Chapter 13 Root Locu 13.1 Introduction In the previou chapter we had a glimpe of controller deign iue through ome imple example. Obviouly when we have higher order ytem, uch imple deign technique will not be ufficient. In thi chapter we will look at a method that help u to determine the poition of the root of the characteritic equation a ome deign parameter varie. Root locu i a fairly general graphical technique ued to determine the migration of the cloed-loop pole a ome parameter, uch a controller gain, i varied. Conider a imple P-controller. Figure 13.1: A imple proportional controller the cloed loop tranfer function i given by G c () = K()G() 1+K()G() = The cloed loop pole are given a the root of, 114 kg() 1+kG()
Lecture Note on Control Sytem/D. Ghoe/2012 115 1+kG() =0 We want to elect k in order to achieve peed, accuracy, and tability by placing the pole in certain region of the -plane. Thi region wa alo hown in the lat ection. An approximate ketch i given below. Figure 13.2: Deirable region of the -plane Let u conider the example of an aircraft. From the notion of tatic tability of an aircraft we know that the CG (center of gravity) of the aircraft mut lie ahead of the AC (aerodynamic center) of the aircraft for tability. Figure 13.3: An aircraft example Let u conider a linearized model of the aircraft dynamic obtained by perturbing the ytem about a teady tate or trim condition. α = Z α α + Z q q + Z e δ e q = M α α + M q q + M e δ e
Lecture Note on Control Sytem/D. Ghoe/2012 116 where, α i the angle of attack, q i the pitch rate, and δ e i the elevator deflection (which i alo the input to the ytem). The coefficient Z and M are the tability derivative. The open-loop tranfer function can be obtained by taking Laplace tranform on both ide and auming initial condition to be zero (thi i true ince the trim condition i taken a the reference). α() = Z α α()+z q q()+z e δ e () q() = M α α()+m q q()+m e δ e () Solving for q(), which the variable of interet to u, q() = M e +(Z e M α Z α M e ) 2 (Z α + M q ) +(Z α M q Z q M α ) δ e() Let u conider an aircraft in the untable configuration at Mach number M =0.9 and altitude 20,000 ft. The correponding value of the coefficient are, Z α = 1.62 ec 1 M α =2.96 ec 1 Z q = 1.00 ec 1 M q = 0.77 ec 1 Z e = 0.17 ec 1 M e = 22.53 ec 1 Uing thee value, q() = 22.53 36.55 2 +2.39 1.71 = 22.53( +1.62) ( +2.97)( 0.575) which how that the open loop ytem i untable a it ha a pole on the RHS of the -plane. So, mall deviation in the elevator deflection will caue the pitch rate to grow without bound. Let u deign a P-control autopilot that tabilize thi ytem. The feedback ytem will be omewhat like that hown in the figure below. which ha the following denominator polynomial, D() = ( +2.97)( 0.575) 22.53k( +1.62) = 2 +2.39 1.71 22.53k 36.55k = 2 +(2.39 22.53k) (1.71+36.55k) Let u form the Routh array, 2 1 (1.71+36.55k) 1 (2.39 22.53k) 0 0 (1.71+36.55k) 0
Lecture Note on Control Sytem/D. Ghoe/2012 117 Figure 13.4: P-control autopilot For the ytem to be table, 2.39 22.53k >0 k<0.106 1.71 36.55k >0 k< 0.047 which mean that a negative k in the order of 1 will tabilize the ytem. However, thi example i fairly imple and when the aircraft dynamic are realitic we will need a much higher order ytem where it will not be very eay to obtain the effect of k or other gain on the root. Root locu technique give u a way by which we can plot approximate locu of the root on the -plane without actually olving for the root. It i eay to find the root locu of firt and econd order ytem. But how about thoe of higher order? Let u conider an example of a third order ytem. Let the open-loop ytem be given by, G() = 1 [( +4) 2 + 16] Let u ue P-control. Then the cloed loop tranfer function i, G c () = kg() 1+kG() = k [( +4) 2 + 16] + k = Let u check for tability uing the Routh array, k 3 +8 2 +32 + k 3 1 32 2 8 k 1 32 k 0 8 0 k 0
Lecture Note on Control Sytem/D. Ghoe/2012 118 Figure 13.5: Root locu for the example For tability, k<32 8 = 256 If we actually plot the root locu for thi ytem, it will look a above. From the figure we can ee that a k increae beyond the value of 256, the complex conjugate root migrate into the RHS of the -plane. With increaing k the root at the origin migrate farther into the LHS of the -plane along the real axi. 13.2 Evan Form Before we can apply the root locu technique, we need to expre the ytem in what i known a Evan form. The Evan form i given a, G() = G 1() 1+kG 2 () where k i the parameter to be varied, and we want to plot the root of a k varie from 0 to. 1+kG 2 () =0
Lecture Note on Control Sytem/D. Ghoe/2012 119 Example: A P-control ytem given below ha the cloed loop repreentation a, Figure 13.6: A P-control ytem which i already in Evan form. Another Example: Conider the following ytem. G c () = kg() 1+kG() Figure 13.7: Another ytem:pd-control The cloed-loop tranfer function i given a, G c () = = (k 1 + k 2 )G() 1+k 2 G()+k 1 G() (k 1 +k 2 )G() 1+k 1 G() 1+k 2 G() 1+k 1 G()
Lecture Note on Control Sytem/D. Ghoe/2012 120 Suppoe the value of k 1 i already fixed, then the ytem i in Evan form and the pole of thi ytem are the root of, [ ] G() 1+k 2 =0 1+k 1 G() Another Example: Conider integral control. Then the cloed loop tranfer function would be, which i in Evan form. k G() k 1+ k G() = G() 1+k G() Yet Another Example: Conider PI-control. The cloed loop tranfer function would be, ( ) k1 + k 2 G() N() 1+ ( 1+k ) = 1 G() [ ] k 1 + k 2 G() G() 1+k 2 {1+k 1 G()} which i now in Evan form for a fixed k 1.