Pole Assignment In Linear Systems Research Articles

Optimal Pole Assignment of Linear Systems by the Sylvester Matrix Equations

The problem of state feedback optimal pole assignment is to design a feedback gain such that the closed-loop system has desired eigenvalues and such that certain quadratic performance index is minimized. Optimal pole assignment controller can guarantee both good dynamic response and well robustness properties of the closed-loop system. With the help of a class of linear matrix equations, necessary and sufficient conditions for the existence of a solution to the optimal pole assignment problem are proposed in this paper. By properly choosing the free parameters in the parametric solutions to this class of linear matrix equations, complete solutions to the optimal pole assignment problem can be obtained. A numerical example is used to illustrate the effectiveness of the proposed approach.

Zero assignment of matrix pencils by additive structured transformations

Matrix pencil models are natural descriptions of linear networks and systems. Changing the values of elements of networks, that is redesigning them, implies changes in the zero structure of the associated pencil and this is achieved by structured additive transformations. The paper examines the problem of zero assignment of regular matrix pencils by a special type of structured additive transformations. For a certain family of network redesign problems the additive perturbations may be described as diagonal perturbations and such modifications are considered here. This problem has certain common features with the pole assignment of linear systems by structured static compensators and thus the new powerful methodology of global linearization [J. Leventides, N. Karcanias, Sufficient conditions for arbitrary Pole assignment by constant decentralised output feedback, Mathematics of Control for Signals and Systems 8 (1995) 222–240; J. Leventides, N. Karcanias, Global asymptotic linearisation of the pole placement map: A closed form solution for the constant output feedback problem, Automatica 31 (1995) 1303–1309] can be used. For regular pencils with infinite zeros, families of structured degenerate additive transformations are defined and parameterized and this lead to the derivation of conditions for zero structure assignment, as well as methodology for computing such solutions. The case of regular pencils with no infinite zeros is also considered and conditions of zero assignment are developed. The results here provide the means for studying problems of linear network redesign by modification of the non-dynamic elements.

Consistency and Pole Assignment in Linear Systems With Incomplete Feedback

New necessary and sufficient conditions in the pole assignment problem are obtained for linear systems with incomplete feedback. The concept of consistency is brought in, which is generalization the concept of complete controllability to systems with incomplete feedback. The conditions are obtained under which the property of consistency is equivalent to global controllability of spectrum for systems with incomplete feedback.

Zero Structure Assignment of Pencils via Static Diagonal Perturbations

The problem of zero assignment of Pencils via static diagonal perturbations is considered here. It has certain common features with the pole assignment of linear systems via diagonal controllers and thus can be tackled via the new powerfull tools that appeared recently in this area. When the pencil has zeroes at infinity we can generically apply the global linearisation technique [2] and construct a solution by improving a certain degenerate solution.

Grassmannian, central projection, and output feedback pole assignment of linear systems

A sufficient rank condition for arbitrary pole assignability by static output feedback of m-input, p-output linear systems of McMillan degree n<mp is given. This condition is easy to check and is satisfied by the generic system. An effective computational method to compute the feedback laws assigning any self-conjugate set of closed-loop poles is also given.

Pole Assignment for Linear Systems by Periodic Output Feedback

It is well known that the poles of a linear system can in general not be arbitrarily placed by memoryless output feedback. On the contrary periodic memoryless output feedback for discrete time invariant linear systems is a powerful tool in the pole-assignment problem. A precise statement is made in this paper. This is then generalised to the context of periodic linear systems.

Robust stability and robust pole assignment of linear systems with structured uncertainty

Sufficient conditions for the robust stability of linear time-varying systems are presented. The stability criteria are extended to the problem of the robust pole assignment of linear time-invariant systems to a specified region. These criteria, both for robust stability and robust pole assignment, improve some results in the literature. Examples are given to demonstrate the new criteria. >

On pole assignment of linear systems by gain output feedback

In this paper, the Geometric Approach is used to derive in a straightforward way a sufficient condition for pole assignability by gain output feedback. This result leads to a pole assignment procedure which reduces to solving a system of min ( n - m, n - p) polynomial equations where n is the number of states, m the number of inputs, p the number of outputs. In the case where m + p > n, this system clearly appears to be linear. The degrees of freedom related to the pole assignment problem are expressed in terms of (right or left) eigenvectors.

A new design method realizing decoupling and arbitrary pole assignment in linear systems.

This paper concerns the decoupling of a linear right invertible system. The system structure of a decoupled system is investigated from the viewpoint of closed loop poles. It is also related with a compensator by a simple equation. The closed loop poles, therefore, can easily be assigned while preserving decoupling by adjusting the system structure; and the corresponding compensator is automatically determined. By taking account of the causality of the compensator, a new decoupling method is derived which simultaneously realises arbitrary pole assignment by a static or dynamic compensator.

A Schur Approach to Pole Assignment Problem

A new approach to the pole assignment in linear systems is proposed which is based on unitary or orthogonal transformation of the closed loop system matrix to its Schur canonical form. The method has a number of advantages over the other known methods. In particular it does not require the computation of the charac-teristic polynomial of the open loop system matrix or the transformation to phase-variable canonical form. The new approach can serve as a basis for development of numerically stable algorithms for pole assignment.

Pole assignment in linear systems with structurally constrained dynamic compensators

This paper is concerned with the problem of designing dynamic compensators for linear time-invariant systems to obtain arbitrary pole placement, while preassigning the poles of the compensator. A sufficient condition for complete pole placement is specified. A practical algorithm for both complete and incomplete pole placement is developed which can easily be programmed on a digital computer.

Addendum to the paper ‘ On the pole assignment in linear systems with fixed order compensators ’

(1976). Addendum to the paper ‘ On the pole assignment in linear systems with fixed order compensators ’. International Journal of Control: Vol. 23, No. 6, pp. 887-888.

Pole assignment in linear systems by virtual model following

It is shown that the problem of pole assignment by gain output feedback in linear time-invariant multivariable systems may be reduced to an algebraic optimisation problem, using the concept of virtual model following. The method requires no transformation of the open-loop system to a canonical form. A number of related problems involving combinations of pole placement with a secondary design objective are also discussed.

Adaptive scheme for pole assignment in linear systems

The problems of arbitrarily assigning the poles of a linear system using state feedback in the presence of slow parameter variations is considered. A method is proposed whereby the varying parameters are rapidly identified, and the feedback vector is continuously updated in a stable fashion using these estimates. An additional requirement of having a zero steady-state error between the plant output and a step-reference input is also incorporated. Digital-computer-simulation results are presented to demonstrate the feasibility of the adaptive scheme.

On the pole assignment in linear systems with fixed order compensators†

The theory of pole assignment in linear time invariant systems with incomplete state measurements is generalized to include the case in which a controllable observable plant of order n is augmented by a compensator of fixed dimensionality p (0≤p<n). The maximum sot of eigenvalues that can be preassigned arbitrarily is specified. The results previously available are shown to be particular cases of the theory proposed in the present paper, and Davison's (1970) algorithm is extended so as to apply to fixed order compensators design.

A note on pole assignment in linear systems with incomplete state feedback

A theorem recently proposed by Davison [1] on pole assignment with incomplete state feedback is extended to noncyclic matrices by using the results of Brasch and Pearson [2]. It is shown that the number of poles that can be arbitrarily assigned is equal to the maximum of the number of nontrivial inputs or outputs.

On pole assignment in linear systems with incomplete state feedback

The following system is considered: <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}= Ax + Bu</tex> <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y = Cx</tex> where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> is an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> vector describing the state of the system, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</tex> is an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> vector of inputs to the system, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y</tex> is an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> vector ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l \leq n</tex> ) of output variables. It is shown that if rank <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C = l</tex> , and if (A,B) are controllable, then a linear feedback of the output variables u = K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> y, where K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> is a constant matrix, can always be found, so that <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> eigenvalues of the closed-loop system matrix A + BK <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> C are arbitrarily close (but not necessarily equal) to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</tex> preassigned values. (The preassigned values must be chosen so that any complex numbers appearing do so in complex conjugate pairs.) This generalizes an earlier result of Wonham [1]. An algorithm is described which enables K <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">*</sup> to be simply found, and examples of the algorithm applied to some simple systems are included.