# Automatic Control Systems (FCS) Lecture- 8 Steady State Error

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1 Automatic Control Systems (FCS) Lecture- 8 Steady State Error

2 Introduction 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. 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.

3 Classification of Control Systems Control systems may be classified according to their ability to follow step inputs, ramp inputs, parabolic inputs, and so on. The magnitudes of the steady-state errors due to these individual inputs are indicative of the goodness of the system.

4 Classification of Control Systems Consider the unity-feedback control system with the following open-loop transfer function It involves the term s N in the denominator, representing N poles at the origin. A system is called type 0, type 1, type 2,..., if N=0, N=1, N=2,..., respectively.

5 Classification of Control Systems As the type number is increased, accuracy is improved. However, increasing the type number aggravates the stability problem. A compromise between steady-state accuracy and relative stability is always necessary.

6 Steady State Error of Unity Feedback Systems Consider the system shown in following figure. The closed-loop transfer function is

7 Steady State Error of Unity Feedback Systems The transfer function between the error signal E(s) and the input signal R(s) is E( s) 1 R( s) 1 G( s) The final-value theorem provides a convenient way to find the steady-state performance of a stable system. Since E(s) is The steady state error is

8 Static Error Constants The static error constants are figures of merit of control systems. The higher the constants, the smaller the steady-state error. In a given system, the output may be the position, velocity, pressure, temperature, or the like. Therefore, in what follows, we shall call the output position, the rate of change of the output velocity, and so on. This means that in a temperature control system position represents the output temperature, velocity represents the rate of change of the output temperature, and so on.

9 Static Position Error Constant (K p ) The steady-state error of the system for a unit-step input is The static position error constant K p is defined by Thus, the steady-state error in terms of the static position error constant K p is given by

10 Static Position Error Constant (K p ) For a Type 0 system For Type 1 or higher systems For a unit step input the steady state error e ss is

11 Static Velocity Error Constant (K v ) The steady-state error of the system for a unit-ramp input is The static position error constant K v is defined by Thus, the steady-state error in terms of the static velocity error constant K v is given by

12 Static Velocity Error Constant (K v ) For a Type 0 system For Type 1 systems For type 2 or higher systems

13 Static Velocity Error Constant (K v ) For a ramp input the steady state error e ss is

14 Static Acceleration Error Constant (K a ) The steady-state error of the system for parabolic input is The static acceleration error constant K a is defined by Thus, the steady-state error in terms of the static acceleration error constant K a is given by

15 Static Acceleration Error Constant (K a ) For a Type 0 system For Type 1 systems For type 2 systems For type 3 or higher systems

16 Static Acceleration Error Constant (K a ) For a parabolic input the steady state error e ss is

17 Summary

18 Example#1 For the system shown in figure below evaluate the static error constants and find the expected steady state errors for the standard step, ramp and parabolic inputs. R(S) - 100( s 2)( s s 2 ( s 8)( s 5) 12) C(S)

19 Example#1 (Steady Sate Errors) K p K v K a

20 Example#1 (evaluation of Static Error Constants) G( s) 100( s s 2 ( s 2)( s 8)( s 5) 12) K K p p lim G( s) s0 100( s 2)( s 5) lim 2 s 0 s ( s 8)( s 12 K p ) K v K v lim sg( s) s0 100s( s 2)( s 5) lim 2 s 0 s ( s 8)( s 12 ) K v K a lim s s0 2 K a G( s) 2 100s ( s 2)( s 5) K a lim 2 s 0 s ( s 8)( s 12) 100( 0 2)( 0 5) ( 0 8)( 0 12)

21 Example#8 (Lecture ) Figure (a) shows a mechanical vibratory system. When 2 lb of force (step input) is applied to the system, the mass oscillates, as shown in Figure (b). Determine m, b, and k of the system from this response curve.

22 Example#8 Figure (a) shows a mechanical vibratory system. When 2 lb of force (step input) is applied to the system, the mass oscillates, as shown in Figure (b). Determine m, b, and k of the system from this response curve.

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