Issue 110, September 1999
Internet Control

CONTROL-SYSTEM DESIGN

Next, you can write a MATLAB script file that reads in the output file of the Maple program and calculates the values of the state space model A and B matrices for a given set of system parameters (equation 18).

In this design, I used a linear quadratic regulator (LQR) controller. The LQR method is essentially a multi-input/multioutput PID controller with an optimization index.

The optimization index requires two matrices, Q and R, which are used to compute a performance index to be minimized. The process is automatic.

You supply the Q and R matrices (which are selected intuitively), and the software computes the feedback gains. After several iterations using the simulation block (described next), we select a set of Q and R matrices that result in equation 19.

As you examine the feedback gains obtained from the LQR design, note that the second row gains have the exact magnitudes of the first row (see equation 20)! The state feedback equation is shown in equation 21. Further examination reveals that the sum of the two rows results in equation 22, which can be rewritten as equation 23.

Equation 23 is a PID controller around the elevation axis, which means that the gains we obtain from LQR design can be used in an elevation control loop (equation 24). Examining the difference between the gains (i.e., Vf – Vb) yields equation 25, which consists of two loops—one for pitch and one for travel.

This equation can be rewritten as equation 26, which is a PID loop to command the pitch to track the desired pitch (Pc). The desired pitch is defined in equation 27, which is another PID loop that controls the travel position. You now have the control equations shown in equation 28, and solving for Vf and Vb, you get the results in equation 29.