EE128 Feedback Control Lecture 5 Fall 2006 Outline Mason s rule 3 9 Models of Electromechanical Systems 2 3 Effects of Pole Locations 3 3 Time Domain specifications 3 4 Mason s Rule section 3 9 G s Y s 1 Gi i U s i Gi path of the ith forward path system determinant 1 all individual loop gains gain products of all possible two loops that do not touch gain products of all possible three loops that do not touch i ith forward path determinant value of for that part of the block diagram that does not touch the ith forward path Mason s Rule Mason s Rule Problem 3 47 Models of Electromechanical Systems section 2 3 Law of motors F Bli newtons Example 2 9 Loudspeaker 2 m 1 26 m 100 F 0 5 1 26 i 0 63i newtons l 20 M x bx 0 63i X s 0 63 M I s s s b M Models of Electromechanical Systems Law of generators e t Blv volts Example 2 10 Loudspeaker with circuit ecoil Blx 0 63x L di Ri va 0 63x dt X s 0 63 Va s s Ms b Ls R 0 63 2 Models of Electromechanical Systems DC motor actuators Torque Back emf Example 2 11 DC motor T K t ia e K e m Models of Electromechanical Systems Problem 2 18 Capacitor microphone C x A x q C x e dielectric constant of the material between the plates q2 fe 2 A A surface area of the plates a Write differential equations that describe the operation of the system it s OK to leave it in non linear form Effects of Pole Locations section 3 3 time constant rate of decay 1 Fast slow pole H s 2s 1 s 2 3s 2 H s 2 s 1 2 s 1 s 2 H s 1 3 s 1 s 2 e t 3e 2t t 0 h t t 0 0 Effects of Pole Locations Time functions associated with points in the s plane Effects of Pole Locations Complex pole s j d a pole has a real negative part if is positive n2 H s 2 s 2 n s n2 n and d n 1 2 damping ratio n undamped natural frequency n2 H s s n 2 n2 1 2 h t a s s j d s j d s 2 d2 n 1 2 e t sin d t 1 t Effects of Pole Locations Impulse responses Step responses Impulse response natural response Stability depends on whether natural response grows or decays Effects of Pole Locations 2nd order system system response with an exponential envelope that multiplies a sinusoid Effects of Pole Locations Example 3 23 oscillatory time response Discuss the correlation between the poles of H s and the impulse response of the system 2s 1 H s 2 s 2s 5 H s n2 5 n 2 24 rad sec 2 n 2 0 447 2s 1 2s 1 s 1 1 2 2 s 2 2 s 5 s 1 2 2 2 s 1 2 2 2 2 s 1 2 2 2 1 h t 2e t cos 2t e t sin 2t 1 t 2 Time Domain specifications section 3 4 Rise time tr time for the system to reach the vicinity of its new set point Settling time ts time for the system transients to decay Overshoot Mp maximum amount the system overshoots its final value divided by its final value Peak time tp time it gets the system to reach the maximum overshoot point Time Domain specifications Rise time For a 2nd order system with no zeros we can relate the polelocation parameters and n by looking at the curves e g for 0 5 y from 0 1 to 0 9 is approx ntr 1 8 tr 1 8 n Time Domain specifications Overshoot when y t reaches maximum value its derivative is 0 The time history of these curves found from the inverse Laplace tansform of H s s is y t 1 e t cos d t sin d t d where d n 1 2 and n Time Domain specifications using A sin B cos C cos y t 1 e t 2 1 2 cos d t d when it reaches max value y t 0 2 t y t e sin d t d sin d t 0 d this occurs when sin d t 0 so d t p t p System output at peak time y t p 1 e d Overshoot M p e 1 2 0 1 d Time Domain specifications Overshoot versus damping ratio for the 2nd order system Overshoot M p e 1 2 0 1 Frequently values used are M p 0 16 for 0 5 and M p 0 05 for 0 7 Time Domain specifications Settling Time when y t is almost the steady state When the decay exponential reaches 1 error with respect of desired output Settling time time required for the transient to decay to a small value 1 error with respect to desired output e nt s 0 01 ts 4 6 n 4 6 Time Domain specifications rise time overshoot settling time all together Time Domain specifications Problem 3 23 Time Domain specifications Problem 3 26 Summary