Regularization of descriptor systems by output feedback

Conditions are given under which a descriptor, or generalized state-space system can be regularized by output feedback. It is shown that under these conditions, proportional and derivative output feedback controls can be constructed such that the closed-loop system is regular and has index at most one. This property ensures the solvability of the resulting system of dynamic-algebraic equations. A reduced form is given that allows the system properties as well as the feedback to be determined. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way.

Regularization of Descriptor Systems by Derivative and Proportional State Feedback

For linear multivariable time-invariant continuous or discrete-time singular systems it is customary to use a proportional feedback control in order to achieve a desired closed loop behaviour. Derivative feedback is rarely considered. This paper examines how derivative feedback in descriptor systems can be used to alter the structure of the system pencil under various controllability conditions. It is shown that derivative and proportional feedback controls can be constructed such that the closed loop system has a given form and is also regular and has index at most 1. This property ensures the solvability of the resulting system of dynamic-algebraic equations. The construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way. The problem of pole placement with derivative feedback alone and in combination with proportional state feedback is also investigated. A computational algorithm for improving the “conditioning” of the regularized closed loop system is derived.

Minimum norm regularization of descriptor systems by mixed output feedback

We study the regularization problem for linear, constant coefficient descriptor systems Ex' = Ax+Bu, y1 = Cx, y2 = Γx' by proportional and derivative mixed output feedback. Necessary and sufficient conditions are given, which guarantee that there exist output feedbacks such that the closed-loop system is regular, has index at most one and E+BGΓ has a desired rank, i.e., there is a desired number of differential and algebraic equations. To resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of E is decreased. Numerical procedures are derived to construct the desired feedback gains. These numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way.

Feedback regulation by Atf3 in the endothelin-1-responsive transcriptome of cardiomyocytes: Egr1 is a principal Atf3 target

Endothelin-1 promotes cardiomyocyte hypertrophy by inducing changes in gene expression. Immediate early genes including activating transcription factor 3 (Atf3), Egr1 and Ptgs2 are rapidly and transiently upregulated by endothelin-1 in cardiomyocytes. Atf3 regulates expression of downstream genes and is implicated in negative feedback regulation of other immediate early genes. To identify Atf3-regulated genes, we knocked down Atf3 expression in cardiomyocytes exposed to endothelin-1 and used microarrays to interrogate the transcriptomic effects. Of upregulated mRNAs, expression of 23 (including Egr1, Ptgs2) was enhanced and expression of 25 was inhibited by Atf3 knockdown. Using quantitative PCR, we determined that knockdown of Atf3 had little effect on upregulation of Egr1 mRNA over 30 min, but abolished the subsequent decline, causing sustained Egr1 mRNA expression and enhanced protein expression. This resulted from direct binding of Atf3 to the Egr1 promoter. Mathematical modelling established that Atf3 can suffice to suppress Egr1 expression. Given the widespread co-regulation of Atf3 with Egr1, we suggest that the Atf3-Egr1 negative feedback loop is of general significance. Loss of Atf3 caused abnormal cardiomyocyte growth, presumably resulting from dysregulation of target genes. Our data therefore identify Atf3 as a nexus in cardiomyocyte hypertrophy required to facilitate the full and proper growth response.