Flight Restrain Arrangements W. -H. Chen Department of Aeronautical and Automotive Engineering Loughborough University 2 Flight Restrain Arrangements by W. -H. Chen, AAE, Loughborough Contents 1 Commencement 1. 1 Overapprehension of the Flight Entdirection 1. 2 Flight restrain arrangements . . . . . . 1. 3 Jurisdictionrn Restrain . . . . . . . . . . 1. 4 Commencement to the manner . . . . 1. 4. 1 Content . . . . . . . . . . 1. 4. 2 Tutorials and mannerwork 1. 4. 3 Impost . . . . . . . . 1. 4. 4 Lecture pur-pose . . . . . . . 1. 4. 5 Relations . . . . . . . . . 7 7 8 8 9 9 10 10 10 11 13 13 16 16 17 17 18 19 19 20 20 20 20 20 24 25 25 25 25 26 27 27 29 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Hankeritudinal confutation to the restrain 2. 1 Hankeritudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2 Nartrounce immeasurableness style . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 Nartrounce waverings . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 2 Unconcealed nartrounce immeasurableness pattern . . . . . . . . . . . . . . . . . . . 2. 3 Hankeritudinal nartrounce immeasurableness pattern . . . . . . . . . . . . . . . . . . . . 2. 3. 1 Numerical copy . . . . . . . . . . . . . . . . . . . . . . . 2. 3. 2 The dainty of nartrounce waverings . . . . . . . . . . . . . . . . . . 2. 4 Zealcraft dynamic behaviour pretence using nartrounce immeasurableness patterns . 2. 4. 1 Zealcraft confutation extinguishededge restrain . . . . . . . . . . . . . . . 2. 4. 2 Zealcraft confutation to restrains . . . . . . . . . . . . . . . . . 2. 4. 3 Zealcraft confutation inferiorneathneath twain jurisdictiontrounce provisos and restrains 2. 5 Hankeritudinal confutation to the elevator . . . . . . . . . . . . . . . . 2. 6 Assign of nartrounce immeasurableness patterns into assign businesss . . . . . . . . 2. 6. 1 From a assign business to a nartrounce immeasurableness pattern . . . . . . . 2. 7 Block diagram fidelity of nartrounce immeasurableness patterns . . . . . . . . . 2. 8 Static possession and dynamic jurisdictions . . . . . . . . . . . . . . . . . . 2. 8. 1 Zealcraft possession . . . . . . . . . . . . . . . . . . . . . . . . 2. 8. 2 Possession with FCS amplification . . . . . . . . . . . . . . . 2. 8. 3 Dynamic jurisdictions . . . . . . . . . . . . . . . . . . . . . . . . . 2. 9 Subsided patterns of hankeritudinal dynamics . . . . . . . . . . . . . . 2. 9. Phugoid appropinquation . . . . . . . . . . . . . . . . . . . . 2. 9. 2 Lacking occasion appropinquation . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Additive confutation to the restrains 3. 1 Additive nartrounce immeasurableness patterns . . . . . . . . . . . . 3. 2 Passing confutation to aileron and rudder . . . . 3. 2. 1 Numerical copy . . . . . . . . . . . . 3. 2. 2 Additive confutation and assign businesss 3. 3 Subsided regulate patterns . . . . . . . . . . . . . . 3. 3. 1 Flatten refluence . . . . . . . . . . . . . . 3. 3. Implication jurisdiction appropinquation . . . . . . . 3. 3. 3 Dutch flatten . . . . . . . . . . . . . . . . . 3. 3. 4 Three degrees of insubservience appropinquation 3. 3. 5 Re-formulation of the additive dynamics . CONTENTS 31 31 33 33 33 35 38 38 39 39 40 43 43 46 46 46 46 48 49 49 55 55 55 58 58 60 60 61 62 65 66 66 67 68 68 68 69 69 69 70 70 71 71 73 73 73 73 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Possession Amplification Arrangements 4. 1 Nartrounce immeasurableness artifice techniques . . . . . . . . . . . 4. 2 Hankeritudinal possession amplification arrangements . . . 4. 2. 1 The dainty of feedend waverings . . . . 4. 2. 2 SAS restraint lacking occasion dynamics . . . . . . 4. 3 Additive possession amplification arrangements . . . . . . 4. 3. 1 Yaw trounce feedend restraint rudder restrain . . . 4. 3. 2 Flatten feedend restraint aileron restrain . . . . . 4. 3. 3 Integration of additive straightforwardional feedend 5 Autopilots 5. 1 Rock calling autoguide . . . . . . . . . . . . . . . . . . . . . . . 5. 1. 1 phugoid stop . . . . . . . . . . . . . . . . . . . . . . 5. 1. 2 Eliminate the well-behaved-regulated fallacy with integration . . . . . . . 5. 1. 3 Better passing work with rock trounce feedend 5. 2 Apex calling autoguide . . . . . . . . . . . . . . . . . . . . . . 5. . 1 An commencemental apex calling autoguide . . . . . . . . . . . 5. 2. 2 Betterd apex calling arrangements . . . . . . . . . . . . . 5. 3 Actuator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 6 Handling Qualities 6. 1 Handing qualities restraint zealcraft . . . . . . . . . . . . 6. 2 Guide-in-loop dynamics . . . . . . . . . . . . . . . . 6. 2. 1 Guide as a restrainler . . . . . . . . . . . . . 6. 2. 2 Estimate confutation of a dynamic arrangement . . 6. 2. 3 Guide-in-loop . . . . . . . . . . . . . . . . . 6. 3 Flying qualities requirements . . . . . . . . . . . . 6. 4 Zealcraft role . . . . . . . . . . . . . . . . . . . . . . 6. . 1 Zealcraft ranki? cation . . . . . . . . . . . . . 6. 4. 2 Flight bearing . . . . . . . . . . . . . . . . . . 6. 4. 3 Smooths of ? ying qualities . . . . . . . . . . . 6. 5 Guide judgment rating . . . . . . . . . . . . . . . . . . 6. 6 Hankeritudinal ? ying qualities requirements . . . . . 6. 6. 1 Lacking occasion rocking vibration . . . . . . 6. 6. 2 Phugoid . . . . . . . . . . . . . . . . . . . . 6. 6. 3 Flying qualities requirements on the s-plane 6. 7 Additive-directional ? ying qualities requirements . . 6. 7. 1 Flatten refluence jurisdiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 6. 7. 2 6. 7. 3 6. 7. 4 5 Implication jurisdiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Dutch flatten jurisdiction . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Additive-directional jurisdiction in s-plane . . . . . . . . . . . . . . . . . 75 77 . . . . . . . . . . . restrain derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 79 79 79 79 79 7 Fly-by-Wire ? ight restrain 8 Appendices 8. Boeing 747-100 grounds . . . . . . . . . . . 8. 2 De? nitions of Aerodynamic possession and 8. 3 Source Locus . . . . . . . . . . . . . . . . 8. 4 Estimate confutation . . . . . . . . . . . . appendices 6 CONTENTS Chapter 1 Commencement 1. 1 Overapprehension of the Flight Entdirection • Flight pur-poseing • Zealcraft checking • Taxi • Take-o? – Rotate, “select” an standing – Clean up (gear, ? aps, anticipation) – Emergencies (engine scarcity, ? re, anticipation) • Climb – Press restrain – Progress (manual, autopilot) • Mission Tasks – Cruise – Combat (zeal to zeal) – Strike (zeal to sphere) – Unconcealed handling (stalling, spinning, aerobatics) – Restraintmation ? ing (Navigation, progress anticipation) – Emergencies – Con? guration (weapons, tanks, fuel inculpate) • Recovery – Descent – Instrument admittance – Landing – Overshoot 7 8 CHAPTER 1. INTRODUCTION Stick – Linkage 6 Trim ? -? Servo Actuator – Zealcraft dynamics Figure 1. 1: Manual guide restrain zealcraft – Restraintmation – Progresss – Emergencies • Taxi Hankeritudinal and additive dynamics thus Flight restrain arrangements are implicated in Take o? , Climb, Mission tasks and Recovery. • Di? eschism zealcraft (aircraft rank) • Di? eschism ? ight bearing Manual– handling qualities/? ight qualities Better the handling qualities of zealplane; Autoguide 1. 2

Flight restrain arrangements Objectives • To better the handling qualities • To discharge the agency bundle of guides to-some-extent or abundantly • To growth the work of zealcraft or missiles Types of Flight Restrain Arrangements (FCS) 1. Known-loop restrain 2. Possession amplification arrangements 3. Autoguide 4. Integrated Navigation arrangements and Autopilots (? ight administration arrangements) 1. 3 Jurisdictionrn Restrain • Rankic restrain– assign business – estimate inclosure • Limitation of rankic artifice process: sole input, sole extinguisheddispose (SISO), ungodlinessgly anxiety the extinguisheddispose behaviour, rectirectilinear arrangements (saturation) • Arrangement style in nartrounce immeasurableness restraintm. 1. 4.

INTRODUCTION TO THE COURSE 9 Stick Trim – Zealcraft dynamics – + ? + -Linkage – ? – ? – Servo Actuator 6 6 Possession Aug. Arrangements Sensor ? Figure 1. 2: Possession Amplification Arrangements Relation Instruct + -? Autoguide – 6 6 + -? 6 – SAS – Actuators – Zealcraft dynamics – Sensor 6 Navigation Arrangements ? ? Figure 1. 3: Autoguide con? guration • Illustrebuke zealcraft or other dynamics arrangements in a be of ? rst regulate di? erential equations. Explicit in a matrix restraintm • Nartrounce immeasurableness segregation and artifice techniques– very potent technique restraint restrain arrangements • Matrix comcollocation enlightenment required 1. 4 1. 4. 1 Commencement to the manner

Content This manner succeed conceal • nartrounce immeasurableness segregation and artifice techniques restraint zealcraft • rudimentary ? ight restrain arrangements including possession amplification arrangements, and rudimentary autopilots • handling qualities 10 CHAPTER 1. INTRODUCTION Flight Administration 6 Arrangements/Autoguide 6 + -? 6 – SAS – Actuators – Zealcraft dynamics – Sensor 6 Navigation Arrangements ? ? Figure 1. 4: Autoguide con? guration • Fly-By-Wire (FBW) 1. 4. 2 Tutorials and mannerwork • Tutorials succeed rouse from Week 3 • Evilgle tutorial minority in each week • Evilgle mannerwork commencementalized on MATLAB/Simulink pretence, must be handed in antecedently 4:00 PM Thursday, Week 11 1. 4. 3

Impost • Mannerwork: 20%; • Examination: 2 hours; endeavor 3 from 5 questions; 80% of the ? nal indication. 1. 4. 4 Lecture pur-pose • Overcomplete ? ight entdirection • Flight restrain arrangements • Jurisdictionrn restrain artifice processology • The commencement of the manner– constituency, impost, exercises, relations 1. Commencement 2. Confutation to the restrains (a) Nartrounce immeasurableness segregation (b) Hankeritudinal confutation to elevator and choke (c) Passing confutation to aileron and rudder 3. Zealcraft possession amplification arrangements 1. 4. INTRODUCTION TO THE COURSE (a) Work evaluation • • • • possession Occasion inclosure requirements Estimate inclosure speci? ations Robustness 11 (b) Hankeritudinal Possession Amplification Arrangements • Dainty of the feedend waverings • Source locus and shape gratification • Phugoid stop (c) Additive possession amplification arrangements • Flatten feedend restraint aileron restrain • Yaw trounce feedend restraint rudder restrain 4. Rudimentary autoguide artifice • Augmented hankeritudinal dynamics • Apex repose arrangements 5. Handling Qualities (a) Occasion retrogression arrangements (b) Guide-in-loop dynamics (c) Handling qualities (d) Estimate inclosure segregation (e) Guide adventitious vibration 6. Flight Restrain arrangement implementation Fly-by-wire technique 1. 4. 5 Relations 1. Flight Dynamics Principles.

M. V. Cook. 1997. Arnold. Chaps. 4,5,6,7,10,11 2. Unreflective Flight Restrain Arrangements. D. McLean. 1990. Prentice Hcomplete International Ltd. Chaps. 2, 3,6,9. 3. Commencement to Avionics Arrangements. Remedy edition. R. P. G. Collinson. 2003. Kluwer Academic Publishers. Chap. 4 12 CHAPTER 1. INTRODUCTION Chapter 2 Hankeritudinal confutation to the restrain 2. 1 Hankeritudinal dynamics From Flight Dynamics manner, we inferiorstand that the rectilinearised hankeritudinal dynamics can be written as mu ? ? ? X ? X ? X ? X u? w? ? w + (mWe ? )q + mg? cos ? e ? u ? w ? ?w ? q ? Z ? Z ? Z ? Z ? u + (m ? )w ? ? w ? (mUe + )q + mg? ungodliness ? e ? u ? w ? ?w ? q ?

M ? M ? M ? M u? w? ? w + Iy q ? ? q ? ?u ? w ? ?w ? q = = = ? X ? t ? Z ? t ? M ? t (2. 1) (2. 2) (2. 3) The tangible meanings of the waverings are de? ned as u: Disturbance abextinguished well-behaved-regulated nartrounce quickness Ue w: Disturbance on well-behaved-regulated nartrounce ordinary quickness We q: Rock trounce ? : Rock propensity Inferiorneathneath the arrogance that the aeroplane is in smooth straightanxious ? ight and the relation axes are curve or possession axes, we own ? e = We = 0 (2. 4) The ocean restrains in hankeritudinal dynamics are the elevator propensity and the engine hope. The feeble disturbance provisos in the equiconsultation edge of the aloft equations can be explicit as ? X ? t ?

Z ? t ? M ? t where 13 = = = ? X ? X ? e + ? ?? e ?? ?Z ? Z ? e + ? ?? e ?? ?M ? M ? e + ? ?? e ?? (2. 5) (2. 6) (2. 7) 14 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL ? e : the elevator de? ection (Voicelessness ? is used in Appendix 1) ? : engine drive disturbance Substituting the aloft indication into the hankeritudinal symmetric disturbance concedes ? X ? X ? X ? X u? w? ? w? q + mg? ?u ? w ? ?w ? q ? Z ? Z ? Z ? Z ? u + (m ? )w ? ? w ? (mUe + )q ? u ? w ? ?w ? q ? M ? M ? M ? M u? w? ? w + Iy q ? ? q ? ?u ? w ? ?w ? q mu ? ? = = = ? X ? X ? e + ? ?? e ?? ?Z ? Z ? e + ? ?? e ?? ?M ? M ? ?e + ?? e ?? (2. 8) (2. 9) (2. 10)

Behind adding the kindred ? ? = q, (2. 11) Eqs. (2. 8)- (2. 11) can be dispose in a past sharp vector and matrix restraintmat. The hankeritudinal dynamics can be written as ? m ? 0 ? ? 0 0 ? ?X ? w ? ?Z m ? ?w ? ? ? M ? w ? 0 0 0 Iy 0 ?? u ? 0 0 ?? w ?? ? ? 0 ?? q ? ? 1 ? ? ? = ? ? ? ? ? ? ? ? ? ?X ? u ? Z ? u ? M ? u ? X ? w ? Z ? w ? M ? w ? Z ? q ? X ? q + mUe ?M ? q 0 0 ?X ?? e ? Z ?? e ? M ?? e 0 ?X ?? ?Z ?? ?M ?? ? ? ? ? 1 ?? ?mg u 0 ?? w ?? 0 ?? q ? 0 ? ? ?+ ? ?e ? (2. 12) 0 Dispose complete waverings in the hankeritudinal dynamics in a vector restraintm as ? ? u ? w ? ? X=? ? q ? ? and suffer m ? ?X ? w ? ? 0 m ? ?Z ? ?w ? = ? 0 ? ?M ? w ? 0 ? ?X ? X ? = ? ? ? B ? = ? ? ? u ? Z ? u ? M ? u ? w ? Z ? w ? M ? w ? Z ? q (2. 13) ? M 0 0 Iy 0 ?X ? q ? 0 0 ? ? 0 ? 1 (2. 14) ? ?mg 0 ? ? 0 ? 0 A + mUe ?M ? q (2. 15) 0 0 ?X ?? e ? Z ?? e ? M ?? e 0 ?X ?? ?Z ?? ?M ?? ? ? ? ? 1 (2. 16) 0 U= ?e ? (2. 17) 2. 1. LONGITUDINAL DYNAMICS Equation (2. 12) becomes 15 ? MX = A X + B U (2. 18) It is hamorsel to transmute the aloft be of equations into a be of ? rst regulate di? erential equations by multiplying twain edges of the aloft equation by the inverse of the matrix M , i. e. , M ? 1 . Eq. (2. 18) becomes ? ? ? ? ? ?? ? u ? xu xw xq x? x? e x? u ? w ? ? zu zw zq z? ? ? w ? ? z? z? ? ? e ? ? ? =? ? ? ?? ? (2. 19) ? q ? ? mu mw mq m? ? ? q ? + ? m? e m? ? ? ? ? ? 0 0 1 0 0 0 ? Suffer xu ? zu A = M ? 1 A = ? ? mu 0 ? ? xw zw mw 0 xq zq mq 1 ? x? z? ? ? m? ? 0 (2. 20) and x? e ? z? e B = M ? 1 B = ? ? m ? e 0 ? x? z? ? ? m? ? 0 (2. 21) It can be written in a sharp restraintmat ? X = AX + BU (2. 22) Eq. (2. 22) with (2. 20) and (2. 21) is referred as the nartrounce immeasurableness pattern of the rectilinearised hankeritudinal dynamics of zealcraft. Appendix 1 produces the kindred betwixt the newlightlightlight possession and restrain derivatives in the matrix A and B, i. e. xu , so on, with the delineational and non-dimensional derivatives, where ?

X ? Xu = ? u (2. 23) delineates delineational derivative and Xu its identical non-dimensional derivative. These kindreds are conservative commencementalized on the Cramer’s administration and repose restraint unconcealed heap axes. In the plight when the derivatives are referred to curve axes, as in this manner, the governthcoming simpli? cations should be made Ue = Vo , We = 0, ungodliness ? e = 0, cos ? e = 1 (2. 24) The style of the hankeritudinal dynamics in the matrix-vector restraintmat as in (2. 19) can be copious to indicate complete unconcealed dynamic arrangements. Consider a arrangement with regulate n, i. e. , the arrangement can be feeling by n regulate di? schismial equation (as it succeed be explained succeeding, this is the resembling as the leading regulate of the denominator polynomial in the assign business is n). In the fidelity (2. 22), A ? Rn? n is the arrangement matrix ; B ? Rn? m is the indispose matrix ; X ? Rn is the nartrounce vector or nartrounce waverings and U ? Rm the indispose or indispose vector. The equation (2. 22) is denominated nartrounce equation. Restraint the possession amplification arrangement, ungodlinessgly the in? uence of the change of the elevator propensity, i. e. the primitive aerodynamic restrain manner, is anxietyed. The aloft equations of disturbance can be simpli? ed. The nartrounce immeasurableness fidelity remains the 6 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL resembling restraintmat as in eq. (2. 22) with the resembling matrix A and nartrounce waverings excepting with a di? eschism B and indispose U as consecrated beneath ? ? x ? e ? z ? B = M ? 1 B = ? ?e ? (2. 25) ? m? e ? 0 and U = ? e (2. 26) Remark: It should be referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attributableiced that in di? eschism textbooks, di? eschism referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attributableations are used. Restraint the nartrounce immeasurableness fidelity of hankeritudinal dynamics, someoccasion expandedtilded derivatives are used as follows ? ? 1 ? X 1 ? X ? ? 1 ? X ? ? ?? 0 ? g u ? u m ? u m ? w m ?? e 1 ? Z 1 ? Z 1 ? Z ? w ? ? 0 ? ? w ? ? m ?? e ? ?+? ? ? ? = ? m ? u m ? w Ue ?? ? ? e (2. 27) ? q ? Mu ? Mw Mq 0 ? ? q ? ? M? e ? ? ? ? 0 0 1 0 0 where Mu = Mw = 1 ? M 1 ? Z 1 ? M + ? Iyy ? u m ? u Iyy ? w ? 1 ? M 1 ? Z 1 ? M + ? Iyy ? w m ? w Iyy ? w ? 1 ? M 1 ? M + Ue ? Iyy ? q Iyy ? w ? (2. 28) (2. 29) (2. 30) (2. 31) Mq = M? e = 1 ? M 1 ? Z 1 ? M + ? Iyy ?? e m ?? e Iyy ? w ? The expandedtilded derivatives and the other derivatives in the matrices are the resembling as the indication of the feeble sufferter derivatives inferiorneathneath vericonsultation arrogances, i. e. using possession axis. 2. 2 2. 2. 1 Nartrounce immeasurableness style Nartrounce waverings A incompleteness be of waverings which, when inferiorstandn at occasion t0 , contemporaneously with the input, are su? ient to illustrebuke the behaviours of the arrangement at any occasion t > t0 . Nartrounce waverings may own no any tangible meanings and may be referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation measurable. Restraint the hankeritudinal dynamic of zealcraft, there are disgusting nartrounce waverings, i. e, ? ? u ? w ? ? X=? (2. 32) ? q ? ? and evilgle indispose or restrain wavering, the elevator de? ection, U = ? e (2. 33) 2. 3. LONGITUDINAL STATE SPACE MODEL Thus n=4 m=1 17 (2. 34) The arrangement matrix and indispose matrix of the hankeritudinal dynamics are consecrated by ? ? xu xw xq x? ? z zw zq z? ? ? A = M ? 1 A = ? u (2. 35) ? mu mw mq m? ? 0 0 1 0 and ? x? e ? z ? B = M ? 1 B = ? ?e ? ? m ? e ? 0 ? (2. 36) respectively. . 2. 2 Unconcealed nartrounce immeasurableness pattern w Ue When the propensity of aggression ? is of anxiety, it can be written as ? = which can be dispose into a unconcealed restraintm as y = CX where y=? = and C= 0 1/Ue 0 0 (2. 40) Eq. (2. 38) is denominated Extinguisheddispose equation; y the extinguisheddispose wavering and C the extinguisheddispose matrix. Restraint past unconcealed plight where there are past than evilgle extinguisheddispose and has a straightanxious route from indispose to extinguisheddispose wavering, the extinguisheddispose equation can be written as Y = CX + DU (2. 41) w Ue (2. 38) (2. 39) (2. 37) where Y ? Rr ,C ? Rr? n and D ? Rr? m . Restraint disturbance of aeroimmeasurableness gaits including zealcraft and missiles, there is no straightanxious route betwixt indispose and extinguishedput.

In this manner ungodlinessgly the plight D = 0 is considered if referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation obviously sharp extinguished. Eq. (2. 22) and (2. 38) (or (2. 41)) contemporaneously indicate the nartrounce immeasurableness style of a dynamic arrangement, which is irreconcilable to the assign business fidelity of a dynamic arrangement elaborebuke in Restrain Engineering manner. 2. 3 Hankeritudinal nartrounce immeasurableness pattern When the behaviours of complete the nartrounce waverings are anxietyed, complete those waverings can be selected as extinguisheddispose waverings. In resolution, there are other confutation quantities of concern including the ? ight route propensity ? , the propensity of aggression ? and the ordinary succor az (nz ).

Putting complete waverings contemporaneously, the extinguisheddispose vector can be written as 18 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL ? ? ? ? ? Y =? ? ? ? ? Invoking the kindreds ? = ? ? ? ? ? ? ? ? ? ? u w q ? ? ? az w Ue (2. 42) (2. 43) w Ue (2. 44) the ? ight route propensity ? =??? =?? and the ordinary succor az (nz ) az = = = ?Z/m = ? (Zu u + Zw w + Zq q + Zw w + Z? e ? e )/m ? ? ? (w ? qUe ) ? ?zu u ? zw w ? zq q ? z? e ? e + Ue zq (2. 45) where the remedy resemblingity substituting the indication matrix is consecrated by ? ? ? u 1 ? w ? ? 0 ? ? ? ? q ? ? 0 ? ? ? Y =? ? ? =? 0 ? ? ? ? ? ? ? 0 ? ? ? ? ? ? ? 0 az ? zu ollows from (2. 9) and the latest resemblingity is obtained by of w in its sharp derivative restraintmat. Hereafter the extinguisheddispose ? 0 1 0 0 1/Ue ? 1/Ue ? zw 0 0 1 0 0 0 ? zq + Ue 0 0 0 1 0 1 0 ? ?? ? ? ?? ?? ?? ? ? ? ? ? ? ? u ? ? ? w ? ? +? q ? ? ? ? ? 0 0 0 0 0 0 ? z? e ? ? ? ? ? ? ? e ? ? ? ? (2. 46) There is a straightanxious route betwixt the extinguisheddispose and input! The nartrounce immeasurableness pattern of hankeritudinal dynamics consists of (2. 22) and (2. 46). 2. 3. 1 Numerical copy Boeing 747 jet rapture at ? ight vocable cruising in even ? ight at approximately 40,000 ft at Mach estimate 0. 8. Bearing grounds are consecrated in Consultation 2. 1 and 2. 2.

Using consultations in Appendix 1, the sharp feeble derivatives can be adapted and then the arrangement matrix and indispose matrix can be conservative as ? ? ? 0. 006868 0. 01395 0 ? 32. 2 ? ?0. 09055 ? ?0. 3151 774 0 ? A=? (2. 47) ? 0. 0001187 ? 0. 001026 ? 0. 4285 ? 0 0 0 1 0 ? ? ? 0. 000187 ? ?17. 85 ? ? B=? (2. 48) ? ?1. 158 ? 0 Resemblingly the parameters matrices in extinguisheddispose equation (2. 46) can be solid. It should be referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attributableiced that English ace(s) is used in this copy. 2. 4. AIRCRAFT DYNAMIC BEHAVIOUR SIMULATION USING STATE SPACE MODELS19 Consultation 2. 1: Boeing 747 rapture grounds 636,636lb (2. 83176 ? 106 N) 5500 ft2 (511. m2 ) 27. 31 ft (8. 324 m) 195. 7 ft (59. 64 m) 0. 183 ? 108 slug ft2 (0. 247 ? 108 kg m2 ) 0. 331 ? 108 slug ft2 (0. 449 ? 108 kg m2 ) 0. 497 ? 108 slug ft2 (0. 673 ? 108 kg m2 ) -0. 156 ? 107 slug ft2 (-0. 212 ? 107 kg m2 ) 774 ft/s (235. 9m/s) 0 5. 909 ? 10? 4 slug/ft3 (0. 3045 kg/m3 ) 0. 654 0. 0430 W S c ? b Ix Iy Iz Izx Ue ? 0 ? CL0 CD Consultation 2. 2: Delineational Derivatives– B747 jet X(lb) Z(lb) M(ft. lb) u(f t/s) ? 1. 358 ? 102 ? 1. 778 ? 103 3. 581 ? 103 w(f t/s) 2. 758 ? 102 ? 6. 188 ? 103 ? 3. 515 ? 104 q(rad/sec) 0 ? 1. 017 ? 105 ? 1. 122 ? 107 2 w(f t/s ) ? 0 1. 308 ? 102 -3. 826 ? 103 5 ? e (rad) -3. 17 ? 3. 551 ? 10 ? 3. 839 ? 107 2. 3. 2 The dainty of nartrounce waverings The nartrounce immeasurableness fidelity of a dynamic arrangement is referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation sole, which depends on the dainty of nartrounce waverings. Restraint engineering contact, nartrounce waverings, in unconcealed, are selected commencementalized on tangible meanings, delineation, or comforconsultation to artifice and segregation. Restraint the hankeritudinal dynamics, in resolutional to a be of the nartrounce waverings in Eq. (2. 32), another expandedly used dainty (in American) is ? u ? ? ? ? X=? ? q ? ? ? (2. 49) Veritablely, when the logitudinal dynamics of the zealcraft are indicateed in provisos of the aloft nartrounce waverings, di? schism A, B and C are effected (discern Tutorial 1). 2. 4 Zealcraft dynamic behaviour pretence using nartrounce immeasurableness patterns Nartrounce immeasurableness pattern exposed aloft provides a very potent instrument in question dynamic behavious of an zealcraft inferiorneathneath multishape vocable. The effect of using nartrounce gait patterns restraint predicting zealcraft dynamic behavious or numerical pretence can be explained by 20 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL the governthcoming indication X(t + ? t) = X(t) + dX(? ) ? |? =t ? t = X(t) + X(t)? t d? (2. 50) ? where X(t) is curschism narrate, ? t is tramp bigness and X(t) is the derivative adapted by the nartrounce immeasurableness equation. . 4. 1 Zealcraft confutation extinguishededge restrain ? X = AX X(0) = X0 (2. 51) 2. 4. 2 Zealcraft confutation to restrains ? X = AX + BU ; X(0) = 0 (2. 52) where U is the guide instruct 2. 4. 3 Zealcraft confutation inferiorneathneath twain jurisdictiontrounce provisos and restrains ? X = AX + BU ; X(0) = X0 (2. 53) 2. 5 Hankeritudinal confutation to the elevator Behind the hankeritudinal dynamics are feeling by the nartrounce immeasurableness pattern, the occasion histories of complete the waverings of concerns can be adapted. Restraint copy, the occasion confutations of the restraintward quickness u, ordinary quickness w (propensity of aggression) and ? ight route propensity ? inferiorneathneath the tramp change-of-place of the levator are displayed in Fig 2. 1–2. 5 Examineion: If the infer restraint tender the elevator is to organize a newlightlightlight well-behaved-regulated nartrounce ? ight vocable, then this restrain resuscitation can narrowly be apprehensioned as lucky. The hanker lightly damped vibration has seriously interfered with it. A amiable-tempered-tempered agency work canreferable attribuconsultation attribuconsultation be achieved by solely changing the propensity of elevator. Clearly, hankeritudinal restrain, whether by a anthropological guide or unreflective guide, demands a past obstructed restrain courage than known-loop temporization. 2. 6 Assign of nartrounce immeasurableness patterns into assign businesss Induction Laplace transconsequence on twain edges of Eq. (2. 2) inferiorneathneath the naught jurisdictiontrounce arrogance concedes sX(s) = Y (s) = where X(s) = L{X(t)}. AX(s) + BU (s) CX(s) (2. 54) (2. 55) 2. 6. TRANSFER OF STATE SPACE MODELS INTO TRANSFER FUNCTIONS21 Tramp confutation to elevator: Quickness 90 80 70 60 Quickness(fps) 50 40 30 20 10 0 0 1 2 3 4 5 Occasion(s) 6 7 8 9 10 Figure 2. 1: Hankeritudinal confutation to the elevator Tramp confutation to evelator: propensity of aggression 0 ?0. 005 ?0. 01 Propensity of aggression(rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 1 2 3 4 5 Occasion(s) 6 7 8 9 10 22 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Tramp respnse to elevator: Flight route propensity 0. 1 0. 08 0. 06 0. 04 Flight route propensity (rad) 0. 02 0 0. 02 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 1 2 3 4 5 Occasion(s) 6 7 8 9 10 Figure 2. 2: Hankeritudinal confutation to the elevator Tramp Confutation to elevator: hanker vocable 90 80 70 60 Quickness (fps) 50 40 30 20 10 0 0 100 200 300 Occasion (s) 400 500 600 Figure 2. 3: Hankeritudinal confutation to the elevator 2. 6. TRANSFER OF STATE SPACE MODELS INTO TRANSFER FUNCTIONS23 Tramp confutation to elevator: hanker vocable 0 ?0. 005 ?0. 01 Propensity of aggression (rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 100 200 300 Occasion (s) 400 500 600 Figure 2. 4: Hankeritudinal confutation to the elevator Tramp confutation to elevator: hanker vocable 0. 1 0. 08 0. 06 0. 04 Flight route propensity (rad) 0. 02 0 ?0. 2 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 100 200 300 Occasion (s) 400 500 600 Figure 2. 5: Hankeritudinal confutation to the elevator 24 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Y (s) = C[sI ? A]? 1 BU (s) Hereafter the assign business of the nartrounce immeasurableness fidelity is consecrated by G(s) = C[sI ? A]? 1 B = C(Adjoint(sI ? A))B det(sI ? A) (2. 56) (2. 57) Copy 1: A lacking occasion disturbance of a zealcraft is feeling by ? ? q ? = ? 0. 334 ? 2. 52 1. 0 ? 0. 387 ? q + ? 0. 027 ? 2. 6 ? e (2. 58) where ? e delineates the elevator de? ection. The assign business from the elevator de? ection to the propensity of aggression is solid as follows: ? (s) ? 0. 27s ? 2. 6 = 2 ? e (s) s + 0. 721s + 2. 65 (2. 59) # The hankeritudinal dynamics of zealcraft is a sole-indispose and multi-outdispose arrangement with evilgle indispose ? e and diverse extinguishedputs, u, w, q, ? , ? , az . Using the technique in Minority (2. 6), the assign businesss betwixt each extinguisheddispose wavering and the indispose elevator can be conservative. The referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attributableation u(s) Gue = (2. 60) ? ?e (s) is used in this manner to delineate the assign business from indispose ? e to extinguisheddispose u. Restraint the hankeritudinal dynamics of Boeing 747-100, if the extinguisheddispose of concern is the restraintward quickness, the assign business can be solid using restraintmula (2. 56) as u(s) ? e (s) ? 0. 00188s3 ? 0. 2491s2 + 24. 68s + 11. 6 s4 + 0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 0041959 (2. 61) Gue ? = = Resemblingly, complete other assign businesss can be conservative. Restraint a arrangement with scanty regulate love the remedy regulate arrangement in Copy 1, the dissectition of the identical assign business from its nartrounce immeasurableness pattern can be completed manually. Restraint intricate arrangements with excellent regulate, it can be dsingle by computer software love MATLAB. It can be establish that although the assign businesss from the elevator to di? eschism extinguishedputs are di? eschism excepting they own the resembling denominator, i. e. s4 + 0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959 restraint Beoing 747-100. Ungodlinessgly the numerators are di? erent. This is consequently complete the denominators of the assign businesss are solid by det(sI ? A). 2. 6. 1 From a assign business to a nartrounce immeasurableness pattern The estimate of the nartrounce wavering is resembling to the regulate of the assign business, i. e. , the regulate of the denominator of the assign business. By choosing di? eschism nartrounce waverings, restraint the resembling assign business, di? eschism nartrounce immeasurableness patterns are consecrated. 2. 7. BLOCK DIAGRAM REPRESENTATION OF STATE SPACE MODELS 25 2. 7 Block diagram fidelity of nartrounce immeasurableness patterns 2. 8 2. 8. 1 Static possession and dynamic jurisdictions

Aircraft possession Consider zealcraft equations of disturbance indicateed as ? X = AX + BU (2. 62) The possession segregation of the primary zealcraft dynamics anxietys if there is no any restrain e? ort,whether the stormy disturbance is unwavering. It is so referred as knownloop possession in unconcealed restrain engineering. The zealcraft possession is solid by the eigenvalues of the arrangement matrix A. Restraint a matrix A, its eigenvalues can be solid by the polynomial det(? I ? A) = 0 (2. 63) Eigenvalues of a nartrounce immeasurableness pattern are resembling to the sources of the individuality equation of its identical assign business.

An zealcraft is unwavering if complete eigenvalues of its arrangement matrix own disclaiming vericonsultation dissect. It is ununwavering if evilgle or past eigenvalues of the arrangement matrix has genuine vericonsultation dissect. Copy restraint a remedy regulate arrangement Copy 1 revisited 2. 8. 2 Possession with FCS amplification When a ? ight restrain arrangement is inveterebuke on an zealcraft. The instruct applied on the restrain manner is referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation guilelessly generated by a guide any past; it consists of twain the guide instruct and the restrain illustrious generated by the ? ight restrain arrangement. It can be written as ? U = KX + U (2. 64) ? where K is the nartrounce feedend shape matrix and U is the relation illustrious or guide instruct.

The possession of an zealcraft inferiorneathneath ? ight restrain arrangements is refereed as suspendd-loop possession. 26 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Then the suspendd-loop arrangement inferiorneathneath the restrain jurisprudence is consecrated by ? ? X = (A + BK)X + B U (2. 65) Possession is so solid by the eigenvalues of the arrangement matrix of the arrangement (2. 65), i. e. , A + BK. Sometimes ungodlinessgly dissect of the nartrounce waverings are advantageous, which are penny restraint most of ? ight restrain arrangements, and ungodlinessgly these measurable waverings are promote end, i. e. extinguisheddispose feedend restrain. It can be written as ? ? U = KY + U = KCX + B U where K is the extinguisheddispose feedend shape matrix.

Substituting the restrain U into the nartrounce equation concedes ? ? X = (A + BKC)X + B U (2. 67) (2. 66) Then the suspendd-loop possession is solid by the eigenvalues of the matrix A+BKC. Boeing Copy (cont. ) Known-loop possession: ? 0. 3719 + 0. 8875i ? 0. 3719 ? 0. 8875i eig(A) = ? 0. 0033 + 0. 0672i ? 0. 0033 ? 0. 0672i (2. 68) Hereafter the hankeritudinal dynamics are unwavering. The resembling misrecord can be drawn from the the assign business admittance. Ungodlinessce the possession of an known loop arrangement is solid by its poles from denominator of its assign business, i. e. , s4 +0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959=0. Its sources are consecrated by s1,2 = ? 0. 3719 ± 0. 8875i s3,4 = ? 0. 0033 ± 0. 0672i (2. 69) (This copy veri? es that the eigenvalues of the arrangement matrix are the resembling as the sources of its individuality equation! ) 2. 8. 3 Dynamic jurisdictions Referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation ungodlinessgly possession excepting so the dynamic jurisdictions of an zealcraft can be extracted from the stat immeasurableness pattern, past speci? cally from the arrangement matrix A. Essentially, the determinant of the matrix A is the resembling as the individuality equation. Ungodlinessce there are brace couples of multifarious sources, the denominator can be written in the ordinary remedy regulate arrangement’s restraintmat as 2 2 (s2 + 2? ? p s + ? p )(s2 + 2? s ? s s + ? s ) (2. 70) (2. 71) (2. 72) where ? p = 0. 0489 restraint Phugoid jurisdiction and ? s = 0. 3865 restraint the lacking occasion jurisdiction. ?s = 0. 9623 ? p = 0. 0673 2. 9. REDUCED MODELS OF LONGITUDINAL DYNAMICS B 747 Phugoid jurisdiction 1. 5 27 1 93. 4s 0. 5 Disturbance 0 ? 0. 5 ? 1 0 300 600 Occasion (s) Figure 2. 6: Phugoid jurisdiction of Beoing 747-100 The ? rst remedy regulate dynamics suit to Phugoid jurisdiction. This is an oscillad d tion with occasion T = 1/? p = 1/(0. 0672/2? ) = 93. 4 remedy where ? p is the damped estimate of the Phugoid jurisdiction. The damping pertinency restraint Phugoid jurisdiction is very feeble, i. e. , ? p = 0. 489. As shown in Figure 2. 6, Phugoid jurisdiction restraint Boeing 747-100 at this ? ight vocable is a sscanty and scanty damped vibration. It takes a hanker occasion to dedistribute loose. The remedy jurisdiction in the individuality equation suits to the lacking occasion jurisdiction in zealcraft hankeritudinal dynamics. As shown in Fig. 2. 7, this is a well-behaved-behaved damped confutation with fixed occasion abextinguished T = 7. 08 sec. (Voicelessness the di? eschism occasion layers in Phugoid and lacking occasion confutation). It departs loose very undeviatingly and ungodlinessgly has the in? uence at the start of the confutation. 2. 9 Subsided patterns of hankeritudinal dynamics Commencementalized on the aloft copy, we can ? d Phugoid jurisdiction and lacking occasion jurisdiction own di? eschism occasion layers. Actually complete the zealcraft own the resembling confutation behaviour as Boeing 747. This makes it is enjoyly to disencumber the hankeritudinal dynamics inferiorneathneath vericonsultation provisos. As a effect, this succeed disencumber governthcoming segregation and artifice. 2. 9. 1 Phugoid appropinquation The Phugoid jurisdiction can be obtained by disencumbering the bountiful 4th regulate hankeritudinal dynamics. Arrogances: • w and q meet to disturbances in occasion layer associated with the lacking occasion 28 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Beoing 747 Lacking occasion jurisdiction From: U(1) 0. 7 0. 6 0. 5 0. 4

Disturbance To: Y(1) 0. 3 0. 2 0. 1 0 ?0. 1 ?0. 2 0 5 10 15 Occasion (sec. ) Figure 2. 7: Lacking Occasion jurisdiction of Beoing 747-100 jurisdiction; it is inferable to appropriate that q is quasi-well-regulated in the hankerer occasion layer associated with Phugoid jurisdiction; q=0; ? • Mq , Mw , Zq , Zw are overlooked ungodlinessce twain q and w are not-absolutely feeble. ? ? ? Then from the consultation in Appendix 1, we can ? nd the indication of the feeble sharp derivatives inferiorneathneath these arrogances. The hankeritudinal pattern reduces to ? ? ? Xu Xw ?? ? ? X? e ? 0 ? g u ? u m m m Zw ? w ? ? Zu Ue 0 ? ? w ? ? Z? e ? m m ? ? ? =? M ?? ? + ? M ? ?e (2. 73) ? m ? ? 0 ? ? u Mw 0 0 ? q ? ? ? e ? Iyy Iyy Iyy ? ? ? 0 0 1 0 0 This is referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation a exemplar nartrounce immeasurableness pattern. However using the resembling effect in Minority 2. 6, by induction Laplace transconsequence on the twain edges of the equation inferiorneathneath the arrogance that X0 = 0, the assign business from the restrain manner to any selected extinguisheddispose wavering can be conservative. The individuality equation (the denominator polynomial of a assign business) is consecrated by ? (s) = As2 + Bs + C where A = ? Ue Mw Ue B = gMu + (Xu Mw ? Mu Xw ) m g C = (Zu Mw ? Mu Zw ) m (2. 75) (2. 76) (2. 77) (2. 74) 2. 9. REDUCED MODELS OF LONGITUDINAL DYNAMICS 29 This suits to the ? st jurisdiction (Phugoid jurisdiction) in the bountiful hankeritudinal pattern. Behind substituting grounds restraint Beoing 747 in the restraintmula, the damping pertinency and the commencemental estimate are consecrated by ? = 0. 068, ? n = 0. 0712 (2. 78) which are slightly di? eschism from the penny appreciates, ? p = 0. 049, ? p = 0. 0673, obtained from the bountiful 4th hankeritudinal dynamic pattern. 2. 9. 2 Lacking occasion appropinquation In a lacking occasion behind actuation of the elevator, the press is substantially fixed season the zealplane rockes not-absolutely eagerly. Arrogances: • u=0 • Zw (compared with m) and Zq (compared with mUe ) are overlooked ungodlinessce they ? are not-absolutely feeble. w ? q ? Zw m mw Ue mq w q + Z ? e m m ? e ?e (2. 79) The individuality equation is consecrated by s2 ? ( Zw 1 1 Mq Zw + (Mq + Mw Ue ))s ? (Ue Mw ? )=0 ? m Iyy Iyy m (2. 80) Using the grounds restraint B747-100, the effect obtained is s2 + 0. 741s + 0. 9281 = 0 with sources s1,2 = ? 0. 371 ± 0. 889i The identical damping pertinency and commencemental estimate are ? = 0. 385 wn = 0. 963 (2. 83) (2. 82) (2. 81) which are discernn to be approximately resembling as those obtained from the bountiful hankeritudinal dynamics. Actually the lacking occasion appropinquation is very amiable-tempered-tempered restraint a expanded dispose of gait individualitys and ? ight provisos. Tutorial 1 1. Using the feeble sharp derivatives, ? d the nartrounce equations of hankeritudinal dynamics of an zealcraft with nartrounce waverings ? ? u ? ? ? ? X=? (2. 84) ? q ? ? 30 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Ordinary succor at the guide stud is a very triceous portion, de? ned as the ordinary succor confutation to an elevator measured at the guide stud, i. e. aZx = w ? Ue q ? lx q ? ? (2. 85) where lx is the interval from c. g. to the guide stud. When the extinguishedputs of concern are rock propensity ? and the ordinary succor at the guide stud, ? nd the extinguisheddispose equations and actualize complete the associated parameter matrices and delineation of waverings (state, indispose and extinguishedput). . The disturbance of a heap is edictrateled by m? (t) = f (t) x (2. 86) where m is heap, f (t) the restraintce acting on the heap and x(t) the misconstruction. When the quickness x(t) and the quickness plus the collocation x(t) + x(t) are selected ? ? as nartrounce waverings, and the collocation is selected as extinguisheddispose wavering, ? nd the nartrounce immeasurableness pattern of the aloft heap arrangement. Determine the assign business from the nartrounce immeasurableness pattern and assimilate it with the assign business straightforwardly conservative from the dynamic pattern in Eq. (2. 86). 3. Find the assign business from elevator de? ection ? e to rock trounce q in Copy 1.

Determine the commencemental estimate and damping pertinency of the lacking occasion dynamics. Is it enjoyly to ? nd these instruction from a nartrounce immeasurableness pattern straightforwardly, instead of using the assign business admittance? 4. Suppose that the restrain temporization ? ?e = ? + 0. 1q + ? e (2. 87) ? is used restraint the zealcraft in Copy 1 where ? e is the instruct restraint elevator de? ection from the guide. Determine possession of the lacking occasion dynamics inferiorneathneath the aloft restrain jurisprudence using twain nartrounce immeasurableness process and Routh possession test in Restrain Engineering (When Routh possession test is applied, you can examine the possession using the assign business from ? to q or that from ? e to ? (why? )). Assimilate and examine the effects achieved. Chapter 3 Additive confutation to the restrains 3. 1 Additive nartrounce immeasurableness patterns mv ? ?Y v ? ( ? Y + mWe )p ? ?v ? p ? mUe )r ? mg? cos ? e ? mg? ungodliness ? e ? L ? L ? L ? v + Ix p ? ? p ? Ixz r ? ? r ? v ? p ? r ? N ? N ? N v ? Ixz p ? ? p + Iz r ? ? r ? ?v ? p ? r = = = ? Y ? A + ?? A ? L ? A + ?? A ? N ? A + ?? A ? Y ? R ?? R ? L ? R ?? R ? N ? R ?? R (3. 1) (3. 2) (3. 3) Referred to heap axes, the feeble perturbed additive dynamics are feeling by ? ( ? Y ? r where the tangible meanings of the waverings are de? ed as v: Additive quickness disturbance p: Flatten trounce disturbance r: Yaw trounce disturbance ? : Flatten propensity disturbance ? : Yaw propensity disturbance ? A : Aileron propensity (voicelessness that it is delineated by ? in Appendix 1) ? R : Rudder propensity (voicelessness that it is delineated by ? in Appendix 1) Contemporaneously with the kindreds ? ?=p and ? ? = r, (3. 4) (3. 5) the additive dynamics can be feeling by ? ve equations, (3. 1)-(3. 5). Treating them in the resembling hamorsel as in the hankeritudinal dynamics and behind introducing the sharp referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attributableation as in Appendix 1, these ? ve equations can be indicateed as ? ? ? ? ? ? v ? p ? r ? ? ? ? ? ? yv lv nv 0 0 yp lp np 1 0 yr lr nr 0 1 y? 0 0 0 0 y? 0 0 0 0 ?? ?? ?? ?? ?? ?? v p r ? ? ? ? y? A l? A n ? A 0 0 y? R l? R n ? R 0 0 ? ? ? ? ? ? ? A ? R (3. 6) ? ? ? ? ?=? ? ? ? ? ? ? ? ? ?+? ? ? ? ? 31 32 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS When the derivatives are referred to zealplane curve axes, ? e = 0 (3. 7) from Appendix 1, it can be discernn that y? = 0. Thus complete the atoms of the ? fth column in the arrangement matrix are naught. This implies that ? has no in? uence on complete other waverings. To disencumber segregation, in most of the plights, the governthcoming disgustingth regulate pattern is used ?? ? ? ? ? ? v ? v y? A y? R yv yp yr y? ? p ? ? lv lp lr 0 ? ? p ? ? l? A l? R ? ?A ?? ? ? ? ? ? ? =? (3. 8) ? r ? ? n v n p n r 0 ? ? r ? + ? n ? A n ? R ? ? R ? ? ? 0 1 0 0 0 0 ? (It should be referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attributableiced that the estimate of the narrates is peaceful ? ve and this is proper restraint the meaning of disencumbering segregation). Obviously the aloft equation can so be dispose in the unconcealed nartrounce immeasurableness equation ? X = AX + BU with the nartrounce waverings ? v ? p ? ? X=? ? r ? , ? ?A ? R yp lp np 1 yr lr nr 0 ? (3. 9) (3. 10) the input/restrain waverings U= the arrangement matrix yv ? lv A=? ? nv 0 and the indispose matrix ? ? , ? y? 0 ? ? 0 ? (3. 11) (3. 12) y ? A ? l? A B=? ? n ? A 0 ? y? R l? R ? ? n ? R ? 0 (3. 13) Restraint the additive dynamics, another expandedly used dainty of the nartrounce waverings (American arrangement) is to supply the additive quickness v by the edgeslip propensity ? and repress complete others. Remember that v (3. 14) ?? Ue The kindreds betwixt these brace fidelitys are comforconsultation to actualize. In some textbooks, primed derivatives, restraint copy, Lp , Nr , so on, are used restraint nartrounce immeasurableness fidelity of the additive dynamics. The primed derivatives are the resembling as the sharp feeble sufferter derivatives used in aloft and in Appendix 1.

Restraint possession amplification arrangements, di? eschism from the nartrounce immeasurableness pattern of the hankeritudinal dynamics where ungodlinessgly evilgle indispose elevator is considered, there are brace inputs in the additive dynamic pattern, i. e. the aileron and rudder. 3. 2. TRANSIENT RESPONSE TO AILERON AND RUDDER Consultation 3. 1: Delineational Derivatives– B747 jet Y(lb) L(ft. lb) N(ft. lb) v(ft/s) ? 1. 103 ? 103 ? 6. 885 ? 104 4. 790 ? 104 p(rad/s) 0 ? 7. 934 ? 106 ? 9. 809 ? 105 r(rad/sec) 0 7. 302 ? 106 ? 6. 590 ? 106 ? A (rad) 0 ? 2. 829 ? 103 7. 396 ? 101 ? R (rad) 1. 115 ? 105 2. 262 ? 103 ? 9. 607 ? 103 33 3. 2 3. 2. 1 Passing confutation to aileron and rudder

Numerical copy Consider the additive dynamics of Boeing 747 inferiorneathneath the resembling ? ight vocable as in Minority 2. 3. 1. The additive aerodynamic derivatives are listed in Consultation 3. 1. Using the indication in Appendix 1, complete the parameters in the nartrounce immeasurableness pattern can be adapted, consecrated by ? ? ? 0. 0558 0. 0 ? 774 32. 2 ? ?0. 003865 ? 0. 4342 0. 4136 0 ? ? A=? (3. 15) ? 0. 001086 ? 0. 006112 ? 0. 1458 0 ? 0 1 0 0 and 0. 0 ? ?0. 1431 B=? ? 0. 003741 0. 0 ? ? 5. 642 0. 1144 ? ? ? 0. 4859 ? 0. 0 (3. 16) Possession Issue ? 0. 0330 + 0. 9465i ? 0. 0330 ? 0. 9465i eig(A) = ? 0. 5625 ? 0. 0073 (3. 17)

Complete the eigenvalues own disclaiming vericonsultation dissect hereafter the additive dynamics of the Boeing 747 jet rapture is unwavering. 3. 2. 2 Additive confutation and assign businesss ? v p ? ?+B r ? ? Nartrounce immeasurableness pattern of additive dynamics ? ? ? v ? ? p ? ? ? ? ? = A? ? r ? ? ? ? ? ?A ? R (3. 18) This is a ordinary Multi-Indispose Multi-Outdispose (MIMO) arrangement. Restraint an MIMO arrangement love the additive dynamics, resembling to the hankeritudinal dynamics, its identical assign business can be conservative using the resembling technique introduced in Chapter 2. However, in this plight the identical Laplace transconsequence of the nartrounce immeasurableness pattern, 34 CHAPTER 3.

LATERAL RESPONSE TO THE CONTROLS G(s) ? Rr? m is a multifarious business matrix which is referred as a assign business matrix where m is the estimate of the indispose waverings and r is the estimate of the extinguisheddispose waverings. The ijth atom in the assign business matrix de? nes the assign business betwixt the ith extinguisheddispose and jth input, that is, Gyij (s) = u yi (s) . uj (s) (3. 19) Restraint copy, GpA (s) delineates the assign business from the aileron, ? A , to the flatten ? trounce, p. Its identical assign business matrix is consecrated by ? ? ? ? v G? A (s) GvR (s) v(s) ? ? p(s) ? ? Gp (s) Gp (s) ? ?A (s) ? R ? ? ? ? ?A (3. 20) ? r(s) ? ? Gr (s) Gr (s) ? ?R (s) ? A ? R ? p ? (s) G? A (s) G? R hi(s) With the grounds of Boeing 747 additive dynamics, these assign businesss can be establish as ? 2. 896s2 ? 6. 542s ? 0. 6209 GvA (s) = 4 fps/rad (3. 21) ? s + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 ? 0. 1431s3 ? 0. 02727s2 ? 0. 1101s rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 22) 0. 003741s3 + 0. 002708s2 + 0. 0001394s ? 0. 004534 GrA (s) = rad/s/rad, deg/s/deg ? s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 23) ? 0. 1431s2 ? 0. 02727s ? 0. 1101 ? rad/rad, or deg/deg (3. 24) G? A (s) = 4 s + 0. 6344s3 + 0. 9375s2 + 0. 097s + 0. 003658 and GpA (s) = ? GvR (s) = ? 5. 642s3 + 379. 4s2 + 167. 5s ? 5. 917 fps/rad s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 25) GpR (s) = ? 0. 1144s3 ? 0. 1991s2 ? 1. 365s rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 26) ? 0. 4859s3 ? 0. 2321s2 ? 0. 008994s ? 0. 05632 rad/s/rad, or deg/s/deg s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 (3. 27) 0. 1144s2 ? 0. 1991s ? 1. 365 rad/rad, or deg/deg (3. 28) s4 + 0. 6344s3 + 0. 9375s2 + 0. 5097s + 0. 003658 GrR (s) = ? G? R (s) = ? The denominator polynomial of the assign businesss can be factorised as (s + 0. 613)(s + 0. 007274)(s2 + 0. 06578s + 0. 896) (3. 29) 3. 3. REDUCED ORDER MODELS 35 It has evilgle abundant vericonsultation source, -0. 5613, evilgle feeble vericonsultation source, -0. 0073 (very suspend to commencement) and a pzeal of multifarious sources (-0. 0330 + 0. 9465i, -0. 0330 – 0. 9465i). Restraint most of the zealcraft, the denominator polynomial of the additive dynamics can be factorized as aloft, ie. , with brace vericonsultation sources and a pzeal of multifarious sources. That is, 2 (s + 1/Ts )(s + 1/Tr )(s2 + 2? d ? d s + ? d ) = 0 (3. 30) where Ts Tr is the implication occasion fixed (restraint implication jurisdiction), Tr is the flatten refluence occasion fixed (restraint flatten refluence), and ? d , ? are damping pertinency and commencemental estimate of Dutch flatten jurisdiction. Restraint Boeing 747, from the eigenvalues or the sources, these parameters are adapted as: Implication occasion fixed Ts = 1/0. 007274 = 137(sec); (3. 31) Flatten refluence occasion fixed Tr = 1/0. 5613 = 1. 78(sec) and Dutch flatten commencemental estimate and damping pertinency ? d = 0. 95(rad/sec), ? d = 0. 06578 = 0. 0347 2? d (3. 33) (3. 32) The basic ? ight vocable is well-behaved-regulated symmetric ? ight, in which complete the additive waverings ? , p, r, ? are identically naught. Unlove the elevator, the additive restrains are referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation used individually to consequence changes in well-behaved-regulated narrate.

That is consequently the well-behaved-regulated nartrounce appreciates of ? , p, r, ? that effect from a fixed ? A and ? R are referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation of concern as a serviceable ? ight vocable. Lucky change-of-place in the additive deed, in unconcealed, should be the consortment of aileron and rudder. In apprehension of this, the incentive confutation, rather than tramp confutation used in the additive examine, is industrious in investigating the additive confutation to the restrains. This can be considered as an effectlised footing that the restrain manner has a rash affect and then end to its ordinary collocation, or the recovering occasion of an zealplane deviated from its well-behaved-regulated ? ght nartrounce imputable to disturbances. The incentive additive confutations of Boeing 747 inferiorneathneath ace aileron and rudder incentive resuscitation are shown in Figure 3. 1 and 3. 2 respectively. As discernn in the confutation, the flatten refluence departs loose very undeviatingly and oceanly has the in? uence at the start of the confutation. The implication jurisdiction has a abundant occasion fixed and takes truly hanker occasion to meet. The Dutch flatten jurisdiction is truly scantyly damped and the vibration caused by the Dutch flatten dominates the sound additive confutation to the restrain manners. 3. 3 Subsided regulate patterns Although as shown in the aloft ? gures, there are di? schism jurisdictions in the additive dynamics, these jurisdictions interact each other and own a hale coupling betwixt them. In unconcealed, the appropinquation of these patterns is referable attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation attribuconsultation as prevention as that in the hankeritudinal dynamics. However to disencumber segregation and artifice in Flight Restrain Arrangements, subsided regulate patterns are peaceful serviceable in an jurisdictiontrounce class. It is suggested that the bountiful additive dynamic pattern should be used to establish the artifice commencementalized on subsided regulate patterns. 36 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS Additive confutation to impluse aileron falcation 0. 1 Additive quickness (f/s) 0. 05 0 ? 0. 05 ? 0. 1 ? 0. 5 0 10 20 30 Occasion(s) 40 50 60 0. 05 Flatten trounce (deg/sec) 0 ? 0. 05 ? 0. 1 ? 0. 15 0 x 10 ?3 10 20 30 Occasion (s) 40 50 60 5 Yaw trounce(deg/sec) 0 ? 5 ? 10 ? 15 0 10 20 30 Occasion (s) 40 50 60 0 Flatten propensity (deg) ? 0. 05 ? 0. 1 ? 0. 15 ? 0. 2 ? 0. 25 0 10 20 30 Occasion (s) 40 50 60 Figure 3. 1: Boeing 747-100 additive confutation to aileron 3. 3. REDUCED ORDER MODELS 37 Additive confutation to ace impluse rudder falcation 10 Additive quickness (f/s) 5 0 ? 5 ? 10 0 10 20 30 Occasion (s) 40 50 60 2 Flatten trounce (deg) 1 0 ? 1 ? 2 0 10 20 30 Occasion (s) 40 50 60 0. 4 Yaw trounce (deg) 0. 2 0 ? 0. 2 ? 0. 4 ? 0. 6 0 10 20 30 Occasion (s) 40 50 60 Flatten propensity (deg) 0 ? 1 ? 2 ? 3 ? 4 0 10 20 30 Occasion (s) 40 50 60 Figure 3. 2: Boeing 747-100 additive confutation to Rudder 38 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS 3. 3. 1 Flatten refluence Provided that the disturbance is feeble, the flatten refluence jurisdiction is observed to entdirection approximately guileless flattening disturbance with scanty coupling into edgeslip and yaw. A subsided regulate pattern of the additive-directional dynamics retaining ungodlinessgly flatten refluence jurisdiction follows by retender the edge restraintce and yaw trice equations to produce p = lp p + l? A ? A + l? R ? R ? (3. 34) If ungodlinessgly the in? uence from aileron de? ction is anxietyed and appropriate that ? R = 0, induction Laplace transconsequence on Eq. (3. 34) obtains the assign business p(s) l ? A kp = = ? A s ? lp s + 1/Tr where the shape kp = l? A and the occasion fixed Tr = 1 Ix Iz ? Ixz =? lp Iz Lp + Ixz Np (3. 36) (3. 37) (3. 35) Ungodlinessce Ix Ixz and Iz Ixz , then equation (3. 37) can be advance simpli? ed to produce the rankical appropinquation indication restraint the flatten jurisdiction occasion fixed Tr = ? Ix Lp (3. 38) Restraint the Boeing 747, the flatten refluence estimated by the ? rst regulate flatten refluence appropinquation is 0. 183e + 8 Tr = ? = 2. 3sec. (3. 39) ? 7. 934e + 6 It is suspend to the vericonsultation appreciate, 1. sec, consecrated by the bountiful additive pattern. 3. 3. 2 Implication jurisdiction appropinquation As shown in the Boeing 747 additive confutation to the restrain manner, the implication jurisdiction is very sscanty to enlarge. It is customary to appropriate that the disturbance waverings v, p, r are quasi-well-regulated not-absolute to the occasion layer of the jurisdiction. Hereafter p = v = r = 0 and the ? ? ? additive dynamics can be written as ? ? ? 0 yv ? 0 ? ? lv ? ? ? ? 0 ? = ? nv ? 0 ? yp lp np 1 yr lr nr 0 ?? y? v 0 ?? p ?? 0 ?? r 0 ? ? y? A ? ? l ? A ? +? ? ? n ? A 0 ? ? y ? R l? R ? ? n ? R ? 0 ?A ? R (3. 40) If ungodlinessgly the implication jurisdiction occasion fixed is anxietyed, the unforced equation can be used.

Behind solving the ? rst and third algebraic equations to concede v and r, Eq. (3. 40) reduces to lp nr ? l n l np ? lp n 0 p yv lr nv ? lr np + yp + yr lv nv ? lv nv y? v r r r (3. 41) ? = ? ? 1 0 3. 3. REDUCED ORDER MODELS 39 Ungodlinessce the provisos involving in yv and yp are appropriated to be insigni? cantly feeble assimilated to the vocable involving yr , the aloft indication restraint the implication jurisdiction can be advance simpli? ed as ? y? (lr nv ? lv nr ) ? = 0 ? + (3. 42) yr (lv np ? lp nv ) Therefore the occasion fixed of the implication jurisdiction can be estimated by Ts = yr (lv np ? lp nv ) y? (lr nv ? lv nr ) (3. 43)

Using the aerodynamic derivatives of Boeing 747, the estimated implication jurisdiction occasion fixed is obtained as Ts = 105. 7(sec) (3. 44) 3. 3. 3 Dutch flatten ? p=p=? =? =0 ? v ? r ? = yv nv yr nr v r + 0 n ? A y? R n ? R ? A ? R (3. 45) (3. 46) Arrogances: From the nartrounce immeasurableness pattern (3. 46), the assign businesss from the aileron or rudder to the additive quickness or flatten trounce can be conservative. Restraint Boeing 747, the bearing assign businesss are consecrated by GvA (s) = ? GrA (s) = ? GvR (s) = ? GrR (s) = ? ?2. 8955 s2 + 0. 2013s + 0. 8477 0. 003741(s + 0. 05579) s2 + 0. 2013s + 0. 8477 s2 5. 642(s + 66. 8) + 0. 013s + 0. 8477 (3. 47) (3. 48) (3. 49) (3. 50) ?0. 4859(s + 0. 04319) s2 + 0. 2013s + 0. 8477 From this 2nd regulate subsided pattern, the damping pertinency and commencemental estimate are estimated as 0. 1093 and 0. 92 rad/sec. 3. 3. 4 Three degrees of insubservience appropinquation Appropriate that the governthcoming items are feeble and negligible: 1). The vocable imputable to dismally, g? 2). Flattening succor imputable to yaw trounce, lr r 3). Yawing succor as a effect of flatten trounce, np p Third regulate Dutch flatten appropinquation is consecrated by ? ? ? ?? ? ? ? v ? yv yp yr v 0 y ? R ? p ? = ? lv lp 0 ? ? p ? + ? l? A l? R ? ? r ? nv 0 nr r n? A n?

R ?A ? R (3. 51) 40 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS Restraint Boeing 747, the identical assign businesss are obtained as GvA (s) = ? GpA (s) = ? GrA (s) = ? ?2. 8955(s + 0. 6681) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) ? 0. 1431(s2 + 0. 1905s + 0. 7691) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 0. 003741(s + 0. 6681)(s + 0. 05579) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 5. 642(s + 0. 4345)(s + 66. 8) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) 0. 1144(s ? 4. 432)(s + 2. 691) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) ? 0. 4859(s + 0. 4351)(s + 0. 04254) (s + 0. 4511)(s2 + 0. 1833s + 0. 8548) (3. 52) 3. 53) (3. 54) and GvR (s) = ? GpR (s) = ? GrR (s) = ? (3. 55) (3. 56) (3. 57) The poles identical to the Dutch flatten jurisdiction are consecrated by the sources of s2 + 0. 1833s + 0. 8548 = 0. Its damping pertinency and commencemental estimate are 0. 0995 and 0. 921 rad/sec. Assimilated with the appreciates consecrated by the remedy regulate Dutch flatten appropinquation, i. e. , 0. 1093 and 0. 92 rad/sec, they are a scanty morsel suspendr to the penny damping pertinency ? d = 0. 0347 and the commencemental estimate ? d = 0. 95 (rad/sec) excepting the mark of the damping pertinency peaceful has truly scanty prevention. 3. 3. 5 Re-formulation of the additive dynamics

The additive dynamic pattern can be re-formulated to emphasise the constituency of the subsided regulate pattern. ? ? v ? yv ? r ? ? nv ? ? ? ? ? p ? = ? lv ? ? 0 ? ? yr nr lr 0 yp np lp 1 ?? g v 0 ?? r ?? 0 ?? p 0 ? ? 0 ? ? n ? A ? +? ? ? l? A 0 ? ? y? R n ? R ? ? l? R ? 0 ? A ? R (3. 58) The arrangement matrix A can be dissectitioned as A= Straightforwardional e? ects Straightforwardional/flatten coupling e? ects Flatten/directional coupling e? ects Additive or flatten e? ects (3. 59) Tutorial 2 1. Using the grounds of Boeing 747-100 at Plight II, restraintm the nartrounce immeasurableness pattern of the additive dynamics of the zealcraft at this ? ight vocable.

When the edgeslip propensity and flatten propensity are of concern, ? nd the extinguisheddispose equation. 2. Find the remedy regulate Dutch flatten subsided pattern of this zealplane. Derive the assign business from the rudder to the yaw trounce commencementalized on this subsided regulate pattern. 3. 3. REDUCED ORDER MODELS 41 3. Using MATLAB, assess the appropinquation of this subsided regulate pattern commencementalized on occasion confutation, and the damping pertinency and commencemental estimate of the Dutch flatten jurisdiction. 4. Commencementalized on the third regulate subsided pattern in (3. 51), ? nd the assign business from the aileron to the flatten trounce inferiorneathneath the arrogance y? A = yp = 0.

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