INTRODUCTION TO CONTROL SYSTEM PERFORMANCE MEASUREMENTS
CONTENTS 1 INTRODUCTION 1 1 1 PART 1 KINEMATICS DYNAMICS AND NONLINEAR CONTROL OF THE MOBILE AIR DEFENCE SYSTEM SUBJECT TO HOLONOMIC AND NONHOLONOMIC VELOCITY CONSTRAINTS 1 1 2 PART 2 PERFORMANCE EVALUATION OF THE MOBILE AIR DEFENCE SYSTEM AGAINST AN ATTACKING AERIAL TARGET 10 2 OVERVIEW OF THE MOBILE AIR DEFENCE SYSTEM 15 2 1 MULTIBODY SYSTEM REPRESENTATION OF THE MOBILE AIR DEFENCE SYSTEM 15 2 2 VEHICLE BODY BODY 5 19 2 2 1 ELECTRIC MOTORS EMC0 EMAS AND EMDV 21 2 3 WHEELS 1 2 3 AND 4 BODY 1 TO BODY 4 23 2 3 1 VELOCITY CONSTRAINTS 25 2 4 TURRET BODY 6 27 2 4 1 ELECTRIC MOTOR EMF0 28 2 5 ANTI AIRCRAFT GUN BODY 7 29 2 6 DRIVE SYSTEM AND DIFFERENTIAL GEARBOX BODY 8 32 2 6 1 VELOCITY CONSTRAINT 33 2 7 STEERING SYSTEM BODY 9 34 2 7 1 GEOMETRIC CONSTRAINTS 36 2 7 2 VELOCITY CONSTRAINTS 38 2 8 ROTOR OF THE TURRET ELECTRIC MOTOR EMC0 BODY 10 39 2 8 1 VELOCITY CONSTRAINT 40 2 9 ROTOR OF THE AA GUN ELECTRIC MOTOR EMF0 BODY 11 41 2 9 1 VELOCITY CONSTRAINT 42 2 10 DE NITION OF GENERALIZED CO ORDINATES Q AND GENERALIZED VELOCITIES P 43 2 11 INERTIAL AZIMUTH AND ELEVATION ANGLES OF A VECTOR 43 2 12 LINE OF SIGHT VECTOR FROM THE MOBILE ADS TO THE AAT 44 2 13 AIMING VECTOR OF THE AA GUN 47 2 14 INERTIAL POSITION AND VELOCITY OF THE AA GUN MUZZLE 48 3 KINEMATIC MODEL OF THE MOBILE AIR DEFENCE SYSTEM 51 3 1 VELOCITY CONSTRAINTS OF THE MOBILE ADS 51 3 2 KINEMATIC MODEL OF THE MOBILE ADS 56 3 3 DECOUPLED KINEMATIC MODELS OF THE VEHICLE SYSTEM AND OF THE TURRET AND AA GUN SYSTEM 60 4 DYNAMIC MODEL AND NONLINEAR CONTROL OF THE MOBILE AIR DEFENCE SYSTEM 63 4 1 BASIC DYNAMIC MODEL OF THE MOBILE ADS 64 4 2 REDUCED DYNAMIC MODEL OF THE MOBILE ADS 71 4 3 NONLINEAR FEEDBACK CONTROL OF THE REDUCED DYNAMIC MODEL OF THE MOBILE ADS 72 4 3 1 ANALYSIS OF THE ZERO DYNAMICS 75 4 4 CONSTRAINED MOTION OF THE CONTROLLED DYNAMIC MODEL OF THE MOBILE ADS 78 4 5 GENERALIZED CONSTRAINT FORCES 80 4 6 LAGRANGE MULTIPLIERS 81 5 OPERATIONAL MODES OF THE MOBILE AIR DEFENCE SYSTEM 83 5 1 ASYMPTOTIC TRACKING RESULTS 83 5 2 VEHICLE BODY TRACKING MODE VM1 VEHICLE BODY IS MANEUVERING 86 5 3 VEHICLE BODY TRACKING MODE VM1A VEHICLE BODY IS MOVING IN A STRAIGHT LINE 87 5 4 VEHICLE BODY TRACKING MODE VM2 VEHICLE BODY IS STATIONARY 88 5 5 AA GUN TRACKING MODE GM1 AA GUN IS ROTATING RELATIVE TO THE VEHICLE BODY 88 5 6 AA GUN TRACKING MODE GM1A AA GUN AIMING VECTOR IS TRACKING THE LOS VECTOR 89 5 7 AA GUN TRACKING MODE GM1B AA GUN AIMING VECTOR IS TRACKING THE FIRE CONTROL VECTOR 90 5 8 AA GUN TRACKING MODE GM2 AA GUN IS NOT ROTATING RELATIVE TO THE VEHICLE BODY 91 5 9 MAIN OPERATIONAL MODES OF THE MOBILE ADS 91 5 10 OPERATIONAL MODE OM1 MOBILE ADS IS MANEUVERING 92 5 11 OPERATIONAL MODE OM1A VEHICLE BODY IS MOVING IN A STRAIGHT LINE VM1A AND THE AA GUN IS ROTATING RELATIVE TO THE VEHICLE BODY GM1 92 5 12 OPERATIONAL MODE OM1B VEHICLE BODY IS MOVING IN A STRAIGHT LINE VM1A AND THE AA GUN IS NOT ROTATING RELATIVE TO THE VEHICLE BODY GM2 92 5 13 OPERATIONAL MODE OM2 VEHICLE BODY IS STATIONARY VM2 AND THE AA GUN IS ROTATING RELATIVE TO THE VEHICLE BODY GM1 93 5 14 OPERATIONAL MODE OM3 VEHICLE BODY IS STATIONARY VM2 AND THE AA GUN IS NOT ROTATING RELATIVE TO THE VEHICLE BODY GM2 93 5 15 FIRING RATE OF THE AA GUN AND FIRING TIMES OF THE AA PROJECTILES 94 6 POINT MASS FLIGHT DYNAMICS MODEL OF THE ANTI AIRCRAFT PROJECTILE 97 6 1 INERTIAL POSITION AND VELOCITY OF THE AA PROJECTILE 98 6 2 POINT MASS FLIGHT DYNAMICS MODEL OF THE AA PROJECTILE 102 6 3 PARAMETER VALUES FOR THE ANTI AIRCRAFT PROJECTILE 106 6 4 PARAMETER VALUES FOR THE MOBILE ADS 108 6 5 COMPUTATION OF THE AA PROJECTILE TRAJECTORY WITH NO WIND 111 6 6 COMPUTATION OF THE AA PROJECTILE TRAJECTORY WITH A CROSS WIND 118 7 THE FIRE CONTROL PROBLEM 121 7 1 FIRE CONTROL PROBLEM FCA 122 7 2 FIRE CONTROL PROBLEM FORMULATION USING FEASIBLE CONTROL 131 7 3 AIR DEFENCE SYSTEM DEPLOYMENT AND ATTACKING AERIAL TARGET ENGAGEMENT SCENARIO 135 7 4 COMPUTATIONAL RESULTS FOR FIRE CONTROL PROBLEM FCA VEHICLE BODY OF MOBILE ADS IS COMPLETELY STATIONARY 138 7 5 COMPUTATIONAL RESULTS FOR FIRE CONTROL PROBLEM FCA VEHICLE BODY OF MOBILE ADS IS MOVING IN A STRAIGHT LINE AT CONSTANT SPEED 143 8 COMPUTATION OF THE IMPACT POINT OF THE AA PROJECTILE ON THE BODY OF THE ATTACKING AERIAL TARGET 153 8 1 GEOMETRY OF THE ATTACKING AERIAL TARGET 153 8 2 INERTIAL TRAJECTORY OF THE ATTACKING AERIAL TARGET 156 8 3 COMPUTATION OF THE IMPACT POINT OF THE AA PROJECTILE ON THE BODY OF THE ATTACKING AERIAL TARGET 157 8 4 VULNERABILITY MODEL OF THE ATTACKING AERIAL TARGET 162 8 5 COMPUTATIONAL RESULTS 162 9 COMPUTATION OF THE PROBABILITY THAT THE AA PROJECTILE WILL IMPACT THE BODY OF THE ATTACKING AERIAL TARGET 169 9 1 STOCHASTIC MODEL OF THE DISPERSION OF THE AA PROJECTILES FIRED BY THE AA GUN 170 9 1 1 REMARKS ON APPLICATIONS OF STOCHASTIC OPTIMAL CONTROL 178 9 2 COMPUTATION OF THE PROBABILITY THAT THE AA PROJECTILE WILL IMPACT THE BODY OF THE ATTACKING AERIAL TARGET PHITT K 180 9 3 COMPUTATIONAL METHOD NM1 FOR THE COMPUTATION OF PHITT K 186 9 4 COMPUTATIONAL METHOD NM2 FOR THE COMPUTATION OF PHITT K 189 9 5 PROBABILITY OF DESTROYING THE AAT FOR THE CASE WHERE A BURST OF AA PROJECTILES IS FIRED 192 9 6 ACCUMULATIVE PROBABILITY OF DESTROYING THE AAT FOR THE CASE WHERE A BURST OF AA PROJECTILES IS FIRED 194 9 7 COMPUTATIONAL RESULTS 196 9 8 VERI CATION METHOD VMB FOR THE COMPUTATION OF THE PROBABILITY PHITT K 199 9 8 1 COMPUTATIONAL RESULTS 219