4.3. Time Response Specification in Design Settling time, Ts ,is the time required for the output to settle within a certain percent of its final value. Common values are 5% and 2%. Settling time is proportional to the , k Ts = k  = ——— n If 2% is used to specify the settling value then k=4. Tr, Ts,and css, are equally meaningful for, over, critically, or under-damped cases, while Tp,and Mp,are meaningful only for under-damped cases. Under-damped cases. To find Tp,we differentiate c(t) and equalizing the result to zero and solve the equation for t. We will find that The typical of unit step response of a system is as in figure below nt c(t) Mpt 1.0 0.9 0.1 Tr Tp 1+ d 1 d css Ts Rise time Tr is the time required for the response to rise from 10% to 90% of the final value css. Mpt is the peak value, and Tp is the time required to reach Mpt. Mpt  css percent overshoot = ————— 100% css

4.3. Time Response Specification in Design Example: Servomotor is used to control the position of plotter pen as in the following figure. Recall that 0.5/s(s+2) motor R(s) Ka Amplifier  + and Both %over-shoot and nTp are functions only of  and can be plotted as in the following figure %over-shoot nTp 100 80 60 40 20 5.0 4.6 4.2 3.8 3.4 3.0 0.2 0.4 0.6 0.8 1  Here we have Suppose that we want to have =1, the fastest response with no overshoot then n= 1 and Ka=2. Note that the settling time is Tp= 4/  n= 4s It is not fast enough. If we want the faster response First we have to choose a different motor, then Redesign the compensator.

4. 3. Time Response Specification in Design 4.3. Time Response Specification in Design. Step response parameters relation in terms of pole location. The poles of the TF is s = n (12) And is plotted in the following figure If we draw a line from origin to s1 then the line and the real axis make angle . Percent over-shoot can be expressed in term of this angle. s   s1 s2 If we want that the %over-shoot to be less then specified values then  must be less then a value that can be calculated. The poles is then restricted in the region shown in the following figure.. s The settling time is Ts= k/n and is inversely related to the real part of the poles. If it is specified that the settling time to be less then Tsm then n>k/Tsm k/Tsm s

4.4. Frequency response of Systems. If input of a system is sinusoidal r(t)=A cos (t), then the output is still sinusoidal, c(t)=A|G(j)| cos(t+) with amplitude = A|G(j)| phase shift =  =  G(j). We define the frequency response to be the complex function G(j). First Order System We are interested primarily in the magnitude. We plot it |G(j)| K B 0.707K  System bandwidth B is  at which the gain 1/2K, B can be found to be B = 1/. It is convenient to have a plot in normalized frequency v = , vB = 1 an the normalized time is tv = t/. Consequently the normalized plot of freq and time response is 0.9K 0.1K t/=tv c(t) K Tr/ normalized step response |Gn(jv)| K vB=1 0.707K v=  normalized frequency response where |G(j)| is also called the gain

4.4. Frequency response of Systems. = 0.1 Second Order System The transfer function of second order system is |Gn(j)| 0.25 The frequency response can be written as (1) 0.5 0.707 1 The magnitude is B / n 0.707 v =/ n Define the normalized frequency as v= /n we have the normalized magnitude |G(j)| The bandwidth varies with  and n . Recall that then for constant  increasing n will decrease Tp with the same factor Plots of this frequency response for various  is given in the following figure

4.4. Frequency response of Systems. |Gn(j)v| B/n 0.707 v =/n 0.5 0.25 = 0.1 1 4.4. Frequency response of Systems. The maximum magnitude denoted by Mp  The peak value of the magnitude of frequency response is function only of , as is the peak value of the step response. The peak value of |Gn(j)v| can be determine by differentiating |Gn(j)v| and set this to zero, this yields The peaking frequency response indicates the condition called resonance, and r is called the resonant frequency

4.5. Time and Frequency Scaling Time is often scaled to set unity . Then the T.F is said to be time scaled. Scaling the time will scales the frequency also. Let t denote the time before scaling, tv denotes time after scaling, and  denotes the scaling parameter thus t = tv and tv = t/ Let c(t) the function to be scaled and cs(tv) be the scaled function, thus cs(tv) = c(t) Let  = 10 then typical c(t) and cs(tv) are For >1 the signal will be sped up, hence will increase the frequency and vice versa. Consequently the frequency will be scaled as  = v and v = / Time derivative will be affected by time scaling as follow Time scaling will make: The amplitude form is unaffected The time constant  will be changed to / The damping ratio  is unaffected. 0 1 2 3 4 c(t) t 0 0.4 cs(tv) tv

4.6. Response of Higher-Order System First consider non-unity second order system Where z and p denotes the zero and pole. Since (2) is real T.F then if there are complex zeros or poles they must be present in conjugate pairs. Using partial fractional expansion The T.F (2) can be expressed as the sum of non- unity gain first order terms and second order terms. (1) The time and freq responses of this system will be K times of those of unity second order system. Now consider the response of Gs(s) = sG(s) From L.T properties we see that gs(t) = g’(t) where g’(t) represent the derivative of g(t) The general T.F of Higher order system is 4.6. Reduced order models Reduced order model is obtained by ignoring insignificant poled Dominant poles and zeros Insignificant poles and zeros unstable region (2) Eq. (2) can be expressed as (3)