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Steady-State Errors | Control Systems - Electrical Engineering (EE)

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Introduction

The deviation of the output of control system from desired response during steady state is known as steady state error. It is represented as ess. We can find steady state error using the final value theorem as follows.
Steady-State Errors | Control Systems - Electrical Engineering (EE)Where,

E(s) is the Laplace transform of the error signal, e(t)

Let us discuss how to find steady state errors for unity feedback and non-unity feedback control systems one by one.

Steady State Errors for Unity Feedback Systems

Consider the following block diagram of closed loop control system, which is having unity negative feedback.
Steady-State Errors | Control Systems - Electrical Engineering (EE)

Where,

  • R(s) is the Laplace transform of the reference Input signal r(t)
  • C(s) is the Laplace transform of the output signal c(t)

We know the transfer function of the unity negative feedback closed loop control system as
Steady-State Errors | Control Systems - Electrical Engineering (EE)
Steady-State Errors | Control Systems - Electrical Engineering (EE)
The output of the summing point is -
E(s)=R(s)−C(s)
Substitute C(s) value in the above equation.
Steady-State Errors | Control Systems - Electrical Engineering (EE)
Steady-State Errors | Control Systems - Electrical Engineering (EE)
Steady-State Errors | Control Systems - Electrical Engineering (EE)
Substitute E(s) value in the steady state error formula
Steady-State Errors | Control Systems - Electrical Engineering (EE)
The following table shows the steady state errors and the error constants for standard input signals like unit step, unit ramp & unit parabolic signals.

Steady-State Errors | Control Systems - Electrical Engineering (EE)

Where, Kp, Kv and Ka are position error constant, velocity error constant and acceleration error constant respectively.

Note

  • If any of the above input signals has the amplitude other than unity, then multiply corresponding steady state error with that amplitude.
  • We can’t define the steady state error for the unit impulse signal because, it exists only at origin. So, we can’t compare the impulse response with the unit impulse input as t denotes infinity.

Example

Let us find the steady state error for an input signal Steady-State Errors | Control Systems - Electrical Engineering (EE) of unity negative feedback control system with Steady-State Errors | Control Systems - Electrical Engineering (EE)
The given input signal is a combination of three signals step, ramp and parabolic. The following table shows the error constants and steady state error values for these three signals.
Steady-State Errors | Control Systems - Electrical Engineering (EE)We will get the overall steady state error, by adding the above three steady state errors.
Steady-State Errors | Control Systems - Electrical Engineering (EE)

Therefore, we got the steady state error ess as 1 for this example.

Steady State Errors for Non-Unity Feedback Systems

Consider the following block diagram of closed loop control system, which is having nonunity negative feedback.
Steady-State Errors | Control Systems - Electrical Engineering (EE)We can find the steady state errors only for the unity feedback systems. So, we have to convert the non-unity feedback system into unity feedback system. For this, include one unity positive feedback path and one unity negative feedback path in the above block diagram. The new block diagram looks like as shown below.

Steady-State Errors | Control Systems - Electrical Engineering (EE)Simplify the above block diagram by keeping the unity negative feedback as it is. The following is the simplified block diagram.

Steady-State Errors | Control Systems - Electrical Engineering (EE)This block diagram resembles the block diagram of the unity negative feedback closed loop control system. Here, the single block is having the transfer function Steady-State Errors | Control Systems - Electrical Engineering (EE) 

 instead of G(s). You can now calculate the steady state errors by using steady state error formula given for the unity negative feedback systems.

Note: It is meaningless to find the steady state errors for unstable closed loop systems. So, we have to calculate the steady state errors only for closed loop stable systems. This means we need to check whether the control system is stable or not before finding the steady state errors. In the next chapter, we will discuss the concepts-related stability.

The document Steady-State Errors | Control Systems - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Control Systems.
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54 docs|45 tests
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