12.3.   HOVER TO FORWARD FLIGHT EXAMPLE

12.3.1.   Model of Helicopter’s Forward Dynamics

The proposed adaptation scheme is illustrated on the following model
representing the longitudinal channel dynamics of an Apache helicopter
constrained to have no vertical motion; only longitudinal and pitch rotation
motions are allowed [18]:

where , , and δ_e represent the forward acceleration (ft/s²), pitch angle
acceleration (rad/s²), and longitudinal cyclic input (deg), respectively. X
represents the aerodynamic force along the ‘‘X axis,’’ and M represents the
pitching moment about the ‘‘Y axis.’’ The parameters, X_trim, X_x, X_θ, X_δe,
M_trim, M_x, M_θ, M_δe, x_trim, _trim, and δ_{e, trim} are functions of . X_trim and M_trim
are the trim values of the aerodynamic force X and the pitching moment M,
respectively. The variables, X_x, X_θ, X_δe, M_trim, M_x, and M_θ are the
partial derivatives of X and M with respect to , , and δ_e, respectively.
The physical constants m and IY are the mass of the helicopter and the moment
of inertia along the Y axis. The state vector of the helicopter model is
[x₁x₂ x₃ x₄]^T = [θ ]^T. It is assumed that the output vector of the
model is the same as the state vector.

12.3.1.1.   Blending Local Mode Controllers Approach

Linear quadratic regulators are designed that regulate the vehicle about
specified operating points. Scalar gains for the hover and forward flight
controllers are calculated first such that the closed-loop system transitions
from one state to the next in minimum time while satisfying physical constraints
on the vehicle’s velocity and acceleration profiles.

Afterwards, the blending gains K_hov(, , θ, ) and   K_FF (, , θ, ) are
realized via a fuzzy neural network construct described in a previous section.

12.3.1.2.   Gain Scheduling Approach

A gain scheduled controller is designed via dynamic linearization about the
minimum time trajectory determined in the previous section:

• 67 equidistant frozen times t₁, . . . , t₆₇ are chosen where t_{k + 1} - t_k =
  (t_f - t₀)/66.
• 67 linear autonomous open loop systems are obtained via Lyapunov
  linearization of the helicopter model about the minimum time state and
  control trajectories at frozen times t₁, . . . , t₆₇.
• 67 linear quadratic regulators are designed for each time-frozen linear
  model of the helicopter created in the previous step.
• The 67 linear control laws are blended according to how near the
  current state is to each of the frozen operating states determined in the
  first step. The interpolated control law is applied to the system to be
  controlled.
  

12.3.1.3.   Least Squares Adaptation Scheme. A sample time of T_S =
0.05 s is chosen for the adaptation scheme. The desired minimum time
trajectory and control are resampled such that they occur every T_S:

Afterwards, the desired transition model of the following mapping is determined
offline:

The active plant model is initially determined offline for the following
mapping:

Also, the linear model information defined at

is incorporated into the consequent part of the active plant model.

The plant adaptation mechanism adapts the active plant model with the
following parameters:

where δ, β, and σ^U are lower thresholds for membership value, the desired
overlap degree between membership functions, and the upper limit of the
width of each membership function, respectively.

The active controller model is the hover to forward flight mode controller
determined in Section 12.3.1.1.

The controller adaptation mechanism adapts the blending weights of the
active controller model with the following parameters:

12.3.2.   Simulation Results

Figures 12.3 and 12.4 show the desired , , θ, and trajectories. For small
parametric changes and wind disturbances, the controller exhibits good
tracking performance of the desired transition trajectory. However, as the
magnitude of the parametric changes and wind disturbances increases, the
tracking performance of the controller degrades. As expected, if the approximate
plant accurately captures the local model information and the input/output
behavior of the system to be controlled, the adapted controller exhibits excellent
tracking performance when encountering parametric changes and wind disturbances.