1. Control engineering (2151908) Time response of first order system PREPARED BY: 140113119007 Patel Ravindra

2. introduction  Time response of a system is defined as the output of a system when subjected to an input which is a function of time.  Time response analysis means subjecting the control system to inputs that are function of time and their outputs which are also function of time.

3. The order of a system is defined as being the highest power of derivative in the differential equation, or being the highest power of s in the denominator of the transfer function. A first-order system only has s to the power one in the denominator, while a second-order system has the highest power of s in the denominator being two types of the input functions (or test input signals) commonly used are: Impulse function: In the time domain, u(t) = cd(t). In the s domain, U(s) = c.

4. Step function: In the time domain, u(t) = c. In the s domain, U(s) = c/s. Ramp function: In the time domain, u(t) = ct. In the s domain, U(s) = c/s 2. Sinusoidal function: In the time domain, u(t) = csin(wt). In the s domain, U(s) = cw/(s 2 + w 2 ). where c is a constant in all the above.

5.  With these test signals, mathematical and experimental analyses of control systems can be carried out easily since the signals are very simple functions of time.  Which of these typical signals to use for analysing system characteristics may be determined by the form of the input that the system will be subjected to most frequently under normal operation.  If the inputs to a control system are gradually changing functions of time, then a ramp function of time may be a good test signal.

6.  Similarly, if a system is subjected to sudden disturbances, a step function of time may be a good test signal, and for a system subjected to a shock input, a pulse or an impulse function may be best.  The time response of a control system consists of two parts: the transient response and the steady state response.  The transient response is defined as the part of the time response which goes from the initial state to the final state and reduces to zero as time becomes very large.

7. The steady-state response is defined as the behaviour of the system as t approaches infinity after the transients have died out. Thus the system response y(t) may be written as: y(t) = yt(t) + ys(t) where yt(t) denotes the transient response, and ys(t) denotes the steady-state response.

8. Roadmap (Time Responses)

9. Why Study Time Responses?? Modelling: – Some parameters in the system can be estimated or identified by time responses. Analysis: – Evaluate transient and steady-state responses to see if they meets performance requirement. Design: – Given design specifications in terms of transient and steady state responses, design controllers satisfying all the design specification.

10. First-Order System General form: Output response:

11.  Transient Response: Gradual change of output from initial to the desired condition. Block diagram representation: K : Gain  : Time constant  By definition itself, the input to the system should be a step function which is given by the following:

12. Response Analysis of First-Order Systems Many systems are approximately first-order. The important feature is that the storage of mass, momentum and energy can be captured by one parameter. Examples of first-order systems are velocity of a car on the road, control of the velocity of a rotating system, electric systems where energy storage is essentially in one capacitor or one inductor, incompressible fluid flow in a pipe, level control of a tank, pressure control in a gas tank, temperature in a body with essentially uniform temperature distribution (e.g. steam filled vessel).

13. Standard test signal  The characteristics of actual input signals are a sudden shock, a sudden change, a constant velocity, and constant acceleration.  The dynamic behavior of a system is therefore judged and compared under application of standard test signals – an impulse, a step, a constant velocity, and constant acceleration.  Another standard signal of great importance is a sinusoidal signal

14. Unit Impulse input  The impulse signal imitate the sudden shock characteristic of actual input signal. If A=1, the impulse signal is called unit impulse signal.

15. The unit impulse is defined as a function of time that is zero everywhere, except for an infinitesimally small neighborhood around the origin, in which the function attains unbounded values. However, its time integral from −∞ to +∞ is exactly +1.Thus, if we denote by 0− and 0+ the instants just before and just after 0, respectively. As a consequence, the unit-impulse function has units of s−1, i.e., of frequency. This function is also called the delta function or the Dirac function, and is represented as a vertical arrow at the origin, as in Fig. of unit length in the scale adopted.

16.  Note that, if the unit-impulse function is multiplied by a constant A to obtain Aδ (t), it follows from Eq. 2.31 that its time integral from −∞ to +∞ is equal to A. We then say that Aδ (t) is an impulse function of magnitude A, the “magnitude” being, in fact, the area under the impulse.  Such a non-unit impulse is thus represented as an arrow of height A in the scale adopted. The height of the arrow, then, denotes the time integral of the associated impulse function on the whole real axis.

17. Unit Step input  The step signal imitate the sudden change characteristic of actual input signal. If A=1, the step signal is called unit step signal

18.  Note that the unit-step function is undefined in the interval 0− <t <0+.  This does not bother us, because the values of this function in that interval are never needed, except for the basic assumption that this function remains bounded everywhere, including that interval.  The unit-step function is needed to represent abrupt changes of variables upon which a function jumps instantaneously from one value to another by a finite amount.  This corresponds to physical situations such as a constant, finite force applied suddenly onto a mass, the sudden closing of a switch in a circuit driven by a battery, the sudden exposure of a body at a given temperature to a constant, finite temperature, different from that of the body, and so on.

19. Unit Ramp input  The ramp signal imitate the constant velocity characteristic of actual input signal. If A=1, the ramp signal is called unit ramp signal 0 t r(t)

20.  The ramp signal is the integral of the step input, and the parabola is the integral of the ramp input. The unit impulse function is also useful for test signal purposes.  The responses due to these inputs allow the prediction of the system’s performance to other more complex inputs.  Ramp- for track a constant angular position (first derivatives are constant).