TECHNICAL FIELD 
     This disclosure relates to the control of automatic transmission clutches. More particularly, the disclosure pertains to a method of adaptively modifying a mathematical model of the clutch and utilizing the model to control clutch engagement. 
     BACKGROUND 
     Many vehicles are used over a wide range of vehicle speeds, including both forward and reverse movement. Some types of engines, however, are capable of operating efficiently only within a narrow range of speeds. Consequently, transmissions capable of efficiently transmitting power at a variety of speed ratios are frequently employed. Transmission speed ratio is the ratio of input shaft speed to output shaft speed. When the vehicle is at low speed, the transmission is usually operated at a high speed ratio such that it multiplies the engine torque for improved acceleration. At high vehicle speed, operating the transmission at a low speed ratio permits an engine speed associated with quiet, fuel efficient cruising. 
     Many automatic transmissions implement a discrete number of different transmission ratios in which each ratio is establish by engaging a particular subset of clutches. A clutch that selectively holds a gearing element against rotation may be called a brake. Some clutches may be actively controlled such as by hydraulic actuation. Other clutches may be passive devices such as one way clutches. To shift from one speed ratio to another speed ratio, one clutch is engaged and another clutch is released. The engagement of the oncoming clutch must be coordinated with the release of the off-going clutch. If the off-going clutch is released before engaging the oncoming clutch, the transmission would temporarily be in a neutral state in which no power is transferred. On the other hand, if there is too much overlap, the transmission will be in a tie-up state. Coordination is much easier when the off-going clutch is a passive device such as a one-way clutch which automatically releases as the oncoming clutch attains sufficient torque capacity. 
     SUMMARY OF THE DISCLOSURE 
     A transmission includes a hydraulically actuated clutch and a controller programmed to estimate a position of a clutch piston based on a model and to adaptively adjust the model based on measured time delays. The model may be adjusted by changing the values of model parameters such as lumped flow coefficients or spring preload forces. The updates may be based on an estimate of the sensitivities of the delays to changes in model parameters. The sensitivity estimates may themselves also be updated based on the measured delays. The delays may be measured, for example, using a torque sensor that measures a shaft torque which changes in response to changes in the torque capacity of the clutch. 
     A method of controlling a clutch includes using a model to estimate a set of delays between initiating engagement and changes in the relationship between torque capacity and commanded pressure, then measuring the delays, then updating model parameters, and then controlling subsequent shifts based on the revised model to reduce the delays. The parameters may be updated based on the differences between the estimated and the measured delays and on an estimate of the sensitivities of the delays to changes in model parameters. Following subsequent engagements, the estimates of the sensitivities may also be updated. Using this method, the controller can update the model parameters following an upshift that does not include an engine flare such that the delay during a subsequent engagement is reduced. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic of a vehicle powertrain suitable for use with the disclosed method. 
         FIG. 2  is a schematic diagram of an exemplary transmission gearing arrangement suitable for use with the disclosed method. 
         FIG. 3  is a cross sectional view of a transmission clutch in a fully open position. 
         FIG. 4  is a cross sectional view of the clutch of  FIG. 3  in a first partially stroked position. 
         FIG. 5  is a cross sectional view of the clutch of  FIG. 3  in a second partially stroked position. 
         FIG. 6  is a cross sectional view of the clutch of  FIG. 3  in a fully stroked position. 
         FIG. 7  is a graph of commanded clutch pressures during a synchronous upshift. 
         FIG. 8  is a flow chart illustrating the disclosed method. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the present disclosure are described herein. It is to be understood, however, that the disclosed embodiments are merely examples and other embodiments can take various and alternative forms. The figures are not necessarily to scale; some features could be exaggerated or minimized to show details of particular components. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a representative basis for teaching one skilled in the art to variously employ the present invention. As those of ordinary skill in the art will understand, various features illustrated and described with reference to any one of the figures can be combined with features illustrated in one or more other figures to produce embodiments that are not explicitly illustrated or described. The combinations of features illustrated provide representative embodiments for typical applications. Various combinations and modifications of the features consistent with the teachings of this disclosure, however, could be desired for particular applications or implementations. 
     A powertrain of a vehicle  10  is illustrated schematically in  FIG. 1 . Solid lines indicate mechanical connections. Broken lines represent the flow of signals. Double lines represent the flow of fluid. Engine  12  provides power to rotate crankshaft  14 . Transmission  16  transits the power from crankshaft  14  to driveshaft  18  while potentially modifying the speed and torque to be more suitable to the present vehicle requirements. Differential  20  distributes the power to a left wheel  22  and a right wheel  24  while permitting slight speed differences between the wheels such as when the vehicle turns a corner. 
     The transmission includes a torque converter  24  and a gearbox  26 . The torque converter transmits the power from crankshaft  14  to turbine shaft  28 . Gearbox  26  transmits the power from turbine shaft  28  to driveshaft  18 . Controller  30  sends signals to valve body  32  causing valve body  32  to send pressurized fluid to clutches in gearbox  26 . The gear ratio of gearbox  26  depends upon which subset of the clutches are provided with pressurized fluid. Controller  30  utilizes many inputs to determine what commands to send to valve body  32  including signals from output torque sensor  34  and turbine torque sensor  36 . 
     An example gearbox is schematically illustrated in  FIG. 2 . The proposed method is applicable to a wide variety of gearbox arrangements. The gearbox utilizes four simple planetary gear sets  40 ,  50 ,  60 , and  70 . Sun gear  42  is fixed to sun gear  52 , carrier  44  is fixed to ring gear  76 , ring gear  56  is fixed to sun gear  62  by shaft  80 , ring gear  66  is fixed to sun gear  72 , turbine shaft  28  is fixed to carrier  54 , and driveshaft  18  is fixed to carrier  74 . Ring gear  46  is selectively held against rotation by brake  88  and sun gears  42  and  52  are selectively held against rotation by brake  90 . Turbine shaft  28  is selectively coupled to ring gear  66  and sun gear  72  by clutch  92 . Intermediate element  82  is selectively coupled to carrier  64  by clutch  94 , selectively coupled to carrier  44  and ring gear  76  by clutch  96 , and selectively coupled to shaft  80  by clutch  98 . 
     As shown in Table 1, engaging the clutches and brakes in combinations of four establishes ten forward speed ratios and one reverse speed ratio between turbine shaft  28  and driveshaft  18 . An X indicates that the corresponding clutch is engaged to establish the speed ratio. 
     
       
         
               
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                   
                 88 
                 90 
                 92 
                 94 
                 96 
                 98 
                 Ratio 
                 Step 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Rev 
                 X 
                 X 
                   
                 X 
                 X 
                   
                 −4.79 
                 102% 
               
               
                 1st  
                 X 
                 X 
                 X 
                 X 
                   
                   
                 4.70 
                   
               
               
                 2nd 
                 X 
                 X 
                   
                 X 
                   
                 X 
                 2.99 
                 1.57 
               
               
                 3rd  
                 X 
                   
                 X 
                 X 
                   
                 X 
                 2.18 
                 1.37 
               
               
                 4th 
                 X 
                   
                   
                 X 
                 X 
                 X 
                 1.80 
                 1.21 
               
               
                 5th 
                 X 
                   
                 X 
                   
                 X 
                 X 
                 1.54 
                 1.17 
               
               
                 6th 
                 X 
                   
                 X 
                 X 
                 X 
                   
                 1.29 
                 1.19 
               
               
                 7th 
                   
                   
                 X 
                 X 
                 X 
                 X 
                 1.00 
                 1.29 
               
               
                 8th 
                   
                 X 
                 X 
                 X 
                 X 
                   
                 0.85 
                 1.17 
               
               
                 9th 
                   
                 X 
                 X 
                   
                 X 
                 X 
                 0.69 
                 1.24 
               
               
                 10th  
                   
                 X 
                   
                 X 
                 X 
                 X 
                 0.64 
                 1.08 
               
               
                   
               
             
          
         
       
     
     All single step and two step shifts are performed by gradually engaging one clutch, called an oncoming element, while gradually releasing a different clutch, called the off-going element. During each of these shifts, three clutches, called holding elements, are maintained fully engaged while one element is maintained fully disengaged. In other gearbox arrangements, the number of holding elements may be different. 
       FIG. 3  shows a cross section of clutch  98 . Clutch housing  82 , the intermediate element in  FIG. 2 , is supported to rotate around shaft  80 , which is, in turn, supported to rotate around turbine shaft  28 . A set of separator plates  100  are splined to housing  82  such that they rotate with housing  82  but are free to slide axially. Snap ring  102  restrains the axial movement toward the right. A set of friction plates  104  are splined to shaft  80  and are interleaved with the separator plates  100 . The friction plates and separator plates may collectively be referred to as a clutch pack. When pressurized hydraulic fluid is routed to apply chamber  106 , piston  108  slides axially with respect to housing  82 . After piston  108  moves into contact with the clutch plate, the force squeezes the friction plates and separator plates together. Friction between the friction plates and separator plates transmits torque between housing  82  and shaft  80 . The maximum amount of torque that can be transmitted at a given moment is called the torque capacity. Isolation spring  110  contacts the clutch pack slightly before the piston itself smoothing the transition. When the pressure in apply chamber  106  is relieved, return spring  112  pushes piston  108  away from the clutch pack to release the clutch. When clutch housing  82  rotates, centrifugal force tends to increase the pressure of fluid in apply chamber  106  which tends to engage the clutch. To avoid unintentional engagement, unpressurized fluid is routed to balance chamber  112 . Centrifugal force pressurizes the fluid in the balance chamber, counteracting the force generated by centrifugal force in the apply chamber. 
     Controller  30  regulates the current (or pulse width) to a solenoid in valve body  32  such that the pressure in a particular channel within valve body is regulated to a commanded pressure. The fluid then flow from the valve body to apply chamber through passageway  114 . Passageway  114  goes through front support  116 , through turbine shaft  28 , through shaft  80 , and into housing  82 . Since these components rotate at different speeds, seals  118  are used to route fluid from one component to another component. Similarly, the unpressurized fluid is routed to balance chamber  112  through passageway  120 . Due to the resistance of passageways  114  and  120 , the pressure in apply chamber  106  is not always equal to the commanded pressure in the valve body. When fluid is flowing from the valve body to the apply chamber, the pressure in the apply chamber will be less than the commanded pressure. The pressure is higher in the apply chamber when the fluid is flowing the opposite direction. During a preparatory phase of a shift, the controller adjust the commanded pressure to move the piston into contact with the clutch pack. In the absence of a measured feedback signal during this phase, the controller must utilize a mathematical model to determine the axial position of the clutch piston. 
     Although many factors influence the axial position of the clutch piston, the axial position of the clutch piston can be modeled with sufficient accuracy using a lumped parameter model. This model is described in detail in U.S. patent application Ser. No. 13/766,829 which is incorporated by reference herein. Specifically, the flow rate, Q, may be predicted based on pressure difference, ΔP, using the formula: 
             Q   =         K   1     *   Δ   ⁢           ⁢   P     +       K   2     *       Δ   ⁢           ⁢   P                 
where K 1  and K 2  are lumped flow coefficients. The pressure difference is defined by the formula:
 
Δ P=P   cmd   −P   apply  
 
where P cmd  is the commanded pressure and P apply  is the pressure in the apply chamber. The flow rate is related to the rate of movement of the clutch piston by the formula:
 
 {dot over (x)}=−Q/A  
 
where A is the area of the clutch piston and x is the distance between the piston and the clutch pack. When the clutch is fully applied, x=0. When the clutch is fully released, x=x max . In between these two conditions, the clutch position may be estimated by integrating Q/A.
 
     The relationships among x, P apply , and the clutch torque capacity, T, vary based on clutch position. In region  1 , when the clutch is fully released as shown in  FIG. 3 : 
               P   apply     =       1   A     *     (       F   0     -       K   rs     *     x   max         )                   T   =   0         
where F 0  is the return spring force when the clutch is fully applied and K rs  is the spring rate of the return spring.
 
     When P cmd  exceeds P apply , the clutch piston moves into region  2 , as illustrated in  FIG. 4 . In region  2 , the clutch has moved from the fully open position but the isolation spring is not in contact with the clutch pack. Region  2  is governed by the equations: 
     
       
         
           
             
               P 
               apply 
             
             = 
             
               
                 1 
                 A 
               
               * 
               
                 ( 
                 
                   
                     F 
                     0 
                   
                   - 
                   
                     
                       K 
                       rs 
                     
                     * 
                     x 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             T 
             = 
             0. 
           
         
       
     
     When x becomes less than the free length of the isolation spring, x is , as shown in  FIG. 5 , then the region  3  equations apply: 
               P   apply     =       1   A     *     (       F   0     +       K   is     *     x   is       -       (       K   rs     +     K   is       )     *   x       )                   T   =       K   is     *     (       x   is     -   x     )     *   2   *   N   *   μ   *   r           
where K is  is the spring rate of the isolation spring, N is the number of friction plates in the clutch pack, μ is the friction coefficient, and r is the effective radius of the clutch pack.
 
     Finally, when x=0 as shown in  FIG. 6 , the piston is in contact with the clutch pack and the region  4  equations apply. In region  4 , the pressure response can be represented with a first order transfer function featuring time constant t p  and a delay t d . 
     
       
         
           
             
               
                 
                   P 
                   apply 
                 
                 
                   P 
                   cmd 
                 
               
               ⁢ 
               
                 ( 
                 s 
                 ) 
               
             
             = 
             
               
                 1 
                 
                   1 
                   + 
                   
                     
                       t 
                       p 
                     
                     * 
                     s 
                   
                 
               
               * 
               
                 e 
                 
                   ( 
                   
                     
                       - 
                       
                         t 
                         d 
                       
                     
                     * 
                     s 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             T 
             = 
             
               
                 ( 
                 
                   
                     A 
                     * 
                     
                       P 
                       apply 
                     
                   
                   - 
                   
                     F 
                     0 
                   
                   - 
                   
                     
                       K 
                       is 
                     
                     * 
                     
                       x 
                       is 
                     
                   
                 
                 ) 
               
               * 
               2 
               * 
               N 
               * 
               μ 
               * 
               r 
             
           
         
       
     
     Many of the parameters of the mathematical model described above will fluctuate between transmissions of the same design due to manufacturing variability. Furthermore, many of the parameters will fluctuate over time due to transmission wear and due to variation in operating conditions. For example, the flow coefficients K 1  and K 2  depend on the viscosity of the fluid, which depends strongly on temperature among other factors. This unknown variation may cause the clutch piston position predicted by the mathematical model to differ slightly from the actual clutch piston position. 
       FIG. 7  illustrates commanded pressure profiles for the oncoming clutch (ONC) and the off-going clutch (OFG) during a synchronous upshift. After the shift scheduling algorithm, or a driver command, indicates that an upshift should be performed, the controller strokes the oncoming clutch during a preparatory phase. At  130 , the pressure to the oncoming clutch is raised to a boost level for a boost duration. The purpose of the boost phase is to move the clutch piston as rapidly as possible from the disengaged position to the stroked position. The boost duration is generally selected such that the boost phase ends slightly before the piston is stroked. Then, at  132 , a holding pressure is commanded. The holding pressure is generally selected to approximate the apply chamber pressure that would overcome the return spring and the isolation spring forces with the piston in the fully stroked position. Then, at  134 , the commanded pressure is gradually increased to gently move the piston into the fully stroked (region  4 ) position. The preparatory phase ends when the piston of the oncoming clutch is fully stroked at  136 . During this time, the commanded pressure for the off-going clutch may be reduced, as shown at  138 , such that the torque capacity is equal to the torque actually transmitted by the off-going clutch. Throughout this preparatory period, neither the torque ratio nor the speed ratio change. However, the controller can predict the position of the oncoming clutch piston using the mathematical model as describe in U.S. patent application Ser. No. 14/036,316 which is incorporated by reference herein. When this preparatory period lasts too long, the transmission appears to the driver to be unresponsive. 
     Once the preparatory phase is complete, the torque transfer phase is executed by gradually reducing the commanded pressure of the off-going clutch as shown at  140  and gradually increasing the commanded pressure to the oncoming clutch as shown at  142 . During this phase, the torque ratio gradually decreases to the upshifted value. Ideally, the speed ratio remains constant, although it can increase if the off-going clutch is released too quickly compared to the rate at which the oncoming clutch is engaged. Once the torque capacity of the off-going clutch reaches zero at  144 , the inertia phase begins. During the inertia phase, the torque capacity of the oncoming clutch acts to slow the input, gradually decreasing the speed ratio to the upshifted value. 
     Shift quality suffers if the boost duration or holding pressure are selected improperly. If the boost duration is too long, it may continue after the clutch is fully stroked. If this happens, the oncoming clutch will have significant torque capacity before the controller begins releasing the off-going clutch. The resulting tie-up condition results in an abrupt drop in the torque ratio and a rough shift. Similarly, if the holding pressure is too high, the torque capacity will increase rapidly as the stroke position is reached, resulting in a tie-up condition and a drop in torque ratio. On the other hand, if the boost duration is too short or the holding pressure is too low, then the piston will not yet be at the stroke position when the controller begins the torque phase. The torque capacity will not ramp up as expected but will instead remain at zero until the clutch reaches the stroke position. In the interim, since the off-going clutch is being released, the input speed will increase in a phenomenon called an engine flare. 
     Traditionally, feedback signals from speed sensors have been used to adapt the boost duration and the holding pressure. If a flare is detected, the boost duration or the holding pressure is increased. Unfortunately, speed sensors do not indicate tie-up conditions, so they do not provide useful feedback for determining that the boost duration or the holding pressure are too high. Instead, the controller must gradually decrease these values until a flare is detected. Use of speed signals to adapt the boost duration and holding pressure requires periodic bad shifts. 
     The transmission is equipped with one or more torque sensors which measure the torque transmitted by one or more shafts, respectively. At least one of these shafts is selected such that the shaft torque changes in response to a change in the torque capacity of the incoming clutch. For example, the torque sensor may be located on a turbine shaft, an output shaft, or on the clutch itself. U.S. Pat. No. 8,510,003 describes a method of computing clutch torque capacity based on torque sensor readings. When the torque capacity begins to change with piston displacement, the controller can conclude that the piston has transitioned from region  2  to region  3 . When the piston transitions from region  3  to region  4 , the torque capacity begins to increase much more rapidly. The time delay between initiating the shift event and transitioning from region  2  to region  3  is called t 23 . Similarly, the time delay between initiating the shift event and transitioning from region  3  to region  4  is called t 34 . 
     A mathematical model of the clutch, as described above, can be used to estimate t 23  and t 34 . The difference between the estimates and the measured values can be used to revise the parameters of the mathematical model. Let the measured values for the nth shift event be called t 23 (n) and t 34 (n). The model based predictions will be called            (n) and          (n), respectively. In matrix notation, the error vector is:
     
       
         
           
             
               
                 
                   
                     e 
                     ⁡ 
                     
                       ( 
                       n 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 ⁢ 
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   t 
                                   23 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ⁢ 
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   t 
                                   34 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   n 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                     
                 
               
             
           
         
       
     
     The set of parameters to be adapted are denoted by the vector u. For example, if the two lumped flow parameters, K 1  and K 2 , are the only parameters to be adapted, then: 
     
       
         
           
             u 
             = 
             
               
                 [ 
                 
                   
                     
                       
                         K 
                         1 
                       
                     
                   
                   
                     
                       
                         K 
                         2 
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
     If these parameters are changed by an amount Au, then the error will change by an amount, Δe as approximated by the equation:
 
Δ e=J*Δu  
 
where J, called the Jacobian, indicates the sensitivity of the delays to changes in the model parameters. In this example,
 
     
       
         
           
             J 
             = 
             
               [ 
               
                 
                   
                     
                       
                         ∂ 
                         
                           t 
                           23 
                         
                       
                       
                         ∂ 
                         
                           K 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         ∂ 
                         
                           t 
                           23 
                         
                       
                       
                         ∂ 
                         
                           K 
                           2 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         ∂ 
                         
                           t 
                           34 
                         
                       
                       
                         ∂ 
                         
                           K 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         ∂ 
                         
                           t 
                           34 
                         
                       
                       
                         ∂ 
                         
                           K 
                           2 
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
     In this example, since the dimensions of Δu(n) and Δe(n) are identical, there is a unique Δu such that e+Δe=0. Theoretically, adjusting u by this amount would completely eliminate the error. However, adjusting model parameters too rapidly may cause instability and excessive variation among shifts. Therefore, a maximum change may be imposed. Also, there may be minimum and maximum values for each parameter. If a solution cannot be found subject to these constraints, then the value of Au that minimizes |e+Δe| subject to the constraints is selected. The parameters are then adjusted accordingly and future shifts are controlled based on the piston position as predicted by the mathematical model. 
     The method is not limited to problems in which the dimensions of Δu and Δe are identical. If there are more parameters to adapt, such as the return spring pre-load force F 0 , then it is possible that multiple solutions would exist such that e+Δe=0. In this case, the solution that minimizes the amount of change in the parameters is selected. As with the case of equal dimensions, there may be no solutions that satisfy the constraints in which case the solution that minimizes the error subject to the constraints is selected. If there are fewer parameters to adapt than measured delays, then constrained optimization is used. 
     An advantage of adapting model parameters using measured delays based on torque measurements is that the system can adapt following a shift that does not induce an engine flare or other degradation in shift quality measurable with a speed sensor. For example, if the current model parameters predict that the clutch will move faster than it actually does move, then the piston will not yet be in the stroke position after the predicted delay. If the controller were relying exclusively on speed sensors for feedback, it would have no way of knowing that the piston was not yet stroked and would begin the torque transfer phase. Since the oncoming clutch piston is not yet stroked, the torque capacity would not increase as expected and an engine flare would result as the off-going clutch is gradually released. However, with the proposed system, the controller will know based on torque sensor feedback that the oncoming clutch piston is not yet stroked. Therefore, it will wait to begin the torque transfer phase until the piston is stroked. Although the torque transfer phase will begin later than optimal, it will not produce an engine flare event. Following the shift, the controller will adapt model parameters to improve the estimation of piston position. The next time the oncoming clutch is engaged, the controller will use a longer boost duration or a higher stroke pressure, reducing the delay. 
     Due to the non-linearity of the system, a change in the model parameters may result in a change in the Jacobian. Even if the sensitivity of the underlying system has not changed significantly, the change in response due to a change in model parameters may provide an opportunity to refine the current estimate of the Jacobian matrix. A method based on Kalman filtering to update the Jacobian based on updated measurements is discussed in detail in U.S. Pat. No. 6,701,193. 
       FIG. 8  illustrates an adaptive clutch control method. Initial values are set for various parameters at  150 . When an upshift is scheduled at  152 , the iteration counter is incremented at  154 . At  156 , the pressure to the oncoming clutch is controlled to stroke the piston based on the position as predicted by the model. During the preparatory phase, the controller keeps track of the starting time of the boost phase, the time that torque capacity is first detected, and the time that torque capacity begins to respond linearly to pressure commands. The differences between these times indicate the delays t 23 (n) and t 34 (n). At  158 , the controller completes the torque transfer phase and inertia phases of the shift. The error vector and change in error vector are computed at  160  and  162 . 
     At  164 , the controller checks to see if the model parameters have changed by enough to justify revising the estimate of the Jacobian. If the parameters have changed by enough, the Jacobian is updated at  166 . If not, the previous Jacobian is carried over at  168 . At  170  and  172 , the controller determines the revised model parameters as described above. 
     While exemplary embodiments are described above, it is not intended that these embodiments describe all possible forms encompassed by the claims. The words used in the specification are words of description rather than limitation, and it is understood that various changes can be made without departing from the spirit and scope of the disclosure. As previously described, the features of various embodiments can be combined to form further embodiments of the invention that may not be explicitly described or illustrated. While various embodiments could have been described as providing advantages or being preferred over other embodiments or prior art implementations with respect to one or more desired characteristics, those of ordinary skill in the art recognize that one or more features or characteristics can be compromised to achieve desired overall system attributes, which depend on the specific application and implementation. These attributes can include, but are not limited to cost, strength, durability, life cycle cost, marketability, appearance, packaging, size, serviceability, weight, manufacturability, ease of assembly, etc. As such, embodiments described as less desirable than other embodiments or prior art implementations with respect to one or more characteristics are not outside the scope of the disclosure and can be desirable for particular applications.
A hydraulically actuated clutch is controlled based on a mathematical model that predicts the position of the clutch piston based on the profile of the commanded pressure over time. During a clutch engagement, a set of delays associated with the preparatory phase of engagement are measured and compared to predicted values. Model parameters are then adaptively adjusted based on the differences between the measured and predicted delays and on an estimate of the sensitivities between model parameters and the delays. Subsequent engagements are controlled based on the updated mathematical model to reduce the delays. Changes in the measured delays following adaptation of the model parameters are utilized to update the estimate of the sensitivities.