Analysis of Linear Control Systems
3.4 Root Locus of Linear Systems
Assume that the feedback control system is established by a unity negative feedback system whose forward path is defined as a static gain K, followed by an open-loop model G(s).
For each value of K, a set of closed-loop poles can be found by solving the characteristic equation 1+ KG(s) = 0. With continuous change in the gain K, the trajectories of closed-loop pole positions can be constructed. The trajectories of the poles of the closed-closed-loop system can be obtained and are referred to as a root locus of the system. It should be noted that the open-loop model G(s) should be used to draw the root locus, and the root locus can be used to describe the pole positions of the closed-loop system.
A MATLAB functionrlocus()is provided in the Control Systems Toolbox to draw the root locus of a given system. The function can be called in one of the following ways:
rlocus(G) % automatic draw of the root locus
rlocus(G,[kmin,kmax]) % root locus over the gain range rlocus(G,K) % root locus for a given gain vector K
[R,K]=rlocus(G) % evaluate the closed-loop pole positions R rlocus(G1,’-’,G2,’-.b’,G3,’:r’) % root locus for several models
It should be noted that this function applies to both continuous- and discrete-time systems.
Only SISO LTI models can be processed in the function. It can also be used in drawing the root locus for discrete-time transfer functions with pure time delays.
On the root locus of the system, one may left click any point on the locus to show the gain, pole position, damping ratio, and overshoot of the system at the same gain K. One can easily find the values of the open-loop gain K for which the closed-loop system is stable.
The commandgridcan be used to superimpose the isodamping and isofrequency lines of the system. These lines may provide useful information in control systems design.
Example 3.23. Let
G(s)= s2+ 4s + 8
s5+18s4+120.3s3+357.5s2+478.5s+306
−7 −6 −5 −4 −3 −2 −1 0 1
−8
−6
−4
−2 0 2 4 6 8
Root Locus
Real Axis
Imaginary Axis
(a) root locus
−7 −6 −5 −4 −3 −2 −1 0 1
−8
−6
−4
−2 0 2 4 6 8
System: G Gain: 772 Pole: −0.0213 + 7.5i
Damping: 0.00284 Overshoot (%): 99.1 Frequency (rad/sec): 7.5
Root Locus
Real Axis
Imaginary Axis
(b) find the critical point
Figure 3.14. Root locus analysis of the system and its inverse.
be an open-loop model of the system under investigation. Using the following MATLAB scripts, the root locus of the system can be easily and accurately drawn, as shown in Fig-ure 3.14(a).
>> num=[1 4 8]; den=[1,18,120.3,357.5,478.5,306];
G=tf(num,den); rlocus(G)
If one left clicks at the point on the intersection with the imaginary axis, the information about the critical point is shown as in Figure 3.14(b), from which it is immediately seen that the critical gain is 772. It can be concluded that when the gain K > 772, the closed-loop system is unstable.
Example 3.24. If a discrete-time open-loop model is given by G(z)= 0.52(z− 0.49)(z2+ 1.28z + 0.4385)
(z−0.78)(z+0.29)(z2+0.7z+0.1586)
with a sampling interval of T = 0.1 seconds, the following statements can be used to input the open-loop system model and draw the root locus of the system, as shown in Figure 3.15(a). It can be seen by clicking the relevant points that the critical gain is K= 2.83.
>> z=tf(’z’,’Ts’,0.1);
G=0.52*(z-0.49)*(zˆ2+1.28*z+0.4385)/(z+0.29)/(zˆ2+0.7*z+0.1586);
rlocus(G), grid
If there exists a pure delay term z−6in the original system, the root locus of the delayed system can be redrawn, as shown in Figure 3.15(b):
>> G.ioDelay=6; rlocus(G), grid
It can be found that the critical gain is reduced to 1.16. It can be seen from the example that the delay term in the discrete-time model reduces the critical gain of the system.
−1.5 −1 −0.5 0 0.5 1
(b) root locus of the delayed system
Figure 3.15. Root locus of a discrete-time system.
−5 −4 −3 −2 −1 0 1 2 3
System: untitled1 Gain: 13.6 Pole: 0.0142 Damping: −1 Overshoot (%): 0 Frequency (rad/sec): 0.0142 Root Locus
Real Axis
Imaginary Axis
Figure 3.16. Root locus for positive feedback systems.
Example 3.25. If
G(s)= s2+ 5s + 6
s5+ 13s4+ 65s3+ 157s2+ 184s + 80
is an open-loop model, the following statements can be used to draw the root locus for the system with unity positive feedback, as shown in Figure 3.16. It can be seen that when 0≤ K ≤ 13.6, the closed-loop system is stable.
>> G=tf([1 5 6],[1 13 65 157 184 80]); rlocus(-G)
Example 3.26. For the open-loop model
G(s)= 0.3(s+ 2)(s2+ 2.1s + 2.23) s2(s2+ 3s + 4.32)(s + a) ,
if one wants to draw the root locus according to variable a, the characteristic equation 1+ G(s) = 0 can be rewritten as
a(s4+ 3s3+ 4.32s2)+ (s5+ 3s4+ 4.62s3+ 1.23s2+ 1.929s + 1.338) = 0
−2 −1.5 −1 −0.5 0 0.5
(b) zoomed root locus
Figure 3.17. Root locus according to variable a.
from which it can be seen that
1+ a s4+ 3s3+ 4.32s2
s5+ 3s4+ 4.62s3+ 1.23s2+ 1.929s + 1.338 = 0.
Let
G(s) = s4+ 3s3+ 4.32s2
s5+ 3s4+ 4.62s3+ 1.23s2+ 1.929s + 1.338.
The characteristic equation can be written as 1+ aG(s)= 0. The root locus according to variable a can be drawn for the G(s)model. The following statements can be given, and the root locus obtained is shown in Figure 3.17(a), and Figure 3.17(b) is the zoomed version:
>> G1=tf([1,3,4.32,0,0],[1,3,4.62,1.23,1.929,1.338]); rlocus(G1)
The root locus can be used in controller design to select an appropriate value for the gain K. If there exists a pair of dominant complex poles on the root locus, which have a relatively low damping ratio for a specific value of K, then selecting this value of K may be appropriate. It is assumed that if the complex poles are dominant and the effects of any zeros can be ignored, then the resulting closed-loop response will approximate that of a second-order system with these complex poles.
Example 3.27. Consider the open-loop model
G(s)= 10
s(s+ 3)(s2+ 2s + 4).
The following statements can be entered into MATLAB, and the root locus of the system can be drawn as shown in Figure 3.18(a). The isodamping lines are also superimposed on the curves.
−5 −4 −3 −2 −1 0 Frequency (rad/sec): 0.943
Root Locus
(b) closed-loop step response
Figure 3.18. Root locus and step response of the system in Example 3.27.
>> s=tf(’s’); G=10/(s*(s+3)*(sˆ2+3*s+4));
rlocus(G), grid
From the isodamping lines, it is easily found by clicking the pole position located at the ζ = 0.707 line that the gain is about K = 1.68, as shown in Figure 3.18(a). It is also seen that this pair of poles is dominant, so selecting the gain K = 1.68 gives the closed-loop step response shown in Figure 3.18(b). This, as expected, is very similar to that of the second-order system with these poles.
>> K=1.68; step(feedback(G*K,1))
Example 3.28. Consider a simple plant model G(s)= 1/(s + 1)3. The root locus of the system can immediately be drawn as shown in Figure 3.19(a) with the following statements:
>> s=tf(’s’); G=1/(s+1)ˆ3; rlocus(G)
However, the root locus drawn with default settings is not complete, since the intersection of the root locus with the imaginary axis is not shown, due to the improper selection of the gain range, by default. One should then enlarge the gain range, for instance, select a range of (0, 20), to modify the root locus drawn. The modified root locus can be obtained as shown in Figure 3.19(b).
>> rlocus(G,[0,20])
Example 3.29. Assume that
G(s)= (s+ 5)(s2+ 2s + 8)
s(s+ 1)(s + 2)(s + 3)(s2+ 6s + 12)
is an open-loop model. The root locus of the system is immediately obtained, as shown in Figure 3.20(a).
−2 −1.5 −1 −0.5 0
(a) default root locus
−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5
(b) modified root locus
Figure 3.19. Problems in automatic root locus drawing.
−8 −6 −4 −2 0 2
(b) root locus of inverse system
Figure 3.20. Root locus of a system and its inverse system.
>> s=tf(’s’);
G=(s+5)*(sˆ2+2*s+8)/s/(s+1)/(s+2)/(s+3)/(sˆ2+6*s+12)
It is interesting to note that the root locus of its inverse system 1/G(s) has exactly the same shape as the original G(s) if it is drawn with the commandrlocus(1/G), shown in Figure 3.20(b). In the inverse system, it is not surprising to note that the poles and zeros are interchanged, and thus the directions of the root loci are all reversed.
Besides, if one reads the gain at a certain point on Figure 3.20(a), one will get the reciprocal of the gain by clicking the same point on Figure 3.20(b).