Robust stability analysis for a cruise missile

Available online at www.sciencedirect.comJournal of Systems Engineering and ElectronicsVol. 19, No. 4, 2008, p.785 -790。ScienceDirectEL SEVIERwww.sciencedirect. com/science/joural/1004132Robust stability analysis for a cruise missile*Shi Yingjing't ，Ma Guangful & Ma Hongzhong+,21. School of Astronautics, Harbin Inst. of Technology, Harbin 150001, P. R. China;2. Third Research Academy of China Aeroepace Science and Industry Corporation, Bejing 100074, P. R China(Received May 12, 2007)Abstract: A global controller design methodology for a fight stage of the cruise missile is proposed. Thismetbodology is based on the method of least square. To prove robust stability in the full airspace with param-eter disturbances, the concepts of convex polytopic models and quadratic stability are introduced. The efect ofaerodynamic parameters on asystem performance is analyzed. The designed controller is applied to track the over-loading signal of the cruise Begment of the cruise misile, avoiding system disturbance owing to ontroller switching.Simulation results demonstrate the validity of the propoeed metbod.Keywords: aircraft control, robust control, least square, quadratic stabilization.sile. Then, a feedforward plus feedback controller1. Introductionis designed for the overloading signal at each char-As stability analysis for time varying systems is veryacteristic point such that the closed-loop poles candificult, for time -invariant systems, one only needs tobe assigned to the desired poles and the closed-loopconsider the real part of the poles of the system ma-system can track the overloading signal. Further-trices. Consequently, the cefficient freezing methodmore, using the least squares principle, we constructhas become a popular tool for stbility analysis in a unifed cotoler.r Finally using the conepts ofengineering applications. However, in Ref. (1], it is .polytopic models and quadratic stability, globalshown that this method is not precise theoretically.parameter robust stability analysis of the closed-loopSystems designed by this method are not guaranteedsystem is presented.to be globally stable. Owing to the recent improve 2. Feedforward plus feedback controllerments in controller precision and the complication ofdesignthe subject, system robustness issues are becomingLemma 16 Consider the linear time -invariantmore importantl2- -3).When designing the control system of a missile, it issystemi= Ax+ Bucommon to approximate the nonlinear system by sev-(1)eral linear systems using the piecewise linearizationy=Cx+ Dv .method. Then, a corresponding controller is designedfor each linear system. On this basis, we designa where A∈RnXn, B∈Rrx1,C∈Rlxn, D∈R.global controller such that the performance of the non- Let λ1, ..... be the n desired closed-loop poles,linear system can be maintained. This is the so-called and f*(a) = s" + a;sn-1 +..+ a; be the desiredgain schedule method reported in Refs. [4-5].closed-loop characteristic polynomial. If (1) is con-Hence, in this note, by analyzing the fight propertytrollal中国煤化工>ack matrix K =of the cruise missile, we first select the character-[ 0MHCNMH(Fat the pole of tbeistic points of the fight course of the cruise mis- closed-iuop sysuemn 18 A1, 2....,.n, whereThis project waB supported by tbe National Natural Science Foundation of China (60674101) and the Education UniversityDoctor Foundation of Chinese Ministry (20050213010).786Shi Yngjing, Ma Guangfu & Ma Hongzhong ;the fight speed of the missile, Ya is the lift coeficient,Uc=[B AB ... An-1B]Y6z is the lift rudder efetiveness, m is the mislemass, and ny is the normal overload of the missile.andf*(A)=A" +qiAn-1 +...+a;I .According to Theorem 1, if the system is controllable,then for arbitrary desired poles, we can design a con-Lemma 2|7] Consider system (1). If there existstroller u= -Kx + Gv such that the closed-loop polessome feedback stabilization controller K, then therecoincide with the desired poles and realize the trackingexists a matrix G such that the output y of the closed-to the desired overloading signal.loop system can track the constant signal 0 under thecontroller u = - Kx+Gv, i.e, ,limy(t) = U. We3. Robustly stable analysis with param-call K the feedback stabilization controller and G theeter variations in full airspacefeedforward tracking controller. The matrix G can beDuring the low altitude cruise course for the cruisedecomposed into U + KV, where K is the feedbackmissile, the effects owing如o air resistance are signif-matrix of the feedback stabilization controller, and U,icant. To overcome this resistance, large amounts ofV are given by the following equationfuel are consumed. Hence, several piecewise switchingcontrollers are often utilized. However, the obtainedAB(2) controller becomes complicated by the introduction ofdisturbances during the course of switching. Thus, aFrom Lemmas 1 and 2, we can easily derive the fol-unifed controller is desired, under which the systemlowing theorem.stability can be maintained. Accordingly, we selectTheorem 1 Consider system (1). If (1) is con-several characteristic points and design a controllertrllable, i.e, det (Uc)≠0, then there exists a con- for each charateritic point using the pole asigmenttroller u= -Kx + Gv such that the closed-loopo poles method. Then, using the least squares method, wecan be arbitrarily assigned and the closed-loop systemderive a common controller. Furthermore, stability iscan track the given constant signal, where, U, K,G proved using the LMI method.are as defined in Lemma 2.3.1 Controller designFor the pitch channel of the cruise missile, under thelttle disturbance linearization bhypothesis, and usingWhen the cruise missile flies horizontally, its fuel massthe coefficient freezing method, we obtain the follow-decreases and its angle of attack diminisbhes. This is agradual changing course, and therefore, we select theing equationsbeginning point and the end point of each horizontal「Ms=fight spell as the characteristic points.| 0wiz= Jz J| AwzJxP+ yay8。△δzThe parameter values of the characteristic pointsmVmV」are given in Table 1 below.Substituting the parameter values of Table 1 into[o P+Ya] Our| y0+(3), we obtain two system equations, denoted byI, IImgOa mg(3)Owz|-2.851 6 -183.95where Sa is tbe disturbance of the angle of attack,Aa」1 -1.071 2▲δz is the elevator deflection disturbance, Owz is thepitching angular velocities disturbance, g is the grav-| -168.46Oδ,ity acceleration, Jz is the moment of inertia of the中国煤化工missile about the z-axis, Mq is the stable moment co-eficient, M? is the operating torque cofficient, My*TH.CNMHG|n=U 103.853|+15.96 x Aδ%is the damp moment cofficient, P is the thrust, V is[a]Robust stability analysis for a cruise missile787Table 1 The parameter values of the characteristic pointsParameterJz/kg . mP/Nm/kg/(ms-1)G/NBegin2 550238009503618 437.823 7007608638 457.4Mg/(deg-1)M2/(deg-)M"*/(ad-1)ya/(deg-1)ys*/(deg-1)-469 063.8- 429 563.7-7 271.512852 355.75134 672.13End-469 193.5- 429 607.5-7418.411858 796.28135 894.26points of the system matrix in the left-part of com-- -2.9092 -184plex plane can exist. Thus, when the coefficient freez-[图1 - 1.345 5|ing method is applied, the robust stability analysis iscompromised.-168.47On the other hand, the aerodynamic parametersII-0.207 2used in the above are obtained by wind tunnel test-ing, ground testing, and dynamic missile fight-testing.ny= 0 104.35|+ 16.068 x OδzHowever, owing to the limitations of the manufactur-| Aa]ing proces and measuring technology, it is dificult toIt is easy to confirm that the two systems are both coD-obtain precise data. In addition, the fight conditionstrollable. Suppose that the desired poles are λ1,2 =are dificult to forecast. Therefore, it is important-10土7i. Then, according to Theorem 1, we can ob-that at each characteristic point, the system designedtain two controllers ui= -K:x + Giv, i= 1,2, whichis robustly stable with respect to perturbations in theparameters.can track the overloading signal, whereHence, this problem has dualism. On one hand, we1: K1=[ - -0.0940 0.3351 ]， G1=-0.011 3need to show robust stability at each characteristicpoint; on the other hand, we need to ilustrate thatII: K2=[. -0.0921 0.3643]， G2=-0.009the robust stability can be achieved in full airspace.We can derive a unifed contrlleru=-Kx + GoThe later requirement is called robust stability in fullfor the above two controllers using the least squaresairspace with parameter variation.methods.The corresponding matrices are R =3.3 Polytopic models and quadratic stability[ -0.0931 0.3497 } and C = -0.0102. In the nextFor convenience, we will introduce the following def-subsection, we will prove that this unified controllernition.can maintain the robust stability in full airspace.Definition 18 A system is called an uncertain3.2 Robust stabilitysystem based on polytopic models if it can be writtenThe controller designed in the previous section is ob-as followstained by considering the linearized model and apA(8(t)=a1(t)A1+...+ak(t)Ak (4)plying the coeficient freezing method. The so calledcofficient freezing method for a linear time-varyingwhereVt∈R,ai(t)≥0,》,a;(t)=1,and.system出= A(t)x involves freezing t in the system证1matrix A(t) and treating the time-varying system 88A....,Ak are given matrices.a time invariant system. In other words, the stabilityGenerally, the selected characteristic points are rep-of the time varying system is analyzed by considering中国煤化工presume that thean associated time invariant system.polyiharacteristic pointsCNMH(However, in Ref. [1], it is proved that an unsta-incluae one open-1oup sysueu uauulily and then coverble time varying linear system with all characteristic the full airspace in the sense of physics. On the otherShi Yingjing, Ma Guangfu & Ma Hongzhonghand, it can be seen from the definition of the dy- quadratic stabilization of parameter variation at char-namics cefficients that the parameter of the missile acteristic points. Suppose that the variation bound foris directly proportional to its disturbance parameters some parameter δ is [omin, 8max]. Then, this problemsuch as the structural parameters, the geometry pa-can be described as follows.rameters, and the aerodynamic parameter. Thus, theFind & symmetric positive definite matrix P, suchvariation scope of the uncertainty is also a polytopic that inequality groupmodel, and thereby, the robust stability problem ofparameter variation about full airspace is transformed(A%egin (omin)" P+ PACegin (omia) <0into a polytopic model system family.(Aegin(omx)" P+ PAEgin(Mmax)<0(7)The fllowing defnition is used in the proof of ro-(Aand (mi))T P+ PAgnd (rmi) <0bust stabilization about full airspace parameter vari-(Aend (Mmax)) P+ PAgnd (8max) <0ation of the designed feedback contrller.Definition 2间Consider the following systemholds, where the subscript begin denotes the begin-ning of the characteristic point, the subscript end de出(t)= A(8)x(t)notes the end characteristic point, and the superscriptSystem (5) is called quadratically stable if there exists c signifies that the quantity is obtained by feedbacka symmetric positive definite matrix P such that the control.matrix inequality4. Simulation resultsAT(8)P+PA(6) <0(6) Since operating torque is the major factor afectingholds for all uncertain real value parameter vectorssystem performance, we only consider the cefficientδ∈0. Here,x∈R^ is the state vector and A(5) is achange about the operating torque. Other situationscan be analyzed similarly. For brevity, we providefunction of δ.If Eq. (5) is quadratically stable, then from (6),the closed-loop system matrices of parameter varia~for any δ∈O, V(x) = xTPx is a quadratic Lya-tion vertex at each characteristic point in the form ofpunov function of (5) that satisfies v < 0. AccordingTable 2.Substituting the data from Table 2 into (7) yieldsto Lyapunov stability theory, this implies that (5) is .asymptotical stable. Thus, ftom the quadratic stabil-the following symmetric positive definite matrix.ity of (5), we can see that, for any uncertain parame-0.0056 -0.002 6terδ∈0, (5) is asymptotical stable. This proves theP=(8)-0.0026 0.651 6robust stability for the system. The converse is notcorrect.Table 2 The closed-loop system matrices of pare-meter variation vertex on each characteristic point3.4 Robust stability analysis(1,1)(1,2)(2,1) (2.2)From subsections 3.2 and 3.3, we know that the ro-Agein(M°xmin) -14.6407 -139.773 0.9805 -1.279 7bust stability problem about the full airspace param-Aegi(Mx max) - 26.4010 -95.5884 0.9805 -1.279 7eter variation can be transformed into the quadraticAd(M. min)-14.6407 -139.773 0.9756 -1.599 5stabilization problem of polytopic models. For conve AS d(Ms. it)_ 26.4010. -5.5884 0.9756 -1.5995nience, we introduce the following theorem.Theorem 28] System (5) with (4) is quadrati-From Theorem 2, we know that the designed con-cally stable, if there exists a symmetric positive defi- troller is robustly stable with parameter variations innite matrix P such that the matrix inequality ATP+ full ai中国煤化工PAi< 0 holds for eachi= ....xe overloading tra-In conclusion, the foregoing feedback stabilization ckingYHC N M H Gon ange curveatproblem of a cruise missile can be transformed intothe the initial time. Figs. 3 and 4 represent the sameRobust stability analysis for a cruise missile789quantities at the terminal time. From the simula-tion results, it is readily seen that the designed sys-0.5tem is robustly stable with parameter variations in full1.5--0.5-1-1.5上0.5-2.51.5一Down-tune 50%; - -:Up tune 50%-0.5七2Fig.4 The pitching deflection angle of the terminal pointTime/8Down tune 50%; --: Up-tune 50%airspace. Under the feedback stabilizing controller deFig.1 The overloading tracking curve of the initial pointsigned by our method, the maximal risetime appearswhen the operating torque is maximal, and at thesame time, the system begins to fly. The maximal risetime is tr = 0.44 8. The maximal overshoot appearswhen the operating torque cofficient is minimum, andat the same time, the system stops fying. The min-imum overshoot is σp = 28.1%. By the simple feed-forward compensator, the control precision obtainedis satisfying.-25. Conclusion0.In this note, we discuss the impreciseness of the co-Time/sefficient freezing method. To compensate for this im-一: Down tune 50%; --Up tune 50%preciseness, we design a control scheme based on the .Fig 2 The pitching defection angle of the initial pointpole assignment method with feedforward plus feed-back structure using the LMI method. The stabilityanalysis is performed using the definition of polytopicmodels and quadratic stability.The control scheme given in this note first sim-plifies the structure of the system, avoiding distur-bance owing to system switching. Then, one canr 0.5+maintain robust stability with parameter variations infull airspace. Finally, the effect of each parameter onthe system can be analyzed and compared using thismethod, which is significant to tbe general design.-0.561.References中国煤化工1] Iond Edition). Harbin:一=: Down-tune 50%; -一Up-tune 50%YHC NMH G, 200 160-12Fig. 3 Tbe overloading tracking curve of the terminal point[2) Ra W s, Whang I H, Ahn J Y. Robust horizontal line790Shi Yingjing, Ma Guangfu & Ma Hongzhongof- sight rate estimator for sea skimming anti-ship missile [8] Yu L. Robust control: an LMI approach. Beijingr Tswith two axis gimballed seeker. IEE Proc. of Radar Sonaringhua University Press, 2002: 75 -79.Navig., 2005 152(1): 9-15.3} Lechevin N, Rabbath C A, Sicard P. A passivity perspec-Shi Yingjing was born in 1975. He isa Ph. D. can-tive for the syothesis of robust terminal guidance. IEEE didate of the School of Astronautics, Harbin InstituteTrans. on Control Systerms Technology, 2005 13(5): 760of Technology. His research interests include aircraft65.control, variable structure control, and robust control.4 Mehrabian A R, Roshanian J. Deig of gain echeduled a- E-mail: Yingjing Shigmail.comtopilot for a highly-agile missile. 1st International Sympo-sium on Systerms and Control in Aerospace and Astronau-Ma Guangfu was born in 1963. He is a professor,tics, 2006: 144-149.doctoral director at the School of Astronautics, Harbin[5] Silva A D, Garran J M, Suarez P A T, et al. Sliding modeInstitute of Technology. His research interests includefuzzy gain scheduling in sampled data nonlinear systems.satellite attitude control and aircraft control. E-mail:American Control Conference, 2005, 7: 4965 -4970.magf@hit.edu.cn[6] Xie K M. Modern control theory foundation. Bejing: Bei-jing University of Technology Pres, 2002: 201-204.7] Tan F, Duan G r, Zhao L J. Robust controller designMa Hongzhong was born in 1970. He is a professorfor autopilot ofa BTT missile. Proc. of the 6th Worldat the Third Research Academy of CASIC. His reCongress on Intelligent Control and Autormation, 2006:search interests include intelligent vehicle control andvariable structure control. E-mail: ma hzh@sina.com6358 6362.中国煤化工MYHCNMHG