Decentralized Pole Assignment and Product Grassmannians

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 simple method is presented on how to generate the state feedback which makes the closed-loop transfer matrix the inverse of the interactor matrix. This can be regarded as a generalization of decoupling control using state feedback. An arbitrary pole assignment which preserves the lower left triangular form of the inverse of the interactor matrix is also considered. >

This short paper deals with the problem of pole assignment with incomplete state observation. It is shown that if the system is controllable and observable, and if n \leq r + m - 1 , an almost arbitrary set of distinct closed-loop poles is assignable by gain output feedback, where n, r , and m are the numbers of state variables, inputs and outputs, respectively. This result improves considerably the ones obtained so far about this problem. Different from the conventional approach using the characteristic equation, an approach based on the properties of the eigenspaces of the closed-loop dynamics is used in this short paper, which gives a new light on the various problems in the linear system theory. It is also shown, as a direct consequence of this result, that the minimum order of the dynamic compensator required for almost arbitrary pole assignment of overall closed-loop system is not greater than n - m - r + 1 .

It is important to know how far non-dominant poles can be located away from the dominant poles since the closed-loop system transient response may not be as expected, if the non-dominant poles are not placed far enough from the dominant pole pair. Even if the arbitrary pole assignment is possible with a full state feedback or an output feedback controller with the order of plant order minus one, low order controllers and especially PID controllers are used in most practical applications. Therefore, the farthest possible non-dominant pole locations with PID controllers are found for all pole systems in this study. Proposed calculation method is demonstrated on two different examples and the closed-loop system performance is investigated with the designed PID controllers.

An extended Howze-Pearson compensator avoiding pole-zero cancellations in a block decoupling system is considered. The basic relation that the compensator satisfies in order to realize block decoupling and arbitrary pole assignment is investigated. A design method to obtain the compensator is derived according to the basic relation. The dynamical order of the compensator is explicitly given by using the coprime fraction description of the system transfer matrix. The results are extended to the case for which partial pole-zero cancellations are allowed.

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.

This article investigates region stabilization issue of switched neural networks (SNNs) with multiple modes (MMs) and multiple equilibria (ME) via a pole assignment method. In such an SNN, every neuron is observed with more than one mode and unstable equilibrium point. First, SNNs with MMs and ME are modeled in terms of switched systems with unstable subsystems and ME. Second, a necessary and sufficient condition and a sufficient condition are, respectively, proposed for arbitrary switching paths pole assignment and arbitrary periodic/quasi-periodic switching paths (PSPs/QSPs) asymptotically region stabilizing pole assignment of switched linear time-invariant (LTI) systems with ME. It is shown that to stabilize a switched LTI system, some/all poles of all/some linear subsystems can be assigned to suitable locations of the right-half side of the complex plane. Third, based on the obtained pole assignment results, an asymptotical-region-stabilizing-control law observed as distributed state feedback controllers of MMs, asymptotical-region-stabilizing PSPs/QSPs, and a corresponding algorithm are all designed for asymptotical region stabilization of switched linear/nonlinear neural networks with MMs and ME. Finally, a numeral example is given to illustrate the effectiveness and practicality of the new results.

A new pole-placement algorithm for multi-input controllable systems is presented. The algorithm can be used to obtain real-valued feedback gains for arbitrarily specified real and complex, distinct or multiple eigenvalues. The multi-input controllable time-invariant system is converted to a single-input system. Pole assignment is achieved by making use of certain properties of the equation TF - AT = D , where A is the open-loop system matrix and F represents a matrix having the desired closed-loop eigenvalues.

This paper considers pole assignment and robust pole assignment problems for discrete-time linear periodic systems by using linear periodic state feedback. The monodromy matrix of the closed-loop system is represented in a special form. By combining this special form with our recent result on solutions to a class of generalized Sylvester matrix equations, a complete parametric approach for pole assignment via periodic state feedback is proposed. The free parameters existing in the solutions to pole assignment are further used to achieve robustness performances. The robust pole assignment problem is converted into a nonconvex optimization problem. Numerical examples illustrate the effectiveness of the proposed approaches.

The problem of pole assignment in linear multivariable systems by means of unity rank output feedback is studied in this paper. It is shown that when the number of specified poles is equal to the number of inputs or outputs of the system, exact pole assignment can be achieved and the feedback matrix is calculated from a set of simple linear equations. When the number of specified poles exceeds this value, the pole assignment problem becomes non-linear. Three different formulations of the problem are considered and three numerical techniques are described for the calculation of the feedback matrix. Numerical examples are given for illustration.

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.

A computational approach is proposed to solve the minimum norm and/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 non-uniqueness of the problem by imposing conditions on the norm of the resulting periodic state feedback and/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 expression 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.

The problem of pole assignment in linear, multivariable systems using an unconstrained-rank feedback matrix is considered. The effect of output feedback of unspecified rank on the characteristic polynomial of a multivariable system is first studied. The results are then used to derive a recursive algorithm for pole assignment without any restrictions on the rank of the output feedback matrix used. The method is based on the pseudoinverse concept for obtaining least-squares solutions of sets of linear equations, and is computationally efficient.

This paper highlights the importance of the co-efficient matrix of the numerator transfer function matrix in pole placement and stabilization assessment of multivariable control systems. The problem of pole assignment with non-unity-rank output feedback, obtained as the sum of unity rank output feedback in two stages, is discussed. The design equations of Seraji [5] formulated in terms of the appropriately partitioned co-efficient matrix of the system numerator transfer function matrix. The pole placement and stabilization assessment procedures are illustrated with numerical examples.

It is wellknown from normal linear systems theory that pole assignment and stabilization are two very basic closely related design problems in linear systems theory. In this chapter, we investigate these two problems for descriptor linear systems.