The invention relates to Flight Management Systems (FMS) providing minimum total cost flight profiles particularly with respect to accounting for arrival error cost functions.

Flight Management Systems are known in the art that utilise an adjustable cost index in providing a minimum-cost flight profile. The ARING characteristic 702 defines Flight Management Systems for commercial aircraft.

A principal objective of an FMS is to minimise the cost of flight. Present day equipment achieves this objective by generating vertical and lateral profiles that minimise direct operating cost (DOC). Direct operating cost is the cost of fuel plus other costs that are proportional to flight time. Flight time costs, such as crew costs, maintenance, repair and replacement of equipment, that may be prorated with flight time, is represented in the FMS by a cost index, which is defined as the ratio of time cost (dollars/hour) to fuel cost (cents/pounds), providing units that are proportional to fuel flow (100 lbs./hour - 45.36 kg/hour). The cost index is selectable by the pilot (usually in the range of 0 to 999), and is intended to remain fixed for a given flight, representing predetermined flight-hour costs. For a given cost index, the prior art FMS provides a speed command at every point in the flight profile that minimises DOC.

A significant disadvantage of the prior art FMS utilising the DOC approach is that such systems do not reflect costs associated with arrival-time error, such as crew overtime, losses due to missed connections in connecting flights and potential losses resulting from customer dissatisfaction with the airline. The cost index is often utilised in prior art Flight Management Systems as a means for adjusting speed to achieve on-time arrival on an average basis. When utilised in this manner, the cost index for a city pair is selected to achieve the desired arrival time under average wind and traffic conditions. Therefore, in the prior art, the cost index is selected to achieve a statistical arrival time performance for a given city pair, without taking into account conditions for an individual flight. The cost index, when so utilised, no longer represents the flight-hour cost as initially intended, but becomes a mechanism for adjusting arrival time. It will be appreciated that not arriving on time normally results in additional cost to a scheduled airline, which cost is not accounted for in present day flight management systems which search for the minimum cost profile in accordance with DOC.

The present invention is defined in the appended claims and provides a Flight Management System for aircraft comprising means for generating a function of direct operating cost (DOC) versus arrival time in accordance with a range of cost index values. Means for combining the DOC function with a function of arrival time error cost versus arrival time error, generates a function of total flight cost versus arrival time. Minimising means determines the minimum of the total flight cost function to provide an optimum arrival time signal and means responsive to the optimum arrival time signal provides an optimum cost index signal corresponding thereto. Speed generator means responsive to the optimum cost index signal generates an airspeed signal corresponding to the optimum cost index signal in accordance with minimum DOC. The system includes a predictor for generating a predicted arrival time signal. Speed adjustor means responsive to the optimum arrival time signal, the predicted arrival time signal and the airspeed signal adjusts the airspeed signal in accordance with the difference between the optimum arrival time signal and the predicted arrival time signal to provide an airspeed command signal for controlling the airspeed of the aircraft.

A Flight Management System in accordance with the present invention will now be described in greater detail, by way of example, with reference to the accompanying drawings, in which:-

• Figure 1 is a schematic block diagram of flight management system apparatus implemented in accordance with the present invention,
• Figure 2 is a schematic block diagram 'illustrating a preferred implementation for a Kopr, T_op_T generator of Figure 1,
• Figure 3 is a diagram illustrating a direct operating cost versus flight time function and a total flight cost versus flight time function usable in describing the apparatus of Figure 2,
• Figure 4 is a diagram of an arrival error cost versus arrival time error function usable in constructing the total flight cost function of Figure 3,
• Figure 5 is a diagram of a cost index versus flight time function usable in describing the apparatus of Figure 2,
• Figure 6 is a schematic block diagram of an alternative embodiment of the tCopr, T_OPT generator of Figure 1, and
• Figure 7 is a diagram illustrating convergence of the speed adjustment performed in accordance with the invention.

Referring to Figure 1, a schematic block diagram of flight management apparatus implemented in accordance with the present invention is illustrated. The apparatus includes a K_oPT, TOPT generator 10 responsive to an input cost index signal K_o that is generally manually selected by the pilot. The value selected is .intended to represent the actual flight-hour cost, excluding fuel cost, as previously described. The generator 10 provides an optimum arrival time signal TOPT and the associated cost index signal K_op_T. T OPT is the optimum arrival time at the flight plan prescribed destination waypoint which minimises total flight cost. K_op_T is the cost index which results in that arrival time. In a manner to be described in detail, total flight cost is the sum of DOC and arrival time error cost. Although the arrival point is normally the destination waypoint, the apparatus of the present invention may also be utilised with respect to intermediate waypoints along the flight plan. The reference point for time is arbitrary but for the embodiment of the invention described herein is assumed to be at take off. In a manner to be described, the generator 10 is enabled to provide values of T_OPT and K_op_T whenever a new flight plan is activated or when a flight plan is altered so as to affect flight time. The details of the structure and operation of the generator 10 will be later described.

The flight management system apparatus of Figure I includes a speed generator 11 which provides a true airspeed signal V_o in response to the Ko_PT signal from the generator 10. The speed generator 11 is a conventional part of present day commercially available flight management systems which, in response to a cost index value K, provides a true airspeed command that minimises DOC. Thus, in the prior art, the output V_o of the speed generator 11 would be the true airspeed command normally provided as an output of the prior art FMS, which output would in the prior art be coupled to the automatic flight control system of the aircraft to control the airspeed thereof. The V_o signal from the speed generator 11 is modified, in accordance with the present invention, in a manner to be described before it is provided as a system output.

In the prior art, one of the inputs to the speed generator 11 would be the cost index K which normally is the value selected manually by the pilot. In the present invention, the cost index input to the speed generator 11 is the K_op_T signal provided by the generator 10. The speed generator 11 conventionally receives inputs schematically represented by a block 12. Such inputs usually include measured values of aircraft gross weight, altitude, wind, temperature and air pressure as well as speed and acceleration limits imposed by the aircraft flight envelope, the airframe and engine limitations, the guidance laws and the flight plan. A speed generator suitable for utilisation in the apparatus of Figure 1 comprises a part of a commercially procurable flight management computer. The inputs from the block 12 are those applied to the speed generator in the prior art apparatus. The cost index input to the speed generator 11 in the present invention is provided by the K_oPTsignal from the generator 10 rather than the K_o signal manually selected by the pilot in the prior art.

The apparatus of Figure 1 includes a predictor 13 which performs a fast time simulation of aircraft flight from the current aircraft position along the prescribed flight plan to the destination waypoint. In a manner similar to that describd above with respect to the speed generator 11, the predictor 13 is a part of prior art flight management systems. The flight management computer discussed above contains a predictor suitable for use as the predictor 13 of Figure 1. As is well appreciated in the art, the predictor 13 receives a variety of conventional data inputs represented schematically by a block 14. The conventional data inputs represented by the block 14 comprise models of the aircraft including the airframe and engines and models of the atmosphere along the flight profile that the aircraft is expected to fly. The atmospheric models include wind, temperature and pressure forecasts. The conventional data inputs 14 include the aircraft flight envelope, the flight plan, the guidance laws as well as speed, acceleration and position, constraints and limitations of the flight plan. The predictor 13 also requires a speed command input on a line 15 identical to the speed command output V_R of the apparatus of Fig. 1, except that instead of measured inputs from aircraft sensor data, the speed command input to the predictor 13 on the line 15 utilises simulated data inputs in a manner to be later described in detail. The predictor 13, in a manner well understood in the art, models the aerodynamics of the aircraft pursuant to the data inputs thereto to simulate the flight of the aircraft from the current aircraft position to the destination waypoint. The predictor 13 accordingly provides an estimate of the flight time from the current position to the destination as well as the fuel burn required therefor. The predictor 13 also provide similar data with respect to the intermediate waypoints along the flight plan. The predictor 13 iteratively and continuously provides successive simulations of the aircraft flight from the current position to the destination waypoint as the aircraft proceeds along the flight plan.

The structure, functions and operations of the predictor 13 are essentially identical to the predictor in prior art flight management systems with the predicted arrival time or flight time to the destination provided on a line 16. Although the predictor 13 provides a fuel burn prediction to the destination point, this signal is not shown as an output from the predictor 13. This output is, however, utilised with respect to a predictor included in the generator 10 in a manner to be described.

The predictor 13 also provides signals denoted as T_UP and T_DN on lines 17 and 18 respectively. The prior art predictor is modified to provide T_UP and T_DN as follows. The prior art predictor includes an accumulator for accumulating predicted arrival time during a prediction. The predictor 13 further includes two additional accumulators for accumulating T_UP and T_DN, respectively, during the prediction. Speed and acceleration limits over segments of the flight plan prevent speed adjustments during these segments necessitating greater speed adjustments over the remaining segments. The data inputs 14 inform the predictor 13 of the flight plan segments during which speed and acceleration limits are imposed. The flight plan, as well as the air frame and engine models, have such speed and acceleration limits included therein. The T_UP accumulator is inhibited from accumulating during segments in which the aircraft is subject to an upper speed or acceleration limit. The T_DN accumulator is inhibited from accumulating during segments in which the aircraft is subject to a lower speed or acceleration limit. Thus, T_UP is the total time during the prediction when the speed is permitted to be upwardly adjusted, and T_DN is the total time during the prediction when the speed is permitted to be downwardly adjusted. The time segments excluded are those where a speed limit, imposed either by the flight plan or by aircraft limits, prevents the speed from being increased or decreased, respectively. Also excluded are the periods where the aircraft is accelerating or decelerating at a limited value. If during a prediction no speed or acceleration limits are applied, T, T_UP and T_DN will be equal.

It will be appreciated from the foregoing, that the predictor 13 provides a fast time simulation of flight from the current aircraft position to the destination waypoint, as well as to the intermediate waypoints of the flight plan, pursuant to a cost index value. The predictor 13 takes into account winds, flight plan constraints, guidance laws and the like at every point along the flight plan, simulating aircraft drag, thrust, fuel flow and the like.

The apparatus of Figure 1 further includes a K_sAgenerator 19 for generating a speed adjustment factor K_SAutilised in fine tuning the speed output V_o from the speed generator 11 so as to achieve high accuracy in the arrival time at the destination waypoint. Each time the predictor 13 performs a prediction pass to provide the estimated flight time T to the destination waypoint, on the line 16, the K_sA generator 19 compares the time T with the optimum arrival time signal T_OPT from the generator 10 and updates the previous value of the speed adjustment factor K_SA based on the comparison between T and T_op_T and on the T_up and T_DN signals on the lines 17 and 18. The K_SA factor remains constant until again updated by a subsequent prediction pass.

Specifically, the K _SA generator 19 updates the previous value of the speed adjustment factor K_SA each time the predictor 13 has completed a prediction to the destination waypoint as follows:Where

• DELTA T = T - T_OPT
• T = predicted arrival time on line 16
• T_A = T_up if DELTA T>0 and T_DN otherwise
• T_up = the signal on line 17
• T _ON = the signal on line 18

Thus it will be appreciated that if at the end of a prediction pass the arrival time at the destination is late, T_up is utilised in updating K_SA, thereby increasing the airspeed during the segments of the flight plan during which speed limitations are not imposed. Similarly, if the arrival time is early, T _DN is utilised in updating K_sA, thereby decreasing the airspeed during those segments of the flight plan where there are no speed limitations.

The K_SA factor is utilised in adjusting the speed V_ofrom the speed generator 11 in a manner to be described. The initial value of K_SA is zero, rendering V_R = V_o until completion of the first prediction pass. The K_sAgenerator 19 initialises the value of K_sA to zero by a signal from an OR gate 20 whenever the generator 10 is activated in a manner to be described.

The speed adjustment factor K_SA from the K_sA generator 19 is applied to a speed adjuster 21 which is also responsive to the airspeed signal V_o from the speed generator 11. The speed adjuster 21 also receives an input Vw representative of the measured value of a tail wind or head wind as - schematically represented by a block 22. It will be appreciated that a head wind is the negative of a tail wind, and that a positive value represents a tail wind. The speed adjuster 21 utilises K_SA, which is a constant between prediction passes of the predictor 13, to adjust the speed V_o which is continuously changing to provide the true airspeed command V_R of the system in accordance with

The V_R signal is applied to the autopilot or autoth- rottle system of the aircraft to control the airspeed thereof.

Alternatively, the speed adjuster 21 may be implemented to embody the following:Where V_GR is the adjusted ground speed at the current prediction position and is equal to:

As discussed above, the input 15 to the predictor 13 is identical to the V_R output of the speed adjuster 21 except that the input 15 to the predictor 13 utilises simulated inputs instead of actual sensor data. Accordingly, a speed generator 23 identical to the speed generator 11 provides an airspeed signal to a speed adjuster 24 which is identical to the speed adjuster 21. The output of the speed adjuster 24 provides the airspeed input to the predictor 13 on the line 15. The speed generator 23 receives the K_OPT signal from the generator 10 in the same manner as the speed generator 11. The K_OPT value remains constant unless the generator 10 is reactivated as described herein.

The speed generator 23 also receives conventional simulated inputs including V_w from a wind forecast model and speed and acceleration limits as schematically represented by a block 25. The inputs provided by the block 25 are identical in type to the inputs provided by the block 12 except that the data from the block 12 is from actual aircraft sensors whereas the corresponding data from the block 25 is simulated.

The speed adjuster 24 receives a simulated V_w signal from the wind forecast model as - schematically represented by a block 26. The simulated V_w from the block 26 corresponds to the measured V_w from the block 22. The speed adjuster 24 also receives an input from a register 27 which is utilised to hold the value of K_SAcorresponding to the previous prediction pass of the predictor 13.

Thus it will be appreciated that the V_R input to the predictor 13 on the line 15 is derived from the previous iteration of the predictor 13 and during the current iteration, the K_oPT value applied to the speed generator 23 and the K_SA value applied to the speed adjuster 24 are constants which do not change while the current prediction pass is being executed.

As discussed above, the generator 10 is activated whenever a new flight plan is introduced or when a flight plan is changed in a manner that affects flight time. As the actual flight progresses without such flight plan change, the generator 10 is also activated when the true airspeed command V_R deviates from V ₀ by a predetermined threshold. Accordingly, the OR gate 20 receives an Activate signal to activate the generator 10 whenever a new flight plan is introduced or when the flight plan is changed so as to affect flight time. A comparator 28 responsive to the V_o signal from the speed generator 11 and the V_R signal from the speed aduster 21 provides an input to the OR gate 20 whenever the difference between V_R and V_o is greater than a predetermined threshold set into the comparator 28. Additionally, whenever the OR gate 20 activates the generator 10, the value of K_SA in the generator 19 is set to zero. The conventional data inputs 14 and the conventional simulated inputs 25 are also applied as inputs to the K_op_T, T_op_T generator 10 for reasons to be described.

As previously discussed, the K_OPT, T_op_T generator 10 provides the optimum arrival or flight time T_op_T that minimises total flight cost and the optimum cost index K_op_T that results in the optimum flight time. The total flight cost for a given cost index may be expressed asWhere

• K = cost index (a variable) in units of fuel flow (100 lbs./hr - 45.36Kg/hr)
• K_o = the value of K that represents actual flight-hour cost (a constant)
• C_F = unit cost of fuel (cents/lb.)
• F(K) = the fuel burn (100-lb - 45.36 kg) resulting with cost index K
• T_F(K) = the flight time (hr.) resulting with cost index K
• AEC(T_E) = arrival error cost function ($)
• T _E = arrival time error (actual arrival time minus - scheduled arrival time)

The predictor of the prior art FMS discussed above provides T_F(K) and F(K), as the result of a prediction pass for a given K. With the constants K_o and C_F given, DOC may be obtained from the above expression

Figure 2 is a schematic block diagram illustrating a preferred implementation for the K_OPT, T_OPT generator 10 of Figure 1. In order to appreciate the interrelationships between the elements of Figure 2, a representative flight is considered. A flight over a 1,000 nautical miles (1853.184 Km) flight plan is undertaken. The initial and final altitudes are 1,000 feet (304.8m) and a cruise altitude of 35,000 feet (10,668m) is selected without step climbs or descents. The International Standard Atmosphere is assumed, and the initial gross weight is 350,000 Ibs (158760 Kg). The 250 kt. ATC speed limit was observed below 10,000 feet (3048m) altitude and the aircraft was at speed at the initial and final altitudes. A head wind was applied of 50 kt. magnitude at 35,000 feet (10,668m) altitude, decreasing linearly to zero at sea level. A clean configuration for the aircraft was assumed throughout the flight. The following values of cost index (100-lb/hr) were considered: -100, -75, -50, -25, -10, 0, 10, 25, 50, 75, 100, 150, 200, 300, 400, 600, 800 and 1,000. The value of cost index that represents actual flight-time cost usually resides in the range of 25 to 75. During the representative flight C_F = 13 cents/lb. (13 cents/0.453 Kg) and Ko = 50. The - scheduled arrival time based on a no-wind condition with K_O = 0, corresponds to a flight time of 2 hours and 20 minutes. Fig. 3 (solid line) illustrates direct operating cost versus flight time for the representative flight described above with the corresponding cost index values K illustrated along the curve.

As described above, total flight cost is the sum of direct operating cost and arrival time error cost. In order to implement K_OPT, T_OPT generator 10 an arrival error cost function for the particular flight plan must be stored in the apparatus. The arrival error cost function is predefined and provided by the airline. The airline decides, based on crew overtime, losses due to. missed connections, potential losses resulting from customer dissatisfaction and airline policies regarding flight time pay, what it will cost for the aircraft to be late (or early) with respect to the scheduled arrival time. The arrival error costs are expectd to be different for each termination and therefore the arrival error cost functions will be generally different for each .arrival situation. The arrival error cost function, AEC(T _E) must be defined for all arrival time error values T_E , should be equal to zero for T_E = 0, and should not decrease as |T_E| increases. It will be appreciated that there is a cost penalty whether the aircraft is late or early with respect to the scheduled arrival time. Arrival error costs continually increase with increasing late arrival. The time points at which step changes in the function, or in its slope, occur must be accommodated. These time points are referred to as the AEC critical points.

The K_OPT, T_op_T generator 10, detailed in Figure 2, conveniently stores the AEC function in the following format:SIGMA(T, , S_I, M_I) is a sloping step function of time which steps up for increasing positive T_E, and also for decreasing negative T_E. SIGMA(T_I, S, M,) executes a step of size S, at step point T_E = T_I and continues with slope M, . The set of time points {T_I} are the AEC critical points. The SIGMA function is defined in terms of T_i, S_I, and M, by:

In the SIGMA format for storing component I of the AEC function, there is a cost penalty for early arrival as well as for late arrival. The upper line of the functional expression is utilised for early arrival, the lower line is used for late arrival and the middle line is utilised for arrival within ± T, of T_op_T . If T, = 0, then the upper line applies when TE is slightly less than ZERO and the lower line applied when T_E is slightly greater than ZERO. Although the illustrated format for constructing the AEC function is convenient, other valid formats may be utilised to the same effect.

Referring to Figure 4, an exemplary AEC function is illustrated comprised of a linear component of $600.00/hr. for T_E<0 and critical points at 0, 10, and 20 minutes. Two steps representing losses of $150.00 each, resulting from missed flight connections, are illustrated at 10 and 20 minutes. No cost penalty is assigned for early arrival. The AEC function of Figure 4 is represented in the SIGMA format byIt will be appreciated that the AEC function is not necessarily comprised of linear segments. The segments between the steps may exhibit any appropriate functional curve. It will be appreciated, however, that in any event, the entire curve can be approximated by a series of piece-wise linear segments which may then be stored in the SIGMA format.

Referring again to Figure 3, the arrival error cost function of Figure 4 is added to the direct operating cost versus flight time function of Figure 3 to provide the total flight cost versus flight time function illustrated in dashed line. The arrival time error T_E of Figure 4 is referenced to the scheduled arrival time T_sof two hours and twenty minutes in deriving the total flight cost curve. The optimum flight time T_OPT, provided by the K_OPT T_op_T generator 10 of Figure 1 is determined from the minimum of the total flight cost curve of Figure 3. In the example illustrated, T_OPT is approximately two hours and twenty-six minutes, or six minutes late. The optimum cost index resulting in the optimum flight time is in the range of 75 to 100 (approximately 85). On-time arrival would require a cost index of 900 and an additional cost of approximately $250.00. Utilising a cost index K of zero, a connecting flight is missed and the total cost is approximately $200.00 more than the optimum. It will be appreciated that the optimum cost index is that which minimises J(K).

Each point of the DOC curve of Figure 3 can be generated by performing a prediction pass for a trial cost index. This is generally an undesirably costly procedure with respect to FMS computer time and cannot be performed for a large number of trial cost index values without excessive response time. It is desirable to minimise the number of prediction passes in generating the true airspeed command V_R in accordance with the invention.

The preferred implementation of the K_OPT, Top-_rgenerator 10 of Figure 1 is predicated on the property that large variations in cost index near the optimum cost index value have a neglibible effect on total cost unless the minimum point of the total flight cost curve (Figure 3) is proximate a critical point. With reference to Figure 3, the cost index can vary from about 30 to 130 with only a $10.00 variation in total cost. This variation in cost index corresponds to over four minutes variation in arrival time which is approximately 3% of the flight time. Therefore, the generation by the generator 10 of Figure 1 of the optimum cost index K_op-_rdoes not have to be very precise and the optimum arrival time T_op_T is only required to be precise if it is proximate a critical point.

The preferred implementation of the generator 10 of Figure 1 for generating the optimum arrival time and cost index comprises utilising cost index values from a reference set thereof from which parabolic interpolation is utilised to estimate intermediate values. A preferred reference set, in units of 100 Ib/hr is:

• (-100, -50, -25, -10, 25, 150, 400, 1000) The generator 10 of Figure 1 is implemented to perform the following procedure to estimate the optimum arrival time and optimum cost index:

  • 1. Select the three cost index values -10, 25 and 150 from the reference set. Alternatively, select three cost index values from the reference set that are expected to reside in the vicinity of the optimum value. The vicinity may be estimated from previous values of cost index that provided approximately correct arrival time on previous flights between the departure and destination points.
  • 2. Perform predictions with the three cost index values to obtain corresponding values for DOC and flight time.
  • 3. Fit a parabla to the three points to approximate DOC versus flight time and then add the predefined arrival error cost function to obtain an approximate function of total flight cost versus flight time.
  • 4. Perform a simple search, such as a linear search on this total flight cost versus flight time function to obtain the minimum point. The search intervals can be relatively large, such as 60 seconds, but must include the AEC critical points. The time at the minimum point is the optimum arrival time T^pp_T.
  • 5. If the resulting T OPT minimum point resides outside the span of the three prediction times, a further adjacent trial cost index from the reference set is selected, in the appropriate direction, and another prediction is performed which replaces the previous prediction furthest removed. The steps from item 3 are then repeated.
  • 6. Determine the optimum cost index, KePT, corresponding to T_OPT by performing parabolic interpolation on the three prediction values of cost index versus flight time. Figure 5 illustrates a curve of flight time versus cost index for the illustrative flight described above.

It will be appreciated from the foregoing that the scheduled arrival time is not necessarily the optimum arrival time. For the illustrative flight arriving six minutes late minimised the total cost.

Referring to Figure 2, a preferred implementation of the K_OPT, T_op_T generator 10 of Figure 1 is illustrated. The generator 10 includes a block 40 for providing a time sequence of trial cost index values. The block 40 stores the reference set of cost index values discussed above (-100 through 1,000) and sequentially provides values -10, 25 and 150. The block 40 is also implemented to provide the next higher or next lower cost index value from the reference set in response to a signal on a line 41. Whether this new trial cost index value is a higher value or a lower value than the initially provided three values is controlled by a signal on a line 42. the operation of the block 40 is activated by the output of the OR gate 20 of Figure 1.

Each trial cost index value from the block 20 is applied to a speed generator 43. The speed gen- rator 43 is identical to the speed generator 23 of Figure 1 and in a manner similar to the speed generator 23 receives the conventional simulated inputs, including simulated wind, and speed and acceleration limits from the block 25. The speed generator 43 provides a true airspeed signal corresponding to the input trial cost index value in the . manner described above with respect to the speed generator 23.

The airspeed signal output of the speed generator 43 provides the airspeed input to a predictor 44. The predictor 44 is identical to the predictor 13 of Figure 1 as described above and receives the same conventional data inputs 14 that are applied to the predictor 13. For each trial cost index value provided by the block 40, the predictor 44 performs a prediction pass and provides a flight time to the destination T_F(K) and the fuel burn to the destination F(K) corresponding to the trial cost index provided by the block 40. As discussed above, the present invention is applicable not only to flight time to the destination but to estimated time of arrival (ETA) at intermediate waypoints. When so utilised, the fuel burn and time outputs of the predictor 44 are with respect to such intermediate waypoints.

The F(K) and T_F (K) outputs of the predictor 44 are applied to a direct operating cost function block 45. The block" 45 also receives the unit cost of fuel constant C _F(cents/lb) as well as the K_o input described above with respect to Figure 1. For each trial cost index provided by the block 40, the direct operating cost function block 45 provides the direct operating cost (DOC) corresponding thereto in accordance with the DOC expression provided above.

The flight time signal T_F(K) from the predictor 44 and the direct operating cost signal (DOC) from the function block 45 are applied to a DOC vs. T_F function generation block 46. The three flight time and DOC values corresponding to the initial three trial cost index values (-10, 25 and 150) are sequentially applied to the block 46 and stored therein. The DOC vs. T_Ffunction block 46 is implemented to generate a quadradic polynomial approximation of the DOC vs. flight time curve passing through the three points stored therein. Standard parabola fitting techniques are utilised implementing an equation such as DOC = A₀+A₁T_F+A₂(T_F)². The three values of T_F and DOC provide three equations from which A_o, A₁ and A₂ may be derived. With the coefficience Ao, A₁ and A₂ now known constants, DOC is provided by the block 46 on a line 47 in response to a T_Finput on a line 48 utilising this parabolic polynomial. It will be appreciated that although parabolic interpolation is utilised to generate the DOC vs. T_Ffunction in the block 46, higher order polynomials may also be utilised to the same effect as well as other known means of curve fitting. As discussed above, Figure 3 illustrates the direct operating cost versus flight time curve for the representative flight discussed above.

The K_OPT, Top_T generator 10 further includes an AEC function storage block 49 for storing the arrival time error cost function discussed above. Conveniently, the AEC function is stored in a piece-wise linear fashion pursuant to the SIGMA construction as explained hereinabove. The AEC function storage block 49 also receives a signal T_s representative of the scheduled arrival time for the flight. A subtractor within the block 49 subtracts T_s from trial flight time inputs T_F to provide arrival time error T_E. T_E generates an AEC cost pursuant to the stored SIGMA function. The trial T_Fvalues are applied on a line 48 and the corresponding AEC cost values are applied to a line 50. The AEC function storage block 49 also provides the critical AEC time points on a line 51. The AEC function block 49 references the AEC critical points in terms of arrival time error T_E to the scheduled arrival time T_s by adding T_E to T_S to provide the AEC critical time points on the line 51. Figure 4 iflustrates a typical AEC function as described above.

The value of DOC on the line 47 and the value of AEC on the line 50 are applied to a summing function 52 to provide the sum thereof. The output of the summing function 52 is the total flight cost corresponding to the trial T_F signal applied on the line 48. Figure 3 (dashed line) illustrates the total flight cost curve for the representative flight discussed above.

The K_OPT, T_OPT generator 10 includes a search function block 53' which searches for the minimum point on the total flight cost curve to provide the optimum flight time T_OPT. The search function 53 provides a sequence of trial T_F values on the line 48 and the summing function 52 provides the total flight cost corresponding to the trial T_F. The search function 53 begins at the earliest reasonable flight time TF that is prior to the scheduled arrival time and receives a total flight cost value that is stored therein. TF is increased by, for example, sixty seconds, or to the next occurring AEC critical point if this is prior to the sixty second increment, and a second value of total flight cost is received by the search function 53 and stored. The search function 53 sets the direction signal on the line 42 to be representative of increasing T_F. The second value of total flight cost is compared to the first value thereof and if total flight cost is decreasing, the search is advanced by an additional sixty second increment or again to the next occurring critical point. The first value of total flight cost is then discarded. The search continues in this manner until total flight cost begins to increase. The previous trial T_Fvalue is then T_OPT. Alternatively, upon reaching the trial T _F value which results in the increased cost, the trial Tp may be decremented by, for example, ten seconds to provide T_op_T.

If, however, the comparison of the first two obtained total flight cost values indicates that increasing the trial Tp values results in increasing cost, the trial Tp values are decremented in increments of sixty seconds, accounting for the AEC critical points as described above, until the minimum value is obtained to provide T_op_T. If the direction of search is reversed, the direction signal on the line 42 is reversed to represent decreasing trial T _F. It will be appreciated that all of the critical points within the search range are utilised in finding the minimum total flight cost value.

If the search function 53 issues a trial Tp value that corresponds to an end point of the DOC vs. TFfunction stored in the block 46 without attaining the minimum value, a signal representative of minimum-not is applied to the line 41. The end point of the DOC vs. T_Ffunction corresponds to the end point of the range of trial cost index values provided by the block 40. If the minimum total flight cost point is not attained, as indicated by the signal on the line 41, the block 40 provides the next occurring cost index value in the reference set and the apparatus adds a point corresponding thereto to the DOC vs. T_F function stored in the block 46 in the manner described above. The search function 53 then searches for the minimum total flight cost value on the extended total flight cost curve. The trial K block 40 selects the next occurring trial cost index value in the direction indicated by the direction signal on the line 42. If the direction signal on the line 42 is representative of increasing T_F, the next smaller cost index value is utilised. If the direction signal on the line 42 is representative of decreasing T_F, the next larger cost index value is issued by the block 40.

When the minimum total flight cost value is attained, the. corresponding trial T_F is issued on a line 54 as the signal T_op_T.

The K_oPT, T_op_T generator 10 further includes a K vs T_F function generation block 55. The block 55 receives as inputs the trial cost index values provided from the block 40 and the corresponding flight time signal T_F(K) from the predictor 44. The function generating block 55 generates a parabolic approximation of the K vs T_F curve in the same manner as described above with respect to the block 46. The minimum-not signal on the line 41 is applied to the function blocks 46 and 55 to control extending the curves stored therein when the search function 53 determines that a minimum has not been attained as described above. The T_op_T signal on the line 54 is applied to the function block 55 which provides the corresponding cost index value on a line 56 as K_OPT. Figure 5 illustrates the flight time versus cost index function corresponding to the representative flight described above.

The apparatus and method described above represents a practical compromise between accuracy and processing load, predicated on the property that the total flight cost curve is relatively insensitive to change in cost index, and that sensitivity to arrival time is significant only in the vicinity of the AEC critical points, such as the - scheduled arrival time or connecting flight times. Since these critical points are predefined and utilised in the search procedure, the accuracy of an optimum point at a critical time is ensured. The procedure will normally require only three prediction passes.

Generally, however, the procedure for obtainng the optimum cost index K_op_T is:

• (1) Determine the time T_op_T which minimises total flight cost from a total flight cost versus flight time curve such as that discussed above with respect to Figure 3 and
• (2) Find the corresponding cost index from a cost index versus flight time curve such as that discussed above with respect to Figure 5.

Referring to Figure 6, an alternative embodiment for the K_OPT, T_OPT generator 10 is illustrated which generates the complete curves to any desired resolution. Like reference numerals represent like components with respect to Figure 2. A block 60 provides a sequence of cost index values with arbitrary resolution over the full range thereof. The generator 10 is activated by the output of the OR gate 20 of Figure 1 as discussed above with respect to Figure 2. Each value of K is processed by the speed generator 43, the predictor 44 and the DOC block 45 in the manner discussed above with respect to Figure 2. The values of DOC and T_F(K) are applied to a DOC vs. T_F function block 61 where the full range of curve values are stored therein in, for example, tabular form. The AEC function is stored in the block 49 in the manner described above with respect to Figure 2 and combined with DOC by the summing function 52. A search function 62 provides the trial T_F values to the blocks 61 and 49 to search for the minimum of the total flight cost curve in the manner described above with respect to Figure 2. Since the entire range of values is initially generated and stored in the block 61, the search function 62 determines the minimum T_op_T without utilising the minimum-not signal on the line 41 and the direction signal on the line 42 as described above with respect to Figure 2. The generator 10 of Figure 6 also includes a K vs T_F function generator 63 that stores the full range of cost index values from the block 60 and the corresponding T_F(K) values from the predictor 44 in, for example, tabular form. The T _OPT signal on the line 54 is then translated by table lookup into the corresponding K_op_Tvalue on the line 56.

It will be appreciated with respect to Figures 1, 2 and 6 that timing circuitry and control signals are applied to the various blocks to control the sequence of events described above. It will be appreciated that the DOC vs. T_F functions in the blocks 46 (Figure 2) and 61 (Figure 6) are first generated and then the search functions 53 (Figure 2) and 62 (Figure 6) perform the search in timed fashion thereafter. The various component blocks of Figures 1, 2 and 6 are preferably functional program segments of a stored program digital computer embodiment of the invention. In such a stored program version of the invention, it will be appreciated that the multiple uses of the predictor and the speed generator would be implemented in the software embodiment by calling the speed generator routine and the predictor routine when required, with appropriate inputs applied thereto. Alternatively, each of the blocks may be implemented, in a conventional manner, by discrete analogue or digital logic circuits.

After the Kpp_T, T_op_T generator 10 provides the otpimal arrival time Tpp_T and the approximately optimal cost index K_op_T, the apparatus of Figure 1 then fine-tunes the speed to obtain the high arrival time accuracy provided by the signal V_R as described above.

Referring to Figure 7, a diagram demonstrating convergence of the speed adjustment performed in the manner described above is illustrated. The previously described representative flight was executed with no wind and a cost index of zero, which results in a flight time of 2:19:58. Fourteen situations of time constraints were performed, corresponding to required arrival time that deviates from the resulting flight time by ± 15 seconds, ± 30 seconds, ± 1 minute, ± 2 minutes, ± 3 minutes, ± 4 minutes and ± 5 minutes. Figure 7 illustrates the result of the speed adjustment of the apparatus of Figure 1 after one and two adjustments. For example, if the non-adjusted speed results in a three- minute late arrival, then the first adjustment reduces the arrival error to 3.6 seconds and the second adjustment to -0.5 seconds. The early arrival situations tend to over adjust on the first adjustment. For example, when the non-adjusted speed corresponded to a five-minute early arrival, the result of the first adjustment was a 43-second late arrival. The second adjustment results in a 4.3- second early arrival. Figure 7 illustrates the rapidity with which the speed adjustment procedure converges. With just one or two prediction passes, the speed adjustment apparatus and procedure of the present invention provides very high arrival time accuracy.

Thus, the present invention provides a practical method for generating the true airspeed command in an FMS that results in minimum total flight cost. The method is comprised of two parts:

• (1) finding the arrival time and associated cost index that results in minimum total flight cost, and
• (2) fine-tuning the speed command obtained with this cost index to achieve high arrival time accuracy. The part one procedure is performed when the flight plan is changed and infrequently thereafter. The part two procedure is performed in connection with the prediction computations that are normally performed in the prior art FMS on a repeated basis.