11th_Class_5-3-24

1. UNIT III ▪ CLASS NO: 5 ▪ DATE:05-03-2024

2. Old Syllabus Updated New Syllabus

3. • E IG E N VA L U E S & E IG E N VE CTORS OF A M A TRIX • M A TRIX IN VE RSE • POL E S OF A TRA N SFE R FU N C TION FEW PRELIMINARIES*** • ZE ROS OF A TRA N SFE R FU N C TION • C H A RA C TE RSTIC S E Q U A TION OF A SYSTE M • ROOTS OF A N E Q U A TION & POL E S || C onnection ??

4. Steady State Errors in Unity Feedback Control System Motivation ▪ Errors in a control system can be attributed to many factors. Changes in the reference input will cause unavoidable errors during transient periods and may also cause steady-state errors. ▪ Any physical control system inherently suffers steady-state error in response to certain types of inputs. A system may have no steady-state error to a step input, but the same system may exhibit nonzero steady-state error to a ramp input. (The only way we may be able to eliminate this error is to modify the system structure.) Objective ▪ To investigate a type of steady-state error that is caused by the incapability of a system to follow particular types of inputs. ▪ Whether a given system will exhibit steady-state error for a given type of input depends on the type of open-loop transfer function of the system, to be discussed in this section.

5. Consider the unity-feedback control system with the following open-loop transfer function ?(?): ?(?) ?(?) + ?(?) − Representing a pole of multiplicity ? at the origin Type 0 system ? = 0 ? = 1 Type 1 system Classification of Control Systems ? = 2 Type 2 system ? = 3 Type 3 system And so on… ***Note • This classification is different from that of the order of a system.

6. Steady State Errors (SSE) Find SSE using Final Value Theorem (A) Static Position Error Constant ?? The steady-state error of the system for a unit-step input is The static position error constant ??is defined by

7. For a type 0 system, the static position error constant ?? is finite, while for a type 1 or higher system, ?? is infinite. For a unit-step input, the steady-state error ???may be summarized as follows:

8. (B) Static Velocity Error Constant ?? The steady-state error of the system with a unit-ramp input is given by The static velocity error constant ??is defined by Thus,the steady-state error in terms of the static velocity error constant ??is given by The term velocity error is used here to express the steady-state error for a ramp input.

9. For a type 0 system, ? ???= = ∞ ?? For a type 1 system, ? =? ???= ?? ? For a type 2 system, ? ???= = ? ??

10. (C) Static Acceleration Error Constant ?? The steady-state error of the system with a unit-parabolic input (acceleration input), which is defined by The static acceleration error constant ??is defined by the equation ? ???= The steady-state error is then ??

12. Summary Ramp Input ? ? = ? Acceleration/Parabolic Input ? ? = ??/? Inputs Step Input ? ? = ? Type ? Type 0 system ∞ ∞ ? + ? ? ? Type 1 system ∞ ? ? ? Type 2 system ? ? ? = 1 Type 0 system Type 1 system ? = 0 ? = 2 Type 2 system Remark: The error constants ??,??, and ??describe the ability of a unity-feedback system to reduce or eliminate steady-state error. Therefore, they are indicative of the steady-state performance.

13. Question 1 What is the Steady State Error (???) of the system ? If Input ? ? is unit ramp.

14. Question 2 What is the Steady State Error (???) of the system ? If Input ? ? is unit step. ? ? ??? ?? = ??+ ??? ?? + ?? + ???? = ? ? Angular speed (RPM) of the motor