1 / 6

4. System Response

4. System Response. This module is concern with the response of LTI system. L.T. is used to investigate the response of first and second order systems. Higher order systems can be considered to be the sum of the response of first and second order system.

apollo
Télécharger la présentation

4. System Response

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. 4. System Response • This module is concern with the response of LTI system. • L.T. is used to investigate the response of first and second order systems. Higher order systems can be considered to be the sum of the response of first and second order system. • Unit step, ramp, and sinusoidal signal play important role in control system analysis. It is therefore we will investigate this signals.

  2. h(t) r(t) c(t) 4. System Response Review of some LTI properties We will express system as in figure below, system input is r(t), output is c(t), and impulse response is h(t) Taking the L.T. of (1) yields C(s) = R(s)H(s) (2) Where C(s)is the L.T. of c(t) R(s) is the L.T. of r(t) H(s) is the L.T. of h(t) H(s) is called the transfer function (T.F) 3. Derivative If the input is r’(t) then the output is c’(t) where r’(t) denotes the derivative of r(t) 1. Impulse response Impulse response, denoted by h(t), is the output of the system when its input is impulse (t). h(t) is called the impulse response of the system or the weighting function 4. Integral If the input is r(t)dt then the output is c(t)dt 5. Poles and Zero T.F.isusually rational and therefore can be expressed as N(s)/D(s). Poles is the values of s resulting T.F to be infinite. Zeroes is the values of s resulting T.F to be zero 2. Convolution Output of LTI system is the convolution of its input and its impulse response: (1)

  3. (4) (1) (5) (2) + + R(s) C(s) (3) c(0) 4.1. Time Response of the First Order Systems. Here we will investigate the time response of the first order systems. The transfer function of a general first order system can be written as: Solving for C(s) yields The eq. can be shown in the block diagram as shown in the figure bellow. We can found the differential equation first we write (1) as The diff. Eq. is the inverse L.T. of (2) Note that the initial condition as an input has a Laplace transform of c(0), which is constant. The inverse L.T of a constant is impulse (t). Hence the initial condition appears as the impulse function Here we can see that the impulse function has a practical meaning, even though it is not a realizable signal Now we take the L.T of (3) and include the initial condition term to get

  4. c(t) τ K (1) t (2) R(s) C(s) (3) 4.1. Time Response of the First Order Systems. Since we usually ignore the initial condition in block diagram, we use the system block diagram as shown bellow. The first term originates in the pole of input R(s) and is called the forced response or steady state response The second term originates in the pole of the transfer function G(s) and is called the natural response Figure below plot c(t) Suppose that the initial condition is zero then Unit step response For unit step input R(s)=1/s, then The final value or the steady state value of c(t) is K that is lim c(t)= K t c(t) is considered to reach final value after reaching 98% of its final value. The parameter  is called the time constant. The smaller the time constant the faster the system reaches the final value. Taking the inverse L.T of (2) yields

  5. (2) c(t) t r(t)=t c(t) (1) 4.1. Time Response of the First Order Systems. A general procedure to find the steady state value is using final value theorem lim c(t) = lim sC(s) = lim sG(s)R(s) t s0 s0 For Unit step input then the final value is css(t)= lim G(s) s0 since R(s) = 1/s System DC gain is the steady state gain to a constant input for the case that the output has a final value. Ramp Response For the input equal to unit ramp function r(t) = t and R(s) = 1/s2, C(s) is therefore • This ramp response is composed of three term • a ramp • a constant • an exponential.

  6. (1) (5) 2 (3) c(t) (2) =0 0.2 0.7 (3) nt (4) 4.2. Time Response of Second Order System The standard form second order system is Case 3: =1 (real equal poles), c(t) is This system is said to be critically damped The poles of the TF is s = n jω(12) Case 4: =0 (imaginary poles), c(t) is Where  = damping ratio n = natural frequency, or undamped frequency. Consider the unit step response of this system This system is said to be undamped For this system we have Time constant =  = 1/n ; frequency = n Case 1: 0<<1 (complex poles), c(t) is This system is said to be underdamped Case 2: >1 (real unequal poles), c(t) is This system is said to be overdamped

More Related