A Kronecker product approach to pole assignment by output feedback

A method is presented which reduces the problem of determining the output feedback modal (eigenvalue or pole) control matrix K of a system to an explicit linear set of equations in the elements of K through the use of the Kronecker product. This set of equations is then partitioned into two subsets, one that constitutes the sufficient conditions for the existence of a solution and, provided this is consistent, a second subset that yields the desired modal control matrix.     

A design procedure for pole assignment using output feedback

A new design procedure is presented for assigning closed-loop eigenvalues of multi-variable, linear systems using output feedback control. Subject to certain mild restrictions, the number of poles which can be assigned to arbitrary, distinct locations is mill [m + r ? 1, n], where m, r and n are the dimensions of the output, control and state vectors, respectively. The design procedure provides an extension of the results of Davison and Chatterjee (1971) and Jameson (1970) but allows a significantly larger number of eigenvalues to be assigned. The simplicity of the design procedure is demonstrated by a numerical example.     

An approach for pole assignment in singular systems

Pole assignment in a singular system Edx/dt=Ax+Bu is discussed. It is shown that the problem of assigning the roots of det(sE-(A +BF)) by applying a proportional feedback u=Fx+r in a given singular system is equivalent to the problem of pole assignment of an appropriate regular system. An immediate application of this result is that procedures and computational algorithms that were originally developed for assigning eigenvalues in regular systems become useful tools for pole assignment in singular systems. The approach provides a useful tool for the combined problem of eliminating impulsive behavior and stabilizing a singular system     

The general problem of pole assignment: A polynomial equation approach

The problem of modifying the invariant polynomials of a linear system by dynamical output feedback is considered. A new necessary condition which the invariant polynomials must satisfy is derived. The sufficiency condition of Rosenbrock and Hayton is proved in an alternative way. The proof is based on polynomial matrix equations and provides a simple construction of the feedback which affects the desirable change.     

A note on pole assignment

This note gives an alternate proof of Davison's theorem [2] on pole placement and further shows that, for a controllable, observable systemdot{x} = hat{A}x + hat{B}u, y = hat{C}x, the number of poles that can be assigned arbitrarily are equal to max (m,p), wheremRankhat{B}andp= Rankhat{C}. In some cases, more than max (m,p) poles can be assigned arbitrarily.     

A parametric solution to the pole assignment problem using dynamicoutput-feedback

A technique is presented for pole placement of linear time-invariant systems using dynamic feedback. A previously developed method for partial pole assignment using constant feedback is generalized to the dynamic output-feedback case. Subject to a mild assumption on the number of complex conjugate poles to be assigned, it is almost always possible to arbitrarily assign all the closed-loop system poles using a compensator of order [(n-φ)/max(m,l)] using this new method. Here, n, m, and l are the order of the system, and the number of inputs and outputs, respectively, and φ Δ/=max(m,l)+[max(m,l)/2]+…+[max(m,l)/min(m,l)] where [x] denotes the nearest integer lower than or equal to x (i.e., floor (x)), and [x] denotes the nearest integer greater than or equal to x (i.e., ceiling (x)). An equivalent result is that using a compensator of order q, it is almost always possible to arbitrarily assign min(n+q,(max(m,l)+1)q+φ) closed-loop system poles. Only the normal procedures of linear algebra are required to implement the technique. Note that φ⩾l+m-1 and, therefore, the result is stronger than previous exact pole assignment results. Since it does not involve iteration or any other numerical techniques, it is possible to implement the method symbolically and, therefore, to obtain general parametric solutions to the pole assignment problem. The freedom in this design approach can also often be used to guarantee the internal stability and/or robustness of the resulting closed-loop system     

A gradient flow approach to on-line robust pole assignment for synthesizing output feedback control systems

Pole assignment is a basic design method for synthesis of feedback control systems. In this paper, a gradient flow approach is presented for robust pole assignment in synthesizing output feedback control systems. The proposed approach is shown to be capable of synthesizing linear output feedback control systems via on-line robust pole assignment. Convergence of the gradient flow can be guaranteed. Moreover, with appropriate design parameters the gradient flow converges exponentially to an optimal solution to the robust pole assignment problem and the closed-loop control system based on the gradient flow is globally exponentially stable. These desired properties make it possible to apply the proposed approach to slowly time-varying linear control systems. Simulation results are shown to demonstrate the effectiveness and advantages of the proposed approach.     

An algorithm for robust pole assignment via polynomial approach

A computational method for designing controllers which attempt to place the roots of the characteristic polynomial of an uncertain system inside some prescribed regions is presented. The analysis is based on transfer functions of characteristic polynomials, and the problem is formulated as one of semi-infinite programming. An example of an application is given to illustrate this approach     

A robust periodic pole assignment algorithm

In this note a robust periodic pole assignment algorithm is proposed for linear, time-invariant, discrete-time systems. The condition numbers of the eigenvector matrices of the closed-loop system are assumed as a robustness measure and a periodic state-feedback law is deduced by the minimization of the condition numbers associated to the eigenvectors of the monodromy matrix of the closed-loop system. The proposed periodic pole assignment algorithm has been tested on a number of examples, giving satisfactory results     

Robust pole assignment

This paper presents new theorems on the theory of interval matrix inequalities and the theory of polynomials with interval roots, and applies them to the problem of robust pole-placement. We formulate optimization problems and derive convergent iterative algorithms which allow the designer to find controllers that place closed-loop poles within desired intervals for plants with unknown-but-bounded parameter uncertainties. The algorithms are computationally reasonable and provide a useful addition to currently existing control CAD tools.     

Simultaneous partial pole placement: A new approach to multimode system design

Simultaneous partial pole placement of a family of single-input single-output plants is proposed as a generalization of the classical pole placement and stabilization problems. This problem finds application in the design of a compensator for a family of linear dynamical systems. In this note we show that the proposed problem is equivalent to a new class of transcendental problem using stable, minimum phase rational functions with real coefficients. A necessary condition for the solvability of the associated transcendental problem is obtained. Finally, a counterexample to the following conjecture is obtained-"pairs of simultaneously stabilizable plants of bounded McMillan degree have simultaneously stabilizing compensators of bounded McMillan degree."     

A geometric framework for pole assignment algorithms

The problem of pole assignment by gain output feedback or by low-order dynamical compensator is considered from a geometric point of view. This makes it possible to unify, in a general framework, most of the existing pole assignment methods formulated in a state-space context, such as the minimal-order observers, the Brasch-Pearson compensator, and the methods proposed by H. Kimura, and to simplify their presentation. Moreover, new pole assignment algorithms may be derived from this general formulation     

A novel approach to the design of unknown input observers

A novel state estimator design scheme for linear dynamical systems driven by partially unknown inputs is presented. It is assumed that there is no information available about the unknown inputs, and thus no prior assumption is made about the nature of these inputs. A simple approach for designing a reduced-order unknown input observer (UIO) with pole-placement capability is proposed. By carefully examining the dynamic system involved and simple algebraic manipulations, it is possible to rewrite equations eliminating the unknown inputs from part of the system and to put them into a form where it could be partitioned into two interconnected subsystems, one of which is directly driven by known inputs only. This makes it possible to use a conventional Luenberger observer with a slight modification for the purpose of estimating the state of the system. As a result, it is also possible to state similar necessary and sufficient conditions to those of a conventional observer for the existence of a stable estimator and also arbitrary placement of the eigenvalues of the observer. The design and computational complexities involved in designing UIOs are greatly reduced in the proposed approach     

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.     

A novel approach to design classifiers using genetic programming

We propose a new approach for designing classifiers for a c-class (c/spl ges/2) problem using genetic programming (GP). The proposed approach takes an integrated view of all classes when the GP evolves. A multitree representation of chromosomes is used. In this context, we propose a modified crossover operation and a new mutation operation that reduces the destructive nature of conventional genetic operations. We use a new concept of unfitness of a tree to select trees for genetic operations. This gives more opportunity to unfit trees to become fit. A new concept of OR-ing chromosomes in the terminal population is introduced, which enables us to get a classifier with better performance. Finally, a weight-based scheme and some heuristic rules characterizing typical ambiguous situations are used for conflict resolution. The classifier is capable of saying "don't know" when faced with unfamiliar examples. The effectiveness of our scheme is demonstrated on several real data sets.     

A note on pole assignment in uncertain systems

A new method for full-rank state-feedback pole assignment is introduced. A particular advantage of this method is that no freedom in the design parameters is lost. Hence, further design problems such as robust pole assignment in uncertain systems are alleviated. An application of this feature is shown for a particular linear model of the F16 aircraft. The connections between state feedback and several concepts in the literature such as incomplete input feedback, incomplete state feedback, output feedback and incomplete pole assignment are also given.     

Central tendency adaptive pole assignment

The concept of central-tendency adaptive control for achieving minimum variance objectives, proposed by the authors in a companion paper (1989), is applied to adaptive pole assignment. At any iteration, given plant parameter estimate and their uncertainty, a controller is designed which is most likely to achieve the pole assignment objectives. Simulations show a factor-of-100 improvement in transient response over certainty equivalent adaptive pole assignment schemes, for one example