control systems - time specification domains

this is II unit control systems notes

1. Ee3503-control systems unit -ii By MRS . R.RAMYA . ME, ap / eee SSMIET.DINDIGUL

2. TIME RESPONSE The time response of the system is the output of the closed loop system as a function of time It is Denoted by c(t) It consists of two parts Transient response Steady state response

3. TRANSIENT RESPONSE: When the input changes from one state to another. STEADY STATE RESPONSE: As a time ‘ t’ approaches infinity

4. FIRST AND SECOND ORDER SYSTEM RESPONSE Transfer Function It is the ratio of Laplace transform of output to Laplace transform of input with zero initial conditions. One of the types of modeling a system Differential equation is obtained L aplace transform is applied to the equation assuming zero initial conditions

5. Order of a system Order of a system is given by the order of the differential equation governing the system Alternatively, order can be obtained from the transfer function I n the transfer function, the maximum power of s in the denominator polynomial gives the order of the system

6. FIRST ORDER SYSTEM A first order control system is defined as a type of control system whose input-output relationship (also known as a transfer function ) is a first-order differential equation. A first-order differential equation contains a first-order derivative, but no derivative higher than the first order. The order of a differential equation is the order of the highest order derivative present in the equation The transfer function (input-output relationship) for this control system is defined as :   

7. Second-Order System The order of a control system is determined by the power of ‘ s’ in the denominator of its transfer function . If the power of s in the denominator of the transfer function of a control system is 2, then the system is said to be second order .

8. Time domain specifications Delay time RISE TIME PEAK TIME MAXIMUM OVERSHOOT SETTLING TIME STEADY STATE ERROR

9. DELAY TIME: It is the time required for the response to reach 50% of the steady state value for the first time. RISE TIME: It is the time taken for response to reach 0 to 100% for the very first time. UNDER DAMPED SYSTEM ( 0 to 100%) OVER DAMPED SYSTEM (10% to 90%) 3. CRITICALLY DAMPED SYSTEM (5% to 95%)

10. PEAK TIME , tp : I t is the time required for the response to reach the maximum or peak value of the response. PEAK OVERSHOOT , Mp : It is defined as the difference between the peak value of the response and the steady state value. It is usually expressed in percent of the steady state value. If the time for the peak is tp , percent peak overshoot is given by, Maximum percent overshoot = 𝑐(𝑡𝑝)−𝑐(∞) / 𝑐 (∞ ) .

11. SETTLING TIME , ts : It is the time taken by the response to reach and stay within a specified error. It is usually expressed as percentage of final value. The usual tolerable error is 2% and 5% of the final value STEADY STATE ERROR ( Ess ) It indicates the error between the actual output and desired output as t tends to infinity

12. STANDARD TEST INPUTS (0R) TYPES OF TEST INPUT 1. Step signal 2. Unit step signal 3. Ramp signal 4. Unit ramp signal 5. Parabolic signal 6. Unit parabolic signal 7. Impulse signal 8. Sinusoidal signal

13. STEP SIGNAL The step signal is a signal whose value changes from zero to A at t=0 and remains constant at A for t>0. The step signal resembles an actual steady input to a system. A special case of step signal is unit step in which A is unity.

14. RAMP SIGNAL The ramp signal is a signal whose value increases linearly with time from an initial value of zero at t=0. The ramp signal resembles a constant velocity input to the system. A special case of ramp signal is unit ramp signal in which the value of A is unity .

15. PARABOLIC SIGNAL In parabolic signal, the instantaneous value varies as square of the time from an initial value of zero at t=0. The sketch of the signal with respect to time resembles a parabola. The parabolic signal resembles a constant acceleration input to the system. A special case of parabolic signal is unit parabolic signal in which A is unity

16. IMPULSE SIGNAL A signal of very large magnitude which is available for very short duration is called impulse signal. Ideal impulse signal is a signal with infinite magnitude and zero duration but with an area of A. The unit impulse signal is a special case, in which A is unity.

17. STABILITY A system is said to be stable, if its output is under control. Otherwise, it is said to be unstable. A  stable system  produces a bounded output for a given bounded input.

18. Types of Systems based on Stability We can classify the systems based on stability as follows. Absolutely stable system Conditionally stable system Marginally stable system

19. Absolutely Stable System If the system is stable for all the range of system component values, then it is known as the  absolutely stable system . The open loop control system is absolutely stable if all the poles of the open loop transfer function present in left half of  ‘s’ plane . Similarly, the closed loop control system is absolutely stable if all the poles of the closed loop transfer function present in the left half of the ‘s’ plane.

20. Conditionally Stable System If the system is stable for a certain range of system component values, then it is known as  conditionally stable system . MARGINALLY STABLE SYSTEM If the system is stable by producing an output signal with constant amplitude and constant frequency of oscillations for bounded input, then it is known as  marginally stable system