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Time Domain analysis

Time Domain analysis. STEADY-STATE ERROR. STEADY-STATE ERROR. The difference between the output and the reference in the steady state was defined as the steady-state error.

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Time Domain analysis

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  1. Time Domain analysis STEADY-STATE ERROR

  2. STEADY-STATE ERROR • The difference between the output and the reference in the steady state was defined as the steady-state error. • In the real world, because of friction and other imperfections and the natural composition of the system, the steady state of the output response seldom agrees exactly with the reference. • Therefore, steady-state errors in control systems are almost unavoidable. • In a design problem, one of the objectives is to keep the steady-state error to a minimum, or below a certain tolerable value.

  3. Steady-State Error of Linear Continuous-Data Control Systems where r(t) is the input; u{t), the actuating signal; b{t), the feedback signal; and y(t), the output. The error of the system may be defined as e{t) = reference signal—y(t) The steady-state error (ess) is defined as

  4. Example: • consider a velocity control system in which a step input is used to control the system output that contains a ramp in the steady state. let K = 10 volts/rad/sec Then, The closed-loop transfer function of the system is

  5. Cont…… • For a unit-step function input, R(s) = l/s. The output time response is • . Thus, the steady-state error of the system is It is known that the referential signal is ramp which is 0.1t. • To establish a systematic study of the steady-state error for linear systems, we shall classify three types of systems and treat these separately. • Systems with unity feedback; H(s) = 1. • Systems with nonunity feedback, but H(s) = KH — constant. • Systems with nonunity feedback, and H(s) has zeros at s = 0 of order N.

  6. Type of Control Systems: Unity Feedback Systems • The steady-state error of the system is written • Clearly, ess depends on the characteristics of G(s). • we can show that ess depends on the number of poles G(s) has at s = 0. This number is known as the type of the control system or, simply, system type. • E.g.

  7. Steady-State Error of System with a Step-Function Input • When the input to the control system with H(s) = 1 is a step function with magnitude R, R(s)=R/s, the steady-state error is let’s say Therefore, the steady-state error is • we can summarize the steady-state error due to a step function input as follows: Type 0 system: =constant Type 1 or higher system: ess = 0

  8. Steady-State Error of System with a Ramp-Function Input • When the input is a ramp function with magnitude R, • where R is a real constant, the Laplace transform of r{t) is • The steady-state error is • Let’s define the ramp-error constant as Therefore, and the ess Type 0 system: ess = Type 1 system: ess = = constant Type 2 system: ess = 0

  9. Steady-State Error of System with a Parabolic-Function Input • When the input is described by the standard parabolic form = • The steady-state error of the system • Defining the parabolic-error constant as • The steady state error become: • Type 0 system: ess = • Type 1 system: ess = • Type2system: ess= =constan t • Type 3 or higher system: ess = 0

  10. Relationship between Steady-State Error and Closed-Loop Transfer Function • the closed-loop transfer function can be used to find the steady-state error of systems with unity as well as non-unity feedback. • let us impose the following condition: • we can define the reference signal as r(t)/KH and the error signal as • in the transform domain, • where M{s) is the closed-loop transfer function

  11. The steady-state error of the system is • Substituting M(s) into the last equation and simplifying, • For the three basic types of inputs for r(t). 1. Step-function input. R{s) = R/s. • the steady-state error due to a step input can be zero only if • or

  12. 2. Ramp-function input. R{s) = R/s . • For a ramp-function input, the steady-state error • The following values of essare possible 3. Parabolic-function input. R(s) = R/s • the steady-state error • The following values of ess are possible

  13. Example • Consider that the system has the following transfer functions: And The closed loop TF is

  14. Steady-State Error of Non-unity Feedback: H(s) Has Nth-Order Zero at s = 0 • In the real world, this corresponds to applying a tachometer or rate feedback. • Thus, for the steady-state error analysis, the reference signal can be defined as • the error signal in the transform domain may be defined as where, • The steady-state error for N = 1 is written as • For a step input of magnitude R

  15. EXAMPLE 5-4-6 Consider that the system has the following transfer functions: And The closed loop transfer function is

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