Lecture 26: Stability of Closed-Loop Systems, Root Locus Plots

Massachusetts Institute of Technology Department of Mechanical Engineering 2.004 Dynamics and Control II Spring Term 2008 Lecture 261 Reading: • Nise: Chapter 6 • Nise: Chapter 8 1 Determining Stability Bounds in Closed-Loop Systems Consider the closed-loop third-order system with proportional controller gain K with openloop transfer function K Gf (s) = 3 s + 3s2 + 5s + 2 shown below. R (s ) P c o n tr o lle r + p la n t K s - 3 + 3 s 2 1 C (s ) + 5 s + 2 The closed loop transfer function is: Gcl (s) = N (s) K = 3 2 D(s) + N (s) s + 3s + 5s + (2 + K) Let’s examine the closed-loop stability by using the pzmap() function in MATLAB: sys = tf(1,[1 3 5 2]); pzmap(sys); hold on; for K = 2:2:30 sys = tf(K,[1 3 5 2+K]); pzmap(sys); end; which superimposes the closed-loop pole/zero plots for K = 0 . . . 30 on a single plot: 1 c D.Rowell 2008 copyright � 26–1 Pole−Zero Map 3 K=30 2 K increasing K=0. Imaginary Axis 1 K increasing K=30 K=0 0 −1 K=0 K increasing −2 −3 −4 K=30 −3.5 −3 −2.5 −2 −1.5 Real Axis −1 −0.5 0 0.5 1 From the plot we note the following: • This system always has two complex conjugate poles and a single real pole. • When K = 0 the poles are the open-loop poles. • As K increases, the real pole moves deeper into the l.h. plane, and the complex con­ jugate poles approach and cross the imaginary (jω) axis, and the system becomes unstable. • Close examination of the plot shows that the system becomes unstable at a value of K between K = 12 and K = 14. We now look at three methods for determining the stability limit of the proportional gain K for this system. Example 1 Use the Routh-Hurwitz method to find the range of proportional controller gain K for which the above system will be stable. The first two rows of the Routh array are taken directly from D(s): s3 s2 1 5 3 2+K 0 0 26–2 and the next two rows are computed as above � � � � 1 �� 1 1 1 �� an an−2 �� 5 �� b1 = − =− � = − (K − 13) � � � 3 2 + K a a an−1 3 3 n−1 n−3 � � � � � � � � 1 � an 1 1 0 � an−4 � b2 = − = − �� =0 � � an−1 an−1 an−5 3 3 0 � Similarly, the s0 row is � 1 �� an−1 c1 = − � b1 b1 � 1 � a c2 = − �� n−1 b1 b1 computed � � � � � 3 an−3 �� 3 24 � �=2+K = − b2 � K − 13 � −(K − 13)/3 2 + K � � � � � 3 an−3 �� 3 0 �� � =− =0 b3 � K − 13 � −(K − 13)/3 0 � and the complete Routh array is s3 s2 s1 s0 1 5 0 3 2+K 0 -(K-13)/3 0 (2+K) 0 We now examine the first column to determine the range of proportional gain for which this system will be stable. In order for there to be no sign changes we require −2 < K < 13 We conclude that if K < −2 there will be one (therefore real) unstable pole, while if K > 13 there will be two unstable poles. When K = 13 the denominator is D(s) = s3 + 3s2 + 5s + 15 = (s + 3)(s + j2.236)(s − j2.236) so that the closed-loop system has a pair of poles on the imaginary axis. The system will be marginally stable (a pure oscillator at a frequency of ω = 2.236 r/s). Example 2 Use the stability criterion for third-order systems developed in Example 3 of Lecture 25 to determine the stability bounds for the above system. In Lecture 25 we showed that for a third-order system with characteristic equa­ tion: D(s) = a3 s3 + a2 s2 + a1 s + a0 = 0 the system is stable only if a1 a2 > a0 a3 26–3 In this case D(s) = s3 + 3s2 + 5s + (2 + K) and therefore for stability we require 15 > 2 + K or K < 13. Example 3 Use the characteristic equation directly to find the closed-loop stability limits for the above system. There are three closed-poles. We conjecture that at the stabil­ ity boundary (marginal stability) there will be a pair of poles on the imaginary axis at s = ±jω, and a single real pole at s = −a. The closed-loop characteristic polynomial will therefore be D(s) = (s + a)(s2 + ω 2 ) = s3 + as2 + ω 2 s + aω 2 Comparing coefficients with the actual closed-loop characteristic polynomial D(s) = s3 + 3s2 + 5s + (2 + K) we determine a = 3 ω2 = 5 → 2 aω = K + 2 2 ω= → √ 5 K = 13 Root Locus Methods We have seen that the closed-loop poles change as controller parameters vary. A root-locus is is an s-plane plot of the paths that the closed-loop poles take as a controller parameter varies. Let’s start with some simple examples. Example 4 Consider the first order plant under proportional control, as shown below: 26–4 P c o n tr o lle r + R (s ) p la n t C (s ) 1 K s + a - The closed-loop transfer function is K s + (a + K) Gcl (s) = with a single pole pc] = −(a + K). The root-locus is simply the path of this pole as K varies from K = 0 to K = ∞. Clearly as K → 0, the closed-loop pole approaches the open-loop pole (s = −a), and as K → ∞, the closed-loop pole p → −∞. This is all the information we need to construct the root-locus for this system. jw s - p la n e K in c r e a s in g K = 0 x -a th e ro in d ic a o f th e v a r ie s s o t lo c u s te s th e p a th p o le a s K . Example 5 Construct the root-locus plot for the first-order system under P-D control with Gc (s) = Kp + Kd s: R (s ) P - D c o n tr o lle r + K p + K d s - 26–5 p la n t 1 s + a C (s ) If we write � Gc (s) = Kd Kp s+ Kd � we have a open-loop pole at s = −a and an open-loop zero at s = −Kp /Kd = −b. The closed-loop transfer function is Gcl (s) = Kd (s + b) (Kd + 1)s + (a + Kd b) with a single pole a + bKd . 1 + Kd We now construct the root-locus as Kd varies from 0 to ∞. Clearly as Kd → 0, the closed-loop pole pcl → −a approaches the open-loop pole at (s = −a), and as Kd → ∞, the closed-loop pole pcl → −b, in other words the closed-loop pole approaches the open-loop zero. There are two possibilities for the root locus based on the relative positions of the open-loop pole and zero: pcl = − jw jw s - p la n e o K in c r e a s in g K = 0 x -a -b s - p la n e K = 0 s -a (a ) b > a x K in c r e a s in g o -b (b ) a > b While the root locus always originates at the pole and terminates at the zero, if b > a the closed-loop pole will move to the left, while if a > b the pole will move to the right. In addition we can note: • There is a closed-loop zero at s = −b. • This is a case when the order of the numerator is equal to the order of the denominator, and there will be direct feed-through from the input to the output, as discussed in Lecture 22. • As K is increased, and the closed-loop pole approaches the zero, the strength of the component epcl t in the response will be diminished (Lecture 23). 26–6 s Example 6 Determine the root locus for the second-order system G(s) = ωn2 s2 + 2ζωn s + ωn2 under proportional control. R (s ) P c o n tr o lle r + p la n t w K s - 2 + 2 z w C (s ) 2 n n s + w n 2 The closed-loop transfer function is G(s) = Kωn2 s2 + 2ζωn s + ωn2 (1 + K) with closed-loop poles p1 , p2 = −ζωn ± ωn � ζ 2 − (1 + K) which will be real only if ζ ≥ 1 and K ≤ ζ − 1, otherwise they will be complex conjugates. We note the following: • As� K → 0, the closed-loop poles approach the open-loop poles −ζωn ± ωn ζ 2 − 1. • If the open-loop poles are real, the closed-loop poles will move together as K → ζ 2 − 1, and then become complex. � • If the closed-loop poles are complex, p1 , p2 = −ζωn ± jωn (1 + K) − ζ 2 , only the imaginary part is affected by K, and as K → ∞ the closed-loop poles p1 , p2 → −ζωn ± j∞. This behavior is summarized in the following root locus plots: 26–7 K in c r e a s in g jw K in c r e a s in g s - p la n e x K = z - 1 x K = 0 K = 0 x K in c r e a s in g K = 0 K in c r e a s in g (b ) z < 1 (a ) z > 1 2.1.1 s - p la n e s s x 2.1 K = 0 jw Some Basic Properties of Root Locus Plots The Number of Branches in the Plot By definition there will be one branch of the plot for each closed-loop pole. For a system with open-loop transfer function Nol (s) Gol (s) = K Dol (s) The closed-loop characteristic polynomial is Dcl (s) = Dol (s) + KNol (s) and provided the order of Nol (s) does not exceed that of Dol (s), the order of Dcl (s) will be the same as that of Dol (s). In other words, the number of closed-loop poles equals the number of open-loop poles, and the number of branches equals the number of open-loop poles.. 2.1.2 Symmetry of the Root Locus Plot Because all closed-loop poles are either real or complex conjugate pairs, the root locus is symmetrical about the real axis. The implication of this is that when we discuss rules for generating a root locus, we only have to consider half of the s-plane. 2.1.3 The Origins of the Branches (K = 0) The closed-loop characteristic polynomial is Dcl (s) = Dol (s) + KNol (s). As K → 0, Dcl (s) ≈ Dol (s), with the result that the n branches of the root locus always originate at the open-loop poles. 26–8 2.1.4 The Terminal Points of the Branches (K → ∞) As K becomes large Dcl (s) ≈ KNol (s) with the result that m of the n closed-loop roots approach the m open-loop zeros. This leaves n − m roots to be accounted for, and we will investigate this later. For now we simply state that these branches diverge away from the origin along a set of n − m straight-line asymptotes, and as K → ∞ these poles approach a distance r = ∞ from the origin. 26–9