Design of Nonlinear Robust Control in a Class of Structurally Stable Functions

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1 World Acdem of Science, Engineering nd Technolog 009 Design of Nonliner Robust Control in Clss of Structurll Stble Functions V. Ten Abstrct An pproch of design of stble of control sstems with ultimtel wide rnges of uncertinl disturbed prmeters is offered. The method relies on using of nonliner structurll stble functions from ctstrophe theor s controllers. Theoreticl prt presents n nlsis of designed nonliner second-order control sstems. As more importnt the integrtors in series, cnonicl controllble form nd Jordn forms re considered. The nlsis resumes tht due to dded controllers sstems become stble nd insensitive to n disturbnce of prmeters. Eperimentl prt presents MATLAB simultion of design of control sstems of epidemic spred, ircrft s ngulr motion nd submrine depth. The results of simultion confirm the efficienc of offered method of design. Kewords Ctstrophes, robust control, simultion, uncertin prmeters. I. INTRODUCTION HERE is lot of methods of design of robust control T which develop with incresing interest nd some of them become clssicl. Commonl ll of them re dedicted to defining the rnges of prmeters (if uncertint of prmeters tes plce) within which the sstem will function with desirble properties, first of ll, will be stble [,]. Thus there re mn reserches which successfull ttenute the uncertin chnges of prmeters in smll (regrding to mgnitudes of their own nominl vlues) rnges. But no one eisting method cn gurntee the stbilit of designed control sstem t rbitrril lrge rnges of uncertinl chnging prmeters of plnt. The pproch tht is offered in the present wor relies on the results of ctstrophe theor [,4,5,6,7], uses nonliner structurll stble functions, nd due to bifurctions of equilibrium points in designed nonliner sstems llows to stbilie dnmic plnt with ultimtel wide rnges of chnging of prmeters. It is nown tht the ctstrophe theor dels with severl functions which re chrcteried b their stble structure. Tod there re mn clssifictions of these functions but originll the re discovered s seven bsic nonlinerities nmed s ctstrophes : + (fold); V. Ten ws with the Eursin Ntionl Universit, Astn, Khstn, (phone: ; f: ; e-mil: vitorten@ mil.ru). He is now with the Center for Applied Mthemtics, Cornell Universit, Ithc, NY 485 USA (cusp); (swllowtil); (butterfl); (hperbolic umbilic); + ( + ) (elliptic umbilic); (prbolic umbilic). 4 A prt of the ctstrophe which does not contin prmeters i is clled s germ of ctstrophe. Adding n of them to dnmic sstem s controller will give effect shown below. On the emple of the ctstrophe elliptic umbilic dded to dnmicl sstems we shll see tht: ) new (one or severl) equilibrium point ppers so there re t lest two equilibrium point in new designed sstem, ) these equilibrium points re stble but not simultneous, i.e. if one eists (is stble) then nother does not eist (is unstble), ) stbilit of the equilibrium points re determined b vlues or reltions of vlues of prmeters of the sstem, 4) wht vlue(s) or wht reltion(s) of vlues of prmeters would not be, ever time there will be one nd onl one stble equilibrium point to which the sstem will ttend nd thus be stble. Let us consider the cses of second-order sstems () nd emples of design of control sstems () of epidemic spred (-), ircrft s ngulr motion (-) nd submrine depth (- ). II. SECOND ORDER SYSTEMS A. Integrtors in series. Let us consider control plnt presented b two integrtors connected in series, s shown in Fig. : u = T S T S Fig. Integrtors in series structure where T nd T re the prmeters of integrtion. This structure is fmous of its instbilit, i.e. no one liner controller cn provide the stbilit to such sstem nd more over with uncertinl chngeble prmeters [8,9]. 660

2 World Acdem of Science, Engineering nd Technolog 009 Let us choose feedbc control lw s following form: u = + ( + ) + +, () nd in order to stud stbilit of the sstem let us suppose tht there is no input signl in the sstem (equl to ero) [0]. Hence, the sstem with proposed controller cn be presented s: d =, T d = T ( + ( + ) + + ). =. The sstem () hs following equilibrium points () s = 0, s = 0; () s =, s = 0. (4) Stbilit conditions for equilibrium point () obtined vi linerition re Let us suppose tht T is positive, or cn be perturbed sting positive. So if we cn set the nd both negtive nd > then it does not mtter wht vlue (negtive or positive) the prmeter T would be (ecept ero), in n cse the sstem () will be stble. If T is positive then equilibrium point () ppers (becomes stble) nd equilibrium point (4) becomes unstble (disppers) nd vice vers, if T is negtive then equilibrium point (4) ppers (become stble) nd equilibrium point () becomes unstble (disppers). Results of MtLb simultion for the first nd second cses re presented in Fig. nd respectivel. In both cses we see how phse trjectories converge to equilibrium points ( 0,0) nd ;0. In the Fig. the phse portrit of the sstem () t constnt =, =-5, =-, T =00 nd vrious (perturbed) T (from to 4500 with step 000) with initil condition =(-;0) is shown. In the Fig. the phse portrit of the sstem () t constnt =, =-, =-, T =000 nd vrious (perturbed) T (from -450 to 450 with step 00) with initil condition =(-0.5;0) is shown. T < 0. TT (5) Stbilit conditions of the equilibrium point (4) re + T > 0. TT (6) Fig. Behvior of designed control sstem in the cse of integrtors in series t vrious T B compring the stbilit conditions given b (5) nd (6) we find tht the signs of the epressions in the second inequlities re opposite. Also we cn see tht the signs of epressions in the first inequlities cn be opposite due to squres of the prmeters nd if we properl set their vlues. Let us suppose tht prmeter T cn be perturbed but remins positive. If we set nd both negtive nd < then the vlue of prmeter T is irrelevnt. It cn ssume n vlues both positive nd negtive (ecept ero), nd the sstem given b () remins stble. If T is positive then the sstem converges to the equilibrium point () (becomes stble). Liewise, if T is negtive then the sstem converges to the equilibrium point (4) which ppers (becomes stble). At this moment the equilibrium point () becomes unstble (disppers). Fig. Behvior of designed control sstem in the cse of integrtors in series t vrious T Another two forms, cnonicl controllble form nd Jordn form re importnt becuse we cn reduce n liner mtri of control plnt to n of them. 66

3 World Acdem of Science, Engineering nd Technolog 009 B. Cnonicl controllble form (CCF). This form is importnt if we would lie to ffect to the lst term of chrcteristic polnomil n which corresponds to generl gin of the sstem. Let us consider the second order sstem which is identicl to CCF: d =, d =. =. It is nown tht the sstem will be stble if nd onl if the prmeters nd re positive. If for emple the smll perturbtion will me the negtive then sstem will become unstble. Let us set the control lw in the form (). Hence we will obtin the following equtions of designed control sstem. d =, d = + ( + ) + +. =. (7) The sstem (7) hs following equilibrium points: s s = 0, s = 0 ; (8) =, = 0 ; s (9) Stbilit conditions for equilibrium points (8) nd (9) respectivel re: > 0. + > 0. ( ) (0) () From inequlities (0) nd () it is es to see tht here it does not mtter wht vlue ecept ero prmeter will be. Similr to bove we cn resume tht sstem (7) will be stble. In the Fig. 4 the motion of the sstem (7) t constnt control prmeters =4, =-4, =-6, constnt plnt prmeter = nd vrious vlues of plnt prmeter which vries from to 9.5 with step.0, with initil condition =(0.05;0) is shown. Fig. 4 Behvior of designed control sstem in the cse of CCF C. Jordn form. Let us consider the model of second order sstem which corresponds to Jordn form, i.e. it is digonl mtri with eigenvlues s prmeters. d = ρ, d = ρ. () Due to n bsence of the reltionship between the phse coordintes here we cn control ech phse coordinte seprtel nd choose the control lw in simplified form (s we sid without germ). Let us choose the control lw in the simplified form of elliptic umbilic ctstrophe without germ nd merging the control prmeters: u + b =, u = () + Hence, the sstem () with set control () is: d = ρ +, b d = ρ +. c c (4) Nonliner control sstem (4) hs the following equilibrium points: 4 s s = 0, s = 0 ; (5) ρ + c = 0, s = ; s (6) ρ + b =, = 0 s s (7) ρ + ρ + b = c, = ; s (8) Stbilit conditions for the equilibrium point (5) re: 66

4 World Acdem of Science, Engineering nd Technolog 009 ρ + b ρ + > 0. c Stbilit conditions for the equilibrium point (6) re: ρ + b ρ + < 0. c Stbilit conditions for the equilibrium point (7) re: ρ + < 0, b ρ + > 0. c Stbilit conditions for the equilibrium point (8) re: ρ + < 0, b ρ + < 0. c (9) (0) () () d = β, d = β γ, d = + γ. () Let us ssume tht the popultion is closed, i.e. the rte t which susceptibles dded to the popultion is equl to 0 nd the rte t which new infectives re dded to the popultion is equl 0. Let us choose the control lw in simplified form of ctstrophe elliptic umbilic without its germ: 0 u = ( ) + + +, B = (4) 0 Hence, the sstem (9) with the offered control is: From inequlities (9-) it is es to see tht here it does not mtter wht vlues ecept ero prmeters ρ nd ρ will be. After n perturbtions whtever vlue ecept ero this pir would not be, ever time there is one nd onl one of the equilibrium points (5-8) to which the sstem will ttend nd t tht moment ll nother equilibrium points will be unstble or will not eist. In the Fig. 5 the motion of the sstem (4) t constnt control prmeters = nd b = c =5 nd plnt vrious prmeters m nd m (ρ nd ρ ) which vr from -50 to 50 with step 500, with initil condition =(50;50) is shown. d = β, d = β γ d = + γ. ( + ) + +, (5) Hence, the sstem (0) hs two equilibrium points: = 0, = 0, = 0 ; (6) = 0, = 0, =. (7) Stbilit conditions of equilibrium point () re: + γ ( + γ )( ( ) + ) ( ) γ γ β β γ ( β γ ) > 0. (8) Fig. 5 Behvior of designed control sstem in the cse of Jordn form III. EXAMPLES OF DESIGN OF CONTROL SYSTEMS A. Epidemic spred. The spred of n epidemic disese cn be described b set of differentil equtions. The popultion under stud is mde up of three groups,, nd, such tht the group is susceptible to the epidemic disese, group is infected with the disese, nd group hs been removed from the initil popultion. The removl of will be due to immunition, deth, or isoltion from. The output of this sstem is the stte vrible. The plnt cn be represented b following equtions [0]: Stbilit conditions of equilibrium point () re: + γ ( + γ )( ( ) + + ) + ( ) γ γ β β γ ( β γ ) > 0. (9) Here we see the opposite not onl for some prmeter but for reltions of severl prmeters. To compre the stbilit with nd without offered controller let us see the Figs. 6 nd 7. Fig. 6 presents the output behvior of the sstem () t constnt vlue = nd vrious vlues of prmeters β nd γ which vr from 4 to 6 with step. 66

5 World Acdem of Science, Engineering nd Technolog 009 In the Fig. 7 the output behvior of the sstem (5) t constnt vlue = nd vrious vlues of prmeters β nd γ which vr from 4 to 6 with step is shown. As it is proposed the output of the sstem (5) ttends to the vlues of the equilibrium points depending on the vlues of prmeters β nd γ, sting ever time stble. 0 0 ω δ A = m, B = m m m, C = ( ), (0) with the following nominl prmeters: δ m =.0 s = 60.7 s ω [ ], = 9.4 [ s ] m, =.8 [ s ] m [ ], u = δ ( t), Fig. 8 Aircrft s motion chrcteristics Fig. 6 Behvior of the epidemic spred t vrious prmeters without controller If we ssume the input δ () t = const = 0 nd stud the dnmic plnt () for stbilit then we see tht it is in the stbilit threshold nd not sufficient for engineering prctice. Let us choose controller in the following form: u = ( ( + ) ). () b Thus, the sstem (0) with the dded controller () will become: Fig. 7 Behvior of the epidemic spred t vrious prmeters with controller B. Aircrft s ngulr motion. Let us consider the dnmics of ircrft s ngulr motion. Often it hs quite complicted structure nd usull is described b high-order sstem of nonliner differentil equtions [8,9]. But commonl it is possible to isolte dnmicl subsstem which vribles nd prmeters chrcterie the ngles nd their reltions in ttitude of flight direction s it is shown in the Fig.8 where ngle of ttc, tilt ngle, pitch ngle, ground speed, nd elevtor control signl re denoted s, θ, ϑ, V, nd δ respectivel [9]. Dnmics of ircrft s isolted ngulr motion is described b the following differentil equtions: = A + Bu, = C. where the mtrices A, B nd C hve the following (nominl) vlues: d =, d ω = m m d =, =. m ( + ) + +, () New nonliner control sstem () hs two equilibrium points: = 0 () = =, 0 (4) Stbilit conditions for the equilibrium point () re: ω m m > 0. ω ( )( ( ω ) + ) m m (5) Stbilit conditions for equilibrium point (4) re: 664

6 World Acdem of Science, Engineering nd Technolog 009 ω + m m > 0. ω ( + )( ( ω ) + + ) m m + (6) from -4., 9.4 nd 0.8 to -0., 49.4 nd 4.8 (devitions from the nominl vlues) with steps, 0 nd respectivel nd δ t = const = is shown. constnt input ( ) If we drw ttention to the lst two inequlities in both stbilit conditions (5) nd (6) then we cn note the opposite requirements for the sign of the prmeter. Let us ssume tht the prmeter stisfies the stbilit conditions of one of the equilibrium points, i.e. the sstem converges to tht. If fter some uncertin perturbtion vlue of the prmeter is chnged such tht sign of it becomes opposite then lthough the current equilibrium point will become unstble or dispper (new vlue will not stisf the current stbilit conditions), nother equilibrium point will pper (become stble) becuse tht new vlue of prmeter will utomticll stisf the stbilit conditions of nother equilibrium point. IOW it does not mtter for the stbilit of the sstem () wht vlue ecept ero the prmeter would be, in n cse the sstem () will be stble. Let us see the results of MATLAB simultion where one nd severl prmeters vried their vlues nd the sstem chnged phse trjectories but sted stble. The Fig. 9 shows the output behvior of the sstem with dded controller t the constnt vlues of the prmeters of plnt m =9.4 nd ω m =.8, prmeter of input δ m =60.7 nd prmeters of control =0., =0., =0.7 (the vlues of prmeters re chosen rbitrril) nd t the vrious vlues of prmeter of the plnt vried from -5.6 to.4 with step 0.5 nd constnt input δ () t = const =. Fig. 0 Behvior of the designed control sstem of ircrft s ngulr motion t vrious, nd ω C. Submrine depth control. Let us consider dnmics of ngulr motion of controlled submrine which is different from the ircrft [0]. This difference results primril from the moment in the verticl plne due to the buonc effect. The importnt vectors of submrine s motion re shown in the Fig.. m m Fig. Angles of submrine s depth dnmics Fig. 9 Behvior of the designed control sstem of ircrft s ngulr motion t vrious In the Fig. 0 the output behvior of the sstem with dded controller t the constnt vlues of the prmeters of control =, =, =7 (the vlues of prmeters re chosen rbitrril), prmeter of input =60.7 nd t the vrious δ vlues of ll prmeters of the plnt m, nd ω m m vried Let us ssume tht θ is smll ngle nd the velocit v is constnt nd equl to 5 ft/s. The stte vribles of the submrine, considering onl verticl control, re = θ, dθ =, =, where is the ngle of ttc nd output. Thus the stte vector differentil eqution for this sstem, when the submrine hs n Albcore tpe hull, is: where 0 0 A =, 0 () t = A + Bδ, (7) 0 B = b, b prmeters of the mtrices re equl to: s 665

7 World Acdem of Science, Engineering nd Technolog 009 =, = , = 0., = 0., = 0. 07, = 0., b = 0.095, b = 0. 07, nd δ s (t) is the deflection of the stern plne. Let us stud the behvior of the sstem (7). In generl form it is described s: d =, d = b δ () t, (8) S d = + + b δ () t. S where input δ s (t)=. B turn let us simulte b MATLAB the chnging of the vlue of ech prmeter devited from nominl vlue. In the Fig. the behvior of the sstem (8) t vrious vlue of (vries from -0.0 to with step 0.005) nd ll left constnt prmeters with nominl vlues is presented. In the Fig. the behvior of the sstem (8) t vrious vlue of (vries from -0.6 to 0.89 with step 0.5) nd ll left constnt prmeters with nominl vlues is presented. In the Fig. 4 the behvior of the sstem (8) t vrious vlue of (vries from to.0 with step 0.) nd ll left constnt prmeters with nominl vlues is presented. In the Fig. 5 the behvior of the sstem (8) t vrious vlue of (vries from -0.4 to 0.57 with step 0.5) nd ll left constnt prmeters with nominl vlues is presented. In the Fig. 6 the behvior of the sstem (8) t vrious vlue of (vries from -. to 0.7 to with step 0.5) nd ll left constnt prmeters with nominl vlues is presented. Fig. Behvior of the dnmics of submrine s depth t vrious Fig. 4 Behvior of the dnmics of submrine s depth t vrious Fig. 5 Behvior of the dnmics of submrine s depth t vrious Fig. Behvior of the dnmics of submrine s depth t vrious Fig. 6 Behvior of the dnmics of submrine s depth t vrious It is cler tht the perturbtion of onl one prmeter mes the sstem unstble. Let us set the feedbc control lw in the following form: 666

8 World Acdem of Science, Engineering nd Technolog 009 ( ) + + u = +. (9) Hence, designed control sstem is: d =, d = b δ () t, (40) S d = + + b δ () t ( + ) + +. S The results of MATLAB simultion of the control sstem (40) with ech chnging (disturbed) prmeter re presented in the Figs. 7, 8, 9, 0, nd. In the Fig. 7 the behvior designed control sstem (40) t vrious vlue of (vries from -0.0 to with step 0.005) nd ll left constnt prmeters with nominl vlues is presented In the Fig. 8 the behvior of the sstem (40) t vrious vlue of (vries from -0.6 to 0.89 with step 0.5) nd ll left constnt prmeters with nominl vlues is presented. In the Fig. 9 the behvior of the sstem (40) t vrious vlue of (vries from to.0 with step 0.) nd ll left constnt prmeters with nominl vlues is presented. In the Fig. 0 the behvior of the sstem (40) t vrious vlue of (vries from -0.4 to 0.57 with step 0.5) nd ll left constnt prmeters with nominl vlues is presented. In the Fig. the behvior of the sstem (40) t vrious vlue of (vries from -. to 0.7 to with step 0.5) nd ll left constnt prmeters with nominl vlues re presented. Fig. 8 Behvior of the submrine depth control sstem t vrious Fig. 9 Behvior of the submrine depth control sstem t vrious Fig. 0 Behvior of the submrine depth control sstem t vrious Fig. 7 Behvior of the submrine depth control sstem t vrious Fig. Behvior of the submrine depth control sstem t vrious IV. CONCLUSION Resuming we cn conclude tht using structurll stble functions from ctstrophe theor s controllers give mn 667

9 World Acdem of Science, Engineering nd Technolog 009 dvntges. The min of them is tht the sfe rnges of prmeters re widened significntl becuse the designed sstem st stble within unbounded rnges of perturbtion of prmeters even the sign of them chnges. The behviors of designed control sstems obtined b MATLAB simultion such tht control of epidemic spred, ircrft s ngulr motion nd submrine depth confirm the efficienc of the offered method. The offered pproch of design cn be pplied not onl for liner but lso for some set or clss of nonliner dnmic plnts. For further reserch nd investigtion mn perspective tss cn occur such tht snthesis of control sstems with specil requirements, design of optiml control, control of chos, etc. REFERENCES [] D.-W. Gu, P.Hr. Petov nd M.M. Konstntinov. Robust control design with Mtlb. London: Springer-Verlg, 005. [] Boris T. Pol, Pvel S. Shcherbov. Robust stbilit nd control. Moscow: Nu, pges. (in Russin). [] Poston, T. nd Stewrt, In. Ctstrophe: Theor nd Its Applictions. New Yor: Dover, 998. [4] Gilmore, Robert. Ctstrophe Theor for Scientists nd Engineers. New Yor: Dover, 99. [5] Arnol'd, Vldimir Igorevich. Ctstrophe Theor, rd ed. Berlin: Springer-Verlg, 99.. [6] [7] [8] Andrievsii B.R., Frdov A.L., Ibrnne glv teorii fvtomtichesogo uprvleni s primermi n e MATLAB (Selected topics of utomtic control theor with emples in the MATLAB lnguge), Petersburg: Nu, p. [9] Bodner V.A. Aircrft control sstems. Moscow: Mshinostroenie, p. (in Russin). [0] Richrd C Dorf, Robert H. Bishop. Modern Control Sstems, /E. Prentice Hll: 008. [] John Dole, Bruce Frncis, Allen Tnnenbum. Feedbc control theor. Mcmilln Publishing Co., 990. [] Hssn K Khlil. Nonliner Sstems Third Edition Prentice Hll, 00. [] M. Beisenbi, V. Ten. An pproch to the increse of potentil of robust stbilit of control sstems. // Theses of the reports of VII Interntionl seminr «Stbilit nd fluctutions of nonliner control sstems». Moscow, Institute of problems of control of Russin Acdem of Sciences, P. -. (in Russin). [4] M. Beisenbi, V. Ten. The designing the control sstems with incresed potentil of robust stbilit in the clss of three-prmetricl structurll stble mps. // The boo of bstrcts of VIII Interntionl conference «The stbilit, control nd rigid bod s dnmics», Donets, P. -. (in Russin). Vitor Ten, Almt, Khstn, Mster s Degree of computer engineering nd softwre Kh Ntionl Technicl Universit (Almt, Khstn) 00, Cndidte s Degree in Technicl Sciences sstem nlsis, dt processing nd control Eursin Ntionl Universit (Astn, Khstn) 005. Senior Lecturer of Mthemtics nd Informtion Technologies Deprtment of Eursin Ntionl Universit (Astn, Khstn). 668