11.4 the Root Locus Method Hitay O ¨ Zbay Introduction the Root Locus Technique Is Agraphical Tool Used in Feedback Control System Analysis and Design

11-34 Systems, Controls, Embedded Systems, Energy,and Machines 11.4 The Root Locus Method Hitay O ¨ zbay Introduction The root locus technique is agraphical tool used in feedback control system analysis and design. It has been formally introduced to the engineering communitybyW.R.Evans [3,4], who received the RichardE.Bellman Control Heritage Award from the American Automatic Control Council in 1988 for this major contribution. In ordertodiscuss the root locus method, we must first review the basic definition of bounded input bounded output (BIBO) stabilityofthe standardlinear time invariant feedback system shown in Figure11.22, wherethe plant, and the controller,are represented by their transfer functions P ( s )and C ( s ), respectively1 The plant, P ( s ), includes the physical process to be controlled, as well as the actuator and the sensor dynamics. The feedback system is said to be stable if none of the closed-loop transfer functions, from external inputs r and v to internal signals e and u ,haveany poles in the closed right half plane, C þ : ¼fs 2 C : Reð s Þ > 0 g . Anecessarycondition for feedback system stabilityisthat the closed right half plane zeros of P ( s )(respectively C ( s )) are distinct from the poles of C ( s )(respectively P ( s )). When this condition holds, we saythat there is no unstable pole–zero cancellation in taking the product P ( s ) C ( s ) ¼ : G ( s ), and then checking feedback system stabilitybecomes equivalent to checking whether all the roots of 1 þ G ð s Þ¼0 ð 11: 45Þ are in the open left half plane, C : ¼fs 2 C : Reð s Þ 5 0 g .The roots of Equation (11.1) are the closed-loop system poles. We would like to understand how the closed-loop system pole locations varyasfunctions of a real parameter of G ( s ). More precisely,assume that G ( s )contains aparameter K ,sothat we use the notation G ( s ) ¼ G K ( s )toemphasize the dependenceon K .The root locus is the plot of the roots of Equation (11.45) on the complex plane, as the parameter K varies within aspecified interval. The most common example of the root locus problem deals with the uncertain (or adjustable) gain as the varying parameter:when P ( s )and C ( s )are fixed rational functions, except for again factor, G ( s )can be written as G ( s ) ¼ G K ( s ) ¼ KF( s ), where K is the uncertain/adjustable gain, and Q m N ð s Þ N ð s Þ¼ ð s z j Þ F ð s Þ¼ where j ¼ 1 n > m ð 11: 46Þ D ð s Þ Q n D ð s Þ¼ ð s p i Þ ; i ¼ 1 with z 1 , ... , z m ,and p 1 , ..., p n being the open-loop system zerosand poles. In this case, the closed-loop system v(t) r(t) e(t) + + u(t) y(t) C (s) P(s) + − FIGURE 11.22 Standard unityfeedback system. 1 Here we consider the continuous time case; thereisessentially no difference between the continuous time case and the discrete time case, as far as the root locus construction is concerned. In the discrete time case the desired closed-loop pole locations are defined relative to the unit circle, whereas in the continuous time case desired pole locations are defined relative to the imaginary axis. Control Systems 11-35 poles are the roots of the characteristic equation w ð s Þ : ¼ D ð s ÞþKNð s Þ¼0 ð 11: 47Þ The usual root locus is obtained by plotting the roots r 1 ( K ),... , r n ( K )ofthe characteristic polynomial w ( s )on the complex plane, as K varies from 0to þ 1 .The same plot for the negative values of K gives the complementaryroot locus.With the help of the rootlocus plot the designer identifies the admissible values of the parameter K leading to aset of closed-loop system poles that are in the desiredregionofthe complex plane. There are several factors to be considered in defining the ‘‘desired region’’ of the complex plane in which all the roots r 1 ( K ),... , r n ( K )should lie. Those are discussed briefly in the next section. Section 11.3 contains the root locus construction procedure, and design examples are presented in section 11.4. The root locus can also be drawn with respect to asystem parameter other than the gain. Forexample, the characteristic equation for the system G ( s ) ¼ G l ( s ), defined by ð 1 l s Þ 1 G l ð s Þ¼P ð s Þ C ð s Þ ; P ð s Þ¼ ; C ð s Þ¼K c 1 þ s ð 1 þ l s Þ T I s can also be transformed into the form given in Equation (11.47). Here K c and T I are given fixed PI (Proportional plus Integral) controller parameters, and l . 0isanuncertain plant parameter.Note that the phase of the plant is p — P ð j o Þ¼ 2tan 1 ð loÞ 2 so the parameter l can be seen as the uncertain phase lag factor (for example, asmall uncertain time delayin the plant can be modeled in this manner,see [9]). It is easy to see that the characteristic equation is 2 1 s ð l s þ 1 ÞþK c ð 1 l s Þ s þ ¼ 0 T I and by rearranging the terms multiplying l this equation can be transformed to 2 1 ð s þ K c s þ K c = T I Þ 1 þ 2 ¼ 0 l s ð s K c s K c = T I Þ –1 2 2 By defining K ¼ l , N ( s ) ¼ ( s þ K c s þ K c / T I ), and D ( s ) ¼ s ( s þ K c s–K c / T 1 ), we see that the characteristic equation can be put in the form of Equation (11.3). The root locus plot can nowbeobtained from the data N ( s )and D ( s )defined above; that shows howclosed-loop system poles move as l –1 varies from 0 to þ 1 ,for agiven fixed set of controller parameters K c and T I .For the numerical example K c ¼ 1and T I ¼ 2.5, the root locus is illustrated in Figure 11.23. The root locus construction procedurewillbegiven in section 11.3. Most of the computations involved in each step of this procedure can be performed by hand calculations. Hence, an approximate graph representing the root locus can be drawn easily.There are also several softwarepackages to generate the root locus automatically from the problem data z 1 , ... , z m ,and p 1 , ... , p n . If anumerical computation program is available for calculating the roots of apolynomial, we can also obtain the root locus with respect to aparameter which enters into the characteristic equation nonlinearly.To illustrate this point let us consider the following example: G ð s Þ¼G ð s Þ where o o ð s 0 : 1 Þ ð s 0 : 2 Þ G ð s Þ¼P ð s Þ C ð s Þ ; P ð s Þ¼ ; C ð s Þ¼ o o 2 2 ð s þ 1 : 2 o o s þ o o Þðs þ 0 : 1 Þ ð s þ 2 Þ 11-36 Systems, Controls, Embedded Systems, Energy,and Machines Root Locus for N ( s )=s 2 + s +0.4, D ( s )=s ( s 2 − s − 0.4) 1.5 arrows show the increasing direction of 1 K =1/ λ from 0to+∞ 0.5 0 Imag Axis − 0.5 − 1 − 1.5 − 2 − 1.5 − 1 − 0.5 0 0.5 1 1.5 2 Real Axis FIGURE 11.23 The root locus with respect to K ¼ 1/l . Here o o $ 0isthe uncertain plant parameter.Note that the characteristic equation o ð 1 : 2 s þ o Þðs þ 0 : 1 Þðs þ 2 Þ 1 þ o o ¼ 0 ð 11: 48Þ s 2 ð s þ 0 : 1 Þðs þ 2 Þþð s 0 : 2 Þðs 0 : 1 Þ cannot be expressed in the form of D ( s ) þ KN( s ) ¼ 0with asingle parameter K .Nevertheless, for each o o we can numerically calculate the roots of Equation (11.48) and plot them on the complex plane as o o varies within arange of interest. Figure11.24 illustrates all the four branches, r 1 ( K ),... , r 4 ( K ), of the root locus for Root Locus 2 1.5 x : ω o =zero o : ω o =infinity 1 0.5 0 − 0.5 − 1 − 1.5 − 2 − 2.5 − 2 − 1.5 − 1 − 0.5 0 FIGURE 11.24 The root locus with respect to o o . Control Systems 11-37 this system as o o increases fromzerotoinfinity. The figure is obtained by computing the roots of Equation (11.48) for aset of values of o o by using M ATLAB. Desired Pole Locations The performanceofafeedback system depends heavily on the location of the closed-loop system poles r i ( K ) ¼ 1,... , n .First of all, for stabilitywewant r i ð K Þ 2 C for all i ¼ 1,... , n .Clearly,having apole ‘‘close’’ to the imaginaryaxis poses adanger,i.e., ‘‘small’’perturbations in the plant mightlead to an unstable feedback system. So the desiredpole locations must be such that stabilityispreservedunder such perturbations (or in the presence of uncertainties) in the plant. Forsecond-ordersystems, we can define certain stabilityrobustness measures in terms of the pole locations, which can be tied to the characteristics of the step response. For higher ordersystems, similar guidelines can be used by considering the dominant poles only.

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