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We consider linear discrete-time descriptor systems, i.e., systems of linear equations of the form $E_{k+1} x^{k+1} = A_k x^k + f^k$ for $k \in \IZ$, where all $E_k$ and $A_k$ are matrices, $f_k$ are vectors and $x_k$ are the vectors of the solution we are looking for. We study the existence and uniqueness of solutions. A strangeness index is defined for such systems. Compared to the continuous-time case, it turns out, that in the discrete-time case it makes a difference, if one has an initial condition and one wants a solution in the future or if one has an initial condition and one wants a solution into the past and future at the same time.
The behavior approach and the problem of dissipativity have both been introduced and studied extensively by Willems et al. However, a computationally feasible method to check dissipativity is missing. Current methods will mostly rely on symbolic representations of rational functions. We will discuss a new characterization for linear systems in behavior form that allows to check dissipativity via the solution of a para-Hermitian, polynomial eigenvalue problem. Thus, we can employ standard methods of cubic complexity.
In this paper a general form of the infinite-horizon linear quadratic control problem is considered. We will discuss quadratic cost functionals which involve not only the state and input-variables but also derivatives of the state and input-variables of arbitrary order under constraints given by linear systems of higher order. We will examine two results that relate the linear quadratic control problem to an optimality system, which is given through a para-Hermitian matrix polynomial. The results can be applied to general rectangular descriptor systems (see Subsection 6.1) to obtain results which so far were only known for quadratic descriptor systems. Also we will see that the notion of dissipativity (when introduced in the proper way) is equivalent to the solvability of the linear quadratic control problem.
The well-known Kalman-Yakubovich-Popov Lemma establishes an equivalence between dissipativity and the solvability of a linear matrix inequality. In this paper we strengthen this result by showing the equivalence of dissipativity to the solvability of a so-called Lur'e equation, which mainly is a linear matrix inequality with a rank minimizing condition. Finally, we apply the result to standard systems to obtain the well-known result about the solvability of the algebraic Riccati equation.
Dissipativity is an important property of individual systems
that guarantees a stable interconnected system.
However, due to errors in the modeling process weakly non-dissipative
models may be constructed.
In this paper we introduce a method to perturb a non-dissipative LTI system in order to enforce dissipativity using spectral perturbation results for
para-Hermitian pencils.
Compared to earlier algorithms the new method
is applicable to a wider class of problems,
it utilizes a simpler framework, and
employs a larger class of allowable perturbations
resulting in smaller perturbations. Moreover, system stability can be enforced as well.
Numerical examples are provided to show the effectiveness of the new approach.