Versatile Symbolic and Numeric Environment
Control System Professional provides symbolic solutions to a wide variety of control problems and at the same time handles numerical control problems with ease. We illustrate this with a few examples of feedback design. The feedback design tools are represented in Control System Professional by the robust and Ackermann algorithms for the eigenvalue assignment problem and the linear-quadratic optimal control design functions. To begin with, we design the discrete-time state feedback controller for a third-order integrator system using the pole assignment algorithm. Here is a state-space realization of the system:
We can design the controller that places the poles of the closed-loop system into the pre-defined locations
Here is the system after closing the loop.
It is easy to see that the poles of the system are indeed placed as required.
The linear-quadratic optimal design is among the problems whose complexity typically calls for numerical methods. Control System Professional provides the tools for solving the infinite-horizon state and output regulator problems, Kalman estimator and filter design, finding the discrete equivalent to the continuous regulator and estimator, as well as direct access to the continuous and discrete algebraic Riccati equation solvers.
As a final example, we show how Control System Professional solves the linear-quadratic-Gaussian (LQG) problem that arises in the design of the linear time-invariant dynamic compensator for the line-of-site pointing and stabilization of the Falcon Eye System [Haessig 1996]. The compensator consists of a steady-state LQ regulator and Kalman filter. The designer adjusts the coefficients of the regulator and filter weighting matrices Q, R, v, and w until the compensator performs satisfactorily. The system is represented by a tenth-order model.
The small negative entries ε are added for numerical reasons.
The control weighting matrix R for the quadratic criterion has only one coefficient.
The state weighting matrix Q is calculated from the vector L.
The controller gain matrix is found by solving the algebraic Riccati equation.
In designing the Kalman filter, we represent the system by the state-space equations
= A x + B u + v
y = C x + w
where the vectors v and w are chosen to be
We append the identity matrix of the appropriate dimensions to the matrix B to create the expanded system.
We then find the Kalman filter gains, reproducing the result of the design in [Haessig 1996].