gt(s) and H(s)

were combine to give the transfer function T(s) in which the steady state errors were calculated

from. A step by step diagram can be seen below.

Figure 1: Step by step process of simplify system 2


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1.4. Test Results:

Table 1:


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1.5. Discussion & Conclusions:

From the data recorded in the rest results section it is apparent that the system type and input have

a direct affect on the type of error in the system.

With both systems being type 0 systems we can observe that the system will only have a positionstatic error constant, Kp. All other input types result in 0 static error and steady state value of

infinity. A correlation between the steady state error and static error is clear. From the derived

formulas we can state that as the value of the steady state error decreases, the static error will

increase.

Given the values of Kp, Kv and Ka you can determine what type the system is and the input of the

system. If any value of the error constants is a constant the related system type and input is know.

Kp is a constant - System is type 0 with step input

Kv is a constant - System is type 1 with ramp input

Ka is a constant - System is type 2 with parabolic input.

From the systems in this experiment these statements are proved to be true.

The steady state error indicates the error between the input response and actual output. This can be

either a positive or negative value. A positive value indicating the system has fallen below the

expected output. On the other hand if the value is negative the system has overshot the desired

output of the system and the output is larger than the input.

This experiment has shown that there are many methods errors that may occur in real life

applications of systems. While these errors can be calculated and compensated for they also can

easily represent the system and input type. Furthermore a direct relationship between the type oferror present in a systems characteristics can be formed.


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1.6. References:

Goodwin, G.C., Virtual Laboratories for Control System Design Laboratory Book : Laboratory 1

Electromechanical Servomechanism University of Newcastle Research Associates (TUNRA), 2007.


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1.7. Appendix:

Matlab code 1: System 1:

clear allclose allclc

num = [10 70];den = poly([0 -4 -8 -12]);

num2 = [1 10];den2 = poly([-5 -15 -20]);

G=tf(num, den)H=tf(num2, den2)

Ge=feedback(G,(H-1)); %tf with unity

%Ge=tf(Ge)%T=feedback(Ge,1); %tf with closed loop

%step inpitKp=dcgain(Ge) % Evaluate Kp=numg/deng for s=0.essstep= 30/(1+Kp) % Evaluate ess for step input.%ramp intputs = tf([1 0], 1);sGe = Ge*s;

sG=minreal(sGe); % Cancel common 's' in

numerator(numsg)

Kv=dcgain(sG) % Evaluate Kv=sG(s) for s=0.essramp=30/Kv % Evaluate steady-state error for

% parabolic input.s2 = tf([1 0 0], 1);s2Ge = Ge*s2;

s2G=minreal(s2Ge); % Cancel common 's' in

numerator(numsg)

Ka=dcgain(s2G) % Evaluate Kv=sG(s) for s=0.esspara=30/Ka


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Matlab code 2: System 2:

clear allclose allclc

G1num = [1 7];G1den = poly([0 -4 -8 -12]);

G2num = 5*(poly([-9 -13]));G2den = poly([-10 -32 -64]);

Hnum2 = 1;Hden2 = [1 3];

G1=tf(G1num, G1den)G2=tf(G2num, G2den)G3=feedback(G2, 10)

Gt=G1*G3H=tf(Hnum2, Hden2)

Ge=feedback... %tf with unity(Gt,(H));

%Ge=tf(Ge)%T=feedback(Ge,1); %tf with closed loop

%step inpitKp=dcgain(Ge) % Evaluate Kp=numg/deng for s=0.essstep= 30/(1+Kp) % Evaluate ess for step input.%ramp intputs = tf([1 0], 1);

sGe = Ge*s;

sG=minreal(sGe); % Cancel common 's' in

numerator(numsg)

Kv=dcgain(sG) % Evaluate Kv=sG(s) for s=0.essramp=30/Kv % Evaluate steady-state error for

% parabolic input.s2 = tf([1 0 0], 1);s2Ge = Ge*s2;

s2G=minreal(s2Ge); % Cancel common 's' in

numerator(numsg)

Ka=dcgain(s2G) % Evaluate Kv=sG(s) for s=0.esspara=30/Ka


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Laboratory 2 Experiment 2A2.1 Requirements

Obverse the steady state errors of a transfer function using MATLAB and the ESVL.

Use MATLAB to determine the ideal transfer function with unity feedback. From this determine the

system type, Kposition, Kvelocity and Kacceleration which in result calculate the steady state error of

step, ramp and parabolic functions.

Use ESVL to describe the non-ideal system behaviour on the steady state error and obverse the

output of the step signal, increase in input voltage and changes in the gain, K, of the system.

2.2 Introduction

ESVL

In this and future laboratories, we make use of Virtual Laboratory software developed by the

University of Newcastle. This series of programs aims to expose students to real world control

systems engineering problems, within the time and physical constraints of a typical teaching

laboratory. Further, it gives students the capacity to test ideas in a realistic setting but without fear

of costly failure. The Virtual Laboratory used in this experiment, the ELECTROMECHANICAL

SERVOMECHANISM VIRTUAL LABORATORY (ESVL), is designed to emulate the operation of a DC

motor servomechanism within a feedback control system. Many of the previously described non-

ideal features of a physical setup have been replicated in this virtual laboratory including the power

amplifier output limits, potentiometer wrap-around and signal noise.1

Steady State Error

Steady state error, in closed feedback loop, is the difference in the output signal to the desired input

signal. The steady state error is defined by equation 1.

Equation 1: Steady State Error

From equation 1, it is clear that the steady state error depends on the input of R(s) and the transfer

function G(s).There are three types of inputs that will be investigated, Step, Ramp and Parabolic.

For the step function, R(s) = R/s we can define the equation 1 as:

Equation 2: Steady State Error of Step Function

1(Vlacic, 2015)


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For the ramp function, R(s) = R/s2we can define the equation 1 as:

Equation 3: Steady State Error for Ramp Function

For the parabolic function, R(s) = R/s3we can define the equation 1 as:

Equation 4: Steady State Error for Parabolic Function

Kp, Kv and Ka are static error constants and can be used to determine what system type the transfer

function (see table 1).

Table 1: System Type and Static Error Constant Relationship


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2.3 Solution description

Using MATLAB a forward transfer function with unity feedback was determined using figure _. To

achieve this, first unity gain was placed into the function by adding -1 and +1 feedback. The new

function was now: the

Kp, Km and Ktheta were multiplied into a new variable Gt and Gp and -1 were added into a new

variable H. Gt and H were then combined to obtain a transfer funtion, Ge, with unity feedback by

using the MATLAB function feedback (appendix). Kp, Kv and Ka were obtained using MATLAB

function dcgain with the appropriate transfer function Ge for Kp, s*Ge for Kv and s2*Ge for Ka.

From there the Error steady states could be easily be determined using equations _.

The system used in MATLAB was placed into the ESVL to give an indictation of the effect of non-ideal

system behaviour on steady-state error. The function Ge was placed into the ideal second order

system by using the calculated Natural Frequency and Damping Ratio obtained in MATLAB. These

parameters were also placed into the ESVL:


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1 volt was placed into the signal generator and changed to 2 volts to obtain a step response. Also

whilst there was a constant 1 volt input, the value of Kp was changed to 1, 2, 4, 8 and 16. The output

was observed.

2.4 Results

Table 2: MATLAB Results of Gp = Kp = 2

Figure 1: ESVL Response of 1V Input


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Figure 2: ESVL Step Response of 2V Input

Figure 3: ESVL Response of Changing Gain (Kp)


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2.5 Discussion and Conclusion

From the transfer function in table 1 we can determine that the system type is type 0, as the value of

n is 0 and in result can determine the static error constants. Kp is a constant, Kv is zero and Ka is zero

from table 1. These values determine the steady state error of each input, step, ramp and parabolic.

From table 2, Kp is 1 therefore the Essstep is 0.5. Which is also consistent with the ESVL as in figure

1. The blue line is the input of 1V and the potentiometer is the green output on the CRO which is

stable at 0.5V giving error of 0.5V.

Although the potentiometer was stable around 0.5V there was a lot of noise to the signal. Which

could be due to other disturbances such as potentiometers (which wrap around their value with

each rotation), will generate and be susceptible to signal and measurement noise, and may beeffected by other external disturbances. These all have an effect on the response to an input signal.

As the input signal magnitude was increased, we can see in the figure 2 that the step response; an

increase of 1V to 2V increased the steady state error by 0.5V to 1V. This trend continued with input

of 3V increasing the steady state error again by 0.5V to 1.5V and 4V input, Ess was 2V. It can be

concluded that the steady state error increases linearly with the increase of the magnitude input

signal. This means for any given input signal, the error is input signal times 0.5.

As the gain of the transfer function is increased from 1 to 4 to 8 to 16 it is clear that the stability of

the function is becoming less stable, oscillating greatly at Kp = 16. The peak to peak value of thesignal at Kp = 16 is much greater than that of Kp = 4. If this trend continues, it can predicted that as


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Kp increases the transfer function will become less stable and more stable as Kp decreases. Although

the steady state error does not change with the gain (Kp) and remains the same at 0.5V per volt.

This transfer function is of order zero and therefore has constant error with a step signal. The gain of

the transfer function determines the stability of the function but does not change the steady state

error of 0.5V per input volt.

Appendix


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Figure 4: MATLAB Code to determine Steady State Error


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ReferencesVlacic, P. L. (2015). Laboratory No.2 - Steady State Error. 3304ENG - Control Systems, 1-9.

Lab 2 Report control systems

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