On the zeros of a minimal realization

AbstractIn an earlier work, the authors have introduced a coordinate-free, module-theoretic definition of zeros for the transfer function G(s) of a linear multivariable system (A,B,C). The first contribution of this paper is the construction of an explicit k[z]-module isomorphism from that zero module, Z(G), to V∗/R∗, where V∗ is the supremal (A,B)-invariant subspace contained in kerC and R∗ is the supremal (A,B)-controllable subspace contained in kerC, and where (A,B,C) constitutes a minimal realization of G(s). The isomorphism is developed from an exact commutative diagram of k-vector spaces. The second contribution is the introduction of a zero-signal generator and the establishment of a relation between this generator and the classic notion of blocked signal transmissions

On the zeros and poles of a transfer function

AbstractThe poles and zeros of a linear transfer function can be studied by means of the pole module and the transmission zero module. These algebraic constructions yield finite dimensional vector spaces whose dimensions are the number of poles and the number of multivariable zeros of the transfer function. In addition, these spaces carry the structure of a module over a ring of polynomials, which gives them a dynamical or state space structure. The analogous theory at infinity gives finite dimensional spaces which are modules over the valuation ring of proper rational functions. Following ideas of Wedderburn and Forney, we introduce new finite dimensional vector spaces which measure generic zeros which arise when a transfer function fails to be injective or surjective. A new exact sequence relates the global spaces of zeros, the global spaces of poles, and the new generic zero spaces. This sequence gives a structural result which can be summarized as follows: “The number of zeros of any transfer function is equal to the number of poles (when everything is counted appropriately).” The same result unifies and extends a number of results of geometric control theory by relating global poles and zeros of general (possibly improper) transfer functions to controlled invariant and controllability subspaces (including such spaces at infinity)

On Minimal-Variance Control of Linear Systems with Quadratic Loss

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryDepartment of the Army / DA-28-043-AMC-00073(E)Ope

On Minimal-Variance Control of Linear Systems with Quadratic Loss

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryDepartment of the Army / DA-28-043-AMC-00073(E)Ope

Linear Cumulant Control and Its Relationship to Risk-Sensitive Control

Matrix differential equation descriptions of the cumulants of an integral quadratic cost associated with a linear system with white-noise input were derived in the mid-70s using generalized Karhunen-Loeve expansion techniques. Here, these same descriptions are derived directly from the cumulant generating function of the cost. A generalization of the k-cumulant control problem class introduced in 1998 is also presented. The solution to this more general class of optimal cumulant control problems is given, and the risk sensitive control problem of optimizing the cumulant generating function of the LQG cost is shown to be included in this cumulant control class