A necessary condition for pole assignment by constant output feedback

In this paper we shall investigate the pole assignment problem by constant output feedback using transfer function techniques. In particular, a necessary condition is given for the existence of constant output feedback which assigns the eigenvalues of closed-loop system to any desired positions.     

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.     

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 ofm _i outputs, p_i 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.This research has been supported by EPSRC Grant GR/H 20466, U.K.     

Pole assignment for linear time-invariant systems by periodic memoryless output feedback

It is well known that the poles of a linear time-invariant controllable and observable system can be assigned arbitrarily by state feedback. When only the output is available, pole assignment is still possible by means of dynamic output feedback. In this paper the potential of time-varying memoryless output feedback is considered. It is shown that, up to some technical conditions, it is indeed possible to allocate the poles of a linear time-invariant discrete-time system by memoryless output feedback with periodic gains. The period of the gains is (n + 1) with n the order of the system. The power of the design technique is proved to be comparable to what can be achieved by the classical dynamic feedback approach.     

Pole assignment by output feedback: A solution for 2 × 2 plants

The problem of pole assignment by static output feedback is solved in this paper for 2 × 2 plants. It is shown that the problem solvability is equivalent to the solvability of some quadratic equation and the latter can be checked through the affine transformation of quadratic functions. All solutions are given in a parametric form if a solution exists. An example is provided to illustrate the method.     

Least squares pole assignment by memoryless output feedback

In this paper, a pole assignment problem for a linear time-invariant control system by memoryless output feedback is posed as a least squares optimisation problem and analysed. The cost functions are appropriately modified so as to obtain convergence of the corresponding gradient flow and existence of the global minimum. This approach is also extended to accommodate the output pole assignment problem with insensitivity against disturbances in system parameters. The relation between the modified cost functions and the original pole assignment problem is revealed. The proposed approach is compared with other existing approaches and illustrated by numerical results.     

Some results on the genericity of the pole placement problem

In this note, the pole placement problem for a linear MIMO systems with p outputs and m inputs is studied from the algebraic point of view. A formulation is proposed, that allows to analyze both theoretical and numerical aspects of the case min(m,p)=2 with more sharpness. Moreover, here it is shown that arbitrary pole placement by static output feedback of unitary rank is generically not possible even if m+p>n holds true.     

Pole assignment for uncertain systems in a specified disk by output feedback

In this paper the problem of pole assignment in a disk by output feedback for continuous-or discrete-time uncertain systems is addressed. A necessary and sufficient condition for quadratic d stabilizability by output feedback is presented. This condition is expressed in terms of two parameter-dependent Riccati equations whose solutions satisfy two extra conditions. An output d stabilization algorithm is derived and a controller formula given.     

Parameter-insensitive pole assignment

A problem of parameter-insensitive pole assignment for a kind of finite-dimensional linear time-invariant (FDLTI) system is considered. Conditions for the existence of some parameter-insensitive poles are obtained. Further, an approach for constructing the feedback law is suggested.     

Pole assignment with output feedback

For pole assignment with output feedback, appropriate partitioning of the coefficient matrix of the system numerator transfer function matrix leads to simplified design equations to solve for the vectors of the unity rank output feedback matrix. This concept is applied to derive the pole placement equations for the proportional-derivative output feedback dynamic compensator. The simplicity of the design procedure is illustrated with a numerical example.     

Generic properties and control of linear structured systems: a survey

In this survey paper, we consider linear structured systems in state space form, where a linear system is structured when each entry of its matrices, like A,B,C and D, is either a fixed zero or a free parameter. The location of the fixed zeros in these matrices constitutes the structure of the system. Indeed a lot of man-made physical systems which admit a linear model are structured. A structured system is representative of a class of linear systems in the usual sense. It is of interest to investigate properties of structured systems which are true for almost any value of the free parameters, therefore also called generic properties. Interestingly, a lot of classical properties of linear systems can be studied in terms of genericity. Moreover, these generic properties can, in general, be checked by means of directed graphs that can be associated to a structured system in a natural way. We review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms. By properties we mean here system-specific properties like controllability, the finite and infinite zero structure, and so on, as well as, solvability issues of certain classical control problems like disturbance rejection, input-output decoupling, and so on. In this paper, we do not try to be exhaustive but instead, by a selection of results, we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool. We emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only. Moreover, the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well-known polynomially bounded combinatorial techniques for larger systems.     

A continuation approach to eigenvalue assignment

This paper describes a continuation approach to eigenvalue assignment. The method is a homotopy technique which embeds the control problem into a parameterized family of control problems. This parameterization describes a continuous deformation of a system with the desired spectrum into the original system. Based on this deformation, a differential equation is constructed whose solution trajectory has an end-point which is a constant output feedback matrix assigning the desired spectrum to the original system. The derivation of this differential equation and conditions which guarantee the existence of a solution are given. Also, two examples of its numerical implementation are detailed.     

Numerical algorithms for eigenvalue assignment by constant anddynamic output feedback

Algorithms are proposed for eigenvalue assignment (EVA) by constant as well as dynamic output feedback. The main algorithm is developed for single-input, multioutput systems and the results are then extended to multiinput, multioutput systems. In computing the feedback, use is made of the fact that the closed-loop eigenvalues can almost always be assigned arbitrarily close to the desired locations in the complex plane, provided the system satisfies the condition m+ p>n, where m, p, and n are , respectively, the number of inputs, outputs and states of the system. The EVA problem has been treated as a converse of the algebraic eigenvalue problem. The proposed algorithms are based on the implicitly shifted QR algorithm for solving the algebraic eigenvalue problem. The performance of the algorithms is illustrated by several numerical examples     

Minimum norm output feedback design under specified eigenvalue areas

Starting from a new parametric expression for the state feedback controller of a linear, time-invariant system a procedure is derived which reduces the design of output feedback controllers to an unconstrained optimization problem where all free parameters, both the closed-loop eigenvalues and the invariant parameter vectors, are used for its numerical solution by a gradient-based search technique. Thus minimal norm output feedback controllers can be designed that place the closed-loop eigenvalues within specified regions of the eigenvalue plane.     

A pole-assignment algorithm for linear state feedback

A new algorithm for the pole-assignment problem of a controllable time-invariant linear multivariable system with linear state feedback is presented. The resulting feedback matrix is a least-squares solution and is robust in a loose sense. The method is based on the controllability canonical (staircase) form and amounts to a new proof for the existence of a solution of the pole-assignment problem. Illustrative examples are given.     

Pole assignment for multi-input multi-output systems using output feedback

In this paper the problem of pole assignment using constant gain output feedback is studied for MIMO system with system order n > m + l − 1, where m and l are the number of inputs and outputs, respectively. A new procedure is presented to design a constant gain output feedback matrix which assigns (m + l − 2) poles exactly to the desired locations and shifts all the unassigned poles to suitable locations using root locus techniques.     

Pole assignment for systems over rings

A very simple proof of the pole assignment theorem for systems over a principal ideal domain (and other rings) is given. Furthermore, an algorithm is presented. Extensions are also indicated.