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THE APPLICATION OF SINGULAR PERTURBATION METHODS TO OPTIMAL CONTROL PROBLEMS IN FLIGHT MECHANICS BY YUNG-NAN HU A DISSERTATION _ Submitted to Michigan State Uinversity in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and Systems Science 1986 ABSTRACT THE APPLICATION OF SINGULAR PERTURBATION METHODS TO OPTIMAL CONTROL PROBLEKS IN FLIGHT MECHANICS BY YUNG-NAN HU The use of singular perturbation methods in performance Optimization problems in flight mechanics is investigated. The thesis addresses three fundamental issues: modeling of flight mechanics models in the singularly perturbed form. boundary-layer instability associated with the use of open- loop control, and the steering control problem of moving the state of the system from a given initial state to a given final state while minimizing a cost functional. A normalization scheme for identifying the time-scale properties of flight mechanics models is presented. Time-scale properties must be identified before solutions can be obtained using the singular perturbation method. It is shown that this new scheme can rationalize identifying the flight vehicle's dynamic equations in a singularly perturbed form. The use of singular perturbation methods in airplane performance optimization is evaluated. The evaluation is based on a study of the minimum time interception problem using r-a aerodynamic and propulsion data as a base line. Emphasis is placed on the boundary-layer instability problem for real-time, auto-pilot implementation. A feedback stabilization scheme is YUNG-NAN HU proposed to circumvent this instability problem. In order to steer the state of a singularly perturbed system from a given initial state to a given final state, while minimizing a cost functional, a composite control strategy is developed. The composite control comprises three components: a reduced control and two boundary-layer controls. The boundary-layer controls do not optimize cost functionals. It is shown that application of this composite control results in a final state which is 0(6) close to the desired state. Moreover, the cost under the composite control is 0(a) close to the optimal cost of the reduced control problem. Particular attention is given to the minimum time-to-climb problem of an aircraft in a vertical plane. ACKNOWLEDGEMENTS I want to express my deep gratitudes to Dr. Hassan Khalil, my thesis advisor, for all his patience, guidance, and encourgement during the course of my research. His expert advice, useful insight, and stimulating discussions made this work possible. I would like to thank my committee members, Dr. R. Schlueter, Dr. R. O. Barr, Dr. L. Tummala and Dr. R. 0. Hill for their help and valuable suggestions. To my wife who has constantly supported me with her love through my research period, I would like to express my deep appreciation. TABLE OF CONTENTS LIST OF SYMBOLS ............................................... LIST OF TABLES ................................................ LIST OF FIGURES ............................................... I. INTRODUCTION .............................................. II. SINCULARLY PERTURBED MODELS IN FLIGHT MECHANICS .......... 11.1 Historical Survey ....................................... 11.2 Development.of the Seventh-order Aircraft Model ---------- 11.3 Singularly Perturbed Model .............................. 11.3.1 Normalization Scheme and Comparison with Ardema's Results and Kelley's Assumptions ...................... 11.3.2 Energy State Modeling ................................. 11.3.3 A Seventh-order Singularly Perturbed Model ............ 11.3.4 A Fourth-order Singularly Perturbed Model for Steady Level Turning Problem and Comparison with Calise's Results ..... , ............................ 11.4 Concluding Remarks ...................................... III. SINGULAR PERTURBATIONS 1N FLIGHT MECHANICS .............. 111.1 Use of Singular Perturbations to Approximate Optimal Trajectories ........................................... 111.1.1‘ Singularly Perturbed Trajectory Optimization ......... ii 111.1.2 Illustrative Example (A Minimun Time-to-climb Problem) ............................................. 52 111.2 Auto-pilot Implementation .............................. 6O 111.2.1 Boundary-layer Instability Problem ................... 52 111.2.2 Feedback Stabilization ............................... 53 IV. STEERING CONTROL OF SINGULARLY PERTURBED SYSTEMS: A COMPOSITE CONTROL APPROACH ............................... 7] 1V.l Problem Statement and Composite Control Approach ........ 7] 1V.2 Derivation of the Composite Control ..................... 73 IV.2.l The Reduced Control ................................... 73 IV.2.2 Boundary-layer Stabilizing Control .................... 77 IV.2.3 Right boundary-layer Control .......................... 79 IV.3 Asymptotic Validity of the Composite Control ............ 82 IV.4 Numerical Examples ...................................... 100 V. APPLICATION OF THE COMPOSITE CONTROL TO MANEUVERS IN A VERTICAL PLANE ............................. 109 V.l Composite Control ........................................ 109 v.2 Application of the Composite Control to the ' Minimun Time-to-climb Problem ............................ 115 VI. CONCLUSIONS .............................................. 125 REFERENCES .................................................... 127 LIST OF SYMBOLS Only basic conventional symbols are listed. Other symbols used in the monograph are defined during the presentation of the material. c Specific fuel consumption C Drag coefficient C Zeroth-lift drag coefficient Do CL Lift coefficient CL“ Lift curve slope D Drag force D Drag due to lift divided by weight Do Zero-lift drag divided by weight E Specific energy g Acceleration of the gravity h Altitude H Hamiltonian function J Performance index; Jacobian matrix 1'v L Lift force m Mass of the vehicle M Mach no. n Load factor r Radial distance from center of the earth; turning radius 5 Reference area t time, see u Control T Thrust V Speed w Weight of vehicle X, Y, Z Cartesian coordinate x, y, z Cartesian coordinate W “1 Q $9- Angle of attack Thrust angle of attack Bank angle Flight path angle ‘ Longitude; dimensionless time Costate function Difference between thrust and drag per unit weight Density of atmosphere, slug/fts Initial time stretch transformation Terminal time stretch transformation Latitude Heading angle Angular velocity Angular velocity Small ”parastic" parameter -1 Induced drag parameter, rad Linear feedback stabilization control Nonlinear feedback stabilization control Azimuth angle vi EQBEQBIEIS app Approximate solution by singular perturbation methods (SPT) N Normalizing reference data max Normalizing reference data (maximum reference data); maximum value typ Normalizing reference data (typical reference data); region of interest ° Initial condition f Terminal condition SHREBSQEIEIE ° Reduced (slow) solution ~ Normalized variable 11 Inner (left) boundary layer ir Inner (right) boundary layer vii Table Table Table Table Table Table Table Table LIST OF TABLES Estimates of state variables’speeds for F-AC aircraft by using the proposed normalizat- ion scheme ....................................... Estimates of state variables’speeds for F-AC aircraft by Ardema's two methods ................. Estimates of state variables’3peeds for F-hC aircraft by using the proposed normalizat- ion scheme with small bank angle ................. Estimates of state variables’speeds for Missile 11 in steady level turning flight by using the proposed normali- zation scheme .................................... Boundary conditions and parameters for Example 4.1 ..................................... Comparison of exact and composite control solutions ................................ The target point (x(l), 2(1)) for e-.1, .05 and .01 ................................ The simulation results of minimum time-to-climb with composite con- trol for airplane 2 under various boundary conditions viii page 24 26 27 43 102 105 108 120 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure LIST OF FIGURES Coordinate systems .............................. Coordinate systems with aerodynamic forces .......................................... Aerodynamic and propulsive forces --------------- Equilibrium of force in a coordinate turn ............................................ Energy time histories for airplane 2 ............ Velocity time histories for airplane 2 .......... Flight path angle time histories for airplane 2 ...................................... Flight trajectory for airplane 2 ................ Approximate control (capp) for airplane 2 ....... Boundary-layer instability for airplane 2 ....... Boundary-layer stabilization (with linear feedback stabilization control) for airplane 2 ...................................... Boundary-layer stabilization (with linear feedback stabilization control) with initial disturbance for airplane 2 .............. Boundary-layer stabilization (with nonlinear feedback stabilization control) for airplane 2 ...................................... ix 12 15 40 S8 58 59 59 50 63 65 66 69 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure .10 Boundary-layer stabilization (with nonlinear feedback stabilization control) with initial disturbance for airplane 2 Interception geometry in the horizontal plane Range time history (with composite control) for a value of e-.l Azimuth angle and flight path angle time histories (with composite control) for a value of ¢-.l Composite control time histories State trajectory (with composite control) for a value of ¢-.l Energy time history for airplane 2 with composite control Velocity time whistory for airplane 2 with composite comtrol Flight path angle time history for airplane 2 with composite control Flight trajectory for airplane 2 with composite control Comparison of the approximate control and composite control Flight trajectory of Case 5 under modified version (change terminal boundary conditi- ons) for airplane 2 with composite control .. Flight trajectory of Case 8 under modified version (change initial boundary conditions) for airplane 2 with composite control Flight trajectory for airplane l with composite control Flight trajectory for airplane 1 by Bryson's energy-state approximation OOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO OOOOOOOOOOOOOO OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO page 70 100 103 104 104 107 117 117 118 118 119 121 122 124 124 I . INTRODUCTION Many problems in science and technology require choosing the best (or the optimal) solution among all the possible solutions. In this second half of our present century one of the most challenging and fascinating optimization problems is the analysis of optimal space trajectories. It consists of finding the best trajectory, in some sense, for the motion of a vehicle in a three-dimensional space. A general optimization problem in three-dimensional atmospheric flight is a difficult problem to solve. Realistic description of physical plants to be controlled usually result in high—order mathematical models. A straightforward application of the maximum principle always leads to a two-point boundary value problem involving several arbitrary parameters. Optimal control design for high-order systems is computationally cumbersome not only because of high-order but also because such systems invariably exhibit simultaneous slow and fast dynamics which are described by "stiff" differential equations. Many of the quasi-steady-state approximations made in the analysis of transport aircraft and other low-performance vehicles are not valid for the highly dynamic maneuvers typical of high-performance military aircraft. Up to a recent time, to display explicity the characteristics of the optimal controls, the different optimization problems considered were reduced order problems. A reduced order model involves less variables and renders the solution to the problem more manageable. Any high order problem would require pure numerical technique for its solution and the results obtained were restricted to a particular set of end conditions for a specified aircraft model. Experience in actual flights, as well as comparison between various solutions in the analysis of the optimal control problem considered, often display the fact that the improvement in performance is minimal when the exact optimal trajectory is compared with a suboptimal one obtained by a simple analysis. A simple analysis, if properly carried.out. has the added advantage that the resulting solution obtained is close to the optimal solution and hence can be used as a first guess reference solution in any iterative procedure. A simple analysis for a complex problem can be obtained in various ways depending on the physical characteristics of the problem, but the ultimate objective is always to reduce the order of the problem. If, in a problem, a certain variable x varies slowly, then the steady-state approximation dx/dt-O will provide an equilibrium relation which can be used to eliminate one component of the state vector or one component of the control vector. A more sophisticated approximation would involve a combination of different state variables and the elimination of variables that are insensitive to the optimization process. One such efficient technique is the singular perturbation method and it is the subject of analysis in this thesis. Singularly perturbed systems and, more generally, multi-time-scale systems, often occur naturally due to the presence of small "parastic" parameters, typically small time constants, masses, etc., multiplying time derivatives or, in more disguised form, due to the presence of large feedback gains and weak coupling. The chief purpose of singular perturbation approach to analysis and design is the alleviation of the high dimensionality and ill-conditioning resulting from the interaction of slow and fast dynamics. The multi-time-scale approach is asymptotic, that is, exact in the limit as the ratio e of the speeds of the slow versus the fast dynamics tends to zero. When e is small, approximations are obtained from reduced-order models in separate time scales. While singular perturbation methods, a traditional tool of fluid dynamics and nonlinear mechanics, embraces a wide variety of dynamic phenomena possessing slow and fast modes, its assimilation in control theory is recent and rapidly developing. The methods of singular perturbations for initial and boundary value problem approximations and stability were already largely established in the 1960s, when they first became a means for simplified computation of optimal trajectories. Singular perturbation methods also proved useful for the analysis of high-gain feedback systems and the interpretation of other model order reduction techniques. More recently they have been applied to modeling and control of dynamic networks and certain classes of large-scale systems. This versatility of singular perturbation methods is due to their use of time-scale properties which are common to both linear and nonlinear dynamic systems. The motivation for this thesis has been to deal with trajectory optimization of singularly perturbed systems in atmospheric flight. Time-scale properties must be identified before solutions can be obtained by using singular perturbation methods. The first objective of the present effort is to develop a systematic procedure to obtain a singularly perturbed models in flight mechanics. Auto-pilot implementation of the approximate open loop control obtained using singular perturbations may cause boundary-layer instability when unstable modes are present in the uncontrolled system. The second objective of this thesis is to demonstrate this fact and emphasis is placed on deriving a feedback stabilization scheme to circumvent this instability problem. For the sake of steering the state of a singularly perturbed system from a given initial state to a given final state while minimizing a cost functional, a composite control approach is developed. The composite control is calculated using reduced-order models in different time scales. Thus, a great reduction in the on-board computations is achieved. A systematic procedure for the identification of time-scale properties of a nonlinear flight mechanics model is first introduced in Chapter 11. A normalization scheme is developed; based on this scheme a singularly perturbed flight mechanics model is proposed. This scheme does not require that an "exact" optimal trajectory to be known, the dynamic state equations and the normalizing reference data are the only information required. The application of singular perturbation methods to trajectory optimization problems in flight mechanics is presented in Chapter III. The use of singular perturbation methods for airplane performance optimization is applied to.a minimum time-to-climb problem. In this chapter, attention is focused on the boundary-layer instability problem for on-line, auto-pilot implementation. A feedback stabilization scheme is proposed to circumvent this boundary-layer instability problem. In Chapter IV, we develop a composite control approach to steer the state of a singularly perturbed system from a given initial state to a given final state; while minimizing a cost functional. The composite control comprises three components; a reduced control and two boundary-layer control components. The boundary-layer controls do not optimize cost functional as in earlier work. Asymptotic validity of the composite control is established by showing that its application to the singularly perturbed systems results in a final state which is O(¢) close to the desired state, and the cost under this composite control is 0(a) close to the optimal cost of the reduced control problem. Finally, in Chapter V, we apply the composite control strategy to the optimal maneuvers of an aircraft in a vertical plane. The objective of this chapter is to demonstrate the performance of the composite control on a typical problem of interest, namely, the minimum time- to-climb problem. In this chapter, we present a systematic procedure for the identification time-scale properties of a nonlinear flight mechanics model. A normalization scheme is developed in order to improve the methods currently in use. Based on this scheme, a singularly perturbed flight mechanics model is proposed. The model agrees, generally, with previous time-scale studies of flight mechanics models. 11.1 HISTORICAL SURVEY Many authors have discussed and illustrated the application of singular perturbation methods [1-7] to the solution of high performance trajectory optimization problems in flight mechanics. The principal advantage cited is that they reduced the order of individual integrations, so the computational burden is significantly reduced. However, several authors like Kelley [2] and Washburn, et al.[7] have observed that there is presently no rigorous and practical method that can cast this COMPIOX nonlinear trajectory optimization problem in a singularly perturbed form. For linear systems, analysis of ting-seal. separation has been discussed by Chow and Kokotovic [8], and by Syrcos and Sannuti [9]. The time-scale analysis of linear systems can be applied to nonlinear systems but the determined properties will only be valid locally. Moreover, linear analysis assumes that an optimal trajectory of the ”exact” system is known (about which the linearization is to be performed). For nonlinear systems, Kelley [2] has considered transformations of state variables that reduce. system's coupling and expose time-scale characteristics. The transformations involved, however, are given by partial differential equations, making this approach generally impractical for complex systems. Because of the difficulties in the above approaches, almost all singular perturbation analyses of aircraft trajectory optimization have relied on ad hoc methods for the selection of time-scales, based on physical insight and past experience. This procedure has been termed ”forced singular perturbations” by Shinar [10]. In order to improve the ad hoc methods, Ardema and Rajan [11] have proposed two methods for the time-scale separation analysis. Both methods require knowledge of the state equations, bounds on the state and control variables and what control problems are of interest. We develop a new normalization scheme that can rationalize identifying the flight vehicle's dynamic equations in a time-scale separation form, using only normalizing reference data. 11.2 DEVELOPMENT OF THE SEVENTH-ORDER AIRCRAFT MODEL In this section, we give a brief account of the derivation of the equations of motion that are essential in the study of singularly perturbed models in the next section. We follows Vinh's book [12], where more details can be found. The motion of a vehicle considered as a point mass flying over a sphreical, rotating earth, is defined by [12-14] fflt) - position vector (2.1a) ‘V(t) - velocity vector (Z-lb) m(t) - mass (2.16) II. SINGULARLY PERTURBED MODELS IN FLIGHT MECHANICS The total force of the flight vehicle is F-Ti-Tmrm‘g‘ (2.2) where T is thrusting force, A is aerodynamic force and m? is gravitational force. The aerodynamic force can be decomposed into a drag force D opposite to the velocity vector V and a lift force ‘1: orthogonal to it. By Newton's second law, with respect to an inertial system, we obtain the vector equation m__ -3? (2.3) In writing equation (2.3) it is implicitly assumed that the rate of change of mass with time is negligible. This assumption is not explicitly stated in [12]. As it will be seen later on, it is justified since m is much slower than V. Consider a fixed coordinate system ox,Y,z, and another system oxyz which is rotating with respect to the fixed system with angular velocity w. Let B be any arbitrary vector with components Bx, By, and Bz along the rotating axes. Then, the time derivative of B, taken with respect to the fixed system is dB dB dB -‘ -* “ x -. 4‘ z -* di dj dk -dt 111.1—3.3+ k+s +3 +3 (2.4) ale» By Possion's formula, the last three terms on the right-hand side of (2.4) are df' df' JE' __. a. Bx 1?:— + By dt + 32 —dt (0 x B (2.5) The first three terms on the right-hand side o£(2.4) can be interpreted as the time derivative of the vector B.if the vector IZ-j; and k were constant unit vectors. Hence, it is the time derivative of B with respect to the rotating system oxyz. We denote it by 53‘ de -*- dB -*- de -* TF'TIt—1+'EELj+—ci?_k <2-6> and write (2.4) as TIF'-'E'+"”‘B (2.7) This is the formula for transforming the time derivative from fixed system to rotating system. The inertial reference frame OX1YIZ1 is taken such that O is the center of the gravitational field of the spherical earth and the OXIY1 plane is the equatorial plane. The OXYZ reference is fixed with respect to the earth with OZ coinciding with 021. It is assumed that the earth is rotating with constant angular velocity'fi’directed along the Z-axis. 10 The vector equation (2.3) is written with respect to the inertial frame. In deriving the equations of motion, we should use the earth-fixed axes OXYZ as the reference frame since it is a convenient base to follow the motion of the vehicle. Hence, putting B -'?'in (2.7) and taking its derivative, we have a? a“? .- 2. Tc ““3: ““‘r (2.8) and dV 3? 1 a? _. . F-;?-+2wx-a—t—+wx(wxa (2.9) where 5'13 constant The vector equation (2.3) now becomes r A A r A -‘ m—r-F-2mwa-mwxuox'fi (2.10) at t For convenience, we change the notation for the time derivative and write this equation as m-g--F-2ni3xV-m?3x(3x?) (2.11) 11 In this equation, V.is the velocity of the vehicle with respect to the earth-fixed axes and the time derivative is taken with respect to these axes . Now, (2.11) and the kinematic equation d? _aE_ _ V (2.12) constitute the vector equations for the position vector'f’and the velocity vector V: They are equivalent to six scalar equations, three of which are the kinematic equations and the other three are force equations. With respect to the earth fixed system OXYZ. (Fig.2.l) , the position vector‘f’is defined by its magnitude r, its longitude 0, measured from X-axis, in the equatorial plane, positively eastward, and it latitude d, measured from the equatorial plane, along a meridian positively northward. 211 2 Y ‘3 I, V g A T . ‘Y o - I”, ' \{ ¢ Q \\ .M \ I g Fig.2.]: Coordinate systems 12 Let 1 be the angle between the local horizontal plane, that is the plane passing through the vehicle located at the point M and orthogonal to the position vector‘?) and the velocity vector V: The angle 1 is termed the flight path angle and is positive when V is above the horizontal plane (Fig.2.2). let p be the angle between the local parallel of the latitude and the projection of Vion the horizontal plane (Fig.2.2) . The angle u is termed the heading angle and is measured positively in the right handed direction about the X-axis. VERTICAL 'LAN‘ nomzoursL PLAN! Fig.2.2 Coordinate systems with aerodynamic forces Let.i:‘3: and k be the unit vectors along the axes of a rotating system oxyz such that the x-axis is along the position vector (Fig.2.2 ). We then have s- :1 (2.13) and i?- (VSINy)-i.+ (VCOSyCOSll’)? + (vcohsmwp)? (2.14) 13 We resolve all the vector terms in (2.11) and (2.12) into components along the rotating axes oxyz. In order to take the time derivative of the vectors‘? and V.in (2.11) and (2.12) with respect to the earth-fixed system OXYZ using their components along the rotating system oxyz, we need to evaluate the angular velocity a-of the rotating axes. The system Oxyz is obtained from the system OXYZ by a rotation 0 about the positive Z-axis, followed by a rotation ¢ about the negative yeaxis. Hence, the angular velocity a'of the rotating system Oxyz is given by n - (SIN¢-§%—)1 - (-—§%—)j + (2.40) dt VN VmCOS1 Slhfiu where SINfl is normalized as SINfl/SINfiN. The relative speed in equation (2.36e) will become speed of fig - SINfi (2.41) speed of V “N N Taking the same normalizing reference data as given in (2.38) and o (2.39), choosing the small bank angle as 3N - 5 - .087radian for both normalizing reference data and substituting into (2.36), (2.41) respectively, the results are shown in the Table 2.3 Speed Variable 8y maximum reference data By typical reference data m . .0108 .0052 X , Y .0291 .0197 E .4320 .4759 . v .5229 -3922 v 1.0000 ‘-°°°° h 1.4208 1.9660 1 7 5.0000 4-5000 Table 2,3 Estimates of state variables’speeds for F-4C aircraft by using the proposed normalization scheme with small bank angle 28 Both of the normalizing reference data show that the heading angle ¢ becomes slower when 5 is small.'which is in agreement with Kelley's assumption [1]. 11.3.2 ENERGY STATE MODELING The use of energy as a state variable in place of altitude or velocity has been behind many applications of singular perturbation techniques to guidance and control (Kelley [17], Ardema [4]). The fact that energy is a slower variable compared with altitude or velocity can be explained using singular perturbation arguments. This was done in Kokotovic, Khalil and O'Reilly [19] and will be recalled here. Consider the case when B is so small (i.e. for flight in a vertical plane) that m, X, Y and p are much slower than the other variables. Then the dynamics of h, V and 1 can be described by the third-order model .31;— - VSIN1 (2.42:0 -%¥— - g(eA - SIN1) (2-42b) ~33- - 1.1-...” (2.42.) where 6A is the difference of thrust minus drag per unit weight; both L and A are function of h, V and a. The system (2.42) has an equilibrium manifold at e - 0, defined by 1 - 0 and L -WW. The manifold is l-dimensional. This indicates that the system has a slow variable which 29 is constant at e - 0 in the r-scale. A constant quantity at e - 0 in r-scale is provided by the fact that without thrust and drag the energy is conserved. Thus, multiplying (2.42a) by g and (2.42b) by V and adding them together, we get at e - 0, for all r z 0 and all h, V and 1 db dV 1 2 8 ‘EE' + V '32- ' 0. gh + —f- V - constant (2,43) ‘ 2 Hence as our slow variable, we take E - gh + -%— V and obtain in t-scale, where t - er -§%- - gAV (2.44a) c g: - g(eA - SIN1) (2.44b) .91. .5. L 244 t dt - V (1T'-COS1) ( . C) Rewrite (2.44) as dE V dV _ ¢(T - o) _ 6 1t— -————m 831111 (2 . 45b) d1 _ L - WCOS1 6 -EE- my (2.45C) This energy modeling confirms that using energy in place of either h (or V) as a state variable yields a standard singularly perturbed 30 model. Equation (2.45) will be used in Chapter 111. 11.3.3 A SEVENTH-ORDER SINCULARLY PERTURBED MODEL We have seen that the normalization scheme of Section 11.3.1 produces measures of speeds of variables that are in agreement with past experience and practice. In particular, we have seen that the slowness of X and Y relative to V can be attributed to the smallness of (hN/RN) which is typical. The slowness of m relative to V is due to the smallness of (N and nNSINfiN, respectively. These findings will now be used to write the equations of motion in the singularly perturbed form. The seventh-order model will comprise the state variables of the original model (2.31) except for h which is replaced by E. The singular perturbation parameters are taken as CTNhN - 2.46 ‘1 WOVN ( a) .2 - $3. (2.46b) £3 - nNsmpN (2.46c) e‘ - (N (2.46d) In view of our discussion above, and the numerical results of Section 11.3.1, it is clear that :1, e2 << 63, e, 31 Morever, depending on the smallness of EN, :3 may be much smaller than c.. In some situations, :1 may be much small than £2. The seventh-order model takes the singularly perturbed form of equation (2.47) below. fig. — -.,(2..c,)1 (2.47a) dt -9§— - 62(2pc1)VCOS1COS¢ (2.47b) dt -2;- - e2(2uc1)VCOS1SIN¢ (2-47C) dt dt mVCOS1 N dE 2 0 ~ —d—:- - £‘(T-f-Lp c1)—:—g (2.478) t m s .‘f .21. - c,( ~ - SIN1) (2.47f) dt m .2}. - nN¢1(.E§g§E. - .EQEI.) (2.473) dt mV VnN 2 where c1 - gtN/VN and p - VN /2ghN Notice that this singularly perturbed form is expressed in the fastest time scale and not the slowest time scale as it is customary in the literature. The fact that a model takes the singularly perturbed form 32 8 - f(x, 2, u), X 6 R“, u c Rn ez - g(x, z, u), 2 f Rm or if’ - ef(x, z, u), are R“, u e Rn g: - g(x, z, u), z e Rm does not automatically mean that the system has a two-time scale (2.48s) (2.48b) (2.49s) (2.49b) property. Additional assumptions are needed to ensure the two-time scale property. Let us discuss these assumptions for the linear system x - A11): + A122+ Blu Upon setting 6 - 0, (2.50) reduces to X - A11x+ A122 '1' Bl“ (2.50s) (2.50b) (2.51s) (2.51b) A typical assumption in the singular perturbation literature, e.g. Kokotovic, Khalil and O'Reilly [19], is to require A22 to be nonsingular. This assumption guarantees that (2.51) represents a well-defined nth-order reduced model. It also guarantees that the system 33 (2.50) has a two-time scale property in the sense that its n+m eigenvalues cluster into n eigenvalues of order 0(1) and m eigenvalues of order 0(1/4). If A22 is singular but [A22 Ba] has rank m, the equation (2.51) will still yield a well-defined nth-order reduced model by interchanging the roles of some components of z and u. This point will be illustrated in Chapter IV. The eigenvalues of (2.50) will not have m eigenvalues of order 0(l/c) in this case, but the use of feedback from 2 with coefficients of order 0(1) can locate m eigenvalues at locations of order 0(l/e), see Khalil [35]. In summary, we can say that the linear model (2.50) represents a standard singularly perturbed model if A,, is nonsingular or if [A22 B2] has rank m and 2 can be measured for feedback. The nonlinear model (2.48) will represent a standard singularly perturbed model if its linearization about every point, along a certain trajectory or in a certain set, satisfies the assumption A22 nonsingular or rank [A2, B2] - m. ' The singularly perturbed nature of the model (2.47) will be studied by linearization. Let us recall that T - T(E, V), D - _%_chovzse-Kh and L - -%-Chapovzse'm. Hence, 1 - T(E, V), E" - f(E, V. a) and ‘1'. - 1(2, ii, a). The lineraization of (2.47) with a and ,8 treated as control variables is 9'9- (1 where r41 51 #41 “as“ t ' to: <1 Consider the determinant P Cease det where a67 - -c1COS1 and + CL§Ve ae7 V «a»: - v2/2g)_§_y_ COS N ave ‘ nNc1'—:—E' '-' m J 1 37 --..-....r---..--..-------_--- ' 54366377 1 8 ) + ‘ 367316 -K(E - v2/2g) m )1 Y V’ E V 7 + Ve'K(E ' v2 N 2>0 V (2.56) 2 v /2g) 3°12: _av 38 0 Thus, for [1 |< 90 and small e, , the determinant (2.56) is positive and the matrix is nonsingular. In other words, (2.52) is in a standard ~ singularly perturbed form, where m, i, Y, E, ¢ are slow variables and V, 1 are fast variables. Neglecting the transinents of V and 1, equation (2.52) can be reduced to the fifth order model ale. ('1 larger than e1, :2, (2.57) Can then be partitioned as 5215 e317 €419 eldl 62d3 ‘2ds ‘sd1 ‘4do ‘idz ‘2d4 ‘2de ‘sds eedio 62324 62334 e216 Gale E4110d '41 F” (2.57) Now, if e, and e4are of the same order (in general it is) and much 39 [a F ; 1 1 a F 2 x : x 2 x [a 7 d ~ 1 .. dt Y I Y + V E 0 6310 V cad, ‘sde B X l - _ EJ : 0 6‘1“, E e‘d, e‘dli 4.. 7 5, ’ (2.58) The matrix A“. is not nonsingular. but [A22 B,] has rank 2; hence equation (2.58) still yields a well-defined third-order reduced model- If e, is much smaller than e‘, we should partition (2.58) with (E, R, Y, ‘6 ) as slow variables and E as a fast variable. Finally, if :1 and :2 are of the same order, then ii, 31, Y are grouped in the same time-scale. If e] is smaller than :2, again we can partition the third—order model of (3, Y, Y) so that E is slower than (it, 1). The above linearization analysis confirms that the model (2.47) is a well-defined singularly perturbed model with (V, 1) as the fastest variables, followed by (V. E) and then by (i, Y, Y) as the slowest variables. This generally agrees with our previous analysis. 40 11.3.4 A FOURTH-ORDER SINCULARLY PERTURBED MODEL FOR STEADY LEVEL TURNING PROBLEM AND COMPARISION WITH CALISE'S RESULTS In this section, we consider a steady-level turning problem for two-dimensional flight in the horizontal plane. In order to maintain the flight in the horizontal plane, the bank angle 5 should satisfy (see Fig;2.4) LCOSfi - v (2.59) Fig.2.4 Equilibrium of force in a coordinate turn The motion can be adequately described by a fourth-order model derived from the seventh-order model (2.31) by assuming constant mass, 4l constant altitude and small flight path angle (1 3 0). Hence equation (2.31c), (2.3le) and (2.3lg) can be dropped yielding the fourth-order model {1%- - vcosw (2.60a) -%-:- - vsmw (2.6%) .91.}. .. 11572.2. (2.60c) 43%- _ LSEQ (2.60d) In Calise's paper [6], he takes D as 2 D - quD + kL /qs (2,61a) o 1 2 q - Tpv (2.61b) 2 2 2 L - L.n + W (2.61c) where Ln is the component of total lift (L) in the horizontal plane, i.e., Ln - LSINB, CD is the zero lift drag coefficient and k is the o lift induced drag coefficient. 'Upon substituting the normalization (2.34) into (2.60), the dimensionless equations of motion are 42 t V .5?. - N N vcosm (2.62a) dt RN ~ t V .91. - N N VSIN¢ (2.62b) dc RN ~ t g; .— dV N N .7 .. __‘7_ .5. (2.62c) dt N m dd gthNSINBN ESINfi _.‘___ .. V . ~~ (2.62d) dt N mVCOS1SINflN The speeds relative to the speed of V are v2 speed of X or Y N ~ - .______. (2.63s) speed of V gRNg’N speed of p _ nNSINfiN (2 63b) speed of V (N Q2m2aria2n_si£h_§alissis_rssults= Let us choose the air-launched Missile 11 as defined in Calise [6] which is a steady level turning problem under the assumptions(of constant altitude, constant mass and small flight path angle 7. The normalizing reference data given by Calise are 43 5 RN - 12.2 x 10 m, VN - 304.8m/sec 2 2 D - 1 pV2 sC + kLN - 939 99k - 25 T N Do 2 ° 3’ nN (2.64) TN - D TN - 227312;, (N - T - 14.67. LN - WonN - 2272.523 2 'VN, o rN - gnN - 379.2m, 8N - 80 other values needed are 2 3 W0 - 90.9kg, s - .016m , p - .876lkg/m (2.65) 2 CDOI- 1.2, k - .02, g-9.8m/sec These data are substituted in (2.62) and the results, given in Table 2.4, show that (X,Y) are slow variables, while (0, V) are much faster than (X, Y). Variable 50820 by normalizing reference data X ' y .00053 V 1.00000 9 1.67827 Table 2.4 Estimates of state variables’speeds for Missile II in steady level turning flight by using the proposed normalization scheme 44 Calise [6] classifies the speeds of variables as follow: (X, Y) are in the same time scale and are the slowest variables, the heading angle p is faster and the velocity V is even much faster than 0. The nondimensional variable equations he obtained are shown below ax 7t— - vcose dY 71?- - vsmup 1'mm at _ _1_._ 11;; T: v 1 ( rmin ) dV _ T - 0 {max RN dt “max 61118.3( where rN - rmin - 3;!- , {max - Tmax /N and rmin is the (2.66s) (2.66b) (2.66c) (2.66d) radius of curvature (see Fig.2.4) which is defined as a horizontal arc (centre C, radius r). Substituting the normalizing reference data given by (2.64) into Calise's nondimensional equation (2.66) yields dX .32. - VCOS¢ dY '3' " VSIW’ (2.67s) (2.67b) (2.670) 45 av .0003 71;- - T - D (2.67d) Equation (2.67) shows that even in Calise's normalization, (p, V) are in the same time scale and they are much faster than (X, Y) which agrees with the conclusion of our normalization. It also shows that Calise's assumption that V is much faster than u is not justified. 11.4 CONCLUDING REMARKS A normalization scheme approach for time scale separation analysis of nonlinear dynamic systems has been proposed. The main point of this approach is the decomposition of a high order, nonlinear complex dynamic flight vehicles systems into several (multiple) lower order systems in an easy and globally valid way. The dynamic state equations and the normalizing reference data are the only information required. This approach was applied to a typical class of flight vehicles dynamic equations. The numerical examples showed that the time scale separation as computed by this approach generally agree with previous practice and assumptions. 111. SINGULAR PERTURBATIONS 1N FLIGHT MECHANICS 46 In this chapter, the application of singular perturbation techniques (SPT) to trajectory optimization problems in flight mechanics is discussed. It is emphasized that auto—pilot implementation of the open loop control, derived using singular perturbation approximation, may cause boundary-layer instability when unsatble modes are present in the uncontrolled system. In other words, real-time implementations generally require the optimal solution to be expressed in a feedback form. The purpose of this chapter is to propose feedback stabilization schemes. Linear and nonlinear feedback stabilization controls are used to circumvent this instability problem. 111.1 USE OF SINGULAR PERTURBATIONS TO APPROXIMATE OPTIMAL TRAJECTORIES In this section, we review the use of singular perturbation techniques to approximate optimal trajectories in flight mechanics problems. As an illustration, we obtain an approximate solution to the aircraft minimum time-to-climb problem. Outer (reduced), boundary-layer (inner) and composite solutions are shown. Originally, the method of singular perturbations was applied in initial value problems like Cole [20], Tihonov [21] and Wasow [22]. It was first introduced into optimal control theory by Kokotovic and Sannuti [35]. The two-point boundary value problems (TPBVP'S) arising in optimal control were investigated by Chow [23] , Freedman and Granoff [24], Vasile'va [25], Wilde and Kokotovic [26]. The singular perturbation method was also applied widely to aircraft and missile performance optimization problems as in Kelley [2, 3, l7], 47 Ardema [4‘ 27], Calise [5, 6, 18, 28], Shiner et a1. [29. 30]. Washburn, et a1. [7] and. Chakvarty [46]. Two point boundary value problems,which arise in the application of optimal control theory to nonlinear control problems in flight mechanics, are known to be of a computational complexity prohibitive practical applications, especally for on-board real-time implementation. For this reason model order reduction concepts, i.e. neglecting fast dynamics which are thought to have small effect on the solution behavior, have received much attention in the past. One of the pioneering methods is "energy state approximation" by Rutowski [15], which has been applied in performance optimization of supersonic aircraft by Bryson ,et al. [16]. The method, however, exhibits undesirable features and may have considerable errors. Considerable effort has been extended in searching for simplification techniques to produce results which are meaningful and attainable at reasonable cost. From the research of the past decade, singular perturbation theory has emerged as the most promising approach to meet the simplification goal. Application of singular perturbations to various performance optimization problems in flight mechanics has been reported by Kelley, Ardema, Calise, etc., in the papers cited above. 48 111.1.1 SINCULARLY PERTURBED TRAJECTORY OPTIMIZATION Consider the singularly perturbed system 3‘ - f(X. 2. t. c. u) (3.1a) 22 - g(x, z, t, e, u) (3.1b) where X and z are n and m dimensional state vectors. and u is an r-dimensional control vector. The small positive parameter c is identified as a time scaling parameter whose presence increases the system order. For e - 0, z ceases to be a state vector and the order of system (3.1) reduces to n. The objective of the design is to find an optimal control u(t) which takes the initial states x(t°) - x0, z(to) - 2° to the final states x(tf) - xf, z(tf) - 2f while minimizing the cost functioanl t f J er V[x(t), z(t), u(t), e, t]dt (3.2) 1:o The functions f, g, and V are assumed to be sufficiently differentiable in all their arguments in an appropriately defined domain. To obtain the necessary conditions for optimal control, we introduce the Hamiltonian 1* T H(x, z, ix, AZ, u, e, c) - - v + Ax f + AZ 3 (3.3) where Ax and )2 are costate variables corresponding to x.and 2 respectively. The maximum principle implies that the costate variables Ax and A: satisfy the equations 49 Ax - - Vx H (3.4a) eAz - - Vz H (3.4b) while 1x(tf) and Az(tf) can not be a priori determined because the terminal states are not free. The maximum principle also implies that along an optimal trajectory vu a - 0 (3.5) Assuming that (3.5) uniquely defines u in terms of X, Z, Xx' and AZ, substitution of u from (3.5) in the state and costate equations (3.1) and (3.4), yields the form x.- f(x, xx, z, 12, e, t) (3-58) 1 - x F(X. XX. 2. AZ. 6. t) (3.6b) c2 - 8(x, XX, 2, AZ, 6, t) (3.6c) €32 ' C(X. Ax' 2. dz. 6. t) (3-5d) Equations (3.6) define a TPBVP of differential order of 2(n+m). We call the solution of (3.6) the "exact solution", and the solution of the system with e set to zero the "reduced" solution- It is obvious that in general the reduced solution will not satisfy both initial and terminal conditions. At least locally, the behavior of the reduced solution will be radically different from that of the exact solution. In fact, 50 the best that can be hoped for is that the reduced solution gives a good approximation for fast variable 2 everywhere except near z(to) - 20 and z(tf) - zf. The phenomenon of boundary-layers occurs in all singular perturbation problems. In such problems, the solution is sought in two (or in some cases, several) separate regions. In an outer region the variables are relatively slowly varying and can be approximated by the reduced solution, which does not generally satisfy all boundary conditions. In an inner(boundary-layer) region, the variables are relatively rapidly varying. They satisfy appropriate boundary condition and converge asymptotically to the reduced solution. W: The solution of the nonlinear two-point boundary values problem (3.6) may be approximated by setting 5 - O in (3.6). This leads to a reduced problem (variable are denote by the superscript °) '0 O O O O x - f(x 9 Ax, 2 v *2: o! t) (3'78) '0 O O O 0 xx- wt, Ax, z , AZ, 0, c) (3.7b) 0 0 0 O o - so: , Ax, z , AZ, 0, c) (3.7c) 0 0 O O o - C(x , Ax, z , AZ, 0, c) (3.7d) 0 O with boundary conditions X (to) ' x0 and x (tf) ' xf 51 The solution of this reduced problem can be an 0(a) approximation for original problem away from the boundary points. but it cannot satisfy the initial and terminal conditions for z. Boundary—layer (geroth-order inner) solution: The zeroth-order initial (left) boundary-layer solution is carried out using a stretched time scale t'to , _ ___?____ (3.8) Substituting (3.8) into (3.6) and again setting c — 0 leads to the following zeroth-order equations (variables in this initial boundary-layer problem are denoted by superscript il) 11 .{E;_ - o , Xil(r) - constant - X0 (3.9a) d‘il 11 dz11 11 11 11 11 11 ‘37"' ' 5 (X0, Ax : z "Az ' O, to) . z (to) ' z0 (3.9c) 11 d) 11 2 Similarly, the zeroth-order terminal (right) boundary-layer equations are formed by introducing the stretching transformation 52 a- __ (3.10) into equation (3.6) and setting 6 - 0, resulting in equations which are similar to the initial boundary-layer equations (3.9) but in the reverse direction (i.e. opposite sign), and with slow variables frozen at their terminal values instead of initial values. The reduced and boundary-layer solutions are combined according to the formula t'to o t't ) - z(co)1+[zu)-—;—5(—->-o (3.13.1) mxy 8V 0 0 E - V r a (3.18b) 55 . A o o o a: 0 1 a L o A - -A Vg(-—- - ( ( a» 3.18 E E as m 6E mV ( c) with the boundary conditions E(to) - E0 and E(tf) - Ef. Since this system is autonomous, H - 0 is a constant of the motion, the relations of (3.18a) and (3.17) lead to o o 0 O 1 and the control law is given by 0 o 1 a - 7 7 . (3.19) 0 0 2(AEnV ) The reduced control (3.19) is the same as the control calculated using the energy state approximation by Bryson et a1. [16]. This follows from the well-posedness of the singularly perturbed control problem, as shown in Section 6.3 of [19]. The left boundary-layer problem is given by 11 a: _ o - (3.20a) 11 dV 11 'E?" ' '331N7 (3.20b) 11 11 11 d L - woosv 'a¥" ' 11 (3.20c) S6 11 dA a _ o (3.21a) 37 11 11 d1 A 11 11 v 11 a 11 11 I - w o ‘37" ' '3‘2 “‘II(V g ) ' m a 11( L 1157 ) (3'21b) av av v 11 d1 . . 11 7 _ i1 11 _ 11 C057 .37—- AV gCOSy A1 -&_;IT_- (3.21C) with initial conditions Eil(ro) - £0, Vi1(ro) - V0 and 111(10) - 10 and the control law is A11 _ 1 (3.22) 2 2011:10,,11 ) il a The right boundary-layer problem is similar to (3.20) and (3.21) but in the reverse direction. The control law is Air air - 1. (3.23) T 20érnvir ) To illustrate the solution, a numerical example is now considered. The aircraft is "airplane 2" of [16]. The boundary conditions are selected as 0 E0 - 1500000 ; Vo - .5 Mach ; 1° - 0 Ef - 5000000 ; Vf - 2.0 Mach ; 7f - free 57 Combining the reduced solution, left boundary-layer and right boundary-layer solutions together, gives the results of energy E, velocity V and flight path angle 1 shown in Figs.3.1-3.3 respectively, the flight trajectory (path) of the various solutions in the (h, V) plane is also shown in Fig.3.4. The minimum time estimate by this SPT approximation is 170 sec which is the time estimate on the slow manifold (reduced solution) 60 sec plus time to dive (left boundary-layer) 60 sec and to zoom (right boundary-layer) 50 sec (see Fig.3.4). It is important to mention here that for this minimum time-to-climb problem, tf is unknown (to be determined). In order to estimate tf, we solve (i.e. integrate) the right boundary-layer in reverse direction until it matches the slow manifold, then tf can be determined. The minimum tf we obtained by this approximation is at about 5% error compared with the exact solution obtained by using steepest descent (162 sec, see [4, 16]). but singular perturbation approximation requires substantially less computational cost. The approximate control which is formed as o 11 t ' to 0 t ' t o aapp(t) - a (t) + [0 (—-¢———-) - 0 (to)] + [air('—f—e—) " 0 (Cf)] is shown in Fig.3.5. \EUKHYoIMWID. 58 m‘ a. ‘ )- ,- i 2 r- o 1 1 L J 0 50 100 150 200 Ifliofln Fig.3.l Energy time histories for airplane 2 an my um 5° “‘ 5° W b- an u: 0.0 l l l l a so too 150 200 ms. sec Fig.3.2 Velocity time histories for airplane 2 mm. 1” F1 59 l.0[ REDUCED SOLUTION FLIGHT PATH ANGLE. rodlan -Lo 1 1 1 , 0 50 100 150 200 nm.ux ' Fig.3.3 Flight path angle time histories for airplane 2 100 r W! )-.5. Mt ). 0 0 «2000 ft (v,.n,) IO )- vxt,)-2.0. u(t,)-00000 Ft w p (Voaho) w >- ”r o 1 1 J 1 1 n l l ,q .I 1.2 1.6 2.0 2J0 2.8 3.2 VELMI". MCI m. Fig.3.4 Flight trajectory for airplane 2 60 20' K - t t't. ° ) - a (t.)l 9 let‘( I e o O‘PP“) - a (c) o [011‘ ‘ ‘ ) - a (ct)! ANGLE or ATTACK, degree 0 *- . U 0 I“ r.— a ——+— C ,V a"- __.. -10 1 1 l I 0 ‘ _50 100 150 200 THE. sec Fig.3.5 Approximate control (aapp) for airplane 2 III.2 AUTO—PILOT IMPLEMENTATION Optimal flight controls are dependent on many factors such as aerodynamics, motor performance, system weight, operational constraints, mission requirements, atmospheric conditions, and the index of performance to be optimized. To establish best possible vehicle performance it is necessary to determine or approximate the optimal control solution. A separate but equally important issue is the real time auto-pilot (on-board) implementation. Recently, a number of flight mechanics optimization problems have been solved using singular perturbation techniques and resulting in 61 state-feedback control laws, suitable for on-board implementation [18, 31, 32]. In the cited work feedback is introduced to reduce on-line computations while improving accuracy. Calise [18] presents a partial evaluation of the use of singular perturbation methods for developing a computer algorithm for on-line optimal control. He expresses the singular perturbation approximation of optimal control in feedback form (near-optimal feedback control). Emphasis is placed on deriving a solution in a form that minimizes on-board computational requirements and improves accuracy. Visser and Shinar [31] introduce a first order correction feedback control law to improve the accuracy of the singular perturbation approximation for real time implementation. Weston J; 31, [32] proposed a feedback control and discusses its auto- pilot implementation. They linearize the fast boundary-layer system about the reduced solution and obtain feedback control for boundary- layer, where feedback coefficients are function of the slow variables. No one investigated the use of feedback for boundary-layer stabilization. We mentioned before that the use of open loop control via singular perturbation approximation for on-line, auto-pilot implementation may cause boundary-layer instability when unstable modes are present in the uncontrolled system. This fact was demonstrated on a second-order example in Wilde and Kokotovic [26]. This instability problem can be overcome by using feedback control to stabilize the boundary-layer system. In this section, we emphasize the role of feedback implementation ixnstabilizing the boundary-layer dynamics and introduce such feedback controls from boundary-layer stabilization viewpoint rather than from 62 near-optimality viewpoint as in earlier work. We propose two feedback stabilization scheme to circumvent this instability problem. III.2.1 BOUNDARY-LAYER INSTABILITY PROBLEM In order to investigate the phenomenon of boundary-layer instability, let us illustrate it by considering the flight mechanics model (3.12), the left boundary-layer equations are dV dr - - gSIN1 - F1 (3.24s) d L - WCOS (1'2" " ""'—"‘"’mv 7 ' F2 (3.2413) The Jacobian matrix. evaluated along the reduced solution, is given by — as, as, - _ T ""‘av T1 0 '8 as, as, L 5" 31 J .8 ° _ where 1 -Kh 1 401 ”In C -Kh xv ) o a "' Wachpose + Wap08V(e W— + be —8 > The chacteristic equation of (3.25) is 2 S + ga - 0 (3-25) The two roots are on the imaginary axis, so, this system does not satisfy the requirement ReA(J)0. Taking k1 - -2 and K1 - 5 for this problem, the simulation result is shown in Fig.3.7. This shows that the linear feedback law indeed stabilizes the boubdary-layer and follows the slow manifold closely. With initial conditions sufficiently close to the slow manifold, the auto-pilot will be able to follow the nominal path over the entire range by using this linear feedback control law. But when the initial 65 conditions are far away from the slow manifold, the trajectory will not follow the nominal path any more. The resulting trajectories. shown in Fig.3.8, illustrate the "local" nature of the linear feedback control. ’F .L..°— K.(v -v°)-K2(r-r°) LINEAR FEEDBACK SOLUTIOMHITH 0L) ----- SPT APPROXIMTE SOLUTION VELOCITY. MCH NO. 5r 8 s ’s‘ g TIE: sec . Fig.3.7 Boundary-layer stabilization (with linear feedback stabilization control) for airplane 2 66 { _ MIMI. PATH I VELOCITY. MACH NO. ‘1r’ 1 1 1 I J I 0 20 ‘40 60 80 100 120 TIE. sec Fig.3.8 Boundary-layer stabilization (with linear feedback stabilization control) with initial disturbance for airplane 2 (b). Non ’ ea ee back ontrol: In order to obtain nonlocal boundary-layer stabilization, we use a nonlinear feedback control law. let us rewrite the flight mechanics model (3.12) as . 2 s - -¥-[T(E, V) - % pV25(CDo(V) + ”ammo. )1 (3.30a) ° 2 2 eV - -‘-§-[I(E, V) - 71— pV s(CDo(V) + "ammo. ] - gSINy (3.30b) a} - ‘21? stchwm - .57. C051 (3.30c) 67 The nonlinear feedback control law is chosen as 2w 2w 2WC081 2Wb'7 a ' z’ ’ 2 + 2 ‘ “"""T‘j N pV sC (V ) pV 50 (V) v sC (V) ”Ssvcm V B La E In p In (3.31) where b. is a positive constant and VB is the energy state approximation solution [see 16] which is the same as the reduced solution V0 we obtained from previous section III.1.2. Again, this control is effective only on the boundary-layer because on the slow a manifold, “N is 0(a) close to a . After substituting (3.31) into (3.30), the boundary-layer equations are given by .gg. - ‘ESINV - 31 (3.32a) d1 _ g, 2 2 , -3;- VVYC (V )[v CLa(V) - VECLa(VE)]- b1 - F2 (3.32b) E In E We assume that vacLa(V) is monotonically increasing (in general it 18): which implies that the first term on the right-hand side of equation (3.3213) is a first-quadrant-third-quadrant function in (V - VE)° Consider a Lyapunov function candidate V-VE 1 2 2 V(V,1)-f [(V +x)C (V +x) -vc (V)]dx 0 (Vs + X)V;CM(VE) s 1.1 a s In a + (I - C051) (3.33) Taking the derivative of v, we have 68 d” I dr - -b1SIN1 < 0 , for -x < 1 < x (3.34) where dy/dr - 0 implies 1 - 0, V - VE' So, by Lasalle's theorem [see 34], the equilibrium point (V - VE’ 1 - 0) is asymptotically stable. Moreover, it was shown by Vasileva [25] and Tihonov [21] that if the domain of interest is bounded and closed, then the property of asymptotic stability for every fixed slow variable, implies the property of asymptotic stability uniformly in the slow variable. 80, this system is asymptotically stable uniformly in slow 0 solution VE (i.e. V ). Therefore, all the conditions of Tihonov theorem [21] are satisfied. The Jacobian matrix of the boundary-layer equation (3.32) is ”as, as,‘ _ __ - 7 av a1 [.0 5 as, as, . ’ L—av Ty _“ “b _ where b is positive constant and acmw) ' 2 E and its characteristic equation is 2 I o S + b S + a g - 0 (3,36) All roots of (3.36) are in the open left-half complex plane. 69 Fig.3.9 shows the simulation of the trajectory when the nonlinear feedback control (3.31) is applied to the full singularly perturbed system (3.30). The simulation shows that the nonlinear feedback control law for on-line, auto-pilot implementation will also stabilize the boundary-layer. Again, tests were performed with initial conditions up to Mach number 1.5. The resulting trajectories are shown in Fig.3.10. These show that the nonlinear feedback control law is able to control the aircraft so that it approaches the neighborhood of the nominal path for initial perturbations larger than those of the linear feedback control. Notice, in particular, the trajectory starting at Mach number 1.5 and compare FigureS3.8 and 3.10. This comparision emphasizes the nonlocal nature of the nonlinear feedback control vs the local nature of the linear one. 011-1110231211! .a—L— -" ." nufifimdu [”cu'Td — “I" m Mlflillfllw ---- 9T MIMI! WHO ’ Ml“. m 00. Fig.3.9 Boundary-layer stabilization (with nonlinear feedback stabilization control) for airplane 2 70 3- NOIIINAL PATH 2 .— 8° 55 , s g 1.s- ------ .>_3 8 ’a—" 3 1-—-—- g 7 SLON MANIFOLD —" ’/ 1 ’I 1 1 1 l 1 I 0 20 00 60 80 100 120 nus. sec Fig.3.10 Boundary-layer stabilization (with nonlinear feedback stabilization control) with initial disturbance for airplane 2 IV. STEERING CONTROL OF SINCULARLY PERTURBED SYSTEMS: A COMPOSITE CONTROL APPROACH 71 In this chapter, we develop a composite control approach to the problem of steering the state of a singularly perturbed system from a given initial state to a given final state, while minimizing a cost functional. Asymptotic validity of the composite control is established by showing that its application to the singularly perturbed system results in a final state which is 0(a) close to the desired state. Moreover, the cost under the composite control is 0(6) close to the optimal cost of the reduced control problem. The performance of the composite control is illustrated by examples. IV.1 PROBLEM STATEMENT AND COMPOSITE CONTROL APPROACH Consider the singularly perturbed system i - f(x, 2, u, e, t) (4.1a) (é - a(x, e, t) + A(x, e, t)z+ B(x, e, t)u + cg1(x, z, u, e,t) (4.1b) where XeRn, zeR , ucRr and e is a small positive parameter. A control u(t) is sought to steer the state x, z from an initial state x(t°) - x0, z(to) - 2° to a terminal state x(tf) - xf and z(tf) - z , while minimizing the cost functional 1: f J -] [V1(x, e, t) + zTV2(x, e, t)z+ uTR(x, e, t)u]dt (4.2) t0 This problem will be studied under the following assumption Assumption 4.1: The functions, f, a, A, B, g,,V,, V2 and R are assumed to be sufficiently smooth in all their arguments, i.e. differentiable a sufficient number of times, in a domain of interest. Furthermore, 72 V1 and V, are positive semidefinite and R is positive definite in the same domain. Other assumptions will be made later on. This optimal control problem has been studied by many researchers; see, for example, O'Malley [36],_Sannuti [37], Chow [23] and Kokotovic, Khalil and O'Reilly [19]. In these studies, asymptotic approximations of the optimal trajectories are obtained via analyzing the singularly perturbed two-point boundary value problem that results from applying the maximum principle. These asymptotic approximations have been extensively used in flight mechanics problems; see, for example, Kelley [3], Ardema [4], Calise [18] and Visser and Shinar [31]. We develop a composite control approach to the steering control problem. Composite control of singularly perturbed systems has been known in the context of stabilizing feedback control. It was first introduced by Chow and Kokotovic [38] for linear systems and later generalized to nonlinear systems by Chow and Kokotovic [39], Suzuki [40] and Saberi and Khalil [41]. According to this approach, a stabilizing feedback control is sought as the sum of two components. The first component is a reduced control that stabilizes the reduced system, obtained by setting 0 - 0 and eliminating fast variables (2 in (4.1)). The second component stabilizes the boundary-layer system. In our steering control problem, the composite control will be sought as the sum of three components. The first one is the reduced control which solves the simplified problem obtained upon setting 8 - 0. The second component is a feedback component that stabilizes the boundary- layer system. The third component is a right boundary-layer component 73 that steerstimafast variable 2 from the reduced solution to the desired terminal state zf. The three components are derived in the next section. The proposed composite control is similar to that of Chow [23] in that it comprises three components, but with different procedures of calculating the boundary-layer components.In our method, the boundary- 1ayer controls ck) not optimize cost functionals as in Chow. The left boundary-layer control is a feedback stabilizing control that ensures boundary-layer stability, while the right boundary-layer control is the well-known minimum energy control that steers the state of the linear boundary-layer system to its target state. Our analysis does not involve asymptotic analysis of the full optimal control. Therefore, our assumptions are weaker than those of Chow. IV.2 DERIVATION OF THE COMPOSITE CONTROL IV.2.1 THE REDUCED CONTROL The reduced (or slow) problem is obtained by setting e - 0 in (4.l)-(4.2) and dropping the requirement z(to) ' zo, z(tf) - zf, that is the reduced problem is defined as ‘0 0 0 0 0 0 x - f(x 9 Z s u 0 00 t) o x (to) -x09 x (tf) - xf (4°38) 0 O 0 0 0 0 - a(x , 0, t) + A(x , 0, t)z + a(x , 0, t)u (4.3b) t o f o o T o 0 o T o o J - [V,(x , 0, t) + (z ) V2(X , 0, t)z + (u ) R(X , 0, t)u ]dt to I (4.4) 74 where the superscript "°" stands for the solution of the reduced problem. For the reduced problem to be well-defined, we must be able to use the algebraic equation (4.3b) to reduced the (n+m)-dimensiona1 o 0 state vector X , Z to an n-dimensional vector. A typical assumption in the singular perturbation literature, e.g., Kokotovic, Khalil and O'Reilly [19], is to require the matrix A(xo, 0, t) to be nonsingular. This assumption, however, is not needed in the asymptotic analysis of the two-point boundary-layer value problem associated with the full problem (4.1) - (4.2). It is also restrictive and eliminates the interesting problems that arise in flight mechanics. Therefore, we make the following weaker assumption. 0 o Assumption 4.2: The mx(m+r) matrix [A(x , 0, t) B(X , 0, t)] has m linearly independent columns in the domain of interest. 0 This assumption implies that rank[A B]-m for all x and t, but not vice-verse. A.weaker assumption would be to require rank[A B]-m, but this would complicate the analysis. Assumption 4.2 guarantees the existence of a permutation matrix P such that the first m 0 columns of [A B]P are linearly independent for all x and t, e.g. 0 0 o o [A201, t) 3201. t)] - [A(x . 0. t) 801.0. t)]P (4.5) o o _o where A,(x , t) is nonsingular. Defining 2. and u by 75 - o1 P -o q -0 z 2 P11 P12 2 - s 9 (4.6) 0 _o _o _u _ _ u _ P21 P22 u . 1 - , the algebraic equation (4 3b) can be rearranged as o 20 o - a + [A B]PPT z - a + (A, 8,] (4.7) o -0 u u Hence, the reduced problem (4.3) - (4.4) can be rewritten as -o — o -o _o o o 0 x-f-xf (4.8a) o o -o o _o 0 - a(x , 0, t) + A,(x, t)z + B,(X , t)u (4.8b) l- o — O O -0 T _O T T J- {V,(x.0.t)+[(z) (11)]? t0 0 0 RO‘ . 0 t) '-o' 2 P )dt (4.9) -o I L“. - o -0 -0 o -0 -0 -o -0 where f(x , z , u , t) - f(x , P11; + P12“ , P212 + P22“ , 0, t),and 0 o A,(x , t) is nonsingular. Note that if A(X , 0, t) is nonsingular to 76 start with, the permutation matrix P is taken to be the identity matrix. Problem (4.8) - (4.9) satisfies the standard assumption of o nonsingularity of A,(X , t). Substitution of -0 ,1 _0 into (4.8a) and (4.9) yields -0 -o o _o o o X - f (X , u , t) , x (to) - xo , x (cf) .xf t 0 U," E o T o _o _o T o o J - [Qo(x , t) + 2Do(x , t)u + (u) Root , t)fi ]dt t0 where -0 o -0 - 0 -1 -0 -0 f(xvuvt)-f(x1 'A2(a+32U),U, t) - -1 - -l .1 Do - B§A2TM11A2 a ‘ M¥2A2 a 1' -1' -1 1' -'r T -1 Ro ' B252 M11A2 B2 ’ 3252 M12 ' M1252 B2 + M22 T T M11 ' P11V2P11 +'P21RP21 T T M12 ' P11V2P12 I P12RP22 (4.10) (4.11) (4.12) (4.13) (4.14s) (4.14b) (4.14c) (4.14d) (4.14e) 77 and 11,, - 9?,V,s,, +9.}, R2,, (4.141) The reduced control problem (4.11), (4.12) is formally correct in the sense that its necessary conditions for optimality coincide with the necessary conditions of the full problem (4.1) - (4.2) upon setting 6 - 0. This fact is shown in section 6.3 of Kokotovic, Khalil and O'Reilly [19] for the reduced problem (4.8) - (4.9), which is the same as the reduced problem (4.11) - (4.12). We assume that Assumption 4.3: The reduced control problem (4.11) - (4.12) has a _o 0 unique optimal solution u (t), X (t), which is continuously o o -0 differentiable on [to, tf]. Once G (t) and X (t) are calculated, 2 (t) 0 can be obtained from (4.10), and the reduced control u (t) is given by 0 -0 -0 u - P212 + s,,., (4.15) IV.2.2 BOUNDARY-LAYER STABILIZING CONTROL A characteristic phenomenon of singularly perturbed systems is the presence of boundary layers during which the fast variable 2 approaches it reduced, or quasi-steady state trajectory. For this phenomenon to take place, the boundary-layer system needs to be asymptotically stable, see Kokotovic, Khalil and O'Reilly [19]. For the system (4.1) this will be the case if the matrix A(X, O, t) is Hurwitz uniformly in xzand t, i.e., ReA[A(X, 0, t)]s-c<0 for all X and t in the domain of interest. 78 If this condition holds, application of the reduced control (4.15) to the system (4.1) will result in trajectories of x, z and u which 0 o o approach X , z and u after a boundary-layer. If A(x, 0, t) is not Hurwitz, feedback must be used to stabilize the boundary-layer system. It is important to notice that without feedback, the boundary-layer will be unstable even when Opvn-loop boundary-layer corrections are 0 . added to u . This fact was demonstrated on a second-order example in Wilde and Kokotovic [26], and on a flight mechanics model in Chapter III. The boundary-layer system is obtained by expressing (4.1b) in the fast time-scale r - (t - t,)/c, t12to, and then setting 8 - 0. It is given by —SE- - a(x, 0, t) + A(X, 0, t)2 + 8(3. 0. t)u (416) o o where u - u + uF.and x, u and t are frozen at their values at t - t1. Notice that the boundary-layer stability should hold along the reduced trajectory and not only at the initial time to. If A(X, 0, t) is not Hurwitz or it is Hurwitz but its stability properties are not adequate, ‘uF can be used to stabilize the system. Let us first shift the equilibrium of (4.16) to the origin. The steady-state of (4.16), with uF.- 0, is 0 o 0 - a(x, O, t) + A(X, 0, t)z + B(x, 0, t)u (4.17) o Substracting (4.17) from (4.16) and setting 2F,- 2 - z , we obtain 79 sz 1;- - 10:, o, t)z, + a(x, o, t)uF (4.18) with X and t treated as fixed parameters. The system (4.18) is a linear system whose stabilizability is stated in the following assumption. Assumption 4.4: The pair [A(X, 0, t) B(X, 0, t)] is stabilizable uniformly in x and t in the domain of interest, e.g., there exists a sufficiently smooth matrix K(x, t) such that ReA(A(x, 0, t) + B(x, O, t)K(x, t)) < -c < 0 (4.19) Thus, the boundaryjlayer stabilizing control is given by uF.- K(x, t)zF - K(x, t)(z - 20) (4.20) IV.2.3 RIGHT BOUNDARY-LAYER CONTROL 0 So far, we have derived the reduced control u and the boundary- 0 layer stabilizing control up. Application of u - u + “F to the system (4.1), (4.1) will result in a trajectory x(t), z(t) that o 0 approaches the reduced state X (t), Z (t) after an 0(eln-%-) boundary- layer and then moves along it. At the terminal time tf, x(tf) and z(tf) o 0 will be in an 0(8) neighborhood of x (tf) - xf and z(tf). Since z°(tf) * 2f. in general, a terminal boundary-layer control ub should be added to 0 u + up" whose function would be to steer z(t) from an 0(8) neighborhood of the desired state 2f. This motion will take place over a time 8O interval [cf-A, tf], A>0. To derive ub, we consider the boundary-layer 0 system (4.16) with u - u + “F'+ ub and with t1 - tf, that is with x, 0 o o 0 z , u and t frozen at X(tf), Z (tf), u (tf) and tf, respectively. Anticipating that x(tf) will be within 0(8) of xf, we freeze X at Xf instead of x(tf). Thus, the right boundary-layer model is given by _%§_ _ a(xf, o, tf) + A(x , o, t)z +-B(x , o, c)[ 0 0 u (cf) + x(xf, cf)(z - z (tf)) + uh] - a(xf, 0, tf) + H(Xf, cf): + B(xf, 0, tf)[uo(tf) - O x) (4.31) IV.3 ASYMPTOTIC VALIDITY OF THE COMPOSITE CONTROL Asymptotic validity of the composite control is established by the following theorem IHEQB£fl_&,1: Suppose that Assumptions 4.1 - 4.5 hold and that the composite control (4.31) is applied to the singularly perturbed system 83 * (4.1). Then, there exsits 8 >0 such that for all 8 6 (0, 6*] x(tf) ' Xf - 0(5) (4.32) z(tf) - zf - 0(8) (4.33) o o J(uc) - J (u ) -O(8) (4.34) The theorem states that application of the composite control (4.31) to the singularly perturbed system (4.1) results in a final state which is 0(6) close to the desired final state and a cost which is 0(8) close to the optimal cost of the reduced control problem. 2129:: Consider the singularly perturbed system (4.1) under the composite control . o x - f(x, 2, u + uF + ub, 8, t) (4.35a) 82 -a (x, 8,t)+A (x, 8, t)Z+B (x, 8, t)(uo +UF+Ub) + 8g1(x, z, u, 8. t) (4.35b) t f J 2,. [V1(x, 8, t) + va,(X, 8, t)z + uTR(X, 8, t)u]dt (4.36) to 0 . where x, z, u are functions of t and u - uc - u + “F‘+ ub is established by (4.31).During the proof we will need to use the property 84 that over the time interval (to, cf] the full solution (X(t),z(t),u(t)) exists and is bounded. This follows from the existence of the reduced solution (Assumption 4.3) and the closeness of the full and reduced solution for sufficiently small 8. This closeness, however, is to be established in the midest of the proof. Therefore, we cannot start by assuming the boundness of the full solution. To circumvent this problem we take a large enough compact set S; defined by S - {x,z,u| “x" sr, "zllsr, |u| 5r) and let T - minttf, first exist time from S). All analysis will be done over the time interval [to, T] for which the full solution is bounded. Then we will show that r can be chosen large enough such that, for sufficiently small 8, the first exist time from S will be greater than tf. Since f and g, are continuous function of x, z, and.u, this implies that f and g, are bounded by a constant which is function of r. Let us introduce the quantities o 0 8a - a(x, 8, t) - a(x , 0, t), 6,A - A(x, 0. t) - A(x , 0, t) 0 0 6A -A(x.c.t)-A(X.0.t).5.B-B(x.o.t)-B(x,o,c) (4.37) 0 6B -B(x,8,t)-B(x,0,t) and d - z(t) - zo(t) (4.38) Substition of (4.37) and (4.38) into (4.35b), yields 85 82 - (6a + 6A2 + 6Bu + 61Azo + 61Buo + 631(X, z, u, 8, t)) + a(xo, 0, t) o 0 o o + A(x , O, t)z + B(x , 0, t)u + [A(x, O, t) + B(x, 0. t)K(X. t)I¢ + B(x, 0. t)ub (4.39) a(é - 8°) - .3 - a(x. c)¢ + B|||| uII + H1301. °- t) - B(xo' 0' t)Hlluoll 87 41.... .... - .83. .. .. + . (1°. ., .. - . |||lull S <3.le - x0l|+ (32‘ (4.47) where the constants C1, i - l, 2, 3, ..... we use in (4.47), and later on, are all dependent on r. Substitution of (4.47) and (4.29) into (4.46), yields 1: H(t - t)/8 ||¢|| sllo|| + {- t Hm. r>|||| n 8T + “W” l|u(t) - u°(t)“ - uo(t) + uF(t) + ub(t) - u°(t) Sllurll+llnbll - "K(x. t)(zm - z°(t»|| +||ubm|| s|]1<(x, t)””z(t) - 20(8)" +"ub(t)" (4.50) Upon substituting (4.49) into (4.50), we obtain 89 '0 (t - t )/8 - (t - t |]u(t) - uo(t)“ 5 C38 + Kse 1 o + K,e a, f )/6 C t a (t r)/8 ' - ' o + 7—5] e 1 "x(r) - X (0” of (41-51) t o for all t 8 [to. T] Now, let H(t) - x(t) - x°(t) (4.52) using (4.35s) and (4.3a), we have a - 1 - 1° - f(x, 2, u, 8, t) - f(xo, so, uo, 0, t) 9 5f (4.53) Integrating both sides of (4.53) yields 1: t [.§.}Lfl .. ”(c) - ”(to) -.[C 6f(r)dr (4-54) to o o Noting that u(to) - x(to) - X (to) - 0. equation (4.54) yields 1: #(t) -I 6f(r)dr (4.55) t o The Lipschitz property of f in all its arguments and the boundedness of the solution for t 8 [to, T] implies C |l#(t)" SUI; (Cv|lfi(r)“ + C.|I¢(r)[|+ C,|]u(r) - “0(7)" + c10¢)d, 0 (4.56) 90 Substituting (4.49) and (4.51) into (4.56), we obtain t CH t ' -a,(r - a)/8 ||p(t)lls 0,}I “p(r)|]dr + -;—:/P e ”u(a)” dadr to to to t -a,(r - to)/€ t -a,(tf - r)/8 + 012 e . dr + C,, e dr + C146 t to o (4.57) The second term on the right-hand side of (4.57) can be written as 0,, ‘t f -a,(r - a)/8 -——- (e lluH da)dr - t to to t t f f ‘01“ ‘ 0)/6 e , dr H140)” d0 ' 1-0 a -to 1' C 1 t Ci: t 'a1(t ' 0)/¢ 1: t <1 ‘ e >Iluuda s 0 ‘a:' ||“(°)||d” (4.58) 91 The third and forth terms on the right-hand side of (4.57) satisfy the inequalities, respectively, t -a,(r - to)/8 C12 e dr 5 8C,,/a1 (4.59) to ‘ ‘ -a=,l| S 07 ”4(1)” or + T ]|p(a)]]da +e(c,, + -;— + a ) t 1 t0 1 1 O t - C158 + C,:/‘ l]p(r)” dr (4.61) to In equation (4.61), C158, C1,, are nonnegative constants and ||p(r)” is a nonnegative valued continuous function. By Bellman- Gronwall inequality we obtain t o . “(Kt)“ -]|x(t) ' x (CHI 5 Cis‘exPj and” to C10(t - to) -0158e S 011‘ (4.62) substituting (4.62) back into (4.49), yields 92 o '01(t - to)/€ -a,(tf - t)/8 ||¢(t)H -]|z(t) - z (c)|| s 0,. + Koe + x73 1 C K It -a (t - 1) 8 1 ‘f e 1 / C178dr t 6 0 -a1(t - to)/8 -a,(tf - t)/8 - C46 + Kse + K7e + __ [1 - e 1 (4.63) “1 Rewrite (4.63) as o -a,(t - to)/€ -a,(tf - t)/8 ll¢(t)H -||Z(t) - z (t)||s C186 + Kae + K76 , for all t 8 [to, T], (4.64) Similarly, substitution of (4.62) into (4.51), yields 0 -a,(t - t°)/8 -a,(tf - t)/8 Ilu - .. K3, and t1 6 [to, T] there exist 8 > 0 such that * . ' every 8 < 8 ,[IXJIS r, for'every t e [t0, t,]. For example, for r - 2 K3, * * v 8 can be taken as 8 - K3/C15exp(C15(t1 - to)). Similarly, z and u are bounded by r. From the above discussions we see that r can be chosen large enough such that for sufficiently small 8 the trajectory does not leave the compact set S for t 8 [to, tf]. This implies that T - tf. From equation (4.62), we have proved that the slow variable x satisfies 0 . IIX(t) - x (t)|| 5 K16 - 0(8) , for all t 8 [80, cf] (4 57) This implies that H x(tf) - x°(8f)]]- 0(8) (4.68.) where x°(tf) - xf Now, let us rewrite (4.35b) as 8z(t) - a (X, 8, t) + B (x, 8, t)uo(t) - B (x, 8, t)K(x, t)zo(t) + 8g,(x, z, u, 8, t) + (A (x, 8, t) + B (x, 8, t)K(X, t))z(t) + B.(X, 8, t)ub(t) (4.69) 94 The transition matrix of (4.69) can be written as (see Kokotovic, Khalil and O'Reilly [19], page 230) ¢/e e B (x(tf))ub(r)dr t O H(x(tf))(tf " t0)/6 -1 - 0(8) - [I - e ]H (x(tf))g2(tf) + 2f - 2°(tf) (4.82) Rewrite (4.82) as -1 z(cf> - 0(8) - H (x(tf))[a (x(tf>> + B >u° + A (X(cf>)z°)tA ) + B (x(cf)>xlz° 0 (4.96) where K is chosen properly in order to satisfy the constraint -1Susl. In this example, the final time tf and the terminal conditions of X(tf) are not specified. Hence, we do not need right boundary-layer 0 control ub to steer x(t) from x (t) to the desired state xf. The composite control is taken as o o uc - u + uF.- - K(x - x ) , K > 0 (4.97) The boundary conditions and fixed parameters are summarized in Table 4.1 [31] Initial flight path angle 10 - 40 Initial azimuth angle x0 - -450 Initial normalized range R, - 1.0 Final normalized range Rf - .1333 Speed ratio Vs/Vs - .6 Performance index weighting parameter p - 2 Table 4.1 Boundary conditions and parameters for Example 4.1 103 Figs.4.2 - 4.3 show the range R, the flight path angle 1 and azimuth angle x time histories for a value of 8 - .1 when the composite control (4.97) is applied to the system (4.90). This shows that the capture time tf obtained by using the composite control is t - 2.13 which is very close to the exact solution f tf - 1.942, ($88 [31]). The composite control uc time histories is also shown in Fig.4.4. Table 4.2 summarized some numerical results for several values of 8. NANG((NONNALIIEDI TINEINORNALIZEOI Fig.4.2 Range time histories (with composite control) for a value of 8 - .1 104 ANGLE. redisn _‘ .0 1 1 1 1 1 0.0 0.5 1.0 1.5 2.0 2.5 TIHE(NORNALIZED) Fig.4.3 Azimuth angle and flight path angle time histories (with composite control) for a value of 8 - .l ‘ ucx-x(x-x°). p.89 0.0 - -0.5' -' .o 1 1 l 4 l 0.0 0.5 1.0 1.5 2.0 2.5 TIN£(NORNALIZEO) Fig.4.4 Composite control time histories 105 and n J , with with t sol with composite contra! ”a“ 501. callous“! “"1"“ ' census control 5"" sol. .10 -.085 .8323 2.130 1.9.; 2.053 2.082 .15 -.885 .8883 2.135 1,” 2.158 2.106 ~20 -.885 .8596 2.200 2.033 2.205 2-255 .' -25 -.885 .8738 2.230 2,077 2.395 2.388 -30 -.885 .8880 2.280 2.121 2.528 2.889 .35 -385 .8965 2.366 2"“ 2.654 2.6" .80 .335 .9078 2.378 2.2" 2.805 2.707 .85 -.885 .9190 2.885 2.2.. 3.038 2.808 .50 -.885 .9320 2.890 2.2,. 3.158 3.033 Table 4.2 Comparision of exact and composite control solutions Example 4.2 Consider the linear system 2 - z (4.98a) 8z - -x.+ z + u (4.98b) which is a special case of (4.1) with A,;Il (not Hurwitz). It is desired to steer the state from x(O) - z(O) - 0 to x(l) - 1, 2(1) - 0. Ihg_zgdg§gg_§9n;;glz The reduced control problem is defined by a} .. x - u , x°(0) - o, x°(1)- 1 (4.99) 106 and the reduced control is given by u°(t) - -.3l3e(1 ' t) (4.100) 0 - u - -K(z - z ) - -K(Z - .157e( t + 1) - .157e(t + 1)) , x > o F (4.101) 0 where z is obtained from reduced solution Bishtshssndarx;laxsr_ssntrsl= The right boundary-layer model is given by dz 0 0 0 0 . .33. - -x + z + u - -x + z + [u - K(z - z ) + ub] (4-102) Defining zb - z - 20, we obtain dzb . V - (1 - K)Zb + ub (4-103) By using the minimum energy control (see(4.23))and the controllability Grammian W (see(4.28)),the right boundary-layer control ub is given by ub - 8‘t ' 1)/‘ w‘1[zf - zo(tf)] - B(t ' 1)/‘ 2(0 - 1.317) - -2.634e(t ' 1)/‘ (4.104) The composite control is taken as 107 ° (1 - t) _ _ _ (-t + 1) _ (t + 1) _ uc u + uF + ub .3l3e e ) - K(z - .157e 2.634e(t ' 1)/‘ (4.105) Application of the control (4.105) to the system brings (x(t), z(t)) to within 0(8) from the target point (1, 0), for sufficiently small 8. To get a better feeling for the deviation from the target point we calculate (x(l), 2(1)) for 8 - .1, .05, and .01 (see Table 4.3). Fig.4.5 shows the trajectory of x(t) and z(t) for a value of 8 - .01. The results confirm that the final point is within an 0(8) neighborhood of the target. If 10 % error is tolerable, then 8 - .05 is small enough for the control to be successful. If 2 8 error is tolerable then 8 - .01 is small enough. The results also show that in this example, for 8 - .1 the error might not be tolerable. U 8-.JIJEINI-U-KU-.ISIEXH-IOI)-.l‘37ilpll'l1] -2.636EIP((t-I)/8) , K-Z, 0‘.0I 1.0 1- ' l 0.5 » I o o l ‘l 1 L A 0.0 0.2 0.8 0.8 0.8 1,0 11‘ Fig.4.5 State trajectory (with composite control) for a value of 8 - .01 108 8=.l 8 =.os 8 8.0] x(l) .7500 .9120 _ .9730 2(1) .1520 .0200 .0036 Table 4.3 The target point (x(l), z(l)) for 8 - .1, .05 and .01 V. APPLICATION OF THE COMPOSITE CONTROL TO MANEUVERS IN A VERTICAL PLANE 109 In this chapter, we apply the composite control strategy to the optimal maneuvers of an aircraft in a vertical plane. The system model will be represented in the singularly perturbed form (4.1) of Chapter IV via a change of variables and state feedback. One of the examples of interest is the minimum time-to-climb problem which is treated in Section III.1.2. v.1 COMPOSITE CONTROL The equations of motion for flight in a vertical plane are given by (see (2.44) of Chapter II or (3.12) of Chapter III) 2 2 “3%“ - %[1(a, V). - g—pv s(CDo(V) + u(vmhww >1 - f(E. V, 1. a, 6. t) (5.13) 2 2 . d: - 735-1102. V) - é—pv s1 -831N1 - -851N1 + 681(3. V. 1. a. e. t) (5.1b) (% - fistcmwm - {—0051 (5.18) A control a(t) is sought to steer the state E(to) - E0, V(to) - V0, 1(to) - 10 to a terminal state B(tf) - Ef, V(tf) - Vf, 1(tf) - 1f; while minimizing the cost functional 110 t J - I;f [(E, V, 1, a, e, t)dt (5.2) The model (5.1) is in the singularly perturbed form but it is not in the form (4.1) of Chapter IV since (5.1b) and (5.1c) are nonlinear in the fast variables V and 1. Our first task is to use state transformations and feedback to bring the model (5.1) into the form (4.1). In other words we want to linearize the boundary-layer system. We introduce the new variable Z - -gSIN1 instead of the variable 1. Taking the derivative of Z, we obtain 62 - '8C081 . 61 substitution of (5.1c) into (5.4) yields 2 2 ' 2 £2 - - -§— 1%- stCIa(V)COS‘7 . a + .5. cos 1 Furthermore, assuming that C081 . cha(v) >0. ‘we set 1 2 2 {-5- C0821 - u] gi- fi-stcmwmos’y a(t) - After substituting (5.6) back into (5.1a), (5.1b) and (5.5), the system (5.1) becomes (5.3) (5.h) (5.5) (5.6) 111 {‘13:— - 7‘3”“. v> - -1fpv2s[cDO0, b2>0 (5.16) F o This control is not active on the slow manifold (e.g., at V - V and 1 - 0). Ihg 11gb; boundary-layer control: The right boundary-layer system is given by ____ _ z (5.17a) dz b ° ‘33’ - . 1(v - v ) - b22 + ub (5 17b) where ub is only effective during the right boundary-layer. Equation ~ 0 (5.17) can be rewritten by setting V - V - V as 114 of: ‘5- '- Z (5.18a) .31.} — 1,17 - 5,2 + ub (5.1%) The right boundary-layer problem is to move V and 2 from V - 0, Z - O _ o 0 at a - -A/e to V - Vf - V (cf), 2 - Zf - Z (tf) - Zf at a - O. The solution of this steering control 11b is H(t: - t)/¢ 2 o T f - u'b .. B (e )TW 10”) [Li] - [Vo(tf)] ] (5.19) where H - [421 422] . B - [2]. and $10») .fen BBeT( “1)de 0 The composite control ac(t) is taken as 2 a - 2 1 {-5—Cosz‘y - (1.10 + uF + ub)] (5.20) ‘Efi' stcm (V)COS-y 0 After substituting u - O, uF.and ub into (5.20), the composite control ac is rewritten as 2 “(C - t) ac - 2 1 {'é'cosz7 + b1(V ' V0) + bzz - BT(e f /‘)P fq-stCmeosy W'l 2f 00 (o)[ vf - V (tf) ]} (5.21) 115 v.2 APPLICATION OF THE COMPOSITE CONTROL TO THE MINIMUM TIME-TO-CLIMB PROBLEM One of the interesting problems of optimal maneuvers of an aircraft in a vertical plane is the minimum time-to-climb (MTC) problem. This problem has been extensively studied in the literature because of the obvious interest in performance and climbing techniques of modern fighter aircrafts . For example, Bryson, Desai and Hoffman [16] used energy-state approximation in performance optimization of supersonic aircraft. Kelley [45] proposed optimum zoom climb techniques in 1959. and Kelley'and.Ede1baum [1] proposed energy climb, energy turn and asymptotic expansion in 1970- Ardema [4] solved the minimum time-to-climb problem by singular perturbations and matched asymptotic expansions. In this section, we apply the composite control ac,which has been obtained in the previous section,to this specific maneuver problem in a vertical plane. In order to illustrate the solution, two numerical examples are now considered. The first example is ”airplane 2" of [16] which is the same model treated in Section III.1.2. The second example is "airplane l” which is also considered in [16]. The model is given by the nonlinear singularly perturbed system equation (5.1). 116 Example 5.1 Given the nonlinear singularly perturbed system equation. (5.1) for ”airplane 2", and all the areodynamic parameters data are given in [16] it is desired to steer the state of the system from the initial state 0 E(t°) - 1500000 , V(to) - .5 Mach and 1(to) - O to the final state B(tf) - 5000000 . V(tf) - 2.00 Mach and 7(tf) - free, while minimizing tf It is important to mention here again that for this minimum time- to-climb problem, tf is unknown (to be determined). In order to obtain the right boundary-layer control ub (see equation (5.17)), we need to estimate tf apriori. This problem can be circumvented by solving (i.e., integrating) the right boundary-layer solution in reverse direction off-line until it matches the slow manifold, then tf can be determined. In this particular example tf - 170sec- The composite control ac is applied to the system equation (5.22), and the resulting energy E, Velocity V, flight path angle 1 and flight trajectory (path) are shown in Figs.5.l-5.4, respectively. In Fig.5.2, it is seen that this composite control steers the state of the aircraft to a final stste (2.06 Mach and altitude 79733 FT) which is about 3% error relative to the given final velocity (2.00 Mach), and about .333% error relative to the given final altitude (80000 PT). The total error is about 3.333% which is O(¢) close to the desired state. The comparison of the composite control do with the approximate control “app which is obtained by off-line singular perturbation approximation of the optimal trajectories (see Section III.1.1) is also shown in Fig.5.5. “1N1". m U. x105 6 4 § § 2 o 117 l .L 1 J 0 50 too 150 200 Till. sec Fig.5.1 Energy time history'for airplane 2 with 2. 5 2..0 I. 5 0..5 composite control V(t,)-2.os n(:,)-79733:r TARGET: V(t')-2_m h(:,)-aoooorr VELOCITY Wzfl MUN“ cm: .3331 WIN. mama: tf-I70sec I l l J 0 so 100 150 ' 200 tilt. sec Fig.5.2 Velocity time history for airplane 2 with composite control .. — \ In! It... ‘. puma. moon FLIGHT PIT" ANGLE. radian 118 1.0 - 0.5- 0.0 -0.5 -1.0 1 + . . 0 50 100 ISO 200 “NE. sec Fig.5.3 Flight path angle time history for airplane 2 with composite control v(t,)-2.06 n(c,)-79733r1 so} 20_ V(to)'.5 \ .6 .8 1.2 1.6 2.0 2.4 2.8 mm. m D. Fig.5.4 Flight trajectory for airplane 2 with composite control 119 20 P r “““ ‘\ \ \ \ \ \ ‘\ ----- woosmmmm. a, 8 10 ~ \ ‘. \ 5 \ — sn mmnmt comm 0. . \ 99 8 2 c d 3 U 3 0 ~ ‘\ 1m _+_ 51011 mxmo'—+— man My um m1 mu "0 I 1 l J ‘0 50 100 150 200 "K. see Fig.5.5 Comparison of the approximate control and composite control In order to demonstrate the composite control strategy, we choose twelve different boundary conditions (the values are chosen such that no saturation occurs), and we use the same airplane 2 as an example. The simulation results are summarized in Table 5.1, which shows that this composite control indeed steers the system from the initial state to a final state which is 0(a) close to the desired final state. 120 case INITIAL POINT TERMINAL POINT FINAL POINT canon c ( L) (gig) h—azooorr h-lOOOOOFT h-99590FT h-.41§ 1 v-.5u v-2.au v-2.ssu V-6.7. 19o h-azooorr h-eoooorr h-aoazzrr h-1.37I 2 V-.5M v-1.ou v-1.osan v—5.a. 103 h-60000FT h-BOOOOFT h-79100FT h-1.125i 3 v-1.ou V-2.0M V-2.11M V-5.5‘ 145 - - -53 000 - h-SOOOOFT h-IOOOOOFT h-100876PT h-.876t a v-2.on v-2.4N v-2.a1zn v.5. Ioo h—azooorr h-80000 h-79733FT h-.333t 5 v-.5n v-2.on v-2.osu v-31 170 h-6000OFT h-lOOOOOFT h-99950FT h—.05. 6 v-1.ou v-2.au v-2.asn V-2.5§ 160 h-lOOOOOPT h-azooorr h-aaooorr h-2.38! 7 v-2.an V-.5H v-.53u v-a. 185 h-80000PT h-azooorr h—a1eoorr h-.68t a v-2.on V-.5M v-.5zn via. 160 h-GOOOOFT h-azooorr h-hl700PT h-.71! 9 v-1.on v-.5u V-.528H v-5.s. 9s 2 h-IOOOOOFT h-60000PT h-599OOFT h-.l7§ 1o v-2.au v.1.on v.1.osu v-e. Isa h-aoooorrr h-aoooorr h-61aoorr h-2.33§ 11 v-2.on v.1.on V-1.05H v-s. 140 h-lOOOOOFT n-aoooorr h-SOIOOFT h-.1250 12 v-2.au v-2.ou 1 v-2.15n a v-7.5. 39 2 Table 5.1 The simulation results of minimum time- to-climb with composite control for airplane 2 under various boundary conditions A characteristic phenomenon of singularly perturbed systems is that, in general, the reduced solution does not satisfy all boundary conditions. It satisfies only a projection of the boundary conditions on the slow manifold, i.e., boundary conditions on the slow variables. Changes in boundary conditions of the fast variables that do not change the boundary conditions of the slow variables will not effect the reduced solution. In order to demonstrate this phenomenon, let us take 121 cases 5 and 8 as examples. First, we keep the same initial conditions and change the terminal conditions of the fast variables to Vf - 2.65 Mach and hf - 20000 FT in Case 5. Second, we change the initial conditions of the fast variables to V0 - 2.65 Mach and ho - 20000 FT and keep the same terminal conditions as in Case 8. In both cases the boundary conditions on energy are unaltered. Simulation results are P shown in Fig.5.6 and Fig.5.7, respectively. The results show different behaviors of boundary-layers but the slow manifolds remain 3! P the same. ' __ 140011110 vtnsxou (CHANGE IERHIIAL (01101110115) 30 1- ----- CASE 5 V(t,)-2.00 /\‘\ t 1111,):11000011 \ § ’0‘ .1 \ —- so 1- ”r \ - \ § \ .. \ Z 1 2‘ l 40 - ,,«’ \\60,,( 6 \\\ ‘- 0," W 20 .. 11111011: v(t,)-2.65 v(1,)-2.64 h(t,)*?0000" n11,)-21121n 1 1 _1 1 1 1 J 0 .4 .8 1.2 1.6 2.0 2.4 2.8 mocm, MC" 110. Fig.5.6 Flight trajectory of Case 5 under modified version (change terminal boundary conditions) for airplane 2 with composite control 122 MODIFIED VERSION (CHANGE INIIIAL CONDITIONS) so 1- " " “5‘ 3 [k “101-2.00 I: I 11(101-8000011 I: 53' I § 60 . u- / A .§- man; V(I,)-.5 (/ 5\ : mfmzooon / 5.x 3 ' f h‘g/ é! 40 .- \ fig)" \ “HI-.54 \ h(t,)-‘IOOOFT \ / \ ) V(to)'2.65 MtthOOOOFT 1 1 J 1 1 l I O .4 .8 1.2 1.6 2.0 2.4 2.8 VELOCIIV. MACH NO. Fig.5.7 Flight trajectory of Case 8 under modified version (change initial boundary conditions) for airplane 2 with composite control Example 5.2 Consider the same nonlinear singularly perturbed system (5.1) for "airplane 1". It is desired to steer the state of the system from V(to) - .38 Mach, E(to) - 89921 secz, h(to) - 0, 7(to) - O to the final state Vth) - 1.00 Mach, E(tf) - 2576000 secz, h(tf) - 65600 FT, 1(tf) - free, whilezminimizing tf. This example is different from the previous one in two aspects. First, the path for "airplane l" exhibits a discontinuity in velocity (a zoom dive), which is not present in "airplane 2" . Second, for this take off example, it is necessary to consider the constraint 123 v s (2191/2 (5.23) where V s (21:)1/2 insures h z 0. In general, we have to consider this constraint all the time during the entire trajectory except when the trajectory is far away from h - 0(586 previous example 5-1)- The composite control ac is applied to this "airplane 1" system equation (5.1) and all the aerodynamic parameters data are given in [16]. The resulting flight path trajectory is shown in Fig.5.8. It is seen that this composite control steers the state of the aircraft to a final state (1.03 Mach and altitude 63177 FT) which is about 3% error relative to the given final velocity (1.00 Mach), and about 3.7% error relative to the given final altitude (65600 FT). The total error is about 6.7% which is O(¢) close to the desired state. The time estimate on this composite control path is 260 sec from h - 0, V'- .38 plus. 40 sec dive and 55 sec zoom, the total time is 355 sec. Use of the energy-state approximation (see [16]) gives the total time 377 sec. The time computed by using steepest descent is 332 sec (see [16]). The flight trajectory which was obtained by Bryson ,et a1. [16] “31118 33913)“ state approximation for this "airplane 1" exhibits a discontinuity in velocity (a zoom dive) near Mach number 1, 1.2, 1.8 which may cause considerable errors. Aummt, 100011 124 IAIEET: v11,1-1.oo MykawUI [fig-1.03 “1“"? [mm hug-53177111 Alum (”3.69: TOTAL «110115.59: 60 r- 40 '- 20 '- VELOCITY. MCI! 110. ' Fig.5.8 Flight trajectory for airplane l with composite control comma or 5’9 ' 801th3 cousmn ENERGY oo-——.1 \\<::: 1151111111111. 110m \~ 1111,1-1 .00 P~\\‘ \ \ Mtg-5560011 \ / mum. ”I D. Fig.5.9 Flight trajectory for airplane l by Bryson's energy-state approximation VI . CONCLUS IONS 125 In this thesis three topics related to nonlinear singularly perturbed optimal control problems were discussed. The results of the analysis were illustrated by the optimal maneuvers of an aircraft in‘a vertical plane which is based on a minimum time intercept problem. The contributions of the thesis are: (l). A normalization scheme for the time-scale modeling of dynamic' systems arising in flight mechanics has been proposed. This scheme is based on the dynamic state equations and the normalizing reference data. It is relatively easy to apply, and is an improvement over the ad hoc methods currently in use. This scheme has been applied to a typical class of aircraft flight dynamics problems. Numerical examples showed that the time-scale separations as computed by this scheme generally agree with previous practice and assumptions. V (2). Application of singular perturbation techniques to trajectory optimization problems in flight mechanics has been studied. It has been demonstrated that for auto-pilot implementation, the open loop control results in a boundary-layer instability. This instability problem has been circumvented by using feedback stabilization schemes. (3).. A composite control approach has been proposed to steer the state of a singularly perturbed system from a given initial state to a given final state, while minimizing a cost functional. Asymptotic validity has been proved by showing that its application to the singularly perturbed system results in a final state which is O(¢) close to the desired stste and the cost under the composite control is 0(a) 126 close to the optimal cost of the reduced control problem. Our analysis does not involve asymptotic analysis of the full optimal control, therefore our assumptions are weaker than earlier assumptions. Application of the composite control strategy to maneuvers of an aircraft in a vertical plane has also been discussed. The attractiveness of the composite control approach has been demonstrated on the minimum time-to- climb problem. Further work should address the following points. First, the multiple-time-scale modeling procedure of Chapter II should be validated using real data of high-performance aircrafts.Second, for relatively large value of e, the composite control will have to be corrected to account for 0(a) terms that have been neglected throughout the derivations. To account for O(¢) terms the slow and fast models will have to be corrected by including higher-order terms and the effect of boundary conditions will be corrected by including higher-order terms. - Also the effect of the fast control on the slow subsystem will have to be taken into consideration. Another possible extension of the results of this thesis is the application the composite control strategy to minimum fuel climb, minimum time turn and maximum range glide problems. Until now it has been applied only to minimum time-to-climb problem. These seem to be challenging problems and are left for future research. LIST OF REFERENCES [1] [2] [3] [4] [5] [5] [7] [3] [9] LIST OF REFERENCES H. J. Kelley, and T. N. Edelbaum, "Energy Climb, Energy Turn, and Asymptotic Expansions," Journal of Aircraft, Vol. 7, No.1, 1970. H. J. Kelley, ”Reduce-Order Modeling in Aircraft Mission Analysis," AIAA Journal, Vol. 9, No. 2, Feb., 1971. H. J. Kelley, "Flight Path Optimization with Multiple Time-Scales," Journal of Aircraft, Vol. 8, Apr. 1971. M. D. Ardema, ”Solution of the Minimum Time-to-Climb Problem by Matched Asymptotic Expansions," AIAA Journal, Vol. 14, 1976. A. J. Calise, "Singular Perturbation Method for Variational Problems in Aircraft Flight,” IEEE Trans. Aut. Control, Vol. AC-21, Jun. 1976. A. J. Calise, "A Singular Perturbation Analysis of Optimal Aerodynamic and Thrust Magnitude Control,” IEEE Trans. Aut. Control, Vol. AC-24, No. 5, Oct. 1979. R. B. 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