ntrol system for stability studies is laborious and time cons rical technique based on simple fast mathematical iteration

-AIDE ED R UMERICAL TE ECHIQUE A. R. Zubair,*, A. Olatunbosun, DEPARTMET OF ELECTRICAL/ELECTROIC EGIEERIG, UIVERSITY OF IBA igerian Journal of Technology (IJOTECH) Vol. 33. o.. January 04, pp.60 7 Copyright Faculty of Engineering, University of igeria, sukka, ISS: 5-8443 www.nijotech.com http://dx.doi.org/0.434/njt.v33i.9 ADA, IGERIA E-mail addresses: ar.zubair@mail.ui.edu.ng, aolatunbosun@yahoo.com Abstract The existing technique of manual calculation lation and plotting of root-locus of a con ntrol system for stability studies is laborious and time cons suming. A computer-aided root-locus numer rical technique based on simple fast maematical iteration is developed to eliminate is rigour. Altern native iteration formulas were formed and tested for conv vergence. A pair of complementary formul las wi highest rate of convergence was selected. The pro ocess of automatic determination of roots and nd plotting of root-locus were divided into steps which were analyzed, simplified and coded into computer programs. The geometric properties of e root- -loci obtained wi is technique are foun nd to conform to ose described in e literature. Root-loci are drawn to scale instead of rough sketches s. eywords: control system, stability, transient response, root-locus, iteration. Introduction A control system is e means by which any parameter of interest in a machine, mechanism or equipment is maintained or altered in accordance wi desired manner. Introduction of feedback into a control system has e advantages of reducing sensitivity of system performance to internal variations in system parameters, improving transient response and minimizing e effects of disturbance signals. However, feedback increases e number of components, increases complexity, reduces gain and introduces e possibility of instability [ 6]. A system is stable if its response to a bounded input vanishes as time t approaches. An unstable system cannot perform any control task. A stable system wi low damping is also not desirable. Therefore, a stable system must also meet e specifications on relative stability which is a quantitative measure of how fast e transients die out wi time in e system. The location of roots of system s characteristic equation determines e stability of e system [ -5, 7, 8]. The roots of e system ss characteristic equation are e same as e poles of e closedbe achieved by loop system. Ideally, a desired performance can a control system by adjusting e location of its roots in e s-plane by varying one or more system parameters. Root-locus Meod is a linear * Corresponding auor, Tel: +34-80378605 time invariant control system design technique at determines e roots of e characteristic equation (or closed-loop pole when e openincreased from zero to loop gain-constant is infinity [9, 0]. The root-locus shows e complete dynamic response of e system and is a measure of e sensitivity of e roots to variation in e parameter (open-loop gain- frequency response of constant ) []. It yields e system and can also be used to solve problems in e time domain. The basic task in root-locus technique is to determine e closed- e configuration of loop pole configuration from e open-loop poles and zeros. The locus of e roots or closed-loop poles are plotted in e s-plane. This is a complex plane, since s σ + jω. The real part, σ, is e index in e exponential term of e time response, and if positive will make e system unstable. Any locus in e right-hand side of e plane erefore represents an unstable system. The imaginary part, ω, is e frequency of transient oscillation. When a locus crosses e imaginary axis, σ 0, e control system is on e verge of instability; where transient oscillations neier increase, nor decay, but remain at a constant value. The design meod requires e closed-loop poles to be plotted in e s-plane as is varied from zero to infinity, and en a value of selected to provide e necessary transient response as required by

OCUS UMERICAL TECHIQUE ECHIQUE, e performance specification. The loci always commence at open-loop poles (denoted by x) and terminate at open-loop zeros (denoted by o) when ey exist [ 5, 9, 0, ]. In e literature, ere exist eight rules based on elementary geometric properties of root-locus which lead to rough sketch of root locus. These rules govern e important features of e rootlocus meod such as asymptotes, roots condition on e real axis, breakaway points, and imaginary axis crossover [3 6]. These rules or steps involve some calculations but avoid determination of actual roots. There are oer approaches which use complex analytic or semianalytic representation at involve e use of equations of e loci [4, 7 5]. There is also a computer aided root-locus meod based on complex conversion of solution of a parameterized family of algebraic problems into solution of a set of associated differential equations []. In is paper, a computer-aided root-locus numerical technique is developed based on simple and fast maematical iteration..0 Root-Locus umerical Technique Figure shows a closed-loop system wi negative feedback; G( is e forward-part gain. H( is e feedback gain. The transfer function of e system is given by Eqn () [ - 5]. The denominator of e transfer function set to zero is referred to as e characteristic equation: + G ( 0. Figure : Closed-Loop System wi egative Feedback C( G( () R( + G( The open-loop transfer function is given as in Eqn. (). B G( A () [ s z()][ s z()]...[ s z( m )][ s z( m)] [ s p()][ s p()]...[ s p( n )][ s p( n)] z(), z(), z(3),, z(m-) and z(m) are m openloop zeros and p(), p(), p(3),, p(n-) and p(n) are n open-loop poles. Given e open-loop poles and zeros, e task is to deduce e configuration of e closed-loop poles as varies from zero to infinity. The characteristic equation becomes: B + 0; A + B 0 (3) A The root-locus technique presented in is paper is divided into four steps.. Step : Expansion for e Formation of Characteristic Equation The first step is e formation of polynomials A and B from e given open-loop poles and zeros respectively. This step involves expansion. A and B are given as A [ s p()][ s p()][ s p(3)]... and [ s p( n )][ s p( n)] a(n + )s n + a( n) s +... + a(3) s + a() s + a() B [ s z()][ s z()][ s z(3)]... [ s z( m )][ s z( m)] b(m + )s m + b( m) s m +... + b(3) s + b() s + b() Suppose an order polynomial C( + ) s + ) s +... + ) s + ) is multiplied wi ( s r) where r is a real open-loop pole or zero. The product is a (+) order polynomial + D( d( + ) s + d( + ) s + d( ) s +... + d() s + d() such at y ) for y ( + ) d( y ) r for y to ( + ) (4) r for y Suppose an order polynomial C( + ) s + ) s +... + ) s + ) is multiplied wi [ s ( r + jh)][ s ( r jh)] where (r+jh) and (r-jh) are complex conjugate openloop poles or zeros. The product is a (+) order polynomial + + D( d( + 3) s + d( + ) s such at: + d( + ) s +... + d () s + d() t(3) y ) for y ( + 3) t(3) y ) + t() y ) for y ( + ) d( t(3) y ) + t() y ) + t() for y 3 to ( + ) t() y ) + t() for y t() for y (5) where [ s ( r + jh)][ s ( r jh)] t(3) s + t() s + t() t ( 3) ; t() r; t() r + h (6) n igerian Journal of Technology, Vol. 33, o., January 04 6 6

OCUS UMERICAL TECHIQUE ECHIQUE, Eqns. (4) and (5) give e general models for expansion involving a single real pole or zero and a pair of conjugate complex pole or zero respectively. As illustrated wi e flow chart of Figure, e expansion of A starts wi C( (order zero) multiplied wi s r for e first real pole. The product D obtained is feedback to e multiplication process as C and e process is repeated for oer real poles. The new product D is feedback as C which is multiplied wi [ s ( r + jh)][ s ( r jh)] t(3) s + t() s + t() for e first pair of conjugate poles. The process is repeated for oer pairs of conjugate poles. A is set equal to e final product D. The whole process is repeated for B using zeros instead of poles. Wi e substitution of A, B and a value for, e characteristic Eqn. (3) is given as in Eqn. (7). The characteristic equation is said to be of order and has roots. E( e( + )s + e( ) s +... (7) + e(3) s + e() s + e() 0. Step : Determination of Roots for a Value of Having obtained e characteristic equation, e next step is to determine e roots for a value of. This step is classified into ree sub-steps... Sub-Step.: Iteration For, e root is: e() r (8) e() For, one of e roots is e() e() 4e() e(3) r (9) e(3) + s + s [( e() e() s ) s )/ e( + ) ] [( e() s e() ) s )/ e( + ) ] [( e() e() s ) s )/ e( + ) ] [( e() s e() ) s )/ e( + ) ] For >, a root of e characteristic Eqn. (7) is obtainable by iterative technique. Iteration computes a sequence of progressively accurate iterates to approximate e solution of an equation [6,7]. Iteration formula is obtainable from Eqn. (7) by making s e subject of formula. Retaining e term e()s in Eqn. (7) on e left hand side (LHS), moving all oer terms to e right hand side (RHS) and dividing bo sides by e() lead to e iteration formula of Eqn. (0). 3 e() s e(4) s... s / e() (0) e( ) s e( + ) s Retaining e term e(+)s in Eqn. (7) on LHS, moving all oer terms to RHS, dividing bo sides by e(+) and evaluating root of bo sides lead to e even version of e iterative formula of Eqn. (). This is used if is even. Retaining e term e(+)s in Eqn. (7) on LHS, moving all oer terms to RHS, dividing bo sides by e(+)s and evaluating root of bo sides lead to e odd version of e iterative formula of Eqn. (). This is used if is odd (- is even). Eqns. () and () are e same except at e result of root and root are taken to be positive in Eqn. () but negative in Eqn. (). For example, e result of 6 ¼ is eier + or -. - - for even for odd for even for odd Eqns. (3) to (6) are obtainable from Eqn. () by breaking down e root into two roots or breaking down e root into two () () roots on e RHS. For example +6 ¼ is e igerian Journal of Technology, Vol. 33, o., January 04 6 6

OCUS UMERICAL TECHIQUE ECHIQUE, same as [(6) ½ ] ½. Eqns. (3), (4), (5) and (6) are e same except at e two roots or e two roots are taken to be ( ), ( + ), ( + + ) and ( ) + respectively. Figure : Flow Chart for Expansion Process [( ) ] e() e() s ) s / e( + ) for even s [( ) ] e() s e() ) s / e( + ) - for odd (3) [( ) ] + e() e() s ) s / e( + ) for even s [( ) ] + e() s e() ) s / e( + ) - for odd (4) + [( ) ] + e() e() s ) s / e( + ) for even s + [( ) ] + e() s e() ) s / e( + ) - for odd (5) + [( ) ] e() e() s ) s / e( + ) for even s + [( ) ] e() s e() ) s / e( + ) - for odd (6) igerian Journal of Technology, Vol. 33, o., January 04 63 63

OCUS UMERICAL TECHIQUE ECHIQUE, The alternative iteration formulas Eqns. (0) to (6) were subjected to tests to determine e most reliable or effective. The results and conclusion of e tests are presented and discussed in section 3.. A guess value of s is substituted in e right side of e iteration formula; a new value of s is computed and substituted back in e right side of e iteration formula and e process is repeated T times. s will converge to a root r of e characteristic equation. The higher e T, e closer e result of iteration to e root and e more time e process takes. Therefore T should not be too small and should not be too large. In some cases, s may not converge to e root [6, 7]. There is need to validate e result obtained as a root... Sub-Step.: Root Validation Test The final value of s ( s r) obtained by iteration is tested by substituting e value of s in e test equation, Eqn. (7) which is adapted from e characteristic equation Eqn. (7). rvt is computed. If rvt is equal to 0 or less an 0.00004, s is accepted and recorded as a root; oerwise it is rejected. rvt e( + )s + e( ) s +... + e(3) s + e() s + e() (7) predicts it s conjugate, r σ jω, and vice versa. Therefore, for a complex root r, Eqn. (7) is divided by ( s σ + jω)( s σ jω) and e order is reduced by. Suppose an order polynomial E( e( + )) s + e( ) s + e( ) s +... + e() s + e() is divided by s r where r is a real root. The quotient a (-) order polynomial 3 D( d( ) s + d( ) s + d( ) s +... + d() s + d() such at e( y + ) for y d ( (8) e( y + ) + rd( y + ) for y ( -) to Suppose a order polynomial E( e( + )) s + e( ) s + e( ) s +... + e() s + e() is divided by a pair of conjugate roots, ( s σ + jω)( s σ jω). The quotient is a (-) order polynomial : 3 D( d ( ) s + d ( ) s such 4 + d ( 3) s +... + d () s + d () at e( y + ) for y - d( e( y + ) td( y + ) for y - e( y + ) td( y + ) pd( y + ) for y ( - 3) to..3 Sub-Step.3: Grouping of Accepted Root (9) into a Sub-branch of Root-Locus where Accepted root is plotted as a standalone dot (.) on ( s σ + jω)( s σ jω) s + ts + p; a root-locus dots-plot. Root-locus for a system t σ ; p σ + ω (0) wi n open-loop poles has n branches. A branch Eqns. (8) and (9) give e general models for of root-locus starts at an open loop pole usually e division process involving a single real root marked wi x and ends at an open loop zero and a pair of conjugate complex root respectively. usually marked wi o or at infinity. At breakaway points, each single branch breaks into..5 Repetition of Sub-Steps. to.4 for ext sub-branches. For e purpose of joining e dots Root wi a curve, each accepted root is checked and The output D of e division process (Sub-Step grouped into a sub-branch of e root-locus..4) is feedback as E (characteristic equation) to Based on e root-locus dots-plot generated by e iteration process (Sub-Step.) which is e algorim, a number of computer program repeated to obtain e next root. The next root is conditional statements guide e placement of subjected to validation test (Sub-Step.); if it is e accepted root into e right sub-branch. An a valid root, it is plotted as a standalone dot (.) on example is presented section 3.3 and illustrated root-locus dots-plot and it is grouped into a subbranch (Sub-Step.3). Sub-Step.4 is also in figure 5...4 Sub-Step.4: Division Process for ext Root repeated to update D. The sub-steps.,.,.3 Having obtained a root, wheer e root is and.4 are repeated until all e roots have acceptable or not in Sub-Step., ere is need to been obtained and e polynomial D is reduced to divide Eqn. (7) to obtain a lower order D (. characteristic equation for e next root. For a real root r, Eqn. (7) is divided by ( s r) and e.3 Step 3: Plotting of Root-Loc Locus order is reduced by. A complex root r σ + jω is varied at some intervals from zero to infinity. The roots corresponding to each selected value igerian Journal of Technology, Vol. 33, o., January 04 64 64

OCUS UMERICAL TECHIQUE ECHIQUE, of are determined as explained in Step. Figure 3 shows e flow chart for e determination and plotting of e root-locus..4 Adding Details to e Root-Locus Having plotted e root-locus, e asymptotes, breakaway points and imaginary axis crossover can be added. For large values of, root-locus branches are parallel to lines called asymptotes. There are (n-m) asymptotes [ - 5]. Point of intersection of asymptotes on e real axis is given as in Eqn. ()[- 5]. (real part of open - loop pole (real part of open - loop zero P asx ( n m) () The angles between asymptotes and e real axis are given as in Eqn. ()[- 5]. (q + ) π θ ( q) where q 0,,,..., ( n m ) () ( n m) Breakaway points are obtained as e solution of d B 0[ - 5]. But + 0, ds A Therefore, da db + A ds ds B d d( A B) 0 ds ds B. da db.i.e B + A 0 (3) ds ds Eqn. (3) is solved for e breakaway points as illustrated in Figure 4. The points of intersection of e root-locus wi e imaginary axis are called imaginary axis crossover. At ese points, e system is said to be marginally stable. Rou-Hurwitz stabilty criterion is used to obtain value( of for marginal stability. Roots of e characteristic equation corresponding to ese values of include e imaginary axis crossover points. The various steps and sub-steps are coded into computer programs. The required inputs to ese programs are e open-loop poles and zeros and e output is e root-locus. 3. Tests and Results 3. Convergence of Alternative Iteration Formulas The alternative iteration formulas Eqns. (0) to (6) discussed in Sub-Step. are tested wi an open-loop transfer function. For each iteration formula, e number of successful iterations which produced valid roots and number of failed iterations which produced non-valid roots are recorded and presented in Table. Percentage success which is e ratio of successful iterations to total iterations expressed in percentage for each formula is also listed in Table. Eqns. () and (4) are found to be similar and displayed e same performance. Eqn. () is found to be most successful wi 83.4% success followed by Eqn. (3) wi 4.9% success. In anoer experiment, Eqns. () and (3) were used complementarily such at Eqn. () is used normally and Eqn. (3) is only used when Eqn. () failed to produce valid root. 97.7% success was achieved as listed in Table. It is erefore concluded at complementary use of iteration formulas Eqns. () and (3), is e choice for best performance. The flowchart of Figure 3 is actually based on complementary use of Eqns. () and (3) as iteration formulas. The effect of use of versions of iteration formulas for even and odd are tested for two different open-loop transfer functions and e results are summarised in Table. Complementary use of bo versions produced e best results. It is erefore concluded at, when is odd, version for odd should be used and when is even, version for even should be used. Figure 4: Flow Chart for e Determination of Breakaway Points. 3. Sample Root-Locus Plots Sample root-loci plots for a number of systems are obtained wi is computer-aided root-locus numerical technique. These plots are presented in Table 3. 3.3 Adding Details to e Root-Locus Plot The root-locus dots-plot obtained for a system wi open-loop transfer function G ( is shown in Figure + 0) 5(a). The details for is system are obtained and are presented in Figure 5(b) and Table 4. igerian Journal of Technology, Vol. 33, o., January 04 65 Start Obtain A & B as in Step Differentiate A & B Obtain e products da B and db A (step ) ds ds Assemble Eqn. (3) Obtain roots of Eqn. 3 Step.,. &.4 Substitute each root in Eqn. (3) to obtain values of corresponding to e breakaway points Start 65

OCUS UMERICAL TECHIQUE ECHIQUE, It is not necessary to find ese details before plotting e root-locus. However if values of for breakaway points are determined first, it will assist e choice of intervals in e values of to be used. For is open-transfer function, is varied from 0 to 6 at incremental interval of. is varied from 63.5 to 64.5 at e incremental interval of 0.05. is varied from 66 to 98 at e incremental interval of. is varied from 99.5 to 00.5 at e incremental interval of 0.05. Finally is varied from 0 to 5000 wi varying incremental interval (initiall to 00 (later) and en finally 500. must vary very slowly near breakaway points. Figure 3: Flow Chart for e Determination and Plotting of Root-Locus as Varies from 0 to Equation Table : Convergence of Alternative Iteration Formulas for open-loop transfer function o of successful iterations G ( + 0) o of failed iterations % success Remark (0) 0 868 0% not reliable () 0 868 0% not reliable () 74 44 83.4% most reliable (3) 4 744 4.9% reliable near break away point (4) 74 44 83.4% same as Eqn. () (5) 0 868 0% not reliable (6) 6 85.84% not reliable ()&(3) 844 4 97.7% Eqn. (3) is used when Eqn. () fails The choice for best performance igerian Journal of Technology, Vol. 33, o., January 04 66 66

OCUS UMERICAL TECHIQUE ECHIQUE, Equation ()&(3) even version only ()&(3) odd version only ()&(3) bo versions ()&(3) even version only ()&(3) odd version only ()&(3) bo versions Table : Convergence of Even and Odd Versions of Alternative Iteration Formulas o of successful o of failed % success Remark iterations iterations G ( H( + 0) 595 73 68.55% Only version for even was used for bo even and odd 0 868 0% Only version for odd was used for bo even and odd 844 4 97.7% Bo versions were used as appropriate ( s + )( s + 3)( s + s + ) G ( s + )( + 3) 088 0.0009% Only version for even was used for bo even and odd 85 904 6.99% Only version for odd was used for bo even and odd 873 6 80.7% Bo versions were used as appropriate There are four branches starting from e four poles and ending at. For e purpose of grouping into sub-branches, six sub-branches are identified. Sub-branch is on e real axis. Subbranch is parallel to e imaginary axis at s. Sub-branches 3 and 4 are above e real axis and are to e right and left of Sub-branch respectively. Sub-branches 5 and 6 are below e real axis and are to e right and left of Sub-branch respectively. Figure 6 shows e flow chart for grouping of root of is system into a sub-branch. 3.4 Comparison wi Existing Technique One of e existing techniques is e rlocus code in Matlab [5]. The results obtained in is work are compared wi e results obtained wi e existing Matlab rlocus code. Table 5 shows at e roots obtained by iteration in is work are exactly e same as ose obtained using an existing Matlab rlocus Code for ree different open-loop transfer functions. The selected pair of iteration formulas developed in is work can be adapted to determine roots of equations for oer applications besides root-locus. (a) Root-Locus Dots-Plot (b) Complete Root-Locus wi details Figure 5: Root-Locus wi details for G ( + 0) igerian Journal of Technology, Vol. 33, o., January 04 67 67

OCUS UMERICAL TECHIQUE ECHIQUE, Table 3: Sample Root-Locus Plots Obtained S/. Sample Plot S/ ( s + 3) G (. s + ) Sample Plot ( s + )( s + 3) G ( s + )( s + s + ) 3. ( s + )( s + 3)( s + s + ) s + )( + 3) ( G 4. G ( s + )( s + 5) 5. ( s + )( s + 4) 6. G ( s ( s + )( s + 3)( s + 5) G ( s + 6)( s + 6s + 8) igerian Journal of Technology, Vol. 33, o., January 04 68 68

OCUS UMERICAL TECHIQUE ECHIQUE, Table 4: Details for S/ Description Open-loop poles Open-loop zeros il 3 4 5 6 7 8 Point of intersection of asymptotes Angles between e four asymptotes and real axis Breakaway points Value of obtained for marginal stability Imaginary crossover Additional roots for 60 G ( + 0) Results p ; p ; 0 j4; p j 3 4 p + s 45 o, 35 o, 5 o and 35 o s ( 64); s j.45 ( 00) s ( 64); and s + j.45( 00) 60 s j3.6and s + j3. 6 s 4 j3.6and s 4 + j3.6 Table 6 compares e root-locus plots obtained in is work and ose obtained using an existing Matlab rlocus Code for ree different open-loop transfer functions. The plots are e same in terms of shape and values of roots. The plots obtained in is work are magnified as e range of e plots along e real axis is controlled and limited to x real( + 0. 5. x is negative and is less an e real part of each of e open-loop poles and zeros. This is sufficient as e system is unstable in e range real( + 0. 5and e plot is predictable in e range real( x. However, e user of e Matlab rlocus code do not have 4 such convenient control over e range of e plot along real axis. The input to e algorim developed in is work can be eier open-loop poles and zeros or transfer function expressed as e ratio of two polynomials. This is because it included an expansion process (section.). This is an advantage over e existing Matlab rlocus code which only accept transfer function expressed as e ratio of two polynomials. Furermore, e developed algorim adds details of asymptotes, breakaway points and imaginary axis crossover to e root-locus plot (section.4). This is anoer advantage over e existing Matlab rlocus code. Figure 6: Flow chart for Grouping of a Root of G s H s into a Sub-Branch. ( ) ( ) s + 4)( s + 4s + 0) Table 5: Comparison of roots obtained in is work and ose obtained using existing Matlab rlocus code for ree different open-loop transfer functions. ( s + 3) ( s + )( s + 3) G( G( G( H( s + ) s + )( s + s + ) + 0). 0. 3.55 Iteration Existing Matlab Iteration Existing Matlab Iteration Results Existing Matlab Results rlocus Code Results rlocus Code rlocus Code st root -.6+j.0 -.6+j.0 -.06+j0.67 -.06+j0.67.0+j3.978.0+j3.978 nd root -.6-j.0 -.6-j.0 -.06-j0.67 -.06-j0.67.0-j3.978.0-j3.978 3 rd root -0.439+j0.755-0.439+j0.755-3.955-3.955 4 root -0.439+j0.755-0.439+j0.755-0.045-0.045 igerian Journal of Technology, Vol. 33, o., January 04 69 69

OCUS UMERICAL TECHIQUE ECHIQUE, Table 6: Comparison of root-locus plots obtained in is work and ose obtained using existing Matlab rlocus code for ree different open-loop transfer functions. S/ Root-Locus Plot Obtained in This Work Root-Locus Plot Obtained Wi Existing Matlab Rlocus Code. ( s + 3) ( s + 3) G ( s + ) G ( s + ). ( s + )( s + 3) G ( s + )( s + s + ) ( s + )( s + 3) G ( s + )( s + s + ) 3. G ( + 0) G ( + 0) 4. Conclusion A computer aided root-locus numerical technique based on simple fast maematical iteration has been developed. The technique has been tested successfully. The geometric properties of e root locus obtained wi is technique are found to conform to ose described in e literature. Using is technique, root-locus is drawn to scale instead of rough sketches. This technique overcome some of e limitations of e existing Matlab rlocus code and will facilitate linear time invariant control system design. igerian Journal of Technology, Vol. 33, o., January 04 70 70

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