John Leventides

Global asymptotic linearisation of the pole placement map: a closed-form solution for the constant output feedback problem

The problem of pole assignment, by constant output feedback controllers, is studied for minimal systems described by a proper transfer function matrix G(s) ϵ Rm × p(s) with McMillan degree n. A new method is presented based on asymptotic linearisation (around a degenerate point) of the pole placement map related to the problem. The essence of the present approach is to reduce the multilinear nature of the problem to one of solving a linear set of equations, and this is achieved without losing any of the degrees of freedom in the controller. The solution is given in closed form in terms of a one-parameter (ϵ) family of static feedback compensators, for which the closed-loop poles approach the required ones as ϵ → 0. Conditions for the generic, as well as exact, solvability of the arbitrary pole placement problem are given in terms of the numbers m, p, n and the rank of a new system invariant, the D-restricted Plucker matrix. It is shown that the method works generically when mp > n, which (along with the boundary case mp = n) is the best possible condition as far as the number of states of the open-loop system is concerned, for achieving arbitrary pole placement via constant output feedback.

A new sufficient condition for arbitrary pole placement by real constant output feedback

Abstract A new sufficient condition for generic pole assignment is given based on the height of the first Whitney class of a certain Grassmannian, this result contains the condition m + p − 1 ≥ n and is stronger than LScat (p, m) ≥ n , this test is equivalent to the factorial test in [7] but it has the advantage that it is testable for most of the cases.

The pole placement map, its properties, and relationships to system invariants

A number of properties of the complex and real pole placement map (PPM) which relate to the dimensions of their images and relate them to known system invariants are derived. It is shown that the two dimensions are equal and that their computation is equivalent to determining the rank of the corresponding differential. A new expression for the differential of the PPM allows the derivation of relationships between the Markov parameters and the Plucker matrix invariant of the system. Conditions for pole assignability are derived, based on the relationships between the rank of the Plucker matrix and the rank of the differential of the PPM. >

Structured squaring down and zero assignment

The problem of zero assignment by squaring down is considered for a system of p-inputs, n-outputs and n-states (m > p), where not all outputs are free variables for design. We consider the case where a k-subset of outputs is preserved in the new output set, and the rest are recombined to produce a total new set of p-outputs. New invariants for the problem are introduced which include a new class of fixed zeros and the methodology of the global linearization, developed originally for the output feedback pole assignment problem, is applied to this restricted form of the squaring down problem. It is shown that the problem can be solved generically if (p − k)(m − p) > δ*, where k (k < p) is the number of fixed outputs and δ* is a system and compensation scheme invariant, which is defined as the restricted Forney degree.

Decentralized dynamic pole assignment with low-order compensators

The problem of arbitrary pole placement via dynamic decentralized output feedback is studied for minimal systems described by a proper transfer function matrix P(s) ∈ R m × p (s) (m = ∑ m i and p = ∑ p i ), with McMillan degree n. The family of controllers to be used includes those decentralized controllers with κ channels whose ith channel has maximum observability index at most d i . The method presented here is based on asymptotic linearization around a decentralized degenerate compensator of the pole placement map related to the problem. It is shown that the method works generically when m + p > n, where m + = min{d i (p i + m i - 1) + m i }, i = 1, …, κ, and the smallest d i of the compensator of the ith channel is the integral part of (n - pm i )/p(p i + m i - 1).

Sufficient conditions for arbitrary pole assignment by constant decentralized output feedback

This paper examines the problem of arbitrary pole assignment by decentralized output feedback. New sufficient conditions for the existence of real solutions are derived in terms of the heights of the first Whitney classes corresponding to the channel pairs ofmi outputs, pi inputs, and some appropriate partitioning of the number of states. These results extend the odd intersection framework approach based on the heights to the decentralized case and provide sufficient conditions for cases not covered by previous results.

GRASSMANN MATRICES, DETERMINANTAL ASSIGNMENT PROBLEM AND APPROXIMATE DECOMPOSABILITY

Abstract The exterior equation an n —dimensional vector space over F , is an integral part of the study of the Determinantal Assignment Problem (DAP) of linear systems and its solvability (decomposability of ) is characterised by the Quadratic Pliicker Relations (QPR). An alternative new test for decomposability of is given, in terms of the rank properties of the Grassmann matrix, , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if dim = m, where . If is decomposable, then the solution space is simply defined by . The linear algebra formulation of the decomposability problem provides an alternative framework (to that defined by the QPRs) for the study of solvability and computation of solutions of DAP and enables the definition and study of “approximate solutions” of exterior equations as a distance problem. For the case of m = 2, n = 4 a solution to approximate decomposability is given and its properties are linked to the singular values of .

The pole placement problem via PI feedback controllers

In this paper we study the pole placement problem using generalized PI controllers of a fixed lag k as compensators. We derive a new strong sufficiency condition which guarantees the arbitrary pole assignability of a given system having m inputs p outputs and McMillan degree n. This sufficiency condition misses the theoretical best possible necessary condition by one degree of freedom. The proof of the main result can be used to derive a numerical procedure. This numerical procedure is however very sensitive in the parameters and further research is required to investigate the conditions under which sensitivity occurs and whether the solution can be made more robust in this regard.

Solution of the determinantal assignment problem using the Grassmann matrices

ABSTRACT The paper provides a direct solution to the determinantal assignment problem (DAP) which unifies all frequency assignment problems of the linear control theory. The current approach is based on the solvability of the exterior equation where is an n −dimensional vector space over which is an integral part of the solution of DAP. New criteria for existence of solution and their computation based on the properties of structured matrices are referred to as Grassmann matrices. The solvability of this exterior equation is referred to as decomposability of , and it is in turn characterised by the set of quadratic Plücker relations (QPRs) describing the Grassmann variety of the corresponding projective space. Alternative new tests for decomposability of the multi-vector are given in terms of the rank properties of the Grassmann matrix, of the vector , which is constructed by the coordinates of . It is shown that the exterior equation is solvable ( is decomposable), if and only if where ; the solution space for a decomposable , is the space . This provides an alternative linear algebra characterisation of the decomposability problem and of the Grassmann variety to that defined by the QPRs. Further properties of the Grassmann matrices are explored by defining the Hodge–Grassmann matrix as the dual of the Grassmann matrix. The connections of the Hodge–Grassmann matrix to the solution of exterior equations are examined, and an alternative new characterisation of decomposability is given in terms of the dimension of its image space. The framework based on the Grassmann matrices provides the means for the development of a new computational method for the solutions of the exact DAP (when such solutions exist), as well as computing approximate solutions, when exact solutions do not exist.

The decentralized Markov parameters and the selection of control structures

The properties of the decentralized pole placement map under constant output feedback are investigated and they are linked to known invariants of the decentralized pole assignment problem. A new expression of the differential of this map allows the derivation of relationships between the decentralized Plucker matrix invariant and the Markov parameters and leads to the definition of the Decentralized Markov Parameters (DMP). The matrix associated with the DMPs provides a new simple test for selection of decentralization schemes using as criteria the avoidance of formation of fixed modes and the preconditioning of the decentralized problem to be linearly assignable which excludes also almost fixed modes and it is a necessary condition for solution of the decentralized control problem. The natural link of this test to the Markov parameters and state space parameters of the models provides the means for affecting the shaping of properties of Plucker matrices by design, redesign of the input, and output structure of the system model. The results which are originally presented for the decentralized constant output feedback are subsequently extended to the case of decentralized PI feedback.