Pole Assignment Problem Research Articles

Variable structure exponential stabilization of chained systems based on the extended nonholonomic integrator

In this paper, a new variable structure control strategy for exponentially stabilizing chained systems is presented based on the extended nonholonomic integrator model, the discontinuous coordinate transformation and the “reaching law method” in variable structure control design. The proposed approach converts the stabilization problem of an n-dimensional chained system into the pole-assignment problem of an ( n−3)-dimensional linear time-invariant system and consequently simplifies the stabilization controller design of nonholonomic chained systems.

Pole assignment for stochastic systems with unknown coefficients

This paper solves the exact pole assignment problem for the single-input stochastic systems with unknown coefficients under the controllability assumption which is necessary and sufficient for the arbitrary pole assignment for systems with known coefficients. The system noise is required to be mutually independent with zero mean and bounded second moment. Two approaches to solving the problem are proposed: One is the iterative learning approach which can be applied when the state at a fixed time can be repeatedly observed with different feedback gains; the other is the adaptive control approach which works when the trajectories satisfy a nondegeneracy condition. Both methods are essentially based on stochastic approximation, and the feedback gains are recursively given without invoking the certainty-equivalency-principle.

Robust and minimum norm pole assignment with periodic state feedback

A computational approach is proposed to solve the minimum norm or robust pole assignment problem for linear periodic discrete-time systems. The proposed approach uses a periodic Sylvester-equation-based parametrization of the periodic pole assignment problem and exploits the nonuniqueness of the problem by imposing conditions on the norm of the resulting periodic state feedback or on the condition numbers of the periodic eigenvector matrices of the closed-loop system. The solution method relies on using gradient search methods on suitably defined cost functions. Explicit expressions of the gradients of cost functions are derived, and the efficient evaluation of the cost functions and gradients is discussed. Numerical examples illustrate the effectiveness of the proposed approach.

POLE ASSIGNMENT IN VIBRATORY SYSTEMS BY MULTI-INPUT CONTROL

The problem of reassigning some poles of a vibratory system, while keeping the other poles unchanged, is considered. The problem may be solved uniquely by single-input state feedback control. A family of solutions to the partial pole assignment problem may be obtained by applying multi-input control forces. An algorithm for determining a multi-input control which is small in some sense is presented. The non-iterative algorithm proposed, defines a closed-form solution to the partial pole assignment problem in its natural second order form, and no first order realization is used. The reduction in the control effort achieved by the proposed method is demonstrated by numerical examples.

An Optimal Differential Sensitivity Approach to Pole Assignment for Structurally Perturbed Systems

This paper introduces a mathematical formula for calculating the eigenvalue differential sensitivities of the closed-loop state matrix with respect to a disturbance matrix when the closed-loop system is subjected to an additive structured perturbation. The robust pole assignment problem is then formulated as an optimization task in which the differential sensitivities of the closed-loop poles are minimized. The case of “multi-perturbations” can be considered naturally in our problem formulation. The gradient of the objective function to be minimized is then derived which permits the minimization to be proceeded effectively. Numerical examples are then employed to illustrate the design technique.

An alternative solution to the problem of pole assignment by static output feedback

This note discusses the problem of pole placement by static output feedback from a geometric point of view. It is shown, without any assumption on the genericity of the system, that the pole placement can be solved by choosing the closed-loop eigenvectors almost freely. We propose a numerical algorithm that allows to fix the desired set of all closed-loop eigenvalues with a small relative error, Moreover, such algorithm seems to be simpler than those known from literature.

A new dynamic output feedback compensator design for pole assignment

This paper proposes a new approach which is applicable for any open loop system either with more outputs than inputs or with at least one stable transmission zero, for the design of dynamic output feedback compensator (DOFC) and for pole assignment. Although this DOFC cannot always guarantee arbitrary pole placement of the feedback system, it has three unique advantages: (1) the DOFC poles are part of the feedback system poles and thus the DOFC stability is guaranteed; (2) this new DOFC can implement state feedback control which is most powerful and whose design is well developed; (3) the design is decoupled into two pole assignment problems of orders r and n, respectively, instead of the existing design of order n + r, where n and r are the orders of open loop system and DOFC, respectively.

On pole assignment of distributed parameter systems with unbounded input elements

In this paper, the pole assignment problem is considered for a class of distributed parameter systems with unbounded input element and with multiple spectral structure. A formula on the spectrum of the closed loop operator is proved and a formula of pole assignment is obtained. Finally, an example concerning a beam vibration is given.

Robust pole assignment in descriptor systems via proportional plus partial derivative state feedback

The problem of eigenvalue assignment with minimum sensitivity in multivariable descriptor linear systems via proportional plus partial derivative state feedback is considered. Different from the purely proportional state feedback case, the number of finite closed-loop eigenvalues is required to be equal to the system dimension. Based on a result in the perturbation theory of generalized eigenvalue problem of matrix pairs, the closed-loop eigenvalue sensitivity measures in terms of the closed-loop normalized right and left eigenvectors are established. By combining these measures and a recently proposed general parametric eigenstructure assignment result for descriptor linear systems via proportional plus derivative state feedback, the robust pole assignment problem is converted into an independent minimization problem. A simple algorithm in sequential order is obtained for solution to the problem of robust pole assignment in descriptor linear systems via proportional plus partial derivative state feedback. The closed-loop eigenvalues may be easily taken as a part of the design parameters and optimized within certain desired fields on the complex plane to improve robustness. Using the proposed algorithm, the optimality of the obtained solution to the robust pole assignment problem is totally dependent on the solution to the independent minimization problem. An example of order six is worked out with multiple solutions, both the indices and the numerical robustness test demonstrate the effect of the proposed approach.

Pole assignment for the vibrating rod

Relocating some poles of an axially vibrating rod by feedback leads to a certain integro-differential eigenvalue problem. An explicit solution to the problem of determining the force needed to assign part of the spectrum while leaving the remaining spectrum unchanged is presented, and the conditions under which this solution is unique are determined. The results are then used to determine a certain self-adjoint control, analogous to symmetric rank-one update in finite-dimensional systems, which solves the partial pole assignment problem with a control which satisfies the reciprocity law relating displacement and force. The results obtained may be used in practical engineering applications of reducing the forced vibration response of harmonically excited systems.

Computation algorithms for disturbance rejection in multivariable systems

In this paper, one way of solving an important problem in linear time-invariant multivariable systems control for synthesizing feedback controllers to make the outputs of a physical system respond in a desirable manner to reference inputs and external disturbances is presented. The proposed algorithms can be regarded as a logical extension of the pole assignment problem in that, measurable or unmeasurable multiple disturbance acting on multivariable systems described by the 4-tuples ( A, B, C, E) or the 6-tuples ( A, B, C, D, E, F) can be rejected at the outputs in steady state. This is done by assignment of the `disturbance blocking zeros' at specified locations using the concept of pole assignment for computing the state feedback controllers. The performance of the algorithms is illustrated by a numerical example.

Block-Partitioned Algorithms for the Pole Assignment Problem of Single-Input Systems

In this paper we present several efficient implementations of the doubleimplicit shift algorithm for the pole assignment problem of single-input systems. Our new implementations obtain a higher performance by reordering the access pattern, while maintaining the same numerical properties. The experimental results report a higher performance of the new implementations over traditional algorithms.

An algorithm for the single-input partial pole assignment problem

An algorithm for translating unstable eigenvalues of single-input partial pole assignment problems based on rank-one transformations is suggested. Special attention is paid to the practical case where translations are constructed using inexact spectral information provided by the Arnoldi procedure. Estimates of the resulting perturbations of stable and translated poles are derived. These estimates depend on the accuracy of spectral information about the unstable poles. The algorithm proposed is illustrated with numerical examples. Bibliography: 11 titles.

Dynamic Pole Assignment Using Global, Blow Up Linearization: Low Complexity Solutions

The problem of pole assignment by low-order dynamics output feedback controllers with McMillan degree n1 is studied for minimal systems described by a proper transfer function matrix G(s)ɛR m×p (s) with McMillan degree n. The new method is based on an asymptotic linearization of the pole placement map related to the problem. In this context, the problem is reduced to solving a set of linear equations, and this is done without losing any of the degrees of freedom of the controller. In terms of composite left matrix fraction description (MFD) representations, the asymptotic solution of the problem as ɛ→0 is given in closed form in terms of a one-parameter family of feedback compensators A(s)+ɛB(s), where A(s) is the so-called degenerate compensator and B(s) is the solution of the linear equations. The arbitrary pole placement property holds true when the linearization matrix has full rank, and it is shown that this occurs genetically when \(n_1 \geqslant n'_1 \), where \(n'_1 \) is the smallest multiple of min(p,m) satisfying \(n'_1 \left( {m + p} \right) + mp \geqslant n + n'\)

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.

線形関数観測器に基づいた動的補償器による任意極配置

In this study, we propose a class of dynamic compensators based on linear function observers which achieve arbitrary pole assignment of the closed loop. The construction procedure directly utilizes Wang's algorithm (1992, 1996) for the static output feedback pole assignment problem, and the same technique can be applied to dual observers also. As a result, assuming the structure of either observer, the sufficient condition of the compensator's order is naturally derived. The condition is more transparent than the one proposed by Rosenthal and Wang (1996), and it is shown that both give the same minimal order under some mild assumptions.

A matrix inverse eigenvalue problem and its application

The problem of generating a matrix A with specified eigenvalues, which maps a given set of vectors of another given set, is presented. An existence theorem is given and proved. A stable algorithm for producing the matrix A is discussed. The relation between this problem and the pole assignment problem in control theory is investigated. The application of this problem in the design of neural networks is discussed.

Robust Control Design for Pole Assignment of Uncertain Systems

This Technical Brief addresses the problem of robust pole assignment for uncertain dynamic systems. A control design methodology is proposed to determine a fixed controller assigning all poles of uncertain systems in a pre-specified disk. The robust pole assignment problem is reduced to a convex optimization problem. The proposed method is applied in design of a flight control system for a small aircraft successfully.

On the distance to uncontrollability and the distance to instability and their relation to some condition numbers in control

According to the methodology of [6], many measures of distance arising in problems in numerical linear algebra and control can be bounded by a factor times the reciprocal of an appropriate condition number, where the distance is thought of as the distance between a given problem to the nearest ill-posed problem. In this paper, four major problems in numerical linear algebra and control are further considered: the computation of system Hessenberg form, the solution of the algebraic Riccati equation, the pole assignment problem and the matrix exponential. The distances considered here are the distance to uncontrollability and the distance to instability.

Orthogonality and partial pole assignment for the symmetric definite quadratic pencil

The eigenvectors of a symmetric matrix can be chosen to form an orthogonal set with respect to the identity and to the matrix itself. Similarly, the eigenvectors of a symmetric definite linear pencil can be chosen to be orthogonal with respect to the pair. This paper presents the three sets of matrix weights with respect to which the eigenvectors of the symmetric definite quadratic pencil are orthogonal. One of these is used to derive an explicit solution of the partial pole assignment problem by state feedback control for a control system modeled by a system of second order differential equations. The solution may be of particular interest in the stabilization and control of flexible, large space structures where only a small part of the spectrum is to be reassigned and the rest of the spectrum is required to remain unchanged.