School of Mechanical Engineering Purdue University. DC Motor Position Control The block diagram for position control of the servo table is given by:

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1 Root Locus Motivation Sketching Root Locus Examples ME375 Root Locus - 1 Servo Table Example DC Motor Position Control The block diagram for position control of the servo table is given by: θ D 0.09 See how the closed-loop loop poles move as proportional gain K P varies from 0 to. Find closed-loop loop characteristic equation: Position Plant G p (s) Controller θ DV K P E i 168 s(0.3s+ 1) θ V 0.09 θ ME375 Root Locus - 2 1

2 Servo Table Example (cont.) ME375 Root Locus - 3 Motivation Example 1 Revisit the DC motor positioning system with proportional control. Its corresponding block diagram is: Plant G p (s) Controller U(s) R(s) K P ss+ ( 57.5) Y(s) Sketch the closed-loop loop poles as the controller gain K P varies from 0 to. Find closed-loop loop characteristic equation: 2 s s K P = 0 ME375 Root Locus - 4 2

3 Example 1 Formulate an expression for the roots of the characteristic equation: Find the roots for K P = 0 and K P : Find K P when the roots are repeated. ME375 Root Locus - 5 Example 1 Sketch the root locus: Img. Axis Real Axis ME375 Root Locus - 6 3

4 Example 2 Using the same plant as in Example 1, try a different controller choice: Plant G p (s) Controller U(s) R(s) K d (s + 80) ss+ ( 57.5) Y(s) Sketch the root locus of the closed-loop loop poles as the controller gain K d varies from 0 to. Find closed-loop loop characteristic equation: ME375 Root Locus - 7 Example 2 Formulate an expression for the roots of the characteristic equation: Find the roots for K d = 0 and K d : Find repeated roots. ME375 Root Locus - 8 4

5 Example 2 Repeated roots (cont.): ME375 Root Locus - 9 Example 2 Sketch the root locus: Imag Axis Real Axis ME375 Root Locus

6 Closed-Loop Characteristic Roots (CL Poles) Reference Input R(s) + Error E(s) K P Control Input U(s) G p (s) Plant Output Y(s) H(s) The closed-loop loop transfer function G CL (s)) is: G ()= CL s Img. jω The closed-loop loop characteristic equation is: Real jω ME375 Root Locus - 11 Definitions Root Locus Root Locus plotting is the method of determining the roots of the e following equation on the complex plane when the parameter K varies from 0 : N( s) 1 + K GOL ( s) = 0 or 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 N ) z D( s) = ( s p )( s p ) ( s p ) 1 2 The N Z roots of the polynomial N(s) ), z 1, z 2,, z Nz, are called the finite open- loop zeros. The N P roots of the polynomial D(s) ), p 1, p 2,, p Np, are called the finite open- loop poles. N P ME375 Root Locus

7 Root Locus Methods of obtaining root locus: Given a value of K, numerically solve the 1 + K G OL (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 K P = 0 1+ K P = 0 2 s( s + 1) s + s >> op_num=[0.48]; >> op_den=[ ]; >> rlocus(op_num,op_den); >> [K, poles]=rlocfind(op_num,op_den); Apply the following root locus sketching rules to obtain an approximate root locus plot. ME375 Root Locus - 13 Root Locus Sketching Rules 1 + K G ( s) = 0 OL Rule 1: There is a branch of the root locus for each root of the characteristic equation. The number of branches is equal to the number of open-loop poles or open-loop zeros, whichever is greater. Rule 2: Root locus starts at open-loop poles (when( K = 0) and ends at open-loop zeros (when( K ). If the number of poles is greater than the number of zeros, roots start at the excess poles and terminate at zeros at infinity. If the reverse is true, branches will start at poles at infinity and terminate at the excess zeros. Rule 3: Root locus is symmetric about the real axis, i.e., closed-loop loop poles appear in complex conjugate pairs. ME375 Root Locus

8 Root Locus Sketching Rules Rule 4: Along the real axis, the root locus includes all points to the left l of an odd number of real poles and zeros. Rule 5: If number of poles N P exceeds the number of zeros N Z, then as K, (N P N Z ) branches will become asymptotic to straight lines. These straight lines intersect the real axis with angles θ k at σ 0. pi zi Sum of open-loop poles Sum of open-loop zeros σ 0 = = N N # of open-loop poles # of open-loop zeros P Z (2k + 1) π (2k + 1) 180 θk = [rad] = [deg], k = 0, 1, 2, NP NZ NP NZ If N Z exceeds N P, then as K 0, (N( Z N P ) branches behave as above. Rule 6: Breakaway and/or break-in (arrival) points can be obtained by solving s in the following equations: d ds ( K s ) ( ) = 0 ME375 Root Locus - 15 Root Locus Sketching Rules Rule 7: The departure (arrival) angle for a pole p i (zero z i ) can be calculated by slightly modifying the following equation: ( s z ) + ( s z ) + + ( s z ) ( s p ) ( s p ) ( s p ) = 1 2 NZ 1 2 N 180 p The departure angle θ n from the pole p n can be calculated by replacing the term ( s p n ) with θ n and replacing all the s s with p n in the other terms. Rule 8: If the root locus passes through the imaginary axis (the stability ty boundary), the crossing point jω and the corresponding gain K can be found as follows: Replace s in the left side of the closed-loop loop characteristic equation with jω to obtain the real and imaginary parts of the resulting complex number. Set the real and imaginary parts to zero, and solve for ω and K.. This will tell you at what values of K and at what points on the jω axis the roots will cross. ME375 Root Locus

9 Steps to Sketch Root Locus Step 1: Formulate the (closed-loop) loop) characteristic equation into the standard form for sketching root locus: K N ( s ) ( s z )( s z ) ( s z N ) Z = 0 or 1+ K = 0 D( s) ( s p1)( s p2) ( s pn ) P Step 2: Find the open-loop zeros, z i, and the open-loop poles, p i. Mark the open- loop poles and zeros on the complex plane. Use to represent open-loop poles and to represent the open-loop zeros. Step 3: Determine the real axis segments that are to be included in the root locus by applying Rule 4. Step 4: Determine the number of asymptotes and the corresponding intersection σ 0 and angles θ k by applying Rules 2 and 5. Step 5: (If necessary) Determine the break-away away and break-in points using Rule 6. Step 6: (If necessary) Determine the departure and arrival angles using Rule 7. Step 7: (If necessary) Determine the imaginary axis crossings using Rule 8. Step 8: Use the information from Steps and Rules to sketch the root locus. ME375 Root Locus - 17 Example 3 A feedback control system is proposed. The corresponding block diagram is: Controller Plant G p (s) R(s) + K U(s) 1 ( s + 4) ss+ ( 2) Y(s) Sketch the root locus of the closed-loop loop poles as the controller gain K varies from 0 to. Find closed-loop loop characteristic equation: ME375 Root Locus

10 Example 3 Step 1: Formulate the (closed-loop) loop) characteristic equation into the standard form for sketching root locus: Step 2: Find the open-loop zeros, z i, and the open-loop poles, p i : Step 3: Determine the real axis segments that are to be included in the root locus by applying Rule 4. ME375 Root Locus - 19 Example 3 Step 4: Determine the number of asymptotes and the corresponding intersection σ 0 and angles θ k by applying Rules 2 and 5. Step 5: (If necessary) Determine the break-away away and break-in points using Rule 6. ME375 Root Locus

11 Example 3 Step 6: (If necessary) Determine the departure and arrival angles using Rule 7. Step 7: (If necessary) Determine the imaginary axis crossings using Rule 8. ME375 Root Locus - 21 Example 3 Step 8: Use the information from Steps and Rules to sketch the root locus. Imag Axis Real Axis ME375 Root Locus