Definitions • Root Locus Root Locus plotting is the method of determining the roots of the following equation on the complex plane when the parameter K varies from 0 → ∞: GOL ( s ) 0 1 + K ⋅= or N (s) ⋅ 1 + K= 0 D( s) where N(s) and D(s) are known polynomials in factorized form: N ( s) = ( s − z1 )( s − z2 )( s − z Nz ) D( s) = ( s − p1 )( s − p2 )( s − pNP ) The NZ roots of the polynomial N(s) , z1 , z2 , …, zNz , are called the finite openloop zeros. The NP roots of the polynomial D(s) , p1 , p2 , …, pNp , are called the finite open-loop poles. School of Mechanical Engineering Purdue University ME375 Root Locus - 8 Root Locus Methods of obtaining root locus: • Given a value of K, numerically solve the 1 + K GOL(s) = 0 equation for a set of roots. Repeat this for a set of K values and plot the corresponding roots on the complex plane. (This is what we did in the last in-class exercise.) • Use MATLAB. In MATLAB use the commands rlocus and rlocfind. You can use on-line help to find the usage for these commands. 1 + K P ⋅ 0.03 ⋅ 16 =0 s(0.0174 s + 1) ⇒ 1 + KP ⋅ 0.48 =0 0.0174 s 2 + s >> op_num=[0.48]; >> op_den=[0.0174 1 0]; >> Gol=tf(op_num,op_den); >> rlocus(Gol) >> [K, poles]=rlocfind(Gol); • Apply root locus sketching rules to obtain an approximate root locus plot. School of Mechanical Engineering Purdue University ME375 Root Locus - 9 Root Locus Sketching Rules Reading material: Palm book 11.1-11.2 Motivation: In the design of feedback controllers, a qualitative understanding is needed on the influence of control parameters on the output performance of the controlled system. We have seen earlier the role played by the poles of the closed-loop transfer functions (CLFTs) on the transient and steady-state response of the output response. The influence of control parameters on the placement of the CLTF poles in the complex plane can been seen by directly solving the characteristic equation of the CLTF. However, this process does not allow for a qualitative understanding. Here, we will use a set of rules for sketching the so-called root locus plots based on information known about the zeros and poles of the open-loop system. From this, we can better understand the trends of parameter influence on response and use this in the design of control systems. School of Mechanical Engineering Purdue University ME375 Root Locus - 10 Rules for sketching root locus 1. The number of branches in the root locus plot equals the number of open-loop poles; 2. The locus starts (K=0) at an open-loop pole, and as K → ∞ , m branches ends at open loop zeros, and n-m branches will end at infinity and asymptotically approach straight lines. (n: number of open-loop poles; m: number of open loop zeros) 3. The locus is symmetric about the real axis. 4. The locus, on the real axis, includes all segments left of an odd number of open loop poles and zeros. 5. As K → ∞ , n-m branches will approach straight lines (asymptotes). These straight lines intersect the real axis at σ 0 with an angle θ k , where: σ0 = ∑ p −∑z i n−m i 180o , k = 0,1,..., n − m − 1 , ( pi : OL poles; zi : OL zeros.) θ k = (2k − 1) n−m 6. A locus leaves (“break away”) or enters (“breaks in”) the real axis at values of K for which repeated roots exist in the closed loop characteristic equation. These points and values of K at these points satisfy: dK =0 ds School of Mechanical Engineering Purdue University ME375 Root Locus - 11 Rules for sketching root locus 7. The departure angle for an OL pole (or the arrival angle for an OL zero) can be calculated from: φl ,dep = ∑ψ − ∑ φ − 180 i i ≠l o i ψ l ,arr = ∑ φi − ∑ψ i + 180o i ≠l where φi is the angle formed by the positive direction of the real axis and the vector connecting another pole to the pole(or zero) of interest; ψ i is the angle formed by the positive direction of the real axis and the vector connecting another zero to the pole(or zero) of interest. 8. If the locus passes through the imaginary axis (boundary between stable and unstable responses), the crossing point and corresponding values of K can be found by: - Replacing s by jω in the closed-loop characteristic equation, and - Setting the real and imaginary parts to zero and solving for K and jω School of Mechanical Engineering Purdue University ME375 Root Locus - 12 Steps for sketching root locus Step 1. Determine the closed-loop transfer function; Step 2. Express the characteristic equation for the closed-loop system in the standard form: 1+ K N (s) 0 = D( s) Step 3. Determine the open-loop poles and zeros using, respectively: D(s)=0 and N(s)=0. Mark the open-loop poles with an “x” and zeros with an “o” in the complex plane. Step 4. Apply Rule 4. Step 5. Find the number of asymptotes and the corresponding values of σ 0 and θ k . (Rule 5) Step 6. If necessary, find the break-away and break-in points. (Rule 6) Step 7. If necessary, find the departure and arrival angles. (Rule 7) Step 8. Sketch the root locus. School of Mechanical Engineering Purdue University ME375 Root Locus - 13