The Kalman-Yakubovich-Popov Lemma in a behavioural framework

New Algorithms for Polynomial J-Spectral Factorization

“…It was also shown in [17] that if Zio > 0 for all o, then the choice P P , i.e., the largest real symmetric solution of the LMI, yields a Hurwitz spectral factor. In fact, we quote from [17]:…”

“…We do not explicitly formulate this result here, but instead concentrate on methods to obtain Hurwitz spectral factors. First note that for a given para-hermitian matrix Z there exists a q  q Hurwitz polynomial matrix H such that Zx H T ÀxHx i¨Zio > 0 for all o A R. It was shown in [17] that if Zio V 0 for all o, then LMI (5.2) has real symmetric solutions P À and P such that any real symmetric solution P satis®es P À U P U P . It was also shown in [17] that if Zio > 0 for all o, then the choice P P , i.e., the largest real symmetric solution of the LMI, yields a Hurwitz spectral factor.…”

Checking dissipativity of linear behavior systems given in kernel representation

“…Brüll (B) Technische Universität Berlin, Sekretariat MA 4-5, Straße des 17. Juni 136, 10623 Berlin, Germany e-mail: bruell@math.tu-berlin.de Known methods to check dissipativity of a linear system will either use KalmanYakubovich-Popov Lemma and test if a certain linear matrix inequality (LMI) is solvable [10], or they will obtain an explicit representation of the Popov function (see Definition 3.4) and then check its positive semi-definiteness along the imaginary axis. In Sect.…”

Energy and power optimization in a behavioural framework

“…A QDF is a quadratic function of a signal and some of its higher-order derivatives, and therefore it is particularly apt to describe expressions involving the variables of a linear differential system, see Willems and Trentelman [13]. The current paper is the sequel of our papers Van der Geest and Trentelman [3] and [4], in which we studied dissipativity of linear differential systems in terms of QDFs. The exchange of energy between a system and its environment is related to the behaviour of the external variable of the system; it does not depend on the partition of this external variable into inputs and outputs.…”

“…As a result, the initial value of the available storage is precisely the amount of energy that may be redeemed from the system (which justifies the name available storage). We use the Kalman-YakubovichPopov (KYP) lemma for linear differential systems from [3] to characterize the available storage and its corresponding dissipation rate in terms of a linear matrix inequality (LMI) in the original coefficients of the differential equation that is used to specify the system. This is in line with the behavioural philosophy of solving control problems in terms of the original problem data.…”