Pole Placement and Matrix Extension Problems: A Common Point of View

Kim, Mee-Young; Rosenthal, Joachim; Wang, Xiaochang Alex

doi:10.1137/s0363012999354429

Cited by 13 publications

(8 citation statements)

References 17 publications

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“…under what conditions a symmetric transfer function is nondegenerate. Similar results were crucial in proving the pole placement results in [3,19,27]. Consider now the coincidence set…”

Section: Resultssupporting

confidence: 61%

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Output Feedback Pole Assignment for Transfer Functions with Symmetries

Helmke¹,

Rosenthal²,

Wang³

2006

SIAM J. Control Optim.

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This paper studies the problem of pole assignment for symmetric and Hamiltonian transfer functions. A necessary and sufficient condition for pole assignment by complex symmetric output feedback transformations is given. Moreover, in the case where the McMillan degree coincides with the number of parameters appearing in the symmetric feedback transformations, we derive an explicit combinatorial formula for the number of pole assigning symmetric feedback gains. The proof uses intersection theory in projective space as well as a formula for the degree of the complex Lagrangian Grassmann manifold.Keywords: Output feedback, Pole placement, inverse eigenvalue problems, Lagrangian Grassmannian, symmetric or Hamiltonian realizations , degree of a projective variety.AMS subject classifications. 14M15, 15A18, 70H14, 70S05, 93B55, 93B60.

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“…under what conditions a symmetric transfer function is nondegenerate. Similar results were crucial in proving the pole placement results in [3,19,27]. Consider now the coincidence set…”

Section: Resultssupporting

confidence: 61%

“…Following [19,26,27,35] we identify a closed loop characteristic polynomial ϕ(s) with a point in P δ . In analogy to the situation of the static pole placement problem considered in [3,35] (compare also with [27, Section 5]) one has a well defined characteristic map (3.9) in the Hamiltonian case.…”

Section: Resultsmentioning

confidence: 99%

Output Feedback Pole Assignment for Transfer Functions with Symmetries

Helmke¹,

Rosenthal²,

Wang³

2006

SIAM J. Control Optim.

Self Cite

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“…Our first class of test problems has its origin in the output pole placement problem in the control of linear systems. We may view this problem as an inverse eigenvalue problem [24]. The polynomial equations arise from minor expansions on det(A|X) = 0, A ∈ C n×m ,…”

Section: Matrix Completion With Pieri Homotopiesmentioning

confidence: 99%

Accelerating polynomial homotopy continuation on a graphics processing unit with double double and quad double arithmetic

Verschelde

2015

Proceedings of the 2015 International Workshop on Parallel Symbolic Computation

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Numerical continuation methods track a solution path defined by a homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of the solution path. With double double and quad double arithmetic, we can solve larger problems that we could not solve with hardware double arithmetic, but at a higher computational cost. This cost overhead can be compensated by acceleration on a Graphics Processing Unit (GPU). We describe our implementation and report on computational results on benchmark polynomial systems.

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“…In the field of controls, algorithms have also been proposed for the related, socalled pole placement problems [14,18,25,36], which arise from the need for the stabilization of dynamic systems. Such pole placement problems are typically of the form: given the state space matrices A∈R n , B∈R n×m , find the static output feedback matrix K∈R m such that spectrum of A + BKB T lies in some desired, closed convex sets, [35] the following problem is considered: the location of a complex conjugate pair of eigenvalues in the left half plane is specified, C 1 ={c}, C 2 ={c}, while requiring the remaining eigenvalues to lie within some convex domain C 3 = · · · = C n = C. For solving such problems, iterative algorithms based on alternating projection have been developed [35].…”

Section: Matrix Inverse Eigenvalue Problemmentioning

confidence: 99%

Inverse Eigenvalue Problems for Exploring the Dynamics of Systems Biology Models

2009

AAMM

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This paper describes inverse eigenvalue problems that arise in studying qualitative dynamics in systems biology models. An algorithm based on lift-andproject iterations is proposed, where the lifting step entails solving a constrained matrix inverse eigenvalue problem. In particular, prior to carrying out the iterative steps, a-priori bounds on the entries of the Jacobian matrix are computed by relying on the reaction network structure as well as the form of the rate law expressions for the model under consideration. Numerical results on a number of models show that the proposed algorithm can be used to computationally explore the possible dynamical scenarios while identifying the important mechanisms via the use of sparsity-promoting regularization.In this paper, we focus exclusively on ODE models of the form,where x(t) denotes the n-dimensional state vector and q the m-dimensional parameter vector. In systems biology modeling, one rarely has a detailed knowledge of the parameter values and sometimes even the knowledge of the network topology is incomplete. However, there often exist a direct relation between the qualitative dynamics of system (1.1) as captured in the bifurcation diagram with the cell physiology [33]. In this paper, we examine an inverse problem associated with the qualitative dynamics of (1.1) which commonly arise at the initial stages of modeling gene regulatory networks, namely: given a set of genes as well as their known and hypothetical interactions, can the network be bistable or oscillatory for some choice of parameter values? If so, what are the minimal sets of parameters that one could vary in order to obtain these dynamical phenotypes? Such questions are of practical relevance to the modelers, who may wish to explore or eliminate different hypothetical reaction network topologies and mechanisms. It needs to be emphasized that the dependence of qualitative dynamics on the choice of parameters cannot be neglected: it has been shown that even with a fixed reaction network topology and biochemical mechanism, different choices of parameters can result in oscillators of various qualitative types [7]. At the initial modeling stage, there is no prior knowledge on the influence of its parameters and inverse eigenvalue analysis could be a useful first step by bringing the system to a relevant parameter regime where a more complete picture of its dynamical characteristics would be obtained by carrying out a (forward) bifurcation analysis [19] followed by possible inverse bifurcation analysis [22,23]. This paper is organized as follows: In Section 2, we describe the underlying methodology and the proposed lift-and-project algorithm. In particular, we discuss in detail the matrix inverse eigenvalue problem that arises and give an illustrative example using a MAP kinase model. In Section 3, we demonstrate the proposed algorithm by applying to a number of systems biology models.The ODE systems for the numerical examples are given in the Appendix.

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Pole Placement and Matrix Extension Problems: A Common Point of View

Cited by 13 publications

References 17 publications

Output Feedback Pole Assignment for Transfer Functions with Symmetries

Output Feedback Pole Assignment for Transfer Functions with Symmetries

Accelerating polynomial homotopy continuation on a graphics processing unit with double double and quad double arithmetic

Inverse Eigenvalue Problems for Exploring the Dynamics of Systems Biology Models

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