Flight Control Systems

Flight Administer Rules W. -H. Chen Department of Aeronautical and Automotive Engineercontemptuous Loughborough University 2 Flight Administer Rules by W. -H. Chen, AAE, Loughborough Contents 1 Portico 1. 1 Overconception of the Flight Contract 1. 2 Flight administer rules . . . . . . 1. 3 Decreern Administer . . . . . . . . . . 1. 4 Portico to the continuity . . . . 1. 4. 1 Content . . . . . . . . . . 1. 4. 2 Tutorials and continuitywork 1. 4. 3 Reprove . . . . . . . . 1. 4. 4 Lecture contrivance . . . . . . . 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 Craveitudinal exculpation to the administer 2. 1 Craveitudinal dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2 Propound intervenience title . . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 1 Propound mutables . . . . . . . . . . . . . . . . . . . . . . . . 2. 2. 2 Public propound intervenience stance . . . . . . . . . . . . . . . . . . . 2. 3 Craveitudinal propound intervenience stance . . . . . . . . . . . . . . . . . . . . 2. 3. 1 Numerical stance . . . . . . . . . . . . . . . . . . . . . . . 2. 3. 2 The rare of propound mutables . . . . . . . . . . . . . . . . . . 2. 4 Essential-qualitycraft dynamic behaviour essential-qualitys uscontemptuous propound intervenience stances . 2. 4. 1 Essential-qualitycraft exculpation externally administer . . . . . . . . . . . . . . . 2. 4. 2 Essential-qualitycraft exculpation to administers . . . . . . . . . . . . . . . . . 2. 4. 3 Essential-qualitycraft exculpation dejecteder twain decreetrounce qualifications and administers 2. 5 Craveitudinal exculpation to the elevator . . . . . . . . . . . . . . . . 2. 6 Translate of propound intervenience stances into translate obey-aparts . . . . . . . . 2. 6. 1 From a translate obey-akeep-abisect to a propound intervenience stance . . . . . . . 2. 7 Block diagram delineateation of propound intervenience stances . . . . . . . . . 2. 8 Static inheritance and dynamic decrees . . . . . . . . . . . . . . . . . . 2. 8. 1 Essential-qualitycraft inheritance . . . . . . . . . . . . . . . . . . . . . . . . 2. 8. 2 Inheritance with FCS enrichment . . . . . . . . . . . . . . . 2. 8. 3 Dynamic decrees . . . . . . . . . . . . . . . . . . . . . . . . . 2. 9 Feeble stances of craveitudinal dynamics . . . . . . . . . . . . . . 2. 9. Phugoid advance . . . . . . . . . . . . . . . . . . . . 2. 9. 2 Inadequate era advance . . . . . . . . . . . . . . . . . . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Vocableinationant exculpation to the administers 3. 1 Vocableinationant propound intervenience stances . . . . . . . . . . . . 3. 2 Passcontemptuous exculpation to aileron and rudder . . . . 3. 2. 1 Numerical stance . . . . . . . . . . . . 3. 2. 2 Vocableinationant exculpation and translate obey-aparts 3. 3 Feeble manage stances . . . . . . . . . . . . . . 3. 3. 1 Flatten firmtlement . . . . . . . . . . . . . . 3. 3. Implication decree advance . . . . . . . 3. 3. 3 Dutch flatten . . . . . . . . . . . . . . . . . 3. 3. 4 Three degrees of immunity advance 3. 3. 5 Re-formulation of the vocableinationant 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 Inheritance Enrichment Rules 4. 1 Propound intervenience delineation techniques . . . . . . . . . . . 4. 2 Craveitudinal inheritance enrichment rules . . . 4. 2. 1 The rare of feedtail mutables . . . . 4. 2. 2 SAS coercion inadequate era dynamics . . . . . . 4. 3 Vocableinationant inheritance enrichment rules . . . . . . 4. 3. 1 Yaw trounce feedtail coercion rudder administer . . . 4. 3. 2 Flatten feedtail coercion aileron administer . . . . . 4. 3. 3 Integration of vocableinationant troddenional feedtail 5 Autopilots 5. 1 Cast avocation autoconvoy . . . . . . . . . . . . . . . . . . . . . . . 5. 1. 1 phugoid destroy . . . . . . . . . . . . . . . . . . . . . . 5. 1. 2 Eliminate the consistent mistake with integration . . . . . . . 5. 1. 3 Better passcontemptuous execution with cast trounce feedtail 5. 2 Acme avocation autoconvoy . . . . . . . . . . . . . . . . . . . . . . 5. . 1 An intuitive acme avocation autoconvoy . . . . . . . . . . . 5. 2. 2 Betterd acme avocation rules . . . . . . . . . . . . . 5. 3 Actuator dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 6 Handlcontemptuous Qualities 6. 1 Handcontemptuous qualities coercion essential-qualitycraft . . . . . . . . . . . . 6. 2 Convoy-in-loop dynamics . . . . . . . . . . . . . . . . 6. 2. 1 Convoy as a administerler . . . . . . . . . . . . . 6. 2. 2 Calculate exculpation of a dynamic rule . . 6. 2. 3 Convoy-in-loop . . . . . . . . . . . . . . . . . 6. 3 Flycontemptuous qualities requirements . . . . . . . . . . . . 6. 4 Essential-qualitycraft role . . . . . . . . . . . . . . . . . . . . . . 6. . 1 Essential-qualitycraft disposei? cation . . . . . . . . . . . . . 6. 4. 2 Flight face . . . . . . . . . . . . . . . . . . 6. 4. 3 Resemblingizes of ? ycontemptuous qualities . . . . . . . . . . . 6. 5 Convoy judgment ratcontemptuous . . . . . . . . . . . . . . . . . . 6. 6 Craveitudinal ? ycontemptuous qualities requirements . . . . . 6. 6. 1 Inadequate era castcontemptuous vibration . . . . . . 6. 6. 2 Phugoid . . . . . . . . . . . . . . . . . . . . 6. 6. 3 Flycontemptuous qualities requirements on the s-plane 6. 7 Vocableinationant-directional ? ycontemptuous qualities requirements . . 6. 7. 1 Flatten firmtlement decree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 6. 7. 2 6. 7. 3 6. 7. 4 5 Implication decree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Dutch flatten decree . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Vocableinationant-directional decree in s-plane . . . . . . . . . . . . . . . . . 75 77 . . . . . . . . . . . administer derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 79 79 79 79 79 7 Fly-by-Wire ? ight administer 8 Appendices 8. Boecontemptuous 747-100 grounds . . . . . . . . . . . 8. 2 De? nitions of Aerodynamic inheritance and 8. 3 Radix Locus . . . . . . . . . . . . . . . . 8. 4 Calculate exculpation . . . . . . . . . . . . appendices 6 CONTENTS Chapter 1 Portico 1. 1 Overconception of the Flight Contract • Flight contrivancecontemptuous • Essential-qualitycraft checkcontemptuous • Taxi • Take-o? – Rotate, “select” an comcomposture – Clean up (gear, ? aps, anticipation) – Emergencies (engine need, ? re, anticipation) • Climb – Expedite administer – Process (manual, autopilot) • Mission Tasks – Cruise – Combat (essential-quality to essential-quality) – Strike (essential-quality to world) – Public handlcontemptuous (stalling, spinning, aerobatics) – Coercionmation ? contemptuous (Navigation, process anticipation) – Emergencies – Con? guration (weapons, tanks, fuel arraign) • Recovery – Descent – Instrument arrival – Landcontemptuous – Overshoot 7 8 CHAPTER 1. INTRODUCTION Stick – Linkage 6 Trim ? -? Servo Actuator – Essential-qualitycraft dynamics Figure 1. 1: Manual convoy administer essential-qualitycraft – Coercionmation – Processs – Emergencies • Taxi Craveitudinal and vocableinationant dynamics thus Flight administer rules are confused in Take o? , Climb, Mission tasks and Recovery. • Di? efissure essential-qualitycraft (aircraft dispose) • Di? efissure ? ight face Manual– handlcontemptuous qualities/? ight qualities Better the handlcontemptuous qualities of essential-qualityplane; Autoconvoy 1. 2
Flight administer rules Objectives • To better the handlcontemptuous qualities • To loose the exercise bundle of convoys in-a-measure or enlightenedly • To acception the execution of essential-qualitycraft or missiles Types of Flight Administer Rules (FCS) 1. Public-loop administer 2. Inheritance enrichment rules 3. Autoconvoy 4. Integrated Navigation rules and Autopilots (? ight address rules) 1. 3 Decreern Administer • Disposeic administer– translate obey-akeep-abisect – calculate inclosure • Limitation of disposeic delineation rule: uncombined input, uncombined extinguisheddispose (SISO), barely share the extinguisheddispose behaviour, trodden rules (saturation) • Rule title in propound intervenience coercionm. 1. 4.
INTRODUCTION TO THE COURSE 9 Stick Trim – Essential-qualitycraft dynamics – + ? + -Linkage – ? – ? – Servo Actuator 6 6 Inheritance Aug. Rules Sensor ? Figure 1. 2: Inheritance Enrichment Rules Relation Instruct + -? Autoconvoy – 6 6 + -? 6 – SAS – Actuators – Essential-qualitycraft dynamics – Sensor 6 Navigation Rules ? ? Figure 1. 3: Autoconvoy con? guration • Naradmonish essential-qualitycraft or other dynamics rules in a firm of ? rst manage di? erential equations. Explicit in a matrix coercionm • Propound intervenience resolution and delineation techniques– very mighty technique coercion administer rules • Matrix manipulation instruction required 1. 4 1. 4. 1 Portico to the continuity

Content This continuity get screen • propound intervenience resolution and delineation techniques coercion essential-qualitycraft • dejectedly ? ight administer rules includcontemptuous inheritance enrichment rules, and dejectedly autopilots • handlcontemptuous qualities 10 CHAPTER 1. INTRODUCTION Flight Address 6 Rules/Autoconvoy 6 + -? 6 – SAS – Actuators – Essential-qualitycraft dynamics – Sensor 6 Navigation Rules ? ? Figure 1. 4: Autoconvoy con? guration • Fly-By-Wire (FBW) 1. 4. 2 Tutorials and continuitywork • Tutorials get rouse from Week 3 • Individual tutorial exception in each week • Individual continuitywork invetereprove on MATLAB/Simulink essential-qualitys, must be handed in precedently 4:00 PM Thursday, Week 11 1. 4. 3
Reprove • Continuitywork: 20%; • Examination: 2 hours; violate 3 from 5 questions; 80% of the ? nal trace. 1. 4. 4 Lecture contrivance • Overperfect ? ight contract • Flight administer rules • Decreern administer delineation ruleology • The portico of the continuity– erection, reprove, exercises, relations 1. Portico 2. Exculpation to the administers (a) Propound intervenience resolution (b) Craveitudinal exculpation to elevator and extinguish (c) Passcontemptuous exculpation to aileron and rudder 3. Essential-qualitycraft inheritance enrichment rules 1. 4. INTRODUCTION TO THE COURSE (a) Execution evaluation • • • • inheritance Era inclosure requirements Calculate inclosure speci? ations Robustness 11 (b) Craveitudinal Inheritance Enrichment Rules • Rare of the feedtail mutables • Radix locus and shape voluptuousness • Phugoid destroy (c) Vocableinationant inheritance enrichment rules • Flatten feedtail coercion aileron administer • Yaw trounce feedtail coercion rudder administer 4. Dejectedly autoconvoy delineation • Augmented craveitudinal dynamics • Acme halt rules 5. Handlcontemptuous Qualities (a) Era relapse rules (b) Convoy-in-loop dynamics (c) Handlcontemptuous qualities (d) Calculate inclosure resolution (e) Convoy ascititious vibration 6. Flight Administer rule 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. Automatic Flight Administer Rules. D. McLean. 1990. Prentice Hperfect International Ltd. Chaps. 2, 3,6,9. 3. Portico to Avionics Rules. Coopeadmonish edition. R. P. G. Collinson. 2003. Kluwer Academic Publishers. Chap. 4 12 CHAPTER 1. INTRODUCTION Chapter 2 Craveitudinal exculpation to the administer 2. 1 Craveitudinal dynamics From Flight Dynamics continuity, we apprehend that the troddenised craveitudinal 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? transgression ? 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 visible meanings of the mutables are de? ned as u: Mobility abextinguished consistent propound rapidity Ue w: Mobility on consistent propound common rapidity We q: Cast trounce ? : Cast determination Dejecteder the impudence that the aeroplane is in resemblingize undeviating ? ight and the relation axes are bend or inheritance axes, we possess ? e = We = 0 (2. 4) The main administers in craveitudinal dynamics are the elevator determination and the engine charge. The minute mobility provisions in the exact plane of the overhead 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 shove mobility Substitutcontemptuous the overhead countenance into the craveitudinal symmetric disturbance surrenders ? 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)
Aftercited addcontemptuous the conformity ? ? = q, (2. 11) Eqs. (2. 8)- (2. 11) can be dispose in a elevate summary vector and matrix coercionmat. The craveitudinal 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 perfect mutables in the craveitudinal dynamics in a vector coercionm 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 practice to turn the overhead firm of equations into a firm of ? rst manage di? erential equations by multiplycontemptuous twain planes of the overhead 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 summary coercionmat ? X = AX + BU (2. 22) Eq. (2. 22) with (2. 20) and (2. 21) is referred as the propound intervenience stance of the troddenised craveitudinal dynamics of essential-qualitycraft. Appendix 1 confers the conformity among the odd inheritance and administer derivatives in the matrix A and B, i. e. xu , so on, with the sizeal and non-dimensional derivatives, where ?
X ? Xu = ? u (2. 23) dramatizes sizeal derivative and Xu its suitcontemptuous non-dimensional derivative. These conformitys are obey-apartial invetereprove on the Cramer’s government and halt coercion public collectiveness axes. In the event when the derivatives are referred to bend axes, as in this continuity, the followcontemptuous simpli? cations should be made Ue = Vo , We = 0, transgression ? e = 0, cos ? e = 1 (2. 24) The title of the craveitudinal dynamics in the matrix-vector coercionmat as in (2. 19) can be distant to delineate perfect public dynamic rules. Consider a rule with manage n, i. e. , the rule can be picturesquely by n manage di? fissureial equation (as it get be explained posterior, this is the selfselfhomogeneous as the coercionemost manage of the denominator polynomial in the translate obey-akeep-abisect is n). In the delineateation (2. 22), A ? Rn? n is the rule matrix ; B ? Rn? m is the indispose matrix ; X ? Rn is the propound vector or propound mutables and U ? Rm the indispose or indispose vector. The equation (2. 22) is denominated propound equation. Coercion the inheritance enrichment rule, barely the in? uence of the deviation of the elevator determination, i. e. the pertinent aerodynamic administer demeanor, is shareed. The overhead equations of disturbance can be simpli? ed. The propound intervenience delineateation dross the 6 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL selfselfhomogeneous coercionmat as in eq. (2. 22) with the selfselfhomogeneous matrix A and propound mutables beparty with a di? efissure B and indispose U as absorbed beneathneath ? ? x ? e ? z ? B = M ? 1 B = ? ?e ? (2. 25) ? m? e ? 0 and U = ? e (2. 26) Remark: It should be referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attributableiced that in di? efissure textbooks, di? efissure referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attributableations are used. Coercion the propound intervenience delineateation of craveitudinal dynamics, someera enlightenedtilded 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 enlightenedtilded derivatives and the other derivatives in the matrices are the selfselfhomogeneous as the countenance of the minute sufferter derivatives dejecteder fired impudences, i. e. uscontemptuous inheritance axis. 2. 2 2. 2. 1 Propound intervenience title Propound mutables A stint firm of mutables which, when apprehendn at era t0 , conjointly with the input, are su? ient to naradmonish the behaviours of the rule at any era t > t0 . Propound mutables may possess no any visible meanings and may be referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard measurable. Coercion the craveitudinal dynamic of essential-qualitycraft, there are foul-mouthed propound mutables, i. e, ? ? u ? w ? ? X=? (2. 32) ? q ? ? and individual indispose or administer mutable, the elevator de? ection, U = ? e (2. 33) 2. 3. LONGITUDINAL STATE SPACE MODEL Thus n=4 m=1 17 (2. 34) The rule matrix and indispose matrix of the craveitudinal dynamics are absorbed 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 Public propound intervenience stance w Ue When the determination of onslaught ? is of share, it can be written as ? = which can be dispose into a public coercionm as y = CX where y=? = and C= 0 1/Ue 0 0 (2. 40) Eq. (2. 38) is denominated Extinguisheddispose equation; y the extinguisheddispose muboard and C the extinguisheddispose matrix. Coercion elevate public event where there are elevate than individual extinguisheddispose and has a trodden footfootroad from indispose to extinguisheddispose mutable, 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 . Coercion disturbance of aerointervenience manners includcontemptuous essential-qualitycraft and missiles, there is no trodden footfootroad among indispose and extinguishedput.
In this continuity barely the event D = 0 is considered if referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard explicitly sharp extinguished. Eq. (2. 22) and (2. 38) (or (2. 41)) conjointly delineate the propound intervenience title of a dynamic rule, which is irreconcilable to the translate obey-akeep-abisect delineateation of a dynamic rule learned in Administer Engineercontemptuous continuity. 2. 3 Craveitudinal propound intervenience stance When the behaviours of perfect the propound mutables are shareed, perfect those mutables can be clarified as extinguisheddispose mutables. In importation, there are other exculpation quantities of share includcontemptuous the ? ight footfootroad determination ? , the determination of onslaught ? and the common succor az (nz ).
Puttcontemptuous perfect mutables conjointly, the extinguisheddispose vector can be written as 18 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL ? ? ? ? ? Y =? ? ? ? ? Invokcontemptuous the conformitys ? = ? ? ? ? ? ? ? ? ? ? u w q ? ? ? az w Ue (2. 42) (2. 43) w Ue (2. 44) the ? ight footfootroad determination ? =??? =?? and the common 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 coopeadmonish parity substitutcontemptuous the countenance matrix is absorbed by ? ? ? u 1 ? w ? ? 0 ? ? ? ? q ? ? 0 ? ? ? Y =? ? ? =? 0 ? ? ? ? ? ? ? 0 ? ? ? ? ? ? ? 0 az ? zu ollows from (2. 9) and the last parity is obtained by of w in its summary derivative coercionmat. Future 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 trodden footfootroad among the extinguisheddispose and input! The propound intervenience stance of craveitudinal dynamics consists of (2. 22) and (2. 46). 2. 3. 1 Numerical stance Boecontemptuous 747 jet ecstasy at ? ight qualification cruiscontemptuous in absolute ? ight at approximately 40,000 ft at Mach calculate 0. 8. Pertinent grounds are absorbed in Board 2. 1 and 2. 2.
Uscontemptuous boards in Appendix 1, the summary minute derivatives can be fitted and then the rule matrix and indispose matrix can be obey-apartial 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 Homogeneously the parameters matrices in extinguisheddispose equation (2. 46) can be firm. It should be referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attributableiced that English obey-apart(s) is used in this stance. 2. 4. AIRCRAFT DYNAMIC BEHAVIOUR SIMULATION USING STATE SPACE MODELS19 Board 2. 1: Boecontemptuous 747 ecstasy 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 Board 2. 2: Sizeal 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 rare of propound mutables The propound intervenience delineateation of a dynamic rule is referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard rare, which depends on the rare of propound mutables. Coercion engineercontemptuous collision, propound mutables, in public, are clarified invetereprove on visible meanings, measurement, or manageable to delineation and resolution. Coercion the craveitudinal dynamics, in importational to a firm of the propound mutables in Eq. (2. 32), another enlightenedly used rare (in American) is ? u ? ? ? ? X=? ? q ? ? ? (2. 49) Firedly, when the logitudinal dynamics of the essential-qualitycraft are delineateed in provisions of the overhead propound mutables, di? fissure A, B and C are vocableinationed (experience Tutorial 1). 2. 4 Essential-qualitycraft dynamic behaviour essential-qualitys uscontemptuous propound intervenience stances Propound intervenience stance plain overhead provides a very mighty dupe in question dynamic behavious of an essential-qualitycraft dejecteder manifold qualification. The conception of uscontemptuous propound gait stances coercion predictcontemptuous essential-qualitycraft dynamic behavious or numerical essential-qualitys can be explained by 20 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL the followcontemptuous countenance X(t + ? t) = X(t) + dX(? ) ? |? =t ? t = X(t) + X(t)? t d? (2. 50) ? where X(t) is curfissure propound, ? t is trudge size and X(t) is the derivative fitted by the propound intervenience equation. . 4. 1 Essential-qualitycraft exculpation externally administer ? X = AX X(0) = X0 (2. 51) 2. 4. 2 Essential-qualitycraft exculpation to administers ? X = AX + BU ; X(0) = 0 (2. 52) where U is the convoy instruct 2. 4. 3 Essential-qualitycraft exculpation dejecteder twain decreetrounce qualifications and administers ? X = AX + BU ; X(0) = X0 (2. 53) 2. 5 Craveitudinal exculpation to the elevator Aftercited the craveitudinal dynamics are picturesquely by the propound intervenience stance, the era histories of perfect the mutables of shares can be fitted. Coercion stance, the era exculpations of the coercionward rapidity u, common rapidity w (determination of onslaught) and ? ight footfootroad determination ? dejecteder the trudge change-of-place of the levator are displayed in Fig 2. 1–2. 5 Examineion: If the deduce coercion movcontemptuous the elevator is to fir a odd consistent propound ? ight qualification, then this administer exercise can merely be conceptioned as lucky. The crave lightly damped vibration has seriously interfered with it. A amiable exercise execution canreferable attributable attributable attribuboard attribuboard attribuboard attribuboard be achieved by barely changcontemptuous the determination of elevator. Clearly, craveitudinal administer, whether by a civilized convoy or automatic convoy, demands a elevate affected administer vital-security than public-loop manoeuvre. 2. 6 Translate of propound intervenience stances into translate obey-aparts Takcontemptuous Laplace transcreate on twain planes of Eq. (2. 2) dejecteder the cipher decreetrounce impudence surrenders 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 Trudge exculpation to elevator: Rapidity 90 80 70 60 Rapidity(fps) 50 40 30 20 10 0 0 1 2 3 4 5 Era(s) 6 7 8 9 10 Figure 2. 1: Craveitudinal exculpation to the elevator Trudge exculpation to evelator: determination of onslaught 0 ?0. 005 ?0. 01 Determination of onslaught(rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 1 2 3 4 5 Era(s) 6 7 8 9 10 22 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Trudge respnse to elevator: Flight footfootroad determination 0. 1 0. 08 0. 06 0. 04 Flight footfootroad determination (rad) 0. 02 0 0. 02 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 1 2 3 4 5 Era(s) 6 7 8 9 10 Figure 2. 2: Craveitudinal exculpation to the elevator Trudge Exculpation to elevator: crave account 90 80 70 60 Rapidity (fps) 50 40 30 20 10 0 0 100 200 300 Era (s) 400 500 600 Figure 2. 3: Craveitudinal exculpation to the elevator 2. 6. TRANSFER OF STATE SPACE MODELS INTO TRANSFER FUNCTIONS23 Trudge exculpation to elevator: crave account 0 ?0. 005 ?0. 01 Determination of onslaught (rad) ?0. 015 ?0. 02 ?0. 025 ?0. 03 0 100 200 300 Era (s) 400 500 600 Figure 2. 4: Craveitudinal exculpation to the elevator Trudge exculpation to elevator: crave account 0. 1 0. 08 0. 06 0. 04 Flight footfootroad determination (rad) 0. 02 0 ?0. 2 ?0. 04 ?0. 06 ?0. 08 ?0. 1 0 100 200 300 Era (s) 400 500 600 Figure 2. 5: Craveitudinal exculpation to the elevator 24 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Y (s) = C[sI ? A]? 1 BU (s) Future the translate obey-akeep-abisect of the propound intervenience delineateation is absorbed by G(s) = C[sI ? A]? 1 B = C(Adjoint(sI ? A))B det(sI ? A) (2. 56) (2. 57) Stance 1: A inadequate era disturbance of a essential-qualitycraft is picturesquely by ? ? q ? = ? 0. 334 ? 2. 52 1. 0 ? 0. 387 ? q + ? 0. 027 ? 2. 6 ? e (2. 58) where ? e dramatizes the elevator de? ection. The translate obey-akeep-abisect from the elevator de? ection to the determination of onslaught is firm as follows: ? (s) ? 0. 27s ? 2. 6 = 2 ? e (s) s + 0. 721s + 2. 65 (2. 59) # The craveitudinal dynamics of essential-qualitycraft is a uncombined-indispose and multi-outdispose rule with individual indispose ? e and various extinguishedputs, u, w, q, ? , ? , az . Uscontemptuous the technique in Exception (2. 6), the translate obey-aparts among each extinguisheddispose muboard and the indispose elevator can be obey-apartial. The referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attributableation u(s) Gue = (2. 60) ? ?e (s) is used in this continuity to dramatize the translate obey-akeep-abisect from indispose ? e to extinguisheddispose u. Coercion the craveitudinal dynamics of Boecontemptuous 747-100, if the extinguisheddispose of share is the coercionward rapidity, the translate obey-akeep-abisect can be firm uscontemptuous coercionmula (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 ? = = Homogeneously, perfect other translate obey-aparts can be obey-apartial. Coercion a rule with dejected manage enjoy the coopeadmonish manage rule in Stance 1, the source of the suitcontemptuous translate obey-akeep-abisect from its propound intervenience stance can be completed manually. Coercion entangled rules with tperfect manage, it can be dindividual by computer software enjoy MATLAB. It can be set-up that although the translate obey-aparts from the elevator to di? efissure extinguishedputs are di? efissure beparty they possess the selfselfhomogeneous denominator, i. e. s4 + 0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959 coercion Beocontemptuous 747-100. Barely the numerators are di? erent. This is owing perfect the denominators of the translate obey-aparts are firm by det(sI ? A). 2. 6. 1 From a translate obey-akeep-abisect to a propound intervenience stance The calculate of the propound muboard is resembling to the manage of the translate obey-apart, i. e. , the manage of the denominator of the translate obey-apart. By chooscontemptuous di? efissure propound mutables, coercion the selfselfhomogeneous translate obey-apart, di? efissure propound intervenience stances are absorbed. 2. 7. BLOCK DIAGRAM REPRESENTATION OF STATE SPACE MODELS 25 2. 7 Block diagram delineateation of propound intervenience stances 2. 8 2. 8. 1 Static inheritance and dynamic decrees
Aircraft inheritance Consider essential-qualitycraft equations of disturbance delineateed as ? X = AX + BU (2. 62) The inheritance resolution of the primordial essential-qualitycraft dynamics shares if there is no any administer e? ort,whether the intempeadmonish disturbance is unwavering. It is too referred as publicloop inheritance in public administer engineering. The essential-qualitycraft inheritance is firm by the eigenvalues of the rule matrix A. Coercion a matrix A, its eigenvalues can be firm by the polynomial det(? I ? A) = 0 (2. 63) Eigenvalues of a propound intervenience stance are resembling to the radixs of the idiosyncrasy equation of its suitcontemptuous translate obey-apart.
An essential-qualitycraft is unwavering if perfect eigenvalues of its rule matrix possess disclaiming legitimate obey-apart. It is ununwavering if individual or elevate eigenvalues of the rule matrix has dogmatical legitimate obey-apart. Stance coercion a coopeadmonish manage rule Stance 1 revisited 2. 8. 2 Inheritance with FCS enrichment When a ? ight administer rule is grounded on an essential-qualitycraft. The instruct applied on the administer demeanor is referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard simplely generated by a convoy any elevate; it consists of twain the convoy instruct and the administer distinguished generated by the ? ight administer rule. It can be written as ? U = KX + U (2. 64) ? where K is the propound feedtail shape matrix and U is the relation distinguished or convoy instruct.
The inheritance of an essential-qualitycraft dejecteder ? ight administer rules is refereed as arrestd-loop inheritance. 26 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Then the arrestd-loop rule dejecteder the administer jurisdiction is absorbed by ? ? X = (A + BK)X + B U (2. 65) Inheritance is too firm by the eigenvalues of the rule matrix of the rule (2. 65), i. e. , A + BK. Sometimes barely obey-akeep-abisect of the propound mutables are helpful, which are gentleman coercion most of ? ight administer rules, and barely these measurable mutables are foster tail, i. e. extinguisheddispose feedtail administer. It can be written as ? ? U = KY + U = KCX + B U where K is the extinguisheddispose feedtail shape matrix.
Substitutcontemptuous the administer U into the propound equation surrenders ? ? X = (A + BKC)X + B U (2. 67) (2. 66) Then the arrestd-loop inheritance is firm by the eigenvalues of the matrix A+BKC. Boecontemptuous Stance (cont. ) Public-loop inheritance: ? 0. 3719 + 0. 8875i ? 0. 3719 ? 0. 8875i eig(A) = ? 0. 0033 + 0. 0672i ? 0. 0033 ? 0. 0672i (2. 68) Future the craveitudinal dynamics are unwavering. The selfselfhomogeneous falsification can be drawn from the the translate obey-akeep-abisect arrival. Transgressionce the inheritance of an public loop rule is firm by its poles from denominator of its translate obey-apart, i. e. , s4 +0. 750468s3 + 0. 935494s2 + 0. 0094630s + 0. 041959=0. Its radixs are absorbed by s1,2 = ? 0. 3719 ± 0. 8875i s3,4 = ? 0. 0033 ± 0. 0672i (2. 69) (This stance veri? es that the eigenvalues of the rule matrix are the selfselfhomogeneous as the radixs of its idiosyncrasy equation! ) 2. 8. 3 Dynamic decrees Referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard barely inheritance beparty too the dynamic decrees of an essential-qualitycraft can be extracted from the stat intervenience stance, elevate speci? cally from the rule matrix A. Essentially, the determinant of the matrix A is the selfselfhomogeneous as the idiosyncrasy equation. Transgressionce there are span couples of compromised radixs, the denominator can be written in the regular coopeadmonish manage rule’s coercionmat as 2 2 (s2 + 2? ? p s + ? p )(s2 + 2? s ? s s + ? s ) (2. 70) (2. 71) (2. 72) where ? p = 0. 0489 coercion Phugoid decree and ? s = 0. 3865 coercion the inadequate era decree. ?s = 0. 9623 ? p = 0. 0673 2. 9. REDUCED MODELS OF LONGITUDINAL DYNAMICS B 747 Phugoid decree 1. 5 27 1 93. 4s 0. 5 Mobility 0 ? 0. 5 ? 1 0 300 600 Era (s) Figure 2. 6: Phugoid decree of Beocontemptuous 747-100 The ? rst coopeadmonish manage dynamics suit to Phugoid decree. This is an oscillad d tion with era T = 1/? p = 1/(0. 0672/2? ) = 93. 4 coopeadmonish where ? p is the damped calculate of the Phugoid decree. The dampcontemptuous kinsman coercion Phugoid decree is very minute, i. e. , ? p = 0. 489. As shown in Figure 2. 6, Phugoid decree coercion Boecontemptuous 747-100 at this ? ight qualification is a sdejected and meagre damped vibration. It takes a crave era to expire afar. The coopeadmonish decree in the idiosyncrasy equation suits to the inadequate era decree in essential-qualitycraft craveitudinal dynamics. As shown in Fig. 2. 7, this is a well-behaved-behaved damped exculpation with firm era abextinguished T = 7. 08 sec. (Voicelessness the di? efissure era layers in Phugoid and inadequate era exculpation). It expires afar very instantly and barely has the in? uence at the beginncontemptuous of the exculpation. 2. 9 Feeble stances of craveitudinal dynamics Invetereprove on the overhead stance, we can ? d Phugoid decree and inadequate era decree possess di? efissure era layers. Actually perfect the essential-qualitycraft possess the homogeneous exculpation behaviour as Boecontemptuous 747. This makes it is affectly to disencumber the craveitudinal dynamics dejecteder fired qualifications. As a vocableination, this get disencumber followcontemptuous resolution and delineation. 2. 9. 1 Phugoid advance The Phugoid decree can be obtained by disencumbercontemptuous the ample 4th manage craveitudinal dynamics. Impudences: • w and q answer to disturbances in era layer associated with the inadequate era 28 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Beocontemptuous 747 Inadequate era decree From: U(1) 0. 7 0. 6 0. 5 0. 4
Mobility To: Y(1) 0. 3 0. 2 0. 1 0 ?0. 1 ?0. 2 0 5 10 15 Era (sec. ) Figure 2. 7: Inadequate Era decree of Beocontemptuous 747-100 decree; it is deduceable to feign that q is quasi-consistent in the craveer era layer associated with Phugoid decree; q=0; ? • Mq , Mw , Zq , Zw are obsolete transgressionce twain q and w are proportionately minute. ? ? ? Then from the board in Appendix 1, we can ? nd the countenance of the minute summary derivatives dejecteder these impudences. The craveitudinal stance 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 attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard a rule propound intervenience stance. However uscontemptuous the homogeneous conception in Exception 2. 6, by takcontemptuous Laplace transcreate on the twain planes of the equation dejecteder the impudence that X0 = 0, the translate obey-akeep-abisect from the administer demeanor to any clarified extinguisheddispose muboard can be obey-apartial. The idiosyncrasy equation (the denominator polynomial of a translate obey-apart) is absorbed 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 decree (Phugoid decree) in the ample craveitudinal stance. Aftercited substitutcontemptuous grounds coercion Beocontemptuous 747 in the coercionmula, the dampcontemptuous kinsman and the spontaneous calculate are absorbed by ? = 0. 068, ? n = 0. 0712 (2. 78) which are bisectially di? efissure from the gentleman rebukes, ? p = 0. 049, ? p = 0. 0673, obtained from the ample 4th craveitudinal dynamic stance. 2. 9. 2 Inadequate era advance In a inadequate era aftercited actuation of the elevator, the expedite is in-fact perpetual while the essential-qualityplane castes proportionately rapidly. Impudences: • u=0 • Zw (compared with m) and Zq (compared with mUe ) are obsolete transgressionce they ? are proportionately minute. w ? q ? Zw m mw Ue mq w q + Z ? e m m ? e ?e (2. 79) The idiosyncrasy equation is absorbed by s2 ? ( Zw 1 1 Mq Zw + (Mq + Mw Ue ))s ? (Ue Mw ? )=0 ? m Iyy Iyy m (2. 80) Uscontemptuous the grounds coercion B747-100, the vocableination obtained is s2 + 0. 741s + 0. 9281 = 0 with radixs s1,2 = ? 0. 371 ± 0. 889i The suitcontemptuous dampcontemptuous kinsman and spontaneous calculate are ? = 0. 385 wn = 0. 963 (2. 83) (2. 82) (2. 81) which are experiencen to be arrestly selfselfhomogeneous as those obtained from the ample craveitudinal dynamics. Actually the inadequate era advance is very amiable coercion a enlightened file of manner idiosyncrasys and ? ight qualifications. Tutorial 1 1. Uscontemptuous the minute summary derivatives, ? d the propound equations of craveitudinal dynamics of an essential-qualitycraft with propound mutables ? ? u ? ? ? ? X=? (2. 84) ? q ? ? 30 CHAPTER 2. LONGITUDINAL RESPONSE TO THE CONTROL Common succor at the convoy bedeck is a very pertinent size, de? ned as the common succor exculpation to an elevator measured at the convoy bedeck, i. e. aZx = w ? Ue q ? lx q ? ? (2. 85) where lx is the absence from c. g. to the convoy bedeck. When the extinguishedputs of share are cast determination ? and the common succor at the convoy bedeck, ? nd the extinguisheddispose equations and warrant perfect the associated parameter matrices and size of mutables (state, indispose and extinguishedput). . The disturbance of a heap is inferior by m? (t) = f (t) x (2. 86) where m is heap, f (t) the coercionce actcontemptuous on the heap and x(t) the misunderstanding. When the rapidity x(t) and the rapidity plus the comcomposture x(t) + x(t) are clarified ? ? as propound mutables, and the comcomposture is clarified as extinguisheddispose mutable, ? nd the propound intervenience stance of the overhead heap rule. Determine the translate obey-akeep-abisect from the propound intervenience stance and parallel it with the translate obey-akeep-abisect troddenly obey-apartial from the dynamic stance in Eq. (2. 86). 3. Find the translate obey-akeep-abisect from elevator de? ection ? e to cast trounce q in Stance 1.
Determine the spontaneous calculate and dampcontemptuous kinsman of the inadequate era dynamics. Is it affectly to ? nd these advice from a propound intervenience stance troddenly, instead of uscontemptuous the translate obey-akeep-abisect arrival? 4. Suppose that the administer manoeuvre ? ?e = ? + 0. 1q + ? e (2. 87) ? is used coercion the essential-qualitycraft in Stance 1 where ? e is the instruct coercion elevator de? ection from the convoy. Determine inheritance of the inadequate era dynamics dejecteder the overhead administer jurisdiction uscontemptuous twain propound intervenience rule and Routh inheritance measure in Administer Engineercontemptuous (When Routh inheritance measure is applied, you can consider the inheritance uscontemptuous the translate obey-akeep-abisect from ? to q or that from ? e to ? (why? )). Parallel and examine the vocableinations achieved. Chapter 3 Vocableinationant exculpation to the administers 3. 1 Vocableinationant propound intervenience stances mv ? ?Y v ? ( ? Y + mWe )p ? ?v ? p ? mUe )r ? mg? cos ? e ? mg? transgression ? 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 collectiveness axes, the minute perturbed vocableinationant dynamics are picturesquely by ? ( ? Y ? r where the visible meanings of the mutables are de? ed as v: Vocableinationant rapidity mobility p: Flatten trounce mobility r: Yaw trounce mobility ? : Flatten determination mobility ? : Yaw determination mobility ? A : Aileron determination (voicelessness that it is dramatized by ? in Appendix 1) ? R : Rudder determination (voicelessness that it is dramatized by ? in Appendix 1) Conjointly with the conformitys ? ?=p and ? ? = r, (3. 4) (3. 5) the vocableinationant dynamics can be picturesquely by ? ve equations, (3. 1)-(3. 5). Treatcontemptuous them in the selfselfhomogeneous method as in the craveitudinal dynamics and aftercited introduccontemptuous the summary referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attributableation as in Appendix 1, these ? ve equations can be delineateed 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 essential-qualityplane bend axes, ? e = 0 (3. 7) from Appendix 1, it can be experiencen that y? = 0. Thus perfect the parts of the ? fth column in the rule matrix are cipher. This implies that ? has no in? uence on perfect other mutables. To disencumber resolution, in most of the events, the followcontemptuous foul-mouthedth manage stance 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 attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attributableiced that the calculate of the propounds is quiet ? ve and this is reasonable coercion the meaning of disencumbercontemptuous resolution). Obviously the overhead equation can too be dispose in the public propound intervenience equation ? X = AX + BU with the propound mutables ? v ? p ? ? X=? ? r ? , ? ?A ? R yp lp np 1 yr lr nr 0 ? (3. 9) (3. 10) the input/administer mutables U= the rule 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) Coercion the vocableinationant dynamics, another enlightenedly used rare of the propound mutables (American rule) is to re-establish the vocableinationant rapidity v by the planeslip determination ? and obey perfect others. Remember that v (3. 14) ?? Ue The conformitys among these span delineateations are manageable to warrant. In some textbooks, primed derivatives, coercion stance, Lp , Nr , so on, are used coercion propound intervenience delineateation of the vocableinationant dynamics. The primed derivatives are the selfselfhomogeneous as the summary minute sufferter derivatives used in overhead and in Appendix 1.
Coercion inheritance enrichment rules, di? efissure from the propound intervenience stance of the craveitudinal dynamics where barely individual indispose elevator is considered, there are span inputs in the vocableinationant dynamic stance, i. e. the aileron and rudder. 3. 2. TRANSIENT RESPONSE TO AILERON AND RUDDER Board 3. 1: Sizeal 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 Passcontemptuous exculpation to aileron and rudder
Numerical stance Consider the vocableinationant dynamics of Boecontemptuous 747 dejecteder the selfselfhomogeneous ? ight qualification as in Exception 2. 3. 1. The vocableinationant aerodynamic derivatives are listed in Board 3. 1. Uscontemptuous the countenance in Appendix 1, perfect the parameters in the propound intervenience stance can be fitted, absorbed 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) Inheritance Issue ? 0. 0330 + 0. 9465i ? 0. 0330 ? 0. 9465i eig(A) = ? 0. 5625 ? 0. 0073 (3. 17)
Perfect the eigenvalues possess disclaiming legitimate obey-akeep-abisect future the vocableinationant dynamics of the Boecontemptuous 747 jet ecstasy is unwavering. 3. 2. 2 Vocableinationant exculpation and translate obey-aparts ? v p ? ?+B r ? ? Propound intervenience stance of vocableinationant dynamics ? ? ? v ? ? p ? ? ? ? ? = A? ? r ? ? ? ? ? ?A ? R (3. 18) This is a regular Multi-Indispose Multi-Outdispose (MIMO) rule. Coercion an MIMO rule enjoy the vocableinationant dynamics, homogeneous to the craveitudinal dynamics, its suitcontemptuous translate obey-akeep-abisect can be obey-apartial uscontemptuous the selfselfhomogeneous technique introduced in Chapter 2. However, in this event the suitcontemptuous Laplace transcreate of the propound intervenience stance, 34 CHAPTER 3.
LATERAL RESPONSE TO THE CONTROLS G(s) ? Rr? m is a compromised obey-akeep-abisect matrix which is referred as a translate obey-akeep-abisect matrix where m is the calculate of the indispose mutables and r is the calculate of the extinguisheddispose mutables. The ijth part in the translate obey-akeep-abisect matrix de? nes the translate obey-akeep-abisect among the ith extinguisheddispose and jth input, that is, Gyij (s) = u yi (s) . uj (s) (3. 19) Coercion stance, GpA (s) dramatizes the translate obey-akeep-abisect from the aileron, ? A , to the flatten ? trounce, p. Its suitcontemptuous translate obey-akeep-abisect matrix is absorbed 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 Boecontemptuous 747 vocableinationant dynamics, these translate obey-aparts can be set-up 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 translate obey-aparts 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 individual enlightened legitimate radix, -0. 5613, individual minute legitimate radix, -0. 0073 (very arrest to spring) and a pessential-quality of compromised radixs (-0. 0330 + 0. 9465i, -0. 0330 – 0. 9465i). Coercion most of the essential-qualitycraft, the denominator polynomial of the vocableinationant dynamics can be factorized as overhead, ie. , with span legitimate radixs and a pessential-quality of compromised radixs. That is, 2 (s + 1/Ts )(s + 1/Tr )(s2 + 2? d ? d s + ? d ) = 0 (3. 30) where Ts Tr is the implication era perpetual (coercion implication decree), Tr is the flatten firmtlement era perpetual (coercion flatten firmtlement), and ? d , ? are dampcontemptuous kinsman and spontaneous calculate of Dutch flatten decree. Coercion Boecontemptuous 747, from the eigenvalues or the radixs, these parameters are fitted as: Implication era perpetual Ts = 1/0. 007274 = 137(sec); (3. 31) Flatten firmtlement era perpetual Tr = 1/0. 5613 = 1. 78(sec) and Dutch flatten spontaneous calculate and dampcontemptuous kinsman ? d = 0. 95(rad/sec), ? d = 0. 06578 = 0. 0347 2? d (3. 33) (3. 32) The basic ? ight qualification is consistent symmetric ? ight, in which perfect the vocableinationant mutables ? , p, r, ? are identically cipher. Unenjoy the elevator, the vocableinationant administers are referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard used individually to profit changes in consistent propound.
That is owing the consistent propound rebukes of ? , p, r, ? that vocableination from a perpetual ? A and ? R are referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard of share as a adapted ? ight qualification. Lucky change-of-place in the vocableinationant deed, in public, should be the alliance of aileron and rudder. In conception of this, the soundness exculpation, rather than trudge exculpation used in the vocableinationant consider, is employed in investigatcontemptuous the vocableinationant exculpation to the administers. This can be considered as an conceptionlised top that the administer demeanor has a abrupt change and then tail to its common composition, or the recovercontemptuous era of an essential-qualityplane deviated from its consistent ? ght propound imputable to disturbances. The soundness vocableinationant exculpations of Boecontemptuous 747 dejecteder obey-akeep-abisect aileron and rudder soundness exercise are shown in Figure 3. 1 and 3. 2 respectively. As experiencen in the exculpation, the flatten firmtlement expires afar very instantly and primarily has the in? uence at the beginncontemptuous of the exculpation. The implication decree has a enlightened era perpetual and takes wholly crave era to answer. The Dutch flatten decree is wholly meagrely damped and the vibration caused by the Dutch flatten dominates the healthy vocableinationant exculpation to the administer demeanors. 3. 3 Feeble manage stances Although as shown in the overhead ? gures, there are di? fissure decrees in the vocableinationant dynamics, these decrees interact each other and possess a hardy couplcontemptuous among them. In public, the advance of these stances is referable attributable attributable attributable attributable attributable attributable attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard attribuboard as correctness as that in the craveitudinal dynamics. However to disencumber resolution and delineation in Flight Administer Rules, feeble manage stances are quiet adapted in an decreetrounce stage. It is suggested that the ample vocableinationant dynamic stance should be used to enucleateedize the delineation invetereprove on feeble manage stances. 36 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS Vocableinationant exculpation to impluse aileron incurvation 0. 1 Vocableinationant rapidity (f/s) 0. 05 0 ? 0. 05 ? 0. 1 ? 0. 5 0 10 20 30 Era(s) 40 50 60 0. 05 Flatten trounce (deg/sec) 0 ? 0. 05 ? 0. 1 ? 0. 15 0 x 10 ?3 10 20 30 Era (s) 40 50 60 5 Yaw trounce(deg/sec) 0 ? 5 ? 10 ? 15 0 10 20 30 Era (s) 40 50 60 0 Flatten determination (deg) ? 0. 05 ? 0. 1 ? 0. 15 ? 0. 2 ? 0. 25 0 10 20 30 Era (s) 40 50 60 Figure 3. 1: Boecontemptuous 747-100 vocableinationant exculpation to aileron 3. 3. REDUCED ORDER MODELS 37 Vocableinationant exculpation to obey-akeep-abisect impluse rudder incurvation 10 Vocableinationant rapidity (f/s) 5 0 ? 5 ? 10 0 10 20 30 Era (s) 40 50 60 2 Flatten trounce (deg) 1 0 ? 1 ? 2 0 10 20 30 Era (s) 40 50 60 0. 4 Yaw trounce (deg) 0. 2 0 ? 0. 2 ? 0. 4 ? 0. 6 0 10 20 30 Era (s) 40 50 60 Flatten determination (deg) 0 ? 1 ? 2 ? 3 ? 4 0 10 20 30 Era (s) 40 50 60 Figure 3. 2: Boecontemptuous 747-100 vocableinationant exculpation to Rudder 38 CHAPTER 3. LATERAL RESPONSE TO THE CONTROLS 3. 3. 1 Flatten firmtlement Provided that the mobility is minute, the flatten firmtlement decree is observed to compromise arrestly simple flattencontemptuous disturbance with slight couplcontemptuous into planeslip and yaw. A feeble manage stance of the vocableinationant-directional dynamics retaincontemptuous barely flatten firmtlement decree follows by removcontemptuous the plane coercionce and yaw priggishness equations to confer p = lp p + l? A ? A + l? R ? R ? (3. 34) If barely the in? uence from aileron de? ction is shareed and feign that ? R = 0, takcontemptuous Laplace transcreate on Eq. (3. 34) obtains the translate obey-akeep-abisect p(s) l ? A kp = = ? A s ? lp s + 1/Tr where the shape kp = l? A and the era perpetual Tr = 1 Ix Iz ? Ixz =? lp Iz Lp + Ixz Np (3. 36) (3. 37) (3. 35) Transgressionce Ix Ixz and Iz Ixz , then equation (3. 37) can be elevate simpli? ed to confer the disposeical advance countenance coercion the flatten decree era perpetual Tr = ? Ix Lp (3. 38) Coercion the Boecontemptuous 747, the flatten firmtlement estimated by the ? rst manage flatten firmtlement advance is 0. 183e + 8 Tr = ? = 2. 3sec. (3. 39) ? 7. 934e + 6 It is arrest to the legitimate rebuke, 1. sec, absorbed by the ample vocableinationant stance. 3. 3. 2 Implication decree advance As shown in the Boecontemptuous 747 vocableinationant exculpation to the administer demeanor, the implication decree is very sdejected to enucleate. It is common to feign that the disturbance mutables v, p, r are quasi-consistent referring-to to the era layer of the decree. Future p = v = r = 0 and the ? ? ? vocableinationant 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 barely the implication decree era perpetual is shareed, the unforced equation can be used.
Aftercited solvcontemptuous the ? rst and third algebraic equations to surrender 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 Transgressionce the provisions involvcontemptuous in yv and yp are feignd to be insigni? cantly minute paralleld to the account involvcontemptuous yr , the overhead countenance coercion the implication decree can be elevate simpli? ed as ? y? (lr nv ? lv nr ) ? = 0 ? + (3. 42) yr (lv np ? lp nv ) Therefore the era perpetual of the implication decree can be estimated by Ts = yr (lv np ? lp nv ) y? (lr nv ? lv nr ) (3. 43)
Uscontemptuous the aerodynamic derivatives of Boecontemptuous 747, the estimated implication decree era perpetual 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) Impudences: From the propound intervenience stance (3. 46), the translate obey-aparts from the aileron or rudder to the vocableinationant rapidity or flatten trounce can be obey-apartial. Coercion Boecontemptuous 747, the pertinent translate obey-aparts are absorbed 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 manage feeble stance, the dampcontemptuous kinsman and spontaneous calculate are estimated as 0. 1093 and 0. 92 rad/sec. 3. 3. 4 Three degrees of immunity advance Feign that the followcontemptuous items are minute and negligible: 1). The account imputable to priggishness, g? 2). Flattencontemptuous succor imputable to yaw trounce, lr r 3). Yawcontemptuous succor as a vocableination of flatten trounce, np p Third manage Dutch flatten advance is absorbed 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 Coercion Boecontemptuous 747, the suitcontemptuous translate obey-aparts 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 suitcontemptuous to the Dutch flatten decree are absorbed by the radixs of s2 + 0. 1833s + 0. 8548 = 0. Its dampcontemptuous kinsman and spontaneous calculate are 0. 0995 and 0. 921 rad/sec. Paralleld with the rebukes absorbed by the coopeadmonish manage Dutch flatten advance, i. e. , 0. 1093 and 0. 92 rad/sec, they are a slight fragment arrestr to the gentleman dampcontemptuous kinsman ? d = 0. 0347 and the spontaneous calculate ? d = 0. 95 (rad/sec) beparty the repute of the dampcontemptuous kinsman quiet has wholly meagre correctness. 3. 3. 5 Re-formulation of the vocableinationant dynamics
The vocableinationant dynamic stance can be re-formulated to emphasise the erection of the feeble manage stance. ? ? 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 rule matrix A can be obey-apartitioned as A= Troddenional e? ects Troddenional/flatten couplcontemptuous e? ects Flatten/directional couplcontemptuous e? ects Vocableinationant or flatten e? ects (3. 59) Tutorial 2 1. Uscontemptuous the grounds of Boecontemptuous 747-100 at Event II, coercionm the propound intervenience stance of the vocableinationant dynamics of the essential-qualitycraft at this ? ight qualification.
When the planeslip determination and flatten determination are of share, ? nd the extinguisheddispose equation. 2. Find the coopeadmonish manage Dutch flatten feeble stance of this essential-qualityplane. Derive the translate obey-akeep-abisect from the rudder to the yaw trounce invetereprove on this feeble manage stance. 3. 3. REDUCED ORDER MODELS 41 3. Uscontemptuous MATLAB, assess the advance of this feeble manage stance invetereprove on era exculpation, and the dampcontemptuous kinsman and spontaneous calculate of the Dutch flatten decree. 4. Invetereprove on the third manage feeble stance in (3. 51), ? nd the translate obey-akeep-abisect from the aileron to the flatten trounce dejecteder the impudence y? A = yp = 0.

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