Essay: Error coefficients

There are two types of error coefficients such as:
1. Static error coefficient
2. Dynamic error coefficient
Static error coefficient
Ability of the system to reduce or eliminate the steady state error is static error coefficient. It is of 3 different types.
1. Position error coefficient:-
Here the reference input signal is a constant (step input), the output signal (position) is a constant in steady-state. The error constant associated with this condition is then referred to as the position error constant. It is given the symbol Kp.
2. Velocity error coefficient:-
Here the reference input is a ramp, then the output position signal is a ramp signal (constant slope) in steady-state. The signal that is constant in this situation is the velocity, which is the derivative of the output position. The error constant is referred to as the velocity error constant. It is given the symbol Kv.
3. Acceleration error coefficient:-
Here the reference input is a parabola, then the output position signal is also a parabola (constant curvature) in steady-state. Therefore, the signal that is constant in this situation is the acceleration, which is the second derivative of the output position. The error constant is referred to as the acceleration error constant. It is given the symbol Ka.
Step input
Ramp input
Parabolic input
Type 0 system
1/(1+Kp)
∞
∞
Type 1 system
1/Kv
∞
Type 2 system
1/Ka
Features of static error are as follows
1. Higher error coefficient increases the steady state performance of the system.
2.It cannot be used to calculate the error of unstable system.
3.It do not indicate the correct manner in which the error changes with time.
4. It is difficult to stabilise the system.
Dynamic error coefficient:
It is used to express dynamic error. It provides error signal as function of time. It is used for determining any type of input. The dynamic error coefficient provides a simple way of estimating error signal.
2.6 Generalized error series
The main disadvantage of defining the steadystate error in terms of error constants is that only one of the constants is finite and non zero for a particular system, whereas the other constants are either zero or infinity. If any error constant is zero, the steady state error is infinity, but we do not have any clue as to how the error is approaching infinity.
If the inputs are other than step, velocity or acceleration inputs, we can extend the concept of error constants to include inputs which can be represented by a polynomial. Many functions which are analytic, can be represented by a polynomial in t. Let the error be given by,
The above eqn may be written as,
Using convolution theorem, it can be written as
Assuming that r(t) has first n derivatives, r(1-) can be expanded into a Taylor series,
Where the primes indicates time derivatives. We have,
To obtain the steady state error, we take the limit t-> on both sides of equation
Where the suffix ss denotes steady state part of the function. It may be further observed that the integrals in equation yield constant values. Hence it may be written as,
Where,
The above equation is known as generalised error series.
The coefficients C0, C1, C2,…. Cn are defined as generalized error coefficients. Generalised error series may be observed that the steady state error is obtained as a function of time, in terms of generalised error coefficients, the steady state part of the input and its derivatives. For a given transfer function G(s), the error coefficients can be easily evaluated as shown in the following.
Let
Taking the derivative of equation with respect to s, we have,
Now taking the limit of equation as s->0, we have,
Thus, the constants can be evaluated using the above equations and the time variation of the steady state error also can be obtained.
Advantages:
1. It gives error signal as a function.
2. The steady state error can be determined for any type of input by using generalized error constants.
2.7 Steady state error
One of the important design specifications for a control system is the steadystate error. The steady state output of any system should be as close to desired output as possible. If it deviates from this desired output, the performance of the system is not satisfactory under steadystate conditions.
The steadystate error reflects the accuracy of the system. Among many reasons for these errors, the most important ones are the type of input, the type of the system and the non-linearities present in the system. Since the actual input in a physical system is often a random signal, the steady state errors are obtained for the standard test signals namely step, ramp and parabolic signals.
Error Constants
Let us consider a feedback control system shown in figure.
Feedback Control System
The error signal E(s) is given by,
We have,
Applying final value theorem, we can get the steady state error ess as
The above equation shows that the steady state error is a function of the input R(s) and the open loop transfer function G(s). Let us consider various standard test signals and obtain the steady state error for these inputs.
1. Unit step or position input
For a unit step input, . Hence from the above equation,
Let us define a useful term, positional error constant KP as,
In terms of the position error constant, ess can be written as,
2. Unit ramp or velocity input
Again, defining the velocity error constant KV as,
3. Unit parabolic or acceleration input
For the special case of unity of feedback system, H(s)=1 and the above eqns are modified as,
Expression to find steady state error of a closed loop control system:
Consider the simple closed loop system, using negative feedback
Where,
G(s) – Forward path transfer function
H(s) – Feedback transfer function
E(s) – Error signal
C(s)H(s) – Feedback signal
The error of the signal E(s) = R(s) – C(s)H(s)
But, C(s) = G(s) E(s)
Then the above equation becomes,
E(s) = R(s) – G(s) E(s) H(s)
E(s) + E(s) G(s) H(s) = R(s)
E(s) [ 1 + G(s)H(s)] = R(s)
Therefore, the error for non unity feedback system is
Therefore, the error for unity feedback system is
By applying final value theorem, the steady state error can be determined as,
The steady state error of the closed loop system is,
Problems:
Let us derive the error coefficients for the system having G(s)H(s) =
Solution:
Error coefficients are Kp , Kv and Ka.
Let us calculate the minimum value of K1, if the steady error is to be less than 0.1 and the input r(t) = (1+6t) is applied to the unity feedback system that has the forward transfer function .
Solution :
Given that, input r(t) = 1 + 6t
On taking Laplace transform of r(t), we get R(s).
The error signal in s-domain E(s) is given by,
The steady state error ess can be obtained from final value theorem,
Another method:
The steady state error for the input r(t) = 1 + 6t is,
We shall discuss the steady state temperature for the block diagram that represents a heat treating oven.
Solution:
The forward path transfer function of the system,
G(s) =
H(s) = 1 and R(s) = 1000/s
Since the given system is type ‘0’, therefore the
Steady state error
Let us calculate K to limit the error of a system for input to 0.8 having .
Solution:
The steady state error for the input 1 + 8t + 18 (t2 /2) is,
Lets calculate the open loop transfer function G(s) whose
Solution:
We know that, for a closed loop, the transfer function
But for a unity feedback system H(s) = 1, then
Lets us determine the type of the system, all the error coefficients and error for ramp input with magnitude 4.
Solution:
The type of the system is 1 (since the number of poles lie at origin =1).
The error coefficients are Kp, Kv and Ka.
The error for ramp input with magnitude 4 is
Let us calculate the position, velocity and acceleration error constants and the steady state error for unit ramp, unit step and unit parabolic inputs for open loop transfer function.
Solution:
The error coefficients are Kp, Kv and Ka.
The steady state errors