Steady State Errors for Unity Feedback Systems

In this Article, We will be going through Steady State Errors for Unity Feedback Systems in control systems, First, we will start our Article with an introduction to Steady State Errors, then we will through its two types, and then we will see mathematical Expression for calculating the Steady-State Error, At last, we will conclude our Article with its Advantages, Disadvantages, Applications and Some FAQs.

What is Steady State Errors?

When a system reaches a stable condition, the difference between the desired and actual outputs is referred to as steady-state error in a unity feedback control system. It indicates the system's ability to produce the desired output without any significant deviation. Steady-state Errors can result from several things, such as disturbances, uncertainties in the system or environment, and intrinsic limitations in the design of the control system. The system's overall performance and capacity to meet requirements can be significantly impacted by these faults. Analyzing the system's response to various inputs, such as step, ramp, or sinusoidal signals, is required for understanding steady-state faults. Common steady-state error types include oscillatory, constant, and zero errors; each has unique properties and causes.

By carefully choosing the control parameters like gain and time constants and using compensating techniques like integral control to reduce or eliminate the effects of steady-state errors, an effective control system design aims to minimize steady-state errors. All things considered, the study of steady-state errors in unity feedback control systems offers important knowledge about the behaviour of the system, enhances performance, and leads choices for design that improve stability, accuracy, and robustness.

Note: Error Analysis is relevant for closed-loop control systems only.

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)

Types of Steady State Error

There are mainly two types of steady state errors

• Actuating Error: It is Also Known as Control Error, refers to the difference between the desired control signal (input to the system) and the actual control signal applied to the system. It represents the discrepancy between the desired behavior of the control system, as specified by the controller, and the actual behavior observed in practice.
• Output Error: It is Also Known as system error or output deviation, refers to the difference between the desired output of the system and the actual output observed in practice. It represents the deviation of the system's response from the desired behavior.

Mathematical Expression For Calculating The Steady State Error

Steady State error can be calculated using final Value theorem

We know the transfer function of the unity negative feedback closed loop control system as,

C(s)/R(s)=G(s)/1+G(s)

⇒C(s)=R(s)*G(s)/1+G(s)...............(1)

The output of the summing point is

e_ss = lim t→ 0 e(t) = lim s→0 s*E(s)

e(t) = r(t) - c(t)

or, Taking Laplace Transform : E(s)=R(s)−C(s)............(2)

Substituting equation 1 in 2, we get

E(s) = R(s) - [ R(s)*G(s)/1+G(s)]

e_ss = lim s→0 s*R(s) [ 1- G(s)/1+G(s)]

e_ss = lim s→0 [ s*R(s)/1+G(s)]

Steady State Errors For a Unity Feedback System

Steady-state errors in a unity feedback control system occurs when the system's actual output after it has reached a stable condition differs from the desired output.

Types of Steady State Errors For a Unity Feedback System

The following three types of steady-state errors frequently occur in unity feedback control systems:

• Steady-state error due to step input (Position Error, ess) : When a system is given a constant input (such as step input) and it is unable to produce the desired result, an error occurs. In simple terms, it's the difference, while the input signal remains constant, between the desired and actual output.
• Steady-state error due to ramp input (Velocity Error, ev ) : When the input to the system is a ramp function  (rising linearly with time), this error occurs. It shows the difference between the intended and actual rates of change of the output.
• Steady-state error due to parabolic input (Acceleration Error, ea ) : This error occurs when the input to the system is a parabolic function (increasing quadratically with time). It represents the difference between the desired acceleration of the output and the actual acceleration of the output.

Expression for Position, Velocity and Acceleration Error Constants

Given Below are the Expression for Position, Velocity and Acceleration error Constants for Type 0, Type 1 and Type 2 Control Systems

Position Error Constant for Unit Step Unit

We know that for a unit step input signal, r(t) = A u(t)

Taking Laplace Transform on both sides we get, R(s)= A/s

According to final Value theorem; e_ss = lim s→0 [ s*R(s)/1+G(s)]

or, e_ss = lim s→0 [s*A/s]/ 1+G(s)

= A lim s→0 [ 1/1+G(s)]

= A / 1 + lim s→0 G(s)

= A/ 1+ Kp

Where Kp = lim s→0 G(s) , Static Position error coefficient

Position Error For Type 0, Type 1 and Type 2 System

We know that transfer function of a control system is the ratio of Laplace transform of the output to Laplace transform of the input: G(s)= N(s)/ D(s)

Type 0 order system : ( System with No poles at origin ⇒ D(s) ≠ 0 : lim s→0 s* G(s) = Kp ≠ 0 ⇒ Static position error coefficient is Finite vale

⇒ e_ss = A/ 1+Kp ⇒ means the steady state position error is finite value which is not desirable.

Type 1 order System or higher order system : G(s) = N(s)/ D(s)

lim s→0 G(s) = lim s→0 N(s)/ D(s)

= A/ sⁿ ( 1+sT) , as s→0; G(s)→∞

Hence Kp= lim s→0 G(s) = ∞

e_ss = A/1+Kp = 1/∞ = 0 , means the position error is zero for type 1 or higher order system, thus it is suitable for designing.

Velocity Error Constant for Ramp Input

We know that for a ramp input signal, r(t) = A t u(t)

Taking Laplace Transform on both sides we get, R(s)= A/s²

According to final Value theorem; ev = lim s→0 [ s*R(s)/ s² (1+G(s))]

or, ev = lim s→0 [s*A/s²]/ 1+G(s)

= A lim s→0 [ 1/ s (1+G(s))]

= A / 1 + lim s→0 s *G(s)

= A/ Kv

Where Kv = lim s→0 s*G(s) , Velocity error coefficient

Velocity Error For Type 0, Type 1 and Type 2 System

We know that transfer function of a control system is the ratio of Laplace transform of the output to Laplace transform of the input: G(s)= N(s)/ D(s)

Type 0 order system : ( System with No poles at origin ⇒ D(s) ≠ 0 : lim s→0 s* G(s) = Kv = 0 ⇒ velocity error coefficient is zero

⇒ e_v = A/ Kv = A/0 = ∞, means the steady state velocity error is infinite which is not desirable

Type 1 order System : G(s) = N(s)/ D(s) = A/ s*(1+sT)

Kv= lim s→0 [ s * (A/ s* (1+sT))] = A

⇒ Kv = Finite value

Hence, e_v = Finite error

Type 2 order System : G(s) = N(s)/ D(s) = A/ s² *(1+sT)

Kv= lim s→0 [ s * (A/ s² (1+sT))] = A/0= ∞

⇒ Kv = Infinite Value

Hence, e_v = A/Kv = A/∞ = 0, The velocity error for a type 2 system is zero therefore it is desirable

Acceleration Error Constant for Parabolic Input

We know that for a Parabolic input signal, r(t) = A t² u(t)

Taking Laplace Transform on both sides we get, R(s)= A/s³

According to final Value theorem; ev = lim s→0 [ s*R(s)/ s³ (1+G(s))]

or, e_a = lim s→0 [s*A/s³]/ 1+G(s)

= A lim s→0 [ 1/ s² (1+G(s))]

= A / 1 + lim s→0 s² *G(s)

= A/ Ka

Where Ka = lim s→0 s² *G(s) , Acceleration error coefficient

Acceleration error for Type 0, Type 1 and Type 2 System

We know that transfer function of a control system is the ratio of Laplace transform of the output to Laplace transform of the input: G(s)= N(s)/ D(s)

Type 0 order system : ( System with No poles at origin ⇒ D(s) ≠ 0 : lim s→0 s² * G(s) = Ka = 0 ⇒ velocity error coefficient is zero

⇒ e_a = A/ Ka = A/0 = ∞, means the steady state acceleration error is infinite which is not desirable

Type 1 order System : G(s) = N(s)/ D(s) = A/ s *(1+sT)

Ka= lim s→0 [ s² * (A/ s (1+sT))] = 0

⇒ Ka = Zero acceleration coefficient

Hence, e_a = A/Ka = A/0 = ∞, means the steady state acceleration error is infinite which is not desirable.

Type 2 order System : G(s) = N(s)/ D(s) = A/ s *(1+sT)

Ka= lim s→0 [ s² * (A/ s2 (1+sT))] = A

⇒ Ka = Finite Value

Hence, e_a = Finite error for type 2 order system.

Applications of Steady State Error Constants in a Unity Feedback

• Steady-state errors quantify the system's ability to track and maintain desired outputs, crucial for assessing control system performance.
• Analysis of steady-state errors aids in fine-tuning controller parameters to minimize errors and enhance system stability and precision.
• In order to reduce mistakes and achieve performance goals, steady-state error analysis helps in the selection of appropriate control algorithms, actuators, sensors, and other system components.
• Steady-state errors can also provide insights into the robustness of a control system against disturbances, uncertainties, and variations in system parameters
• Comparing steady-state errors between different control system designs or tuning strategies allows engineers to evaluate and select the most suitable approach for a given application.

Advantages of Steady State Error Constants in a Unity Feedback

• Steady-state errors can be used as performance parameters to evaluate the effectiveness of the control system.
• Steady-state errors help in tuning the controller parameters to optimize system performance.
• Errors in the steady state may offer information about the stability the system. Large or recurring steady-state errors may be a symptom of instability or insufficiency in the control system design, requiring additional research and improvement.
• Steady-state errors can highlight limitations and constraints in the control system.
• Understanding the trade-offs between different control system parameters and performance metrics, including steady-state errors, allows engineers to make informed design choices.

Disadvantages of Steady State Error Constants in a Unity Feedback

• Steady-state errors indicate the difference between the system's desired and actual outputs, which could affect performance.
• When there are consistent differences between the desired and actual outputs, steady-state errors can cause the system to become unstable. These errors can result in instability or oscillations, especially if the controller's parameters are not adjusted appropriately to reduce them.
• Steady-state errors can affect the overall behavior of the system, such as its settling time, overshoot, and response to disturbances.
• Steady-state errors can result in inaccurate control of the system, leading to suboptimal performance or failure to meet desired specifications.
• Designing a control system to minimize steady-state errors can be challenging and may require sophisticated control strategies or additional components, leading to increased complexity and cost.

Conclusion

In unity feedback control systems, steady-state errors can have a major effect on accuracy, stability, and performance. They may result in less stable systems, incorrect control, poor system performance, and design issues. Long settling times, reduced response to disturbances, and inefficiencies can arise from persistent differences between expected and actual outputs. In order to meet performance requirements and achieve desired system behavior, steady-state error minimization is essential. To guarantee accurate control and reduce errors, this frequently requires accurate analysis, adjustment, and optimization of control parameters. In applications like industrial automation, robotics, and safety-critical systems, where accuracy, stability, and dependability are crucial, addressing steady-state faults is crucial.