The following figure shows a Simulink® block diagram shows a tracking problem in aircraft autopilot design. To open
this diagram, type
lqrpilot at the MATLAB® prompt.
Key features of this diagram to note are the following:
The Linearized Dynamics block contains the linearized airframe.
sf_aerodyn is an S-Function block that contains the nonlinear
equations for .
The error signal between and the is passed through an integrator. This aids in driving the error to zero.
Beginning with the standard state-space equation
The variables u, v, and w are the three velocities with respect to the body frame, shown as follows.
Body Coordinate Frame for an Aircraft
The variables and are roll and pitch, and p, q, and r are the roll, pitch, and yaw rates, respectively.
The airframe dynamics are nonlinear. The following equation shows the nonlinear components added to the state space equation.
Nonlinear Component of the State-Space Equation
To see the numerical values for A and B, type
load lqrpilot A, B
at the MATLAB prompt.
For LQG design purposes, the nonlinear dynamics are trimmed at and p, q, r, and
θ set to zero. Since u, v, and
w do not enter into the nonlinear term in the preceding figure, this
amounts to linearizing around with all remaining states set to zero. The resulting state matrix of the
linearized model is called
The goal to perform a steady coordinated turn, as shown in this figure.
Aircraft Making a 60° Turn
To achieve this goal, you must design a controller that commands a steady turn by going through a 60° roll. In addition, assume that θ, the pitch angle, is required to stay as close to zero as possible.
To calculate the LQG gain matrix,
at the MATLAB prompt. Then, start the
lqrpilot model with the nonlinear
This figure shows the response of to the 60° step command.
Tracking the Roll Step Command
As you can see, the system tracks the commanded 60° roll in about 60 seconds.
Another goal was to keep θ, the pitch angle, relatively small. This figure shows how well the LQG controller did.
Minimizing the Displacement in the Pitch Angle, Theta
Finally, this figure shows the control inputs.
Control Inputs for the LQG Tracking Problem
Try adjusting the
R matrices in
lqrdes.m and inspecting the control inputs and the system states,
making sure to rerun
lqrdes to update the LQG gain matrix
K. Through trial and error, you may improve the response time of this
design. Also, compare the linear and nonlinear designs to see the effects of the
nonlinearities on the system performance.