The Polynomial Toolbox provides several routines to solve typical design tasks. Their modifications as well as polynomial solutions to many other design problems can easily be built with the help of the basic tools of the Polynomial Toolbox. We successively discuss several basic control design routines,H₂ optimization, and HY optimization.

Basic control routines
The Polynomial Toolbox offers several basic functions to

• stabilize the plant and, moreover, to parametrize all stabilizing controllers (function stab)
• place closed-loop poles by dynamic output feedback (pplace)
• design deadbeat controllers for discrete-time systems (debe)

Stabilization
A simple random stabilization can be achieved by applying the macro stab. The resulting closed-loop poles are randomly placed in the stability region, whose shape of course depends on the choice of the variable. Both left and right polynomial matrix fractions (PMF) are accepted.

The same command with two additional outputs is used to obtain the parametrization of all stabilizing controllers according to the Youla-Kučera rule.

Pole-placement
Typically, the closed-loop poles should not only be stable but also located in prescribed positions within the stability region. The routine pplace takes care of this.

We will skip more detailed description of this command since the macro has been introduced and discussed in this corner in the 2nd issue of this newsletter

Deadbeat Control
In discrete-time systems, there is one pole location of particular interest. The closed-loop system can be forced to have a finite time response from any initial condition by making the closed-loop characteristic polynomial equal to a suitable power of z or q . Equivalently, the characteristic polynomial equals 1 for systems described by a delay operator (z^-1 or d ). The resulting performance is called deadbeat control and can be achieved by feeding the Polynomial Toolbox routine .debe with the plant's transfer function numerator and denominator (for MIMO systems, both left and right PMF are acceptable of course).

Also in this case, parametrization of all deadbeat controllers is available, similar to the pole-placement problem.

H₂ optimization
H₂ or linear-quadratic-Gaussian (LQG) control is a modern technique for designing optimal dynamic controllers. It allows to trade off regulation performance and control effort, and to take process and measurement noise into account. The Polynomial Toolbox offers two macros splqg and plqg for H₂ optimization by polynomial methods.

Scalar case
The function splqg results in the solution of the SISO LQG problem defined by

• Response of the measured output to the control input:
• Response of the controlled output to the control input:
• Response of the measured output to the disturbance input:

On one hand, the problem statement is quite general, however, generalization for MIMO case is not easy. That is why another definition of the problem has been taken into account for MIMO systems.

MIMO case
For MIMO systems, the routine plqg can be applied. The resulting controller minimizes the steady-state value of a quadratic criterion involving control inputs and controlled (and, at the same time, measured) outputs. Input and measurement white noises intensities are also considered.

H-infinity optimization
H-infinity optimization is a powerful modern tool. It allows the design of high-performance and robust control systems. Polynomial Toolbox tools concerning this topic were the subject of this corner in the 3rd issue of this newsletter.