Patent Publication Number: US-6220676-B1

Title: Antiskid control of multi-wheel vehicles using coupled and decoupled Kalman filtering incorporating pitch weight transfer

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation-in-part of U.S. patent application Ser. No. 08/853,555, entitled “Antiskid Brake Control System Using Kalman Filtering”, and filed on May 9, 1997, now U.S. Pat. No. 5,918,951. 
    
    
     TECHNICAL FIELD 
     The present invention relates generally to brake control systems, and more particularly to antiskid brake control systems using Kalman filter techniques. 
     BACKGROUND OF THE INVENTION 
     Antiskid brake control systems have been in widespread use for many years. In the simplest sense, an antiskid brake control system compares the speed of a vehicle derived from a wheel speed sensor (and wheel radius) to the vehicle speed derived from a secondary or reference source. If the wheel is determined to be skidding an excessive amount, then brake pressure applied to the wheel is released and the wheel is allowed to spin back up to the appropriate speed. 
     There are, of course, two major problems that immediately become apparent in any such antiskid system. The first relates to determining the appropriate amount of skidding. The second relates to determining from where to obtain the reference velocity. The appropriate amount of skidding is described by the much discussed but seldom measured mu-slip curve. Typically such curve is represented by the coefficient of friction μ between the wheel and the running surface on a vertical axis and the slip ratio on the horizontal axis. A slip ratio of zero is when the wheel is not skidding while a slip ratio equal to one represents a fully locked wheel. 
     The amplitude and peak location of the mu-slip curve unfortunately can vary substantially for different running surfaces or even the same running surface. A lower amplitude mu-slip curve may represent an ice or water patch. Ideally, the antiskid brake control system should allow the wheel to slip at the peak of the mu-slip curve which provides the maximum stopping power. Antiskid brake control systems are commonly accepted to be ninety percent efficient which means that, on average, the control system should be within ten percent of the mu-slip peak regardless of the value or location of the peak. However, since the mu-slip curve depends on so many variables (e.g., and without being limited thereby, tire tread groove pattern, tire tread compound, temperature, tire pressure, running surface material and finish, etc.), the mu-slip curve begins to resemble a random variable. This makes it difficult for conventional antiskid brake control systems to track adequately the peak of the mu-slip curve. 
     Furthermore, it is not always easy to obtain a reference velocity of the vehicle. The braked wheel cannot be used freely because the wheel might be skidding. In the case of an aircraft brake control system, the nose wheel speed could be used for providing a reference velocity. Unfortunately, there could be instances where the nose wheel has not touched down yet or perhaps is worn or has low pressure so as to have a reduced radius. Alternatively, for example, the nose wheel may be locked due to a defective bearing. These instances will result in an erroneous reference velocity. A global positioning satellite (GPS) system or Doppler radar could provide a reference velocity but it may be inoperable or jammed. Alternatively, an inertial navigation system might provide a reference velocity. However, such systems can be prone to measurement error and/or relatively high cost. 
     Recently there have been efforts to utilize optimal state estimation techniques, such as Kalman filters, in antiskid brake control systems. For example, U.S. Pat. No. 4,679,866 to van Zanten et al. discusses a method for ascertaining a set-point braking moment using a Kalman filter. U.S. Pat. No. 4,715,662 to van Zanten et al. describes a method for determining an optimal slip value using a Kalman filter. While such systems attempt to take advantage of state estimation techniques in determining the optimal slip value, etc., there still remain difficulties as far as observability, speed of convergence, process and measurement noise, etc. Matters such as these can greatly affect the ability of such a system to actually operate in practice. 
     In view of the aforementioned shortcomings associated with conventional antiskid brake control systems, there is a strong need in the art for a system capable of accurately and reliably ascertaining the relevant parameters such as the appropriate amount of skidding and reference velocity. Moreover, there is a strong need in the art for a system which can ascertain such information more rapidly and accurately as compared to conventional systems. 
     SUMMARY OF THE INVENTION 
     According to one aspect of the invention, a brake controller is provided for controlling an amount of braking force applied to a plurality of wheels of a vehicle running on a surface. Each of the plurality of wheels has a sensor for providing an output signal indicative of a corresponding speed of the wheel. The controller includes a state estimator which estimates an amount of friction between each of the plurality of wheels and the surface based on the output signals of the sensors, the estimated amount of friction for each wheel being based on the estimated amount of friction for the other wheels. The controller further includes a control output which adjusts the amount of braking force applied to the plurality of wheels based on the estimated amount of friction. 
     According to another aspect of the invention, a method is provided for controlling an amount of braking force applied to a plurality of wheels of a vehicle running on a surface. In such method, each of the plurality of wheels has a sensor for providing an output signal indicative of a corresponding speed of the wheel. The method includes the steps of estimating an amount of friction between each of the plurality of wheels and the surface based on the output signals of the sensors, the estimated amount of friction for each wheel being based on the estimated amount of friction for the other wheels, and adjusting the amount of braking force applied to the plurality of wheels based on the estimated amount of friction. 
     To the accomplishment of the foregoing and related ends, the invention, then, comprises the features hereinafter fully described and particularly pointed out in the claims. The following description and the annexed drawings set forth in detail certain illustrative embodiments of the invention. These embodiments are indicative, however, of but a few of the various ways in which the principles of the invention may be employed. Other objects, advantages and novel features of the invention will become apparent from the following detailed description of the invention when considered in conjunction with the drawings. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a block diagram of an aircraft antiskid brake control system using Kalman filtering in accordance with a preferred embodiment of the present invention; 
     FIG. 2A is a schematic representation of the brake control system of FIG. 1 in accordance with the present invention; 
     FIG. 2B is a block diagram of the Kalman filter based antiskid controller used in the brake control system in accordance with the present invention; 
     FIG. 3 is a simplified flowchart representing the operation of the brake control system in accordance with the present invention; 
     FIG. 4 is an exemplary mu-slip curve illustrating the relationship between the coefficient of friction (μ) and the slip ratio (r slip ) between an aircraft wheel and a runway surface in accordance with the present invention; 
     FIGS. 5A and 5B show a detailed flowchart describing more specifically the operation of the brake control system in accordance with the present invention; 
     FIG. 6 is flowchart representing a routine for computing a brake command pressure signal delivered to the brake actuator in accordance with the present invention; 
     FIG. 7 is a flowchart illustrating a routine for determining whether the brake has begun or ended a skid condition in accordance with the present invention; 
     FIG. 8A is a flowchart representing a routine for detecting if the brake is currently in a skid condition in accordance with the present invention; 
     FIG. 8B is a flowchart illustrating a routine for detecting whether a skid condition is not occurring in accordance with the present invention; 
     FIG. 9 is a graph which illustrates the sensitivity of the peak slip ratio r p  in accordance with the present invention; 
     FIG. 10A is a graph representing a least observable scenario in accordance with the present invention; 
     FIG. 10B is a graph representing a resetting of the initial conditions in the event a skid condition occurs in accordance with the present invention; 
     FIG. 11 is a flowchart which illustrates setting of the observation matrices in accordance with the present invention; 
     FIG. 12 is a flowchart illustrating a routing for initializing/reinitializing the system in accordance with the present invention; 
     FIG. 13 is a flowchart defining the manner in which the covariance matrix is reset in the event a skid condition occurs in accordance with the present invention; 
     FIG. 14 is a flowchart describing a routine for range checking the system parameters in accordance with the present invention; 
     FIG. 15 is a schematic view of a three wheel or five wheel aircraft incorporating a brake control system in accordance with the present invention; 
     FIG. 16 is a schematic diagram of a three/five wheel brake control system using a coupled Kalman filter in accordance with a particular embodiment of the present invention; and 
     FIG. 17 is a schematic diagram of a three/five wheel brake control system using a decoupled Kalman filter in accordance with another embodiment of the present invention. 
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The present invention will now be described with reference to the drawings, wherein like reference labels are used to refer to like elements throughout. 
     Referring initially to FIG. 1, an antiskid brake control system for an aircraft in accordance with the present invention is generally designated  30 . Brake control on an aircraft is usually structured in a paired wheel configuration for functional modularity. For example, if the aircraft has two wheels on the left side of the aircraft and two wheels on the right side, the outer two wheels form a pair and the inner two wheels form another pair. Within a pair there is a right wheel control and left wheel control. The left and right wheel control functions are uncoupled except for locked wheel protection. Locked wheel protection (which compares left and right wheel speed and releases pressure when the wheel speeds differ by 30%) is considered a different function than antiskid control. The basic unit therefore consists of control of a single wheel which can be left or right. Nevertheless, the effect of one wheel on another may further be taken into consideration as is discussed more fully below. As utilized herein, it will be appreciated that the term “wheel” is intended to refer collectively to both the wheel and tire. 
     For sake of simplicity, the brake control system  30  as shown in FIG. 1 represents the basic unit for providing antiskid control of a single wheel (left or right). However, it will be appreciated that control for the other wheel(s) can be provided via corresponding systems  30  or in a single system incorporating the same inventive principles. Moreover, the preferred embodiment of the present invention provides antiskid brake control in connection with an aircraft. Nevertheless, it will be appreciated that the brake control system of the present invention has utility for virtually any type of vehicle and is not limited necessarily to brake control for aircraft. 
     The system  30  includes a pilot brake device  32  for providing operator brake control. In addition, the system  30  includes a Kalman filter based antiskid controller  34 . The controller  34  employs state estimation techniques as is described more fully below in order to provide optimum brake control during antiskid conditions. The system  30  further includes the aircraft electronics  36  for providing measurement data such as aircraft velocity and acceleration to the controller  34 . The controller  34  provides a control signal to a brake actuator  36  also included in the system  30 . The brake actuator  36  may be any conventional type actuator (e.g., hydraulic or electromechanical) for applying pressure to the brake friction material (not shown) in a brake assembly  38 . The brake assembly  38  in turn provides braking action to a wheel  40  as is conventional. 
     The system  30  preferably includes a velocity sensor  42  and brake torque sensor  44 . The velocity sensor  42  senses the angular velocity of the wheel  40 . The brake torque sensor  44  measures the amount of torque exerted by the wheel  40  on the brake assembly  38 . The measured velocity and torque are input to the controller  34 . As will be discussed in detail below, the controller  34  processes the wheel velocity (speed), brake torque, aircraft velocity and acceleration using a Kalman filter in order to provide appropriate brake pressure to the brake assembly  38  to provide optimum braking. 
     Generally describing the operation of the control system  30 , the pilot brake device  32  comprises a pedal or an equivalent thereof. During braking, the pilot of the aircraft activates the pilot brake device  32  by pushing the pedal (or its equivalent). The depression of the pedal is converted to an electrical signal (PILOT PRESSURE) which is provided to the controller  34 . The PILOT PRESSURE signal is indicative of the degree of depression of the pedal, and is related to the amount of brake pressure requested by the pilot as is conventional. 
     The controller  34  receives the request for brake pressure via the PILOT PRESSURE signal and compares it to a maximum brake pressure that the Kalman filter within the controller  34  estimates can be applied without incurring an undesirable skid condition. The lower of the pilot requested pressure and the estimated maximum pressure is then passed to a rate limiter within the controller  34 . Thereafter, the resultant pressure request is passed on to the actuator  36  in the form of a command pressure (P c ) signal. The value of the P c  signal is indicative of the amount of brake pressure to be applied by the actuator  36 . The actuator  36 , using conventional techniques, then applies pressure to the brake friction material in the brake assembly  38  based on the value of the P c  signal. As a result, braking action occurs in the form of a reduction of wheel speed and an increase in brake torque and wheel slippage. 
     As will be described more fully below, the Kalman filter-based antiskid controller  34  receives, at a minimum, the measured wheel speed (ω m ) as an input from the velocity sensor  42 . However, in the preferred embodiment the controller  34  also takes in the measured brake torque (T m ) from the sensor  44  together with the measured aircraft velocity (V m ) and acceleration ({dot over (V)} m ) provided by the aircraft electronics  36 . The aircraft velocity V m  and acceleration {dot over (V)} m  may be provided by a conventional velocity sensor and accelerometer, respectively, included in the aircraft electronics  36 . From such measured data, the Kalman filter within the controller  34  estimates the peak of the mu-slip curve as is discussed in detail below. The mu-slip curve represents the relationship between the coefficient of friction (μ) between the wheel  40  and the runway surface and the amount of slip between the wheel  40  and runway surface. The controller  34  then utilizes the estimated peak of the mu-slip curve to calculate the peak pressure that can be applied by the brake  38  without undesirable skidding. 
     The Kalman filter algorithm carried out within the controller  34  differs from conventional antiskid control algorithms by minimizing the excursions past the peak of the mu-slip curve which can result in excess tire wear on the wheel  40 , and in extreme cases, flat spotting and flats. Conventional antiskid algorithms find the peak of the mu-slip curve by going past the peak and identifying a skid by high wheel deceleration (or other means). The Kalman filter algorithm as implemented in the present invention has many less excursions past the peak because it continuously estimates the location of the peak as discussed below. 
     Generally speaking, Kalman filters are typically used where there are unknown or poorly known parameters or models. In the context of antiskid control, the primary unknown parameter is the peak of the mu-slip curve with respect to both amplitude and location. As is described below, the amplitude and location of the peak are modeled as “constants” in the Kalman filter although two techniques are utilized to track changes in these parameters. Slowly varying changes are accounted for by adding fictitious noise to the friction parameters. Step-like changes in the parameters are handled by resetting the covariances within the filter. 
     Use of the Kalman filter based controller  34  has other advantages in that measurements other than the conventional wheel speed can be incorporated by allowing for the appropriate observation matrix to be set as discussed below. An additional advantage is that since a model of the aircraft, wheel and brake is built into the Kalman filter algorithm, the algorithm is easily adapted to other aircraft simply by incorporating the specific aircraft constants (e.g., wheel and brake inertia). 
     FIG. 2A presents a schematic illustration of the system  30  with emphasis on the controller  34 . Block  50  represents the “real” system hardware including, for example, the wheel  40 , brake assembly  38 , actuator  36 , friction between the brake assembly friction material and the wheel  40 , the aircraft electronics  36 , pilot brake device  32 , velocity sensor  42 , brake torque sensor  44 , as well as any other physical parameters (including the aircraft itself) which may be considered when employing the Kalman filter algorithm. Block  50  provides as an output signals representing measured data including, for example, aircraft velocity V m  and acceleration {dot over (V)} m , PILOT PRESSURE, wheel speed ω m , and brake torque T m . Such measured data is input to the controller  34 . 
     Block  52  within the controller  34  represents a mathematical model of the braking system hardware. Those having ordinary skill in the art of digital filtering will appreciate that the system hardware can be modeled as a set of equations including a set of state equations. These equations are utilized within the controller  34  to provide estimated data relating to the various parameters within the system  30  including, for example, the estimated aircraft velocity, acceleration, wheel speed, etc. The controller  34  compares the measured data to the estimated data via an adder  54 . The difference between the measured data and the estimated data represents a system error which is provided to the Kalman filter  56 . In turn, the Kalman filter  56  computes Kalman gains and outputs updated state estimates. The state estimates are provided to a control laws section  58  which computes a brake pressure set point from the state estimates and outputs a brake command pressure (P c ) The brake command pressure P c  is provided to the actuator  36  included in block  50  for providing the desired amount of braking. In addition, the brake command pressure P c  together with the state estimates from the Kalman filter  56  are fed back to the system model block  52  in order to update the state equations. 
     FIG. 2B is a block diagram illustrating an actual embodiment of the controller  34  in accordance with the present invention. The controller  34  includes a microprocessor  60  responsible for performing the various control and computational functions described herein. The microprocessor  60  is coupled to a memory  62  including, for example, both volatile and non-volatile sections. The memory  62  has operating code stored therein which is executed by the microprocessor  60  in carrying out the various functions described herein. It will be appreciated that those having ordinary skill in the art of programming and digital filtering will be able to program such a microprocessor  60  to perform the various functions described herein without undue effort or experimentation. Such programming can be implemented using any of a number of conventional programming languages and will be readily apparent particularly in view of the various flowcharts described below. Accordingly, specific detail regarding the operating code itself is omitted for sake of brevity. 
     The memory  62  also has stored therein the various system equations, initialization and reset parameters, etc., referred to herein for purposes of carrying out the antiskid control. Moreover, the microprocessor  60  is coupled to an input/output (I/O) interface  64  which allows the microprocessor  60  to receive the time variant measured data (e.g., V m , {dot over (V)} m , PILOT PRESSURE, ω m , and T m ). In addition, the interface  64  serves to couple the command pressure signal P c  from the microprocessor  60  to the brake actuator  36 . 
     System State Equations 
     In the preferred embodiment, the antiskid brake control system  30  uses the following set of equations to model the system: 
     aircraft (A/C) velocity 
     
       
           {dot over (V)} =−μ[( V−ωr )/ V,μ   p   ,r   p   ]g+n+n   f   (1) 
       
     
     wheel speed 
     
       
         {dot over (ω)}=(1 /l )[μ[( V−ωr )/ V,μ   p   ,r   p   ]W   t   r −sign[ω] kP]+n+n   f   (2) 
       
     
     brake pressure 
     
       
           {dot over (P)} =(1 /   T )[ P   c   −P]+n+n   f   (3) 
       
     
     friction peak amplitude 
     
       
         {dot over (μ)} p   =n+n   f   (4) 
       
     
     friction peak location 
     
       
           {dot over (r)}   p   =n+n   f   (5) 
       
     
     torque to pressure ratio 
     
       
           {dot over (k)}=n+n   f   (6) 
       
     
     weight per wheel ratio 
     
       
           {dot over (W)}   t   =n+n   f   (7) 
       
     
     wheel radius 
     
       
           {dot over (r)}=n+n   f   (8) 
       
     
     where: 
     the superscript “ • ” indicates the time derivative of the respective parameters as is conventional; 
     μ is a mathematical function describing the coefficient of friction of between the wheel (tire) and runway surface; 
     g refers to acceleration of gravity; 
     I refers to the inertia of the wheel and brake assembly sign refers to the sign operation; 
       T  represents a time delay associated with the response time of the brake actuator; 
     P c  represents the command pressure; 
     n represents the actual noise for the respective parameters; and 
     n f  represents the fictitious noise for the respective parameters. 
     Equations (1) and (2) for the aircraft velocity V and wheel speed ω are derived by applying force and torque sums (represented by square brackets) to the aircraft and the wheel  40  and brake assembly  38 , respectively. The specific application of force and torque sums as they relate to aircraft velocity and wheel speed will depend on the particular aircraft, wheel type, etc., as will be appreciated. Those having ordinary skill in the art of digital filtering techniques and aircraft braking systems will be able to determine specific state equations for a given application based on the disclosure herein. 
     The brake pressure equation (3) is a simple first order lag which accounts for the finite time it takes the actuator  36  to respond to a command. As is described in more detail below, equations (1)-(3) are solved in accordance with the present invention to find the peak pressure that can be applied to the brake  38  while remaining at the top of the mu-slip curve. Generally speaking, the brake pressure required to operate at the peak of the mu-slip curve may be calculated from the torque sum on the wheel  40 : 
     
       
           I{dot over (ω)}   p =μ p   W   t   r−kP   p   (9) 
       
     
     where 
     P p  represents the peak pressure that can be applied to the brake  38 . 
     The deceleration of the wheel  40  when operating at the peak of the mu-slip curve may be calculated from the force sum on the aircraft as follows:                  V   .     p     =       -     μ   p          g             (10)                   ω   .     p     =       (     1   -     r   p       )              V   .     p     /   r               (11)                            =       -     μ   p              g        (     1   -     r   p       )       /   r                 (12)                         
     where 
     r p  represents the friction peak location. 
     Equations (9) through (12) may then be solved for the peak pressure P p , giving: 
     
       
           P   p =(μ p   /k )[ W   t   r+gI (1 −r   p )/ r]   (13) 
       
     
     Inspection of this equation identifies the variables utilized within the system  30  to calculate the peak pressure and hence the state vector. The state vector is represented as follows: 
       X   T   ={V,ω,P,μ   p   ,r   p   ,k,W   t   ,r}   (14) 
     where the superscript T represents the transpose as is conventional. 
     It will be noted that the inertia I of the wheel and brake assembly is not included in the state vector. In the preferred embodiment, the inertia I is not included as it would take up computer resources unnecessarily. The equation for torque sum on the wheel  40  can be divided by inertia resulting in the terms of W t /I and k/I. These terms could be defined as state vector elements but since inertia is not expected to change much in an aircraft, the state elements are chosen as W t  and k. If inertia is high, the W t  and k term will estimate low, etc. It will be appreciated that Equ. 13 for the peak pressure could be modified to include secondary terms such as rolling resistance and/or other parasitic terms. 
     Since the peak location of the mu-slip curve is least observable when operating at the peak μ p  as discussed more fully below, the initial conditions for the friction states, μ p  and r p , are preferably biased to the high side of the mu-slip curve. The Kalman filter has been found to estimate better (e.g., more quickly) when the true value of the friction parameters were less than the estimated values. 
     System Measurement Vector 
     As mentioned briefly above, the possible measurements within the system  30  as represented in FIG. 1 include: 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 V m   
                 aircraft velocity 
                 (ft/sec) 
               
               
                   
                 ω m   
                 wheel speed 
                 (rad/sec) 
               
               
                   
                 T m   
                 brake torque 
                 (ft.lb) 
               
               
                   
                 
                   m 
                 
                 aircraft acceleration 
                 (ft/sec 2 ) 
               
               
                   
                   
               
            
           
         
       
     
     The measurement vector, z, is represented as follows: 
     
       
           z   T   ={V   m ,ω m   ,T   m   ,{dot over (V)}   m } T   (15a) 
       
     
     where the superscript T represents the transpose as is conventional. 
     Referring now to FIG. 3, a flowchart is provided summarizing the general operation of the brake control system  30  in accordance with the invention. Beginning in step  100 , the controller  34  computes the appropriate command pressure P c  to provide to the actuator  36  for providing the desired amount of braking action. The manner in which the command pressure P c  is computed is described in more detail below in relation to FIG.  6 . Next, in step  102  the controller  34  analyzes and processes the various data within the system  30  via the Kalman filter. (See FIGS. 5A,  5 B discussed below.) Based on such information, the controller  34  in step  104  determines whether a skid condition has started as a result of the braking action. (See FIGS. 7,  8 A,  8 B discussed below). Provided a skid is not occurring or had previously begun, the controller  34  proceeds to step  106  in which the Kalman gain for the filter is computed and the covariances updated in order to maintain operation at or near the peak of the estimated mu-slip curve. (See FIGS. 5A,  5 B). Thereafter, the controller  34  returns to step  100  to recompute the command pressure based on the updated information. 
     If a skid is determined to have begun in step  104 , the controller  34  proceeds instead to step  108 . In step  108 , the controller  34  resets the covariances of the Kalman filter in order that it may attempt to reconverge on the peak of the estimated mu-slip curve so as to provide optimum braking without skidding. (See, FIGS. 5A,  5 B,  10 B). Following step  108  the controller  34  returns to step  100  in which it recalculates the command pressure provided to the actuator  36  based on the new covariances. 
     Referring briefly to FIG. 4, a mu-slip curve  110  as modeled in accordance with the preferred embodiment of the present invention is shown. In FIG. 4, the ordinate represents the coefficient of friction μ between the wheel  40  and the runway surface. The abscissa represents the slip ratio r slip  between the wheel  40  and the runway surface and is given by: 
     
       
           r   slip =( V−ωr )/ V   (15b) 
       
     
     The peak of the mu-slip curve  110  is designated by the friction peak amplitude μ p  and is located at the friction peak slip ratio r p  (also referred to herein as the friction peak location or peak slip ratio r p ) as shown. The mathematical function describing the relationship is: 
     
       
         μ[ r   slip ,μ p   ,r   p ]=(μ p β)( r   slip   /r   p )/{(β−1)+| r   slip   /r   p | β }  (15c) 
       
     
     where β is a parameter which controls the steepness of the mu-slip curve. A value of two has been found to match test data well. This quantity may be scaled up or down to match specific data. The Kalman filter within the controller  34  uses this model of the mu-slip curve in estimating the appropriate braking commands as discussed in greater detail below. 
     Turning to FIGS. 5A and 5B, operation of the Kalman filter based brake control system  30  will now be described in more detail. Operation begins with step  150  upon the system  30  being powered up initially, etc. Following step  150 , the controller  34  proceeds to step  152  in which the controller  34  determines if it is necessary to initialize/reinitialize itself. Specifically, the controller  34  determines if an initialization flag INIT has been set as discussed below. Generally speaking, the INIT flag is set upon the system  30  initially being powered up or upon one or more of the state vector elements falling outside a predefined range. In the event the INIT flag has been set as determined in step  152 , the controller  34  proceeds to step  154  in which it initializes/reinitializes itself. 
     The procedure for initializing/reinitializing the controller  34  is described in detail below in relation to FIG.  12 . Briefly, such procedure involves first measuring the current wheel speed ω m  as provided to the controller  34  from the velocity sensor  42 . The state vector X (Equ. 14) is then initialized by setting the estimated wheel velocity ω equal to the measured wheel velocity ω m . In addition, the initial values of the remaining state vector elements are set as described in more detail below. Also in step  154  the state covariance matrix P for the Kalman filter is initialized to predefined initial values P 0  selected for carrying out the invention. Following step  154  (or step  152  in the event the INIT flag is not set as determined in step  152 ), the controller  34  proceeds to step  156 . 
     In step  156 , the controller  34  computes the command pressure P c  which is to be applied to the brake  38  via the actuator  36 . The command pressure P c  is computed based on the PILOT PRESSURE signal received by the controller  34 , and the command pressure P c  is output to the actuator  36 . The procedure for computing the command pressure P c  based on the estimates provided by the Kalman filter within the controller  34  is discussed below in detail in connection with FIG.  6 . The controller  34  then provides the calculated command pressure P c  to the actuator  36  as represented by step  156 . 
     Following step  156 , the controller  34  proceeds to step  158  in which the state transition matrix φ for the Kalman filter is computed as follows: 
     
       
         φ =I+FΔt+FΔt   2 /2  (15d) 
       
     
     where I is the identity matrix, Δt represents the time increment of the data being processed by the Kalman filter, and F represents the state vector dynamics matrix which is constructed from the state equations 1-8. 
     The controller  34  then proceeds from step  158  to step  160  in which the controller  34  extrapolates the state vector information X n  for discrete time point n based on the following difference equation: 
     
       
           X   n   =φX   0   +Gu   (16) 
       
     
     where X n  represents the new or current state vector X at time point n, X 0  represents the state vector from the old or last processing cycle, u represents the control input, and G represents a matrix arising in the formulation. The process for converting the state equations (1-8) to matrix form (equations 15d and 16) can be found in Gelb et al., Applied Optimal Estimation, TASC (1974). 
     Next, in step  162  the controller  34  computes the real process noise QRK matrix associated with the system state equations 1-8 set forth above. The real process noise matrix Q RK  is represented as follows:                      Q   RK     =       off                 diagonals                     Q   RK          (     i   ,   j     )         =       0                 for                 i     ≠   j                       Q   RK          (     i   ,   j     )       =           (     real                 process                 noise     )     2                   for                 i     ≠   j                   (17a)                         
     Exemplary values for the real process noise matrix Q RK  are discussed below. 
     Following step  162 , the controller  34  proceeds to steps  164  and  166 . Specifically, in step  164  the controller  34  computes the fictitious process noise matrix Q FK  as is described more fully below. Generally, the fictitious process noise matrix Q FK  can be represented as follows:                      Q   FK     =       off                 diagonals                     Q   FK          (     i   ,   j     )         =       0                 for                 i     ≠   j                       Q   FK          (     i   ,   j     )       =           (     fictitious                                process                 noise     )     2                   for                 i     ≠   j                   (17b)                         
     Following step  164 , the controller  34  proceeds to step  168  in which it computes the process noise matrix Q K  as the sum of the real process noise matrix Q RK  determined in step  162  and the fictitious process noise matrix Q FK  determined in step  164 . 
     Concurrent with steps  164  and  168 , the controller  34  in step  166  computes the error state transition matrix φ ε  as follows: 
     
       
         φ ε   =I+F   ε   Δt+F   ε   Δt   2 /2  (18) 
       
     
     where F ε  is the error state dynamics matrix. 
     Again, the process for converting the state equations to error state equations and then to matrix form can be found in Gelb et al., supra. 
     Following steps  166  and  168 , the controller  34  proceeds to step  170  in which it calculates the new or current covariance matrix P N  as follows: 
     
       
           P   N =φ ε   P   0 φ ε   T   +Q   K   (19) 
       
     
     Next, in step  172  the controller  34  determines if N=N MAX , where N is the number of consecutive discrete time point samples which have been processed through the Kalman filter  56  (FIG. 2A) as represented by steps  158 - 170  since last updating the state vector X. N MAX  is a predefined number representing the maximum number of data samples which are to be processed by the Kalman filter prior to each update of the state vector X. If N is not equal to N MAX  in step  172 , the controller  34  returns to step  158  and the state transition matrices, φ and φ ε  are updated in the above described steps  158 - 170  which are repeated for the next data sample. If N does equal N MAX  as determined in step  172 , the controller  34  proceeds to step  174 . 
     In step  174 , the controller  34  analyzes the measured data relating to wheel speed, aircraft velocity and acceleration, and torque in conjunction with the state estimates provided by the Kalman filter to determine whether the wheel  40  is in a skid condition. As will be described in more detail below in connection with FIGS. 7,  8 A and  8 B, the controller  34  in step  174  of a preferred embodiment preforms four tests to see if the wheel  40  is in a skid. In addition, the controller  34  performs two tests to determine when a skid event is over. On the transition from not skidding to skidding, a skid start flag is set. This indicates the covariance matrix P for the Kalman filter  56  should be reset. Summarizing for now, the tests to determine if the wheel  40  is in a skid are based on i) a comparison between the estimated wheel speed ω and the measured wheel speed ω m ; ii) a comparison between the estimated slip ratio and the estimated peak of the mu-slip curve; iii) the estimated slip ratio; and iv) the estimated wheel acceleration {dot over (ω)}. Upon the controller  34  initially detecting a skid occurring based on such tests as described below, a SKIDSTART flag is set internally. The two tests to determine if a skid is over involve i) a comparison of the estimated slip ratio with the estimated peak location r p ; and ii) determining if the estimated slip ratio has dropped below a predefined ratio as discussed below. 
     Next, in step  176  the controller  34  determines if the SKIDSTART flag has been set previously in step  174  so as to indicate the beginning of a skid condition. 
     Ideally, the Kalman filter within the controller  34  is designed to cause the brake system to apply pressure to the brake assembly  38  in order to operate at or near the peak of the mu-slip curve and to avoid causing a skid condition. Nevertheless, in the event a skid occurs due to abnormal runway conditions, wheel conditions, etc., as detected in step  174 , the controller  34  proceeds from step  176  to step  178 . As is described in more detail below in connection with FIG. 13, the controller  34  resets its covariance matrix P for the Kalman filter by setting the initial conditions for the filter to prescribed limits. This has been found to allow the Kalman filter to converge significantly faster compared to conventional techniques as will be described below. In the event the covariance matrix P is reset in step  178 , the controller  34  then returns to step  150  in FIG.  5 A and the above-described steps are repeated. 
     Referring again to step  176 , if the SKIDSTART flag is not set indicating that a skid condition has not initially occurred, the controller  34  proceeds to step  180 . In step  180 , the controller  34  sets the observation matrices H and H ε  for the aircraft wheel speed, velocity, acceleration and brake torque as is discussed in more detail below in connection with FIG.  11 . Next, in step  182  the controller  34  sets the real observation noise matrix R R  as is discussed in more detail below. Following step  182 , the controller  34  in step  184  sets the fictitious observation noise matrix R F  as is also discussed below. Thereafter, in step  186  the controller  34  sets the combined observation noise matrix R equal to R R +R F . 
     Following step  186 , the controller  34  proceeds to step  188  in which it computes the Kalman gain {overscore (K)} using conventional techniques as follows: 
     
       
           {overscore (K)}=PH   ε   T   [H   ε   PH   ε   T   +R]   −1   (20) 
       
     
     Next, in step  190  the controller  34  updates the state vector X, again using conventional Kalman filter techniques, as follows: 
     
       
           X   N   =X   0   +{overscore (K)} ( z ( i )− HX   0 )  (21) 
       
     
     where z represents the measurement vector and z(i) represents the particular element. 
     Following step  190 , the controller  34  goes to step  192  in which it performs a range check with respect to the respective system parameters. As is described more fully below in connection with FIG. 14, the controller  34  determines in step  192  whether any of the system parameters within the state vector X fall outside a predefined range during braking control. If yes, the controller  34  sets an internal initialization flag INIT indicating that the Kalman filter within the controller  34  is to be initialized/reinitialized. As previously described, the controller  34  in step  152  detects whether the INIT flag has been set and initializes the controller  34  based thereon. The controller  34  then proceeds from step  192  to step  194  in which it updates the covariance matrix P as follows: 
     
       
           P   N =( I−{overscore (K)}H   ε ) P   0   (22) 
       
     
     It will be appreciated that steps  180 - 194  as represented in FIG. 5B will be carried out for each z(i) observation. The Kalman filter within the controller  34  processes the data in order to update the system parameters which are then utilized to carry out the prescribed control laws. Following step  194 , the controller  34  returns to step  150  as shown. 
     Observability 
     Prior to describing the various steps represented in FIGS. 5A and 5B in detail, it will be useful to discuss the role of observability in the present invention for optimizing the performance of the controller  34 . In order for a Kalman filter to be effective, the states must be observable as is known. There are analytical expressions for observability, but they are not really of practical use. From a practical perspective, observability means that one system parameter is uniquely distinguishable from the rest. 
     Observability as it pertains to the mu-slip curve will now be discussed with reference to FIG.  10 A. The figure shows two mu-slip curves  200  and  202  intended to represent a transition during operation, e.g., braking during landing of an aircraft. The mu-slip curve  200  represents an initial curve during the transition, and the mu-slip curve  202  represents a final curve as a result of the transition. It is assumed, for sake of example, that the initial operating point (IOP) during the transition is at the peak of the mu-slip curve  200  which coincidentally is also on the mu-slip curve  202 . After such a specifically defined transition between mu-slip curves, if the pressure applied to the brake assembly  38  were to remain constant, no observable information will be delivered to any antiskid algorithm, not just the Kalman filter based algorithm of the present invention. This is because the slip ratio will remain the same which means that the developed friction will stay the same. Therefore, the wheel deceleration remains the same and hence the aircraft deceleration. Any antiskid algorithm has to have the pressure applied to the brake arbitrarily perturbed to generate observable information. 
     According to one aspect of the present invention, it has been noted that the aforementioned unobservable condition occurs when the new operating points of friction peak amplitude μ p  and location r p  (FIG. 4) are greater than the old. Conversely, they are most observable when the new operating points of friction peak amplitude μ p  and location r p  are lesser than the old. Thus, in accordance with the present invention the controller  34  sets the initial conditions for the friction peak amplitude μ p  and location r p  toward the high end of their anticipated range. 
     For example, runway friction is generally considered to have a peak value μ p  between 0.1 and 0.6. As a result, the controller  34  as part of the initialization/reinitialization in step  154  (FIG. 5A) sets the peak amplitude μ p  to 0.6 as discussed below in relation to FIG.  12 . The covariance is then chosen to be 0.5 in order to cover the range as discussed below in connection with FIG.  12 . The value 0.6 represents the upper end of the anticipated range according to known characteristics of runway conditions. However, in another application where the upper end and range are represented by some other values, the initial peak amplitude μ p  and covariance may be set accordingly without departing from the scope of the invention. 
     Similarly, the peak location r p  of the mu-slip curve in typical runway conditions is generally considered to be located between 5% and 15%. Therefore, in step  154  (FIG. 5A) the initial condition for the peak location r p  is set to be 0.15 so as to be towards the upper end of the expected range. 
     It has been found that the convergence speed of the Kalman filter based antiskid controller  34  depends significantly on the initial conditions of the friction peak amplitude μ p  and location r p . By initially setting both of these parameters at or near the higher end of their respective ranges and setting their initial covariances large enough to cover the anticipated range, the controller  34  will converge substantially more quickly as compared to the case where the initial conditions are set in the middle of the respective ranges. This is contrary to the more intuitive filtering approach where the initial conditions would be set towards the middle of their expected ranges with a one sigma covariance. Curve  206  in FIG. 10B illustrates a search area in accordance with such an intuitive approach. 
     The better choice as determined by the present inventor is to set the initial conditions at or near the maximum expected values and approximately double the covariance. Initially it may appear that such an approach would cause the filter to be slower because the search area is quadrupled in area as represented by curve  208  in FIG.  10 B. However, the expansion in search area is more than made up for from the fact that the number of convergence paths is reduced by four. Namely, since the initial condition on the wheel friction parameters is set to the high end of the region where the true parameters lie, the trajectory the estimates take towards the true values will tend to be downward. Furthermore, the large initial covariance of the wheel friction parameters increase the Kalman gain for these parameters and hence increase its convergence speed. Tests have shown the controller  34  to converge over five times faster by setting the initial conditions at their maximum expected values in accordance with the present invention as compared to a more conventional approach. 
     Observability of the Kalman filter based controller  34  is also increased in accordance with the present invention by adding an “excitement” signal. As is discussed below in relation to FIG. 6, an excitement signal is added to a slip ratio set point used in the calculation of the command pressure P c  provided by the controller  34  to the actuator  36 . Referring again to FIG. 10A, assume again that the initial operating point IOP is where the mu-slip curves  200  and  202  intersect as shown. By varying the pressure signal, the slip ratio r slip  is also varied. The Kalman filter within the controller  34  then sees a segment of the current mu-slip curve rather than just a point. This segment is uniquely determined and hence observable. The frequency of this excitement signal is chosen to be different from the aircraft gear walk frequency (e.g., the frequency where the landing gear/wheel combination acts as a pendulous arm and oscillates unduly as is known). As is discussed below, the amplitude in terms of slip ratio is about ±7% which, because the mu-slip curves tend to be flat near the top, results in a pressure variation of about ±2% in the context of aircraft antiskid applications. 
     Referring now to FIG. 6, the procedure for computing the command pressure P c  (step  156 ) will now be discussed. Generally speaking, in the exemplary embodiment the pressure calculation is either zero or the peak pressure μ p  depending on whether a skid is determined to be occurring or not. As will be described below, the pressure is set starting with the peak pressure (Eq. 13) calculation which is augmented with a proportional gain term which increases pressure based upon how far the slip ratio parentheses set point is from the estimated slip ratio. The slip ratio set point (r sp ) is then slightly less than the estimated slip ratio peak with the aforementioned excitement signal imposed. The proportional gain is adaptive as it is dependant on the average slope of the mu-slip curve and the certainty with which this slope is known. The purpose of the proportional gain is two-fold: (1) increase the response speed; and (2) drive the pressure signal to a higher value when the estimate differs from the set point greatly. The pressure signal is then rate limited where the rate limit is a tuning factor for a particular braking system based upon dynamometer testing. 
     Describing now the procedure in detail, step  156  begins with the controller  34  determining in step  250  whether a skid is occurring. Whether a skid is occurring or not is determined in step  250  based on if an internal SKID flag has been set or not. The routine for determining whether a skid is occurring and setting the SKID flag accordingly is described in detail below in relation to the flowchart of FIG.  7 . If the SKID flag=1 as determined in step  250 , indicating that a skid is occurring, the controller  34  proceeds from step  250  to step  252 . In step  252 , the controller  34  sets a pressure set point parameter P sp  equal to zero. The controller  34  then proceeds to step  254  in which the controller  34  compares the current pilot pressure signal (PILOT PRESSURE), indicating the amount of braking requested by the pilot, to the pressure set point P sp . 
     Specifically, in step  254  the controller  34  selects as pressure P the lowest pressure between the pressure set point P sp  and the current PILOT PRESSURE. In this case where step  254  immediately follows step  252 , the controller  34  would select P=P sp =0 since the PILOT PRESSURE could not be a negative value. 
     Following step  254 , the pressure signal P is rate limited in step  256  to ensure that the change in the pressure signal P (ΔP) since the previous observation is within an acceptable predefined range (e.g., between ΔP− and ΔP+). The pressure signal P is then amplitude limited in step  258  to ensure the pressure signal P does not exceed a predefined amplitude range (e.g., between 0 and 3000 psi). The particular ranges in steps  256  and  258  are predetermined so as not to exceed the limitations (e.g., structural and/or response) of the particular wheel  40  and brake assembly  38  Such ranges may be determined from dynamometer testing, for example. Following the rate limiting and amplitude limiting of steps  256  and  258 , the pressure signal P is then output by the controller  34  as the brake command pressure P c  as part of step  258 . 
     Referring back to step  250 , if the flag SKID=0 so as to indicate a skid is not occurring, the controller  34  proceeds to step  260  to determine the appropriate command pressure P c  to be applied to the brake  38  to provide braking action without causing a skid condition. It is noted that although it is referred to herein as there being no skid occurring when SKID=0, this is by definition. Ideally some skid is occurring since the mu-slip curve indicates that maximum friction and hence stopping can occur if there is sufficient slipping or skidding to remain at the peak. Thus, a skid condition is defined by the particular parameters set forth herein. Other parameters could be used in other embodiments as will be appreciated. 
     In step  260 , the controller  34  computes the slip ratio set point r sp . The slip ratio set point r sp  governs the amount of pressure that is applied to the brake assembly  38 . As mentioned above, the slip ratio set point r sp  is the estimated slip ratio peak with an excitement signal imposed. The excitement signal represents a time variation in the set point which allows various parameters in the system to become more observable in the steady state. The particular nature of the time-variant excitement signal is not critical. For example, one could use a sinusoid, sawtooth, ramp, or other periodic waveform as the excitement signal. The maximum amplitude of the excitement signal preferably is limited to the maximum amount of slip a wheel  40  can deliver. In addition, preferably the maximum amplitude of the excitement signal is limited by the amount of loss of braking efficiency which can be tolerated in the system. Also, the frequency of the excitement signal preferably is different from the aircraft gear walk frequency as previously mentioned. 
     In the preferred embodiment, the slip ratio set point r sp  is calculated in step  260  as follows: 
     
       
           r   sp =0.82 r   p +0.07 r   p *cos [4 πt]   (23) 
       
     
     where t represents time and the excitement signal is a sinusoid having a magnitude of 0.07 times the friction peak location r p . The frequency of the excitement signal may be, for example, 2 Hz. 
     Equ. 23 indicates that the slip ratio set point r sp  in the steady state will fluctuate between 0.75r p  and 0.89r p  with an average of 0.82r p . Current antiskid systems are generally accepted to operate at 90% efficiency. This occurs at r slip =0.63r p  as may be ascertained from Equ. 15c. The preferred embodiment is to provide 98% braking efficiency which occurs at r slip =0.82r p . The amplitude of the signal is chosen so as to avoid going over the peak of the mu-slip curve and avoid the excitation of landing gear resonant frequencies unduly. A 2% modulation in pressure has been found to be an acceptable compromise. Modulating the slip ratio 7% will give 2% pressure modulation which translates into a 2% modulation of drag force on the axle, for example. It will be appreciated that the selected average value and excitement signal amplitude may vary depending on the particular aircraft, wheel configuration, etc. For example, if the specific mu-slip curve for the wheel (tire) has a specific β value other than 2, then the 82% value and the 7% value can be modified to achieve the goal of 98% efficiency with 2% modulation of pressure. Also, other efficiencies and modulations may be selectively obtained as desired. 
     Following step  260 , the controller  34  proceeds to step  262  in which it computes the slip ratio r slip  as follows: 
     
       
           r   slip =( V−ωr )/ V,   (24) 
       
     
     where V is the estimated A/C velocity (Equ. 1), ω is the estimated angular velocity of the wheel  40  (Equ. 2), and r is the estimated radius of the wheel  40  (Equ. 8). 
     Next, in step  268  the controller  34  computes an adaptive proportional gain term to augment the slip ratio set point r sp  obtained in step  260 . The proportional gain term is proportional to the estimated average slope of the mu-slip curve. The effect of the proportional gain term is to provide a higher level of control when a steep curve is encountered and more subtle control when low wheel friction occurs. 
     Ordinarily, when the value of μ step changes from low to high (e.g., from the low of its expected range (0.1) to the high of its expected range (0.6)), a Kalman filter will be slow to respond. To speed up the response, the controller  34  in the present invention causes the brake pressure to “sweep” up the mu-slip curve in order to see the peak. In the present embodiment, this is accomplished by providing an adaptive proportional gain term K p  selected to increase pressure applied to the brake  38  based upon how far the slip ratio set point r p  is from the estimated slip ratio r p . 
     Accordingly, in step  268  the adaptive proportional gain term K p  is computed as follows: 
     
       
           K   p =μ p   /r   p   (25) 
       
     
     It will be appreciated that this value may be scaled up or down based up specific conditions and requirements. 
     After the slip ratio set point r sp  and proportional gain K p  have been determined, the controller  34  proceeds to step  270  in which it calculates the friction set point μ sp  as follows: 
     
       
         μ sp   =μ[r   sp ,μ p   ,r   p   ]+K   p (μ p   /r   p )( r   sp   −r   slip ))  (26) 
       
     
     The first term in Equ. 26 represents a constant term and the second term represents a proportional term with respect to the friction set point μ sp . The friction set point μ sp  is then used by the controller  34  to compute the pressure set point P sp  in step  272  as follows: 
     
       
           P   sp =(μ sp   /k )( W   t   r+gI (1 −r   sp )/ r )  (27) 
       
     
     where k represents the estimated torque/pressure ratio. 
     Using the pressure set point P sp  as determined from the Kalman estimates and the various control laws applied thereto as discussed above, the controller  34  then proceeds from step  272  to step  254  where the lowest pressure is selected as previously described. Again, the PILOT PRESSURE command or the pressure set point P sp  calculated in step  272 , whichever is lower, is subjected to rate and amplitude limiting in steps  256  and  258 , respectively. The command pressure P c  is then output to the brake actuator  36 . 
     Thus, the flowchart of FIG. 6 illustrates the manner in which the appropriate brake pressure is applied to the brake actuator  36  using the Kalman filter estimates provided within the controller  34 . Use of the excitement signal and proportional gain significantly increase the observability of the Kalman filter in estimating the system parameters and converging quickly to the “true” system values. 
     Skid Detection 
     Of course a key to any antiskid control system is the detection of a skid condition. According to the present invention, a skid condition is detected based on the process which will now be explained with reference to FIGS. 7-9. Referring initially to FIG. 7, shown is a procedure which the controller  34  carries out in step  174  (FIG. 5B) discussed above. Four tests are made to see if the wheel  40  is in a skid and two tests are made to determine when a skid condition is over. On the transition from not skidding to skidding, the SKIDSTART flag is set internally by the controller  34 . This indicates the covariances should be reset as discussed above in relation to step  178  of FIG.  5 B. Summarizing briefly, the four tests made to determine if the wheel  40  is in a skid are: 
     1. If the wheel speed estimate ω varies from the wheel speed measurement ω m  by 3.5 standard deviations, an INSKID flag is set to on. 
     2. If the estimated slip ratio is 1 standard deviation past peak, the INSKID flag is set to on. 
     3. If the estimated slip ratio is over 0.25, the INSKID flag is set to on. 
     4. If the wheel acceleration is less than −μ p g(1−r p )/r, the INSKID flag is set to on. 
     There are two tests to see if the skid is over. An objective of the present invention in accordance with the preferred embodiment is to not let the wheel speed drop below a predetermined portion (e.g., 40%) of the peak slip ratio r p  to avoid diminishing returns on observability as discussed more fully below. Since it takes a finite amount of time to actuate the brake  38  and decelerate the wheel  40  (due to rotational inertia), it has been found that the brake reapplication should occur when the wheel  40  passes a slip ratio of 0.75 r p , for example. This value may change based upon wheel size, vehicle weight, etc., as will be appreciated. The second condition is meant to be effective at low aircraft speed when fidelity in the peak slip ratio r p  estimate may be low. 
     The skid off conditions denoting the end of a skid are: 
     1. If the slip ratio is less than 0.75 r p , a SKIDOVER flag is set to on. 
     2. If the slip ratio is less than 0.02, the SKIDOVER flag is set to on. 
     Referring now specifically to FIG. 7, in step  290  the controller  34  sets the internal SKIDSTART flag equal to 0 so as to enable the controller  34  to identify the start of a skid. Next, in step  292  the controller  34  determines if the SKID flag (described below) is already true (SKID=1) indicating the wheel  40  had been identified as being in a skid condition based on the previous iteration. If the SKID flag is not true (i.e., SKID=0), the controller  34  proceeds to step  294 . In step  294 , the controller  34  determines whether the wheel  40  is in a skid condition based on the state estimates. 
     The preferred routine for determining whether the wheel  40  is in a skid in step  294  is shown in detail in FIG.  8 A. Beginning in step  294   a , the controller  34  resets an INSKID flag to 0. In concurrent steps  294   b  and  294   c  following step  294   a , the controller  34  sets the real wheel speed observation noise matrix R R (2,2) and the wheel speed observation matrices H and H ε , respectively. With regard to step  294   b , real wheel speed observation noise matrix R R  is set as will be described more fully below. As for step  294   c , the wheel speed observation matrices H and H ε  are set such that H=H ε =[0,1,0,0,0,0,0,0]. 
     Following steps  294   b  and  294   c , the controller  34  proceeds to step  294   d  in which it calculates a wheel speed rejection ratio ω rej  as follows: 
     
       
         ω rej =(| z (2)− HX |)/( H   ε   PH   ε   T   +R   R ) ½   (28) 
       
     
     where z(2) represents the wheel speed measurement. 
     Following step  294   d , the controller  34  in step  294   e  determines if the wheel speed rejection ratio ω rej  is greater than 3.5. In other words, the controller  34  determines if the wheel speed estimate varies from the measured wheel speed by more than 3.5 standard deviations. If yes, the controller  34  sets the INSKID to 1 in step  294   f . Thereafter, the controller  34  proceeds to step  294   g . If no in step  294   e , the INSKID flag remains set to 0 and the controller  34  continues to step  294   g.    
     The controller  34  in step  294   g  sets “past peak” observation matrices H and H ε  as follows: 
     
       
           H=[ωr/V   2   ,−r/V, 0,0,−1,0,0,0] 
       
     
     
       
           H   ε   =[ωr/V   2   ,−r/V, 0,0,−1,0,0,−ω/ V]   
       
     
     Next, in step  294   h  the controller  34  calculates the past peak rejection ratio r prej  using the past peak observation matrices from step  294   g  as follows: 
       r   prej   =HX /( H   ε   PH   ε   T ) ½   (29) 
     The controller  34  next in step  294   i  determines if the past peak rejection ratio r prej  determined in step  294   h  is greater than 1. In other words, the controller  34  determines if the estimated slip ratio is greater than 1 standard deviation past peak. If yes, the INSKID flag is set to 1 in step  294   j  and the controller proceeds to step  294   k . Otherwise, the INSKID flag remains at its current setting and the controller  34  proceeds directly to step  294   k.    
     The controller in step  294   k  determines if the estimated slip ratio (V−ωr)/V is over 0.25. If yes, the INSKID flag is set to 1 in step  294   l . Then the controller  34  proceeds to step  294   m . If no in step  294   k , the controller  34  proceeds directly to step  294   m.    
     The controller  34  in step  294   m  calculates the wheel acceleration based on the state estimates as follows: 
     
       
         {dot over (ω)}=(1 /I )[μ[( V−ωr )/ V, μ   p   , r   p   ]W   t   r−kP]   (30) 
       
     
     Next, in step  294   n  the controller  34  determines if the wheel acceleration {dot over (ω)} is less than −μ p g(1−r p )/r which is representative of the peak acceleration which occurs when operating at the peak of the curve as described in Equ. 12. If yes in step  294   n , the controller  34  sets the INSKID flag to 1 in step  294   o . Thereafter, the controller  34  completes step  294  and returns to step  296  in FIG.  7 . If no in step  294   n , the controller  34  simply proceeds directly to step  296 . It will be appreciated that the quantity μ p g(1−r p )/r may be scaled up or down. 
     Referring back to FIG. 7, following step  294  the controller  34  proceeds to step  296  in which a SKID flag is set to 0. Next, in step  298  the controller  34  determines whether the wheel  40  is in a skid condition by virtue of the INSKID flag being set to 1 in step  294 . If yes, the controller  34  proceeds to step  300 . Otherwise, the controller  34  proceeds to step  176  in FIG.  5 B. In step  300 , the controller  34  sets the SKID flag to 1 and the SKIDSTART flag to 1 to indicate the wheel  40  has been identified as transitioning from a no skid condition to a skid condition as will be appreciated. The controller  34  then returns to step  176  in FIG.  5 B. 
     If in step  292  the SKID flag is already equal to 1 indicating the wheel was already in a skid condition, the controller  34  proceeds to step  310 . In step  310  the controller  34  determines whether the skid condition has ended. The process of step  310  will now be described in detail with reference to FIG.  8 B. Specifically, beginning in step  310   a  the controller  34  sets a SKIDOVER flag to 0. Next, in step  310   b  the controller  34  determines if the slip ratio (V−ωr)/V is less than 0.75 times the peak slip ratio r p . As was discussed above in relation to FIG. 6, brake pressure is temporarily released in the event of a skid condition. By considering various factors, it is possible with the present invention to arrive at an optimum point at which to reapply brake pressure at the end of the skid condition. 
     More particularly, according to the present invention after a skid is detected, brake pressure is relaxed. Pressure is reapplied when the effect of the peak of the mu-slip curve is most prominent. This maximizes observability to the Kalman filter while minimizing loss of braking efficiency. This point has been found to occur at roughly 40% of the peak slip ratio r p . 
     Conventional control philosophy calls for dumping pressure applied to the brake until slip ratio gets to near zero. However, it has been found in accordance with the present invention that there is an optimum slip ratio at which to apply pressure. The curve shown in FIG. 9 illustrates the observability by the Kalman filter based on a comparison of δ(μ)/δ(r p ) in relation to the quantity r slip /r p . The observation matrix entry has a maximum absolute value at about forty percent of r p . This indicates that to maximize observability of r p  one should dump pressure until r slip &lt;0.40 r p , but dumping pressure any further results in diminishing returns, and reduced braking efficiency due to pressure response. 
     The observation matrix entry is zero at r slip /r p  equal to one as shown in FIG.  9 . This indicates that operating exactly at the peak of the mu-slip curve makes the peak location r p  unobservable at that point. Biasing the operating point off peak with the excitement signal (step  260 , FIG. 6) makes this parameter more observable. 
     Assuming it will take some finite amount of time for the actuator  36  and brake  38  to decelerate the wheel  40  upon receiving a command, it is desirable to detect a point ahead of time at which r slip  has not dropped as low as 0.40 r p . In the exemplary embodiment a value of 0.75 times the peak slip ratio r p  has been found to avoid loss of observability within the Kalman filter based controller  34  while taking into account the finite amount of time. This results in the slip ratio r slip  typically being about 0.40 r p  when deceleration actually occurs so as to optimize observability. The value of 0.75 r p  can change based on various parameters such as wheel size, vehicle weight, etc., which would affect the rotational inertia of the system and hence the delay associated with decelerating the wheel. In any case, however, pressure is reapplied at a value substantially greater than a slip ratio r slip  of zero. 
     In the event the slip ratio (V−ωr)/V is less than or equal to 0.75 r p  as determined in step  310   b , this thereby indicates the end of a skid condition. Thus, the controller  34  goes to step  310   c  where the SKIDOVER flag is set to 1 to indicate such. The controller  34  then proceeds to step  310   d . If in step  310   c  the slip ratio is not less than or equal to 0.75r p , the controller  34  proceeds directly to step  310   d.    
     In step  310   d , the controller  34  determines if the slip ratio is less than or equal to 0.02. This condition is effective early in the estimation period when r p  may not be known with high fidelity. If yes in step  310   d , the SKIDOVER flag is set to 1 in step  310   e  to indicate the skid is over. Following step  310   e , the controller  34  then returns to the flowchart shown in FIG.  7  and specifically proceeds to step  312 . 
     In step  312  the controller  34  sets the SKID flag to 1. Next, the controller  34  proceeds to step  314  in which it determines if the skid condition is over as represented by the status of the SKIDOVER flag. If SKIDOVER=1, the controller  34  proceeds to step  316  in which the SKID flag is set equal to 0. Thereafter, the controller  34  returns to step  176  in FIG.  5 B. If SKIDOVER=0 in step  314  so as to indicate the skid is still occurring, on the other hand, the controller  34  proceeds directly from step  314  to step  176  in FIG.  5 B. 
     Referring now to FIG. 11, the manner in which the observation matrices H and H ε  for each of the wheel speed, aircraft velocity, aircraft acceleration and brake torque are set in step  180  of FIG. 5B will now be described. Step  180  may be represented by individual steps  180   a - 180   d  corresponding to the wheel speed, aircraft velocity, acceleration, and brake torque, respectively. Each of steps  180   a - 180   d  are carried out with respect to each observation in steps  180 - 194  in FIG. 5B as represented by block  180   e . Specifically, with regard to step  180   a  the observation matrices for the wheel speed ω are set as follows: 
     
       
         H=H ε =[0,1,0,0,0,0,0,0] 
       
     
     The observation matrices for the aircraft velocity V are set in step  180   b  as follows: 
     
       
         H=H ε =[1,0,0,0,0,0,0,0] 
       
     
     The observation matrices for the aircraft acceleration {dot over (V)} are set in step  180   c  as follows: 
     
       
           H=[ 0,0,0,− gδμ/δμ   p ,0,0,0,0] 
       
     
     
       
           H   ε   =[−gδμ/δV,−gδμ/δω, 0,− gδμ/δμ   p   ,−gδμ/δr   p ,0,0 ,−gδμ/δr]   
       
     
     Finally, the observation matrices for the brake torque are set in step  180   d  as follows: 
     
       
         H=[0,0,k,0,0,0,0,0] 
       
     
     
       
         H ε =[0,0,k,0,0,P,0,0] 
       
     
     Referring briefly to FIG. 12, step  154  (FIG. 5A) relating to initializing/reinitializing the controller  34  is shown in more detail. Specifically, beginning in step  154   a  the controller  34  measures the wheel speed ω m  and then initializes the state vector X to its initial state as represented by X 0 . For example, in step  154   a  the state vector X is initialized as follows: 
     
       
         
           
               
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 X = X 0  = 
                   
                   
               
               
                   
                   
                 V 0  = ω 0 r 0   
                 fr/sec 
               
               
                   
                   
                 ω 0  = ω m   
                 rad/sec 
               
               
                   
                   
                 P 0  = 0 
                 psi 
               
               
                   
                   
                 μ p0  = 0.6 
               
               
                   
                   
                 r p0  = 0.15 
               
               
                   
                   
                 k 0  = 4 
                 ft.lb./psi 
               
               
                   
                   
                 W t0  = 13800 
                 lb. 
               
               
                   
                   
                 r 0  = 1.125 
                 ft 
               
               
                   
                   
               
            
           
         
       
     
     It will be appreciated that the initial values k 0 , W t0  and r 0  are vehicle specific. Such values are likely to change for a different aircraft. 
     It is noted that the wheel speed ω is initialized (ω 0 ) by measuring the output ω m  of the wheel speed velocity sensor  42  (FIG.  1 ). The aircraft velocity V is initialized (V 0 ) by multiplying the wheel speed by the nominal wheel radius r 0 . This has been found to put the initial slip ratio at zero which is in the middle of the linear portion of the mu-slip curve. Mathematical linearization as part of the Kalman filtering can then proceed more efficiently as compared with other possible techniques. For example, it has been found that if the initial velocity V 0  is obtained from the aircraft measurement V m , the initial estimated slip ratio may be over peak of the mu-slip curve due to corruption of the measurement by noise. The subsequent numerical linearization will then be of the wrong sign and numerical difficulties may arise. 
     Following step  154   a , the controller  34  then proceeds to step  154   b  as shown in FIG.  12 . In step  154   b  the controller  34  sets the covariances in the state covariance matrix P to their initial values as represented by the matrix P 0 . In the exemplary embodiment the initial state covariance matrix P 0  is represented as follows: 
     
       
         
           
               
             
               
                   
               
             
            
               
                 P 0  = 
               
            
           
           
               
               
               
            
               
                   
                 off diagonals 
                 P 0 (i,j) = 0  for i ≠ j 
               
               
                   
                 V  
                 P 0 (1,1) = r 0   2 P 0 (2,2) + ω 0   2 P 0 (8,8) (ft/sec) 2   
               
            
           
           
               
               
               
               
            
               
                   
                 ω  
                 P 0 (2,2) = R(2,2) 
                 (rad/sec) 2   
               
               
                   
                 P  
                 P 0 (3,3) = 200 2   
                 (psi) 2   
               
               
                   
                 μ p    
                 P 0 (4,4) = 0.5 2   
               
               
                   
                 r p    
                 P 0 (5,5) = (0.1) 2   
               
               
                   
                 k  
                 P 0 (6,6) = 1 2   
                 (ft.lb./psi) 2   
               
               
                   
                 W t    
                 P 0 (7,7) = 1400 2   
                 (lb) 2   
               
               
                   
                 r  
                 P 0 (8,8) = 0.1 2   
                 (ft) 2   
               
               
                   
                   
               
            
           
         
       
     
     It will be noted that covariances for ω, P, k, W t  and r will tend to be vehicle specific as will be appreciated. Such values are likely to change for different aircraft. Following step  154   b , the controller  34  returns to step  156  in FIG.  5 A. 
     It is recognized that initializing aircraft velocity from wheel speed introduces an error in the aircraft velocity initial estimate because the wheel radius is also unknown with complete certainty. This is accounted for using well know statistical techniques resulting in the equation above for P 0 (1,1). The initial aircraft velocity covariance is then neither too low which might result in divergence nor too high which might result in slow response time. 
     Referring now to FIG. 13, step  178  (FIG. 5B) for resetting the state covariance matrix P will now be described. Specifically, in step  178   a  the controller  34  resets the covariance matrix P to P R  as follows: 
     
       
         
           
               
               
               
             
               
                   
               
             
            
               
                 P R =   
                   
                   
               
               
                 off diagonals P R (i,j) 
                 = 0 
                 for i ≠ j 
               
               
                 V  P R (1,1) 
                 = 5 2   
                 (ft/sec) 2   
               
               
                 ω  P R (2,2) 
                 = 5 2   
                 (rad/sec) 2   
               
               
                 P  P R (3,3) 
                 = 200 2   
                 (psi) 2   
               
            
           
           
               
               
            
               
                 μ p   P R (4,4) 
                 = reset[μ p , xnom(4), sigmin(4), 
               
               
                   
                 sigmax(4)] 
               
               
                 r p   P R (5,5) 
                 = reset[r p , xnom(5), sigmin(5), 
               
               
                   
                 sigmax(5)] 
               
               
                 k  P R (6,6) 
                 = reset[k, xnom(6), sigmin(6), 
               
               
                   
                 sigmax(6)] (ft.lb./psi) 2   
               
               
                 W t   P R (7,7) 
                 = reset[W t , xnom(7), sigmin(7), 
               
               
                   
                 sigmax(7)] (lb) 2   
               
               
                 r  P R (8.8) 
                 = reset[r, xnom(8), sigmin(8), 
               
               
                   
                 sigmax(8)] (ft) 2   
               
               
                   
               
            
           
         
       
     
     The function “reset” determines how far away the current state estimate (e.g., μ p , r p , k, W t  and r) is from the nominal value (e.g., xnom(4) through xnom(8), respectively) for the particular aircraft in question, and adds it to a minimum value (sigmin(4) through sigmin(8)), respectively. A “clip” function, making up part of the reset function, then bounds the value between “sigmin” and “sigmax”. This value is then squared to arrive at the reset state covariance value. Stated mathematically, 
     
       
         
           
               
             
               
                   
               
             
            
               
                 reset[x, xnom, sigmin, sigmax] = clip[sigmin + 
               
               
                 |x-xnom|, sigmin, sigmax] 2   
               
            
           
           
               
               
            
               
                   
                 where       if 
               
            
           
           
               
               
               
               
            
               
                   
                 clip[x, xmin, xmax] 
                 = x 
                 xmin ≦ x ≦ xmax 
               
               
                   
                   
                 = xmin 
                 x &lt; xmin 
               
               
                   
                   
                 = xmax 
                 x &gt; xmax 
               
               
                   
                   
               
            
           
         
       
     
     For the exemplary embodiment, the values of xnom, sigmin and sigmax are as follows: 
     
       
         
           
               
               
               
               
               
             
               
                   
                   
               
             
            
               
                   
                 μ p   
                 xnom(4) = 0.35 
                 sigmin(4) = 0.25 
                 sigmax(4) = 0.5 
               
               
                   
                 r p   
                 xnom(5) = 0.1 
                 sigmin(5) = 0.05 
                 sigmax(5) = 0.1 
               
               
                   
                 k 
                 xnom(6) = 4 
                 sigmin(6) = 1.0 
                 sigmax(6) = 2.0 
               
               
                   
                 W t   
                 xnom(7) = 13800 
                 sigmin(7) = 1400 
                 sigmax(7) = 2800 
               
               
                   
                 r 
                 xnom(8) = 1.125 
                 sigmin(8) = 0.1 
                 sigmax(8) = 0.1 
               
               
                   
                   
               
            
           
         
       
     
     The result of the reset and clip functions is to independently set the limits of the search space for the one sigma covariance. This has been found to improve the response time of the Kalman filter by up to five times. 
     FIG. 14 illustrates in detail the range check procedure for setting the initialization (INIT) flag in step  192  of FIG.  5 B. Beginning in step  192   a , for each system parameter in the state vector X the controller  34  determines whether the respective parameter x(i) for i=1 to 8 falls outside a predefined range xmin(i) to xmax(i). If each of the given parameters x(i) are currently within their respective ranges xmin(i) to xmax(i) such that xmin(i)&lt;x&lt;xmax(i), the controller  34  proceeds to step  192   b . In step  192   b , the controller  34  sets the INIT flag to 0 (or false) so as to indicate that initialization/reinitialization is not required via step  154 . 
     If in step  192   a  each of the given parameters x(i) are not currently within their respective ranges xmin(i) to xmax(i) such that for at least one of the parameters x&lt;xmin(i), or x&gt;xmax(i), the controller  34  proceeds to step  192   c . In step  192   c , the controller  34  sets the INIT flag to 1 (or true) so as to indicate that initialization/reinitialization is required via step  154 . 
     The particular values of xmin(i) and xmax(i) are predefined so as to represent the expected ranges of the particular parameters. Limits for various parameters are known from data provided by the airframe manufacturer. For example, based upon engine power limitations, it is known that aircraft velocity may not exceed certain values. Another example is weight. The aircraft manufacturer knows how much the aircraft will weigh with it&#39;s maximum and minimum contingent of fuel, passengers, arms, etc. 
     Process Noise and Observation Noise 
     Real Noise n 
     Steps  162  (FIG. 5A) and  182  (FIG. 5B) referred to above pertain to computing the real process noise matrix Q RK  and real observation noise matrix R R , respectively. Such values are represented by noise n in the state equations. The real process noise and observation noise will depend on the particular system hardware which is implemented as will be appreciated. Such values can be arrived at using conventional techniques. In the exemplary embodiment, the real process noise matrix Q RK  can be represented as follows: 
     
       
         
           
               
             
               
                   
               
             
            
               
                 Q RK  = 
               
            
           
           
               
               
               
               
            
               
                   
                 off diagonals 
                 Q RK (i,j) = 0 
                 for i ≠ j 
               
            
           
           
               
               
               
               
            
               
                   
                 V  
                 Q RK (1,1) = 0 2   
                 (ft/sec) 2   
               
               
                   
                 ω  
                 Q RK (2,2) = 0 2   
                 (rad/sec) 2   
               
               
                   
                 P  
                 Q RK (3,3) = 20 2   
                 (psi) 2   
               
               
                   
                 μ p    
                 Q RK (4,4) = 0 2   
               
               
                   
                 r p    
                 Q RK (5,5) = 0 2   
               
               
                   
                 k  
                 Q RK (6,6) = 0 2   
                 (ft.lb./psi) 2   
               
               
                   
                 W t    
                 Q RK (7,7) = 0 2   
                 (lb) 2   
               
               
                   
                 r  
                 Q RK (8,8) = 0 2   
                 (ft) 2   
               
               
                   
                   
               
            
           
         
       
     
     Real process noise is determined during testing of the braking system, For example, measurements of pressure have shown that the noise on the hydraulics in the exemplary embodiment is 20 psi rms. 
     With respect to the real observation noise matrix R R  as computed in step  182 , such matrix may be represented as follows: 
     
       
         
           
               
             
               
                   
               
             
            
               
                 R R  = 
               
            
           
           
               
               
               
               
            
               
                   
                 off diagonals 
                 R R (i,j) = 0 
                 for i ≠ j 
               
            
           
           
               
               
               
               
            
               
                   
                 V m    
                 R R (1,1) = 1 2   
                 (ft/sec) 2   
               
               
                   
                 ω m    
                 R R (2,2) = 1 2   
                 (rad/sec) 2   
               
               
                   
                 T m    
                 R R (3,3) = 20 2   
                 (ft.lb) 2   
               
               
                   
                 
                   m  
                 
                 R R (4,4) = .1 2   
                 (ft/sec 2 ) 2   
               
               
                   
                   
               
            
           
         
       
     
     These values come from data sheets provided by manufacturers of the various sensors. If data sheets are not available, the values can be determined by testing. For example, the wheel speed sensor could be spun at a constant rate and data collected. The observation noise could be calculated offline as the variance around the mean value. 
     Fictitious Noise n f    
     Steps  164  (FIG. 5A) and  184  (FIG. 5B) referred to above relate to computing the fictitious process noise matrix Q FK  and real observation noise matrix R F , respectively. Such values are represented by noise n f  in the state equations. 
     As will be appreciated, optimal Kalman filtering refers to the situation when the model being used in the filter (the transition, noise and observation matrices) exactly represent the hardware. In practice, however, this never happens and the suboptimal model has to be made to work with the hardware which by definition is optimal. There are two general methods which involve adding “fictitious” noise to the covariance calculation. The fictitious noise accounts for all unmodeled effects. 
     The unmodeled effects can be in the model or the measurement and are referred to as fictitious process noise and fictitious observation noise. 
     It is often the ability to determine these noises correctly that determines the ability of the filter to operate in practice. Such noises can be determined analytically and/or experimentally. In practice, fictitious observation noise R F  is added to account for modeling errors that cannot be attributed to any specific source while fictitious process noise Q FK  focusses on specific modeling errors. For example, there are some analytical methods outlined to deal with this in “Applications of Minimum Variance Reduced-State Estimators” (C. E. Hutchinson et al.; IEEE Transactions on Aerospace and Electronics Systems; September 1975; pp.785-794). The approach is to assume that a model of the parameter exists and that it is in the optimal state vector. The covariance propagation of the soon to be unmodeled state into any modeled state is computed. These terms are then identified as fictitious process noise. A similar approach of initially assuming a parameter is in the state vector during the measurement process results in fictitious observation noise. 
     The experimental approach is based upon measured data. For example, if one were to observe during testing that the ratio of torque to pressure (k) varies from 4 to 3 during a 10 second stop, fictitious observation noise can be added at such rate. 
     The Kalman filter based antiskid controller  34  can use a variety of fictitious process and observation noises which are determined both experimentally and analytically. The following are examples but should in no way be construed as limiting to the invention. 
     Aircraft Velocity V and Wheel Speed ω 
     The fictitious noise on these two states is zero. 
     
       
         Q FK (1,1)=0 2   
       
     
      Q FK (2,2)=0 2   
     Pressure P 
     The dynamics modeled in a Kalman filter based antiskid controller are dependent on the actuator dynamics being correct. If they are incorrect due to production variations or wear, performance could be less than optimum. According to the present invention, the controller  34  implements a technique for desensitizing the performance of the Kalman filter therein to the dynamics of the actuator  36 . 
     Specifically, the controller  34  takes into account that actuators (hydraulic or electric) have a time response that may change as the unit wears. Since the dynamics of torque, drag and aircraft acceleration are all dependent on the actuator, it is important to account for actuator response. To avoid modeling the response explicitly, the controller  34  models the dynamics of the actuator  36  as a nominal value and changes are modeled as fictitious process noise in the Kalman filter  56 . In the preferred embodiment, this is done by setting Q FK (3,3) as follows: 
     
       
           Q   FK (3,3)=(Δ t ( P   c ( t )− P )σ T /( T   2 )) 2 (psu) 2   
       
     
     where Δt represents the extrapolation time Δt; P c (t) represents the command pressure as a function of time; and σ T  represents the standard deviation of hydraulic response time constant. 
     Friction Peak Amplitude μ p    
     It has been found that by setting Q FK (4,4) as follows, this parameter controls how fast the Kalman filter will respond to a change in μ p  (Δμ p  in ΔT seconds) with an extrapolation time Δt: 
     
       
           Q   FK (4,4)=(Δμ p (Δ t/ΔT )) 2   
       
     
     Friction Peak Location r p    
     Q FK (5,5) has been found to control how fast the Kalman filter will respond to a change in r p  (Δr p  in ΔT seconds) with an extrapolation time Δt when set as follows: 
     
       
           Q   FK (5,5)=(Δ r   p (Δ t/ΔT )) 2 (ft/sec) 2   
       
     
     Torque/Pressure Ratio k 
     The fictitious noise on the torque to pressure ratio has been found to consist of two parts. The first part is a constant which takes into account the change in the brake coefficient of friction as it heats. A second component was found to depend on the rate of change of the brake pressure P. 
     The first term,Q FK1 (6,6), has been found to control how fast the Kalman filter will respond to a change in k (Δk in ΔT seconds) with an extrapolation time Δt when set as follows: 
     
       
           Q   FK1 (6,6)=(Δ k (Δ t/ΔT )) 2 (ft.lb/psi) 2   
       
     
     The amount of change per unit time can be determined experimentally. The second term, Q FK2 (6,6), has been found to control how fast the Kalman filter will respond to a change in k (Δk in ΔT seconds) with an extrapolation time Δt with a concurrent pressure rate of {dot over (P)}: 
     
       
           Q   FK2 (6,6)=(Δ k (Δ t/ΔT )( {dot over (P)}/{dot over (P)}   ave )) 2 (ft.lb/psi) 2   
       
     
     The amount of change per unit time can be determined experimentally. The term {dot over (P)} ave  is the average rate of change of pressure which occurred while k was varying. 
     Weight/Wheel Ratio W t  and Wheel Radius r 
     Q FK (7,7) and Q FK (8,8) in the preferred embodiment are simply set to zero. Similarly, all off diagonal values of Q FK (i,j) where i≠j are set to zero. 
     As for the real observation noise matrix R F , such matrix can be represented as follows: 
     
       
         
           
               
             
               
                   
               
             
            
               
                 R F  = 
               
            
           
           
               
               
               
               
            
               
                   
                 off diagonals 
                 R F (i,j) = 0 
                 for i ≠ j 
               
            
           
           
               
               
               
               
            
               
                   
                 V m    
                 R F (1,1) = 3 2   
                 (ft/sec) 2   
               
               
                   
                 ω m    
                 R F (2,2) = 3 2   
                 (rad/sec) 2   
               
               
                   
                 T m    
                 R F (3,3) = 60 2   
                 (ft.lb) 2   
               
               
                   
                 
                   m  
                 
                 R F (4,4) = .3 2   
                 (ft/sec 2 ) 2   
               
               
                   
                   
               
            
           
         
       
     
     It will be appreciated by those skilled in the art that the plant model is often unknown or poorly known. The values of fictitious observation noise are often determined by trial and error until suitable performance is obtained. 
     Control Incorporating Pitch Plane Weight Transfer 
     The above discussion of the controller  34  focused on control of a single wheel. However, the same principles apply to integrated control of multiple braked wheels. In the following embodiments of the invention, the controller  34  operates to provide antiskid control taking into account the integration of multiple braked wheels and weight transfer to the nose wheel during braking. One embodiment, described in connection with FIG. 16, involves the use of coupled Kalman filter which has all the states for each braked wheel in one large Kalman filter. Another embodiment, discussed with reference to FIG. 17, utilizes a decoupled Kalman filter made up of a separate Kalman filter for each braked wheel. The trade off is performance versus computational burden. The decoupled approach is found to be far less burdensome, and hence is the more preferred design from such perspective. Each embodiment will be described below, emphasizing the primary differences between the single wheel control discussed above. Discussion of those features common to all embodiments is omitted for sake of brevity. 
     Underlying Theory 
     When multiple wheels are present in an aircraft (e.g., three or five including the nose wheel), each braked wheel will produce a retarding force on the corresponding axle. The retarding force will transfer weight off the axle which may tend to reduce the amount of torque which may be applied to the brake. The force and torque sum on the aircraft for a single wheel give the above Equs. (1) and (2) which, ignoring noise for the time being, are repeated as follows as Equs. (31) and (32), respectively: 
     aircraft (A/C) velocity 
     
       
           {dot over (V)} =−μ[( V−ωr )/ V,μ   p   ,r   p   ]g   (31) 
       
     
     wheel speed 
     
       
         {dot over (ω)}=(1/ I )[μ[( V−ωr )/ V,μ   p   ,r   p   ]W   t   r −sign[ω] KP]   (32) 
       
     
     Equs. (31) and (32) can be easily modified to account for multiple braked wheels and weight transfer. The modified equations are: 
     aircraft (A/C) velocity 
     
       
           {dot over (V)}=−C[μ   ave ]μ ave   g   (33) 
       
     
     wheel speed 
     
       
         {dot over (ω)}=(1 /I   i )[μ[( V−ω   i   r   i )/ V   i ,μ pi   ,r   pi   ]C[μ   ave   ]W   t   r   i /imax−sign[ω i   ]kP]   (34) 
       
     
     where: 
     μ ave  is the average developed friction of the i tires, 
     C[μ ave ] is a term accounting for the weight transfer, and 
     imax=2 or 4 for three and five wheel aircraft, respectively, 
     and, the average developed tire friction is given by: 
     
       
         μ ave =(1/imax)Σ i=1 to imax [μ[( V−ω   i   r   i )/ V,μ   pi   ,r   pi ]  (35) 
       
     
     where: 
     i=1 is the left main outer tire 
     i=2 is the right main outer tire 
     i=3 is the left main inner tire (included in five wheel aircraft) 
     i=4 is the right main inner tire (included in five wheel aircraft) 
     FIG. 15 illustrates schematically an exemplary three wheel or five wheel aircraft  400 . The aircraft includes a nose wheel  402  and a set of main wheels  40 . Each wing  404  includes a strut  406  coupled to an axle about which a single wheel  40  rotates in a three wheel aircraft, and a pair of wheels  40  rotate in a five wheel aircraft. The center of mass  408  of the aircraft is predetermined based on the known physical characteristics of the aircraft  400 . The center of mass  408  is located a horizontal distance “b” from the axle of the nose wheel  402  as shown in FIG.  15 . The center of mass  408  is located a horizontal distance “c” from the rear axles of the main wheels as shown in FIG.  15 . Finally, the center of mass  408  is located a vertical distance “h” from the runway surface  410 . 
     The function accounting for the steady state weight transfer is:                  C   ss          [     μ   ave     ]       =     1     1   +       (     c   /   b     )          (     1   +       (     h   /   c     )          μ   ave         )                   (36)                         
     where: 
     c is the horizontal distance from the rear axle of the main wheels to the center of mass  408   
     b is the horizontal distance from the aircraft center of mass  408  to the nose wheel axle; and 
     h is the vertical distance from the center of mass  408  to the runway surface  410 , as noted above. 
     The function ( 36 ) may then be passed through a second order filter representing the aircraft pitch plane dynamics to arrive at the final form of the weight transfer function. The pitch plane filter characteristics are given by:                  C        [   s   ]           C   ss          [   s   ]         =       w   n   2         s   2     +     2      ζ                   w   n        s     +     w   n   2                 (37)                         
     where: 
     s is the Laplace transform variable; 
     w n  is pitch plane rotation natural frequency; and 
     ζ is the damping ratio. 
     The natural frequency w n  is calculated from the pitch plane rotational inertia, the gear spring constants, and geometry, all of which are known physical parameters. 
     Coupled Kalman Filter 
     FIG. 16 schematically represents an embodiment of the invention in which the Kalman filter based controller (now designated  34 ′) includes one large Kalman filter having all the states for each braked wheel. As shown in FIG. 16, the system includes four main wheels including a left pair  412  made up of a left main outer wheel  40  and a left main inner wheel  40  with an axle  413  therebetween. In addition, the system includes a right pair  414  made up of a right main outer wheel  40  and a right main inner wheel  40  with an axle  416  therebetween. In a three wheel aircraft embodiment, the left main inner wheel  40  and the right main inner wheel  40  may be omitted as will be appreciated. 
     Each wheel includes its own brake actuator  36 , wheel speed sensor  42  and brake torque sensor  44 , similar to the single wheel represented in FIG.  1 . The actuator  36  for each wheel  40  receives the command pressure signal P c  output from the controller  34 ′. In addition, each wheel  40  provides individual measured wheel speed ω m  and brake torque T m  to the controller  34 ′ from its respective sensors  42  and  44 . As described above in relation to FIG. 1, the controller  34 ′ also receives the measured aircraft velocity V m  and acceleration {dot over (V)} m  if available. Moreover, the controller  34 ′ receives the PILOT PRESSURE signal from the pilot brake device  32 . 
     The state equations for the controller  34 ′ consist of two for the aircraft  400  (velocity and weight), and six additional for each wheel  40  (one for wheel speed, two for tire friction, one for tire radius and one for brake friction). These state equations are basically processed according the Kalman algorithm described above, with the exception that the various matrices, etc. are expanded to account for the additional states. If the aircraft  400  includes only two rear wheels  40  rather than four, the additional states associated with the extra two wheels  40  are omitted. A complete set of the state equations, taking into account noise (n+n f ), is as follows: 
     Coupled State Equations (FIG. 16) 
     Aircraft States: 
     velocity 
     
       
           {dot over (V)}=−C[μ   ave ]μ ave   g+n+n   f   
       
     
     weight 
     
       
         
           W 
           t 
           =n+n 
           f 
         
       
     
     Left Main Outer Wheel States: (i=1) 
     wheel speed 
     
       
         {dot over (ω)} i =(1 /I   i )[μ[( V−ω   i   r   i )/ V   i ,μ pi   ,r   pi   ]C[μ   ave   ]W   t   r   i /imax−sign[ω i   ]kP]+n   i   +n   fi   
       
     
     brake pressure 
     
       
           {dot over (P)}   i =(1 /   T   i )[ P   ci   −P   i   ]+n   i   +n   fi   
       
     
     friction peak amplitude 
     
       
         {dot over (μ)} pi   =n   i   +n   fi   
       
     
     friction peak location 
     
       
         
           {dot over (r)} 
           pi 
           =n 
           i 
           +n 
           fi 
         
       
     
     torque to pressure ratio 
     
       
         
           {dot over (k)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     wheel radius 
     
       
         
           {dot over (r)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     Right Main Outer Wheel States: (i=2) 
     wheel speed 
     
       
         {dot over (ω)} i =(1 /I   i )[μ[( V−ω   i   r   i )/ V   i ,μ pi   ,r   pi   ]C[μ   ave   ]W   t   r /imax−sign[ω i   ]kP]+n   i   +n   fi   
       
     
     brake pressure 
     
       
           {dot over (P)}   i =(1 /   T   i )[ P   ci   −P   i   ]+n   i   +n   fi   
       
     
     friction peak amplitude 
     
       
         {dot over (μ)} pi   =n   i   +n   fi   
       
     
     friction peak location 
     
       
         
           {dot over (r)} 
           pi 
           =n 
           i 
           +n 
           fi 
         
       
     
     torque to pressure ratio 
     
       
         
           {dot over (k)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     wheel radius 
     
       
         
           {dot over (r)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     Left Main Inner Wheel States: (i=3) 
     wheel speed 
     
       
         {dot over (ω)} i =(1 /I   i )[μ[( V−ω   i   r   i )/ V   i ,μ pi   ,r   pi   ]C[μ   ave   ]W   t   r   i /imax−sign[ω i   ]kP]+n   i   +n   fi   
       
     
     brake pressure 
     
       
           {dot over (P)}   i =(1 /   T   i )[ P   ci   −P   i   ]+n   i   +n   fi   
       
     
     friction peak amplitude 
     
       
         {dot over (μ)} pi   =n   i   +n   fi   
       
     
     friction peak location 
     
       
         
           {dot over (r)} 
           pi 
           =n 
           i 
           +n 
           fi 
         
       
     
     torque to pressure ratio 
     
       
         
           {dot over (k)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     wheel radius 
     
       
         
           {dot over (r)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     Right Main Inner Wheel States: (i=4) 
     wheel speed 
     
       
         {dot over (ω)} i =(1 /I   i )[μ[( V−ω   i   r   i )/ V   i ,μ pi   ,r   pi   ]C[μ   ave   ]W   t   r   i /imax−sign[ω i   ]kP]+n   i   +n   fi   
       
     
     brake pressure 
     
       
           {dot over (P)}   i =(1 /   T   i )[ P   ci   −P   i   ]+n   i   +n   fi   
       
     
     friction peak amplitude 
     
       
         {dot over (μ)} pi   =n   i   +n   fi   
       
     
     friction peak location 
     
       
         
           {dot over (r)} 
           pi 
           =n 
           i 
           +n 
           fi 
         
       
     
     torque to pressure ratio 
     
       
         
           {dot over (k)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     wheel radius 
     
       
         
           {dot over (r)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     Decoupled Kalman Filter 
     FIG. 17 represents an embodiment of the invention in which the Kalman filter based controller (now desginated  34 ″) is decoupled with respect to many states. The controller  34 ″ differs from that discussed above in relation to FIG. 16 in that the computational burden is shared among four separate Kalman filters (KF) each assigned to control a respective braked wheel  40 . Each of the Kalman filters KF maintains a state for the aircraft velocity {dot over (V)} and weight W t . While this may be redundant, the purpose of such partitioning is to allow each filter KF to continue functioning if there is a hardware failure limited to a wheel. 
     As shown in FIG. 17, each filter KF recieves the PILOT PRESSURE command from the brake device  32 , the measured wheel speed ω m  and the measured brake torque T m  for the corresponding wheel  40 . Each filter KF has its own corresponding output for providing a command signal P c  to the respective wheel  40 . The controller  34 ″ further includes registers M 1  thru M 4  which allow the decoupled filters KF to exchange data therebetween. Specifically, registers M 1  and M 2  store the measured aircraft velocity V m  and acceleration {dot over (V)} m  from the aircraft electronics  36 , if available. Each filter KF has access to the registers M 1  and M 2  for carrying out the computations described above in connection with the single wheel embodiment except as modified herein. 
     Furthermore, each filter KF posts its estimated value of the developed friction μ (Equ. 15c) and peak friction μ p  (Equ. 4) for its respective wheel  40  in register M 4  so that the other filters KF may access and use such data as described below. Specifically, such terms are used by each filter KF to calculate the average tire friction μ ave , the weight transfer parameter C[μ ave ], and the respective pressure command P c . In addition, each filter KF posts its corresponding value of P μμ  (representing the covariance of the developed friction for the respective wheel as defined below) in register M 3  which allows each filter KF to access the same from the other filters. 
     Furthermore, since the average tire friction μ ave  and weight transfer parameter C[μ ave ] for a given filter KF use information from the other filters KF which may be in error, and this error is not known to the covariance matrix, fictitious process noise is added to the velocity and wheel speed states to account for this. The process noise terms Q FK  may be calculated from: 
     aircraft velocity {dot over (V)} 
     
       
           Q   FK =Σ i=1 to imax ( d{dot over (V)}   i   /dμ   j   [·]Δt ) 2   P   μμj    i≠j   (38) 
       
     
     wheel speed {dot over (ω)} 
     
       
           Q   FK =Σ i=1 to imax ( d{dot over (ω)}   i   /dμ   j   [·]Δt ) 2   P   μμj    i≠j   (39) 
       
     
     where: 
     μ[·] is short hand notation for the mu slip function (Equ. 15c) 
     P μμj  is the covariance of the developed friction of the jth wheel 
     Δt is the extrapolation time increment 
     The covariance of the developed friction μ is calculated from each of the filters KF respective covariance matrix P, as follows: 
     
       
         P μμ=H   μμPH   μμ   T   (40) 
       
     
     The developed μ observation matrix, H μμ , is calculated from 
     
       
           H   μμ   T   ={dμ[·]/dV,dμ[·]/dω, 0 ,dμ[·]/dμ   p   ,dμ[·]/dr   p ,0,0 ,dμ[·]/dr}   (41) 
       
     
     where: 
     d is the derivative operator. 
     Accordingly, each filter KF maintains six wheel specific states, namely one for wheel speed, two for tire friction, one for tire radius and one for brake friction. The state equations for the decoupled embodiment of FIG. 17 are therefore given by: 
     Decoupled State Equations (FIG. 17) 
     Aircraft States: 
     velocity 
     
       
           {dot over (V)}=−C[μ   ave ]μ ave   g+n+n   f   
       
     
     weight 
     
       
         
           W 
           t 
           =n+n 
           f 
         
       
     
     Wheel States: (i=1, 2, 3, 4) 
     wheel speed 
     
       
         {dot over (ω)} i =(1 /I   i )[μ[( V−ω   i   r   i )/ V   i ,μ pi   ,r   pi   ]C[μ   ave   ]W   t   r   i /imax−sign[ω i   ]kP]+n+n   fi   
       
     
     brake pressure 
     
       
           {dot over (P)}   i =(1 /   T   i )[ P   ci   −P   i   ]+n   i   +n   fi   
       
     
     friction peak amplitude 
     
       
         {dot over (μ)} pi   =n   i   +n   fi   
       
     
     friction peak location 
     
       
         
           {dot over (r)} 
           pi 
           =n 
           i 
           +n 
           fi 
         
       
     
     torque to pressure ratio 
     
       
         
           {dot over (k)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     wheel radius 
     
       
         
           {dot over (r)} 
           i 
           =n 
           i 
           +n 
           fi 
         
       
     
     Effect on Pressure Command 
     With regard to the embodiments of FIGS. 16 and 17, since there is less force on each axle  413 ,  416  (FIG. 15) as pitch plane weight transfer occurs, the pressure command P c  is modified to apply less pressure to each wheel  40 . This may be done by modifying the pressure peak command P p . As discussed above in relation to the embodiment of FIG. 1, the pressure peak command P p  was previously given by Equ. 13 as follows: 
     
       
           P   p =(μ p   /k )[ W   t   r+gI (1 −r   p )/ r]   (13) 
       
     
     The modified peak pressure command is derived as follows: 
     The steady state brake pressure required to operate at the peak of the mu-slip curve may be calculated from the torque sum on the wheel  40  when operating at the peak as follows: 
     
       
           I{dot over (ω)}   p =μ p   C   ss[μ   avep   ]W   t   r /imax− k P   pss   (42) 
       
     
     where the p subscript has been added to indicate peak quantities. 
     The deceleration of the wheel  40 , when operating at the peak of the mu-slip curve, may be calculated from the force sum on the aircraft as follows: 
     
       
           {dot over (V)}   p =−μ avep   C   ss   [μ   avep   ]g   (43) 
       
     
     
       
         {dot over (ω)} p =(1 −r   p ) {dot over (V)}   p   /r   (44) 
       
     
     and substituting Equ. 43 into Equ. 44 for {dot over (V)} p  yields 
      {dot over (ω)} p ==μ avep   C   ss[μ   avep   ]g (1 −r   p ) V   p   /r   (45) 
     Equations 42 thru 45 may then be solved for the peak pressure P p , giving: 
     
       
           P   pssi =( C   ss[μ   avep   ]/k   i )[μ pi   W   t   r   i /imax+μ avep   gI   i (1 −r   p )/ r   i ]  (46) 
       
     
     This weight transfer term based on the friction peaks in this equation is replaced with the instantaneous term to account for a substantial lag in acheiving the weight transfer. Thus, the modified peak pressure for each wheel becomes: 
     
       
           P   pi =( C[μ   ave   ]/k   i )[μ pi   W   ti   r /imax+μ avep   gI   i (1 −r   pi   /r   i ]  (47) 
       
     
     The above computations are carried out by the single Kalman filter based controller  34 ′ in the embodiment of FIG. 16, or by the decoupled Kalman filter based controller  34 ″ in the embodiment of FIG. 17 by virtue of the individual filters KF exchanging relevant data therebetween. The peak pressure P pi  for each ith wheel is then augmented with a set point and excitement signal as discussed previously to arrive at the command pressure P c  for each individual wheel  40 . 
     Accordingly, it is possible to account for pitch plane weight transfer among multiple braked wheels. The coupled Kalman filter of FIG. 16 or the decoupled Kalman filter of FIG. 17 provides for the exchange of information between the respective wheels in order to permit such operation. It will be appreciated that another embodiment may include both a coupled Kalman filter and decoupled filters. For example, another embodiment may utilize a first Kalman filter which couples control of the left main inner wheel  40  and the right main inner wheel  40 . A second Kalman filter couples the control of the left main outer wheel  40  and the right main outer wheel  40 . The first Kalman filter and the second Kalman filter, on the other hand, remain decoupled. Such embodiment represents a hybrid of the embodiments of FIGS. 16 and 17 as will be appreciated. 
     It will be appreciated that the approaches described herein can easily be extended to aircraft with more than five wheels by those with ordinary skill in the art of antiskid braking and Kalman filters. 
     Thus, it will be appreciated that the antiskid control system using Kalman filter techniques in accordance with the present invention provides for quick and efficient braking. Although the invention has been shown and described with respect to certain preferred embodiments, it is obvious that equivalents and modifications will occur to others skilled in the art upon the reading and understanding of the specification. For example, the use of a excitement signal need not be limited to increasing the observability of the slip ratio set point as described above. A excitement signal can be utilized in connection virtually any of the other parameters as will be appreciated. 
     Furthermore, it will be appreciated that state estimation techniques other than a Kalman filter in a preferred embodiment may be used in accordance with the present invention for obtaining the state estimates disclosed herein. 
     The present invention includes all such equivalents and modifications, and is limited only by the scope of the following claims.