Reliable Computing 3: 229–237, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
-Control Using Polynomial Matrices and
URGEN K. WEINHOFER
and WERNER C. HAAS
Department of Automatic Control, Johannes Kepler University, Altenbergerstraße 69, A-4040 Linz,
Austria, e-mail: firstname.lastname@example.org
(Received: 14 October 1996; accepted: 19 February 1997)
Abstract. A design method for H
-compensators is presented for linear multivariable plants. The
advantages of a compensator with two degrees of freedom are applied, and the intuitive method
using transfer and polynomial matrices is pursued. With the introduction of an interval arithmetic,
the numerical obstacles are avoided, and a numerical stable design tool is offered. The tool was then
tested on a single engine VISTA F-16 supersonic test vehicle.
Historically, computer tools use state space methods to calculate compensators in
the frequency domain because of numerical reasons. Today, a more direct approach
using transfer and polynomial matrices seems to be more adequate. Therefore, Sec-
tion 2 presents some well-known results of the factorization approach . These are
the parameterization of all stabilizing controllers and the decoupling of the tracking
and disturbance design. Using these results, the control problem can be transformed
to a model matching problem. In Section 3 the model matching problem is then
reduced to a suboptimal Nehari problem, and the compensator is constructed. For
the implementation of the algorithms, some numerical operations (e.g. coprime
factorization, spectral factorization) must be carried out. Theoretical and numer-
ical considerations for these operations are given in Section 4. For the computer
implementation an interval arithmetic is applied to avoid numerical problems .
Compared to a standard arithmetic, an interval arithmetic offers numerical stable
comparison with zero and dynamic adjustment of the mantissa length. Hence, a
precision bound of the results can be deﬁned in advance. A short introduction in
this arithmetic developed by one of the authors  is given in Section 5. The last
section illustrates the applicability of this approach to ﬂight control systems.
All systems considered in this article are assumed to be linear and time invariant
where R(s)andR[s] denote the set of all rational transfer matrices and polynomial
matrices. Moreover, H
R(s) denotes the set of all transfer matrices, which
are analytic on the closed right half plane (imaginary axis). The adjoint of a matrix
This work was supported in part from the FWF under grant P10518-TEC.