A note on robust pole assignment for periodic systems

In this note a robust pole assignment algorithm is proposed for linear periodic discrete-time systems with time-varying dimensions of the state and/or input spaces. The algorithm deduces a periodic state feedback law by the minimization of the condition numbers of the eigenvector matrices of the closed-loop system. Numerical examples are provided to show the performances of the algorithm     

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.     

Regional pole assignment for uncertain delta-operator systems

The pole assignment in a specified disk by state feedback for uncertain delta-operator systems is studied. By making use of algebra Riccati equations, a sufficient and necessary condition of pole assignment for a kind of parameter uncertain delta-operator system in a specified disk by state feedback is presented. And the design method of state feedback controller is also developed.The proposed method can unify some previous rehted results of continuous and discrete time systems into the delta framework. The efficiency of the design method is illustrated by a numerical example.     

On pole structure assignment in linear systems

The problem of pole structure assignment (PSA) by state feedback in implicit, linear and uncontrollable systems is discussed in the article. It is shown that the problem is solvable if the system is regularisable. Then necessary and sufficient conditions for characteristic polynomial assignment are established. In the case of PSA (invariant polynomials assignment) just necessary conditions have been obtained. But it turns out that these conditions are also sufficient in some special cases. This happens, for example, when the system does not possess any non-proper controllability indexes. A possible application of the achieved results to modelling a constrained movement of a robot arm is outlined, too.     

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     

On pole assignment in multi-input controllable linear systems

It is shown that controllability of an open-loop system is equivalent to the possibility of assigning an arbitrary set of poles to the transfer matrix of the closed-loop system, formed by means of suitable linear feedback of the state. As an application of this result, it is shown that an open-loop system can be stabilized by linear feedback if and only if the unstable modes of its system matrix are controllable. A dual of this criterion is shown to be equivalent to the existence of an observer of Luenberger's type for asymptotic state identification.     

Robust pole assignment for systems with parameters

In this paper we investigate issues related to the control of linear multivariate systems which contain structured, parametric-type uncertainties. The objective is to construct parameter-independent proper compensators that provide ‘robust control’. Specifically, we consider the question of robust pole assignment. Systems are described by matrix fraction representations which involve a structured parametric form of uncertainty. We suggest conditions for robust pole assignability and demonstrate how to construct the required proper compensators. For the problem considered, we show that the class of systems which satisfy these conditions is quite large and we include results on generic solvability. The method is demonstrated by a physical example.     

Note on the pole assignment of q-dimensional linear systems

The method in Chu (1986) for the solution of the pole assignment problem of separable 2-D linear discrete systems with state feedback is improved and extended to q-D linear discrete systems and systems with output feedback.     

A computational algorithm for pole assignment of linear multiinput systems

An efficient computational algorithm for pole assignment of linear multiinput systems is proposed. A preliminary stage of the algorithm is a reduction of the system matrices into orthogonal canonical form. The gain matrix elements are then found by orthogonal transformation of the closed-loop system matrix into upper quasi-triangular form whose diagonal blocks correspond to the desired poles. The algorithm is numerically stable, the computed gain matrix being exact for a system with slightly perturbed matrices. It works equally well with real and complex, distinct, and multiple poles and is applicable to ill-conditioned and high-order problems.     

Comments on "A pole assignment algorithm for multivariable control systems"

In the above paper, the authors claim to have presented an algorithm which constitutes "a simple way of assigning poles." However, it is demonstrated by the presentation of a counterexample that this algorithm leads to erroneous results.     

Robust pole assignment in systems subject to structured perturbations

The problem of robust pole assignment by feedback in a linear, multivariable, time-invariant system which is subject to structured perturbations is investigated. A measure of robustness, or sensitivity, of the poles to a given class of perturbations is derived, and a reliable and efficient computational algorithm is presented for constructing a feedback which assigns the prescribed poles and optimizes the robustness measure.     

Parameter insensitive pole assignment for scalar input systems

Under the assumption that the parameters affect the systems "linearly," arbitrary partial insensitive pole placement problem for scalar input systems have been solved. A new definition for partial pole placement, which takes into account the effect of the unknown varying parameters in the system, has been used. The result is first given for a system with one parameter, and then, it is extended to systems with more than one parameter. The dual of the same result is also given. The paper also emphasizes the structural properties of systems with varying parameters.     

A note on robust pole placement

Pole placement algorithm in single input-single output (SISO) systems is discussed with respect to the corresponding, real stability radius of the resulting closed loop polynomial     

Disk pole location for uncertain systems through convex optimization

In this paper, a necessary and sufficient condition for disk pole location for uncertain continuous or discrete time systems subject to norm-bounded uncertainty is derived. This condition being convex, the control law design is made through convex optimization. The approach adopted follows the quadratic stabilizability philosophy in the problem of pole placement.     

Robustness of regional pole placement for uncertain continuous-timeimplicit systems

In this note, under assumptions that the nominal linear implicit system is regular, impulse-free, with all its generalized eigenvalues lying inside some specified regions, sufficient conditions are proposed to preserve the assumed properties when structured perturbations are added into the nominal system. Explicit bounds on perturbations with different structures are given in terms of the spectral radii of some nonnegative matrices     

Simultaneous pole assignment in linear periodic systems byconstrained structure feedback

The authors present methods for pole assignment by feedback with constrained structure in linear periodic systems using generalized sampled-data hold functions (GSHF). Constrained structure is taken to mean that the input of each channel is restricted to depend only on the measurements of some specific output channels. The basic idea of GSHF control is to use sample and hold, but to consider the hold function as a design parameter. Four strategies are proposed for closing a control loop using GSHF. The key to the results obtained is that, when GSHF control with constrained structure is applied to a periodic system, a discrete-time time-invariant, decentralized system is obtained for which control design methods are available     

On pole assignment in linear systems with incomplete state feedback

The following system is considered:dot{x}= Ax + Buy = Cxwherexis annvector describing the state of the system,uis anmvector of inputs to the system, andyis anlvector (l leq n) of output variables. It is shown that if rankC = l, and if (A,B) are controllable, then a linear feedback of the output variables u = K^*y, where K^*is a constant matrix, can always be found, so thatleigenvalues of the closed-loop system matrix A + BK^*C are arbitrarily close (but not necessarily equal) tolpreassigned 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^*to be simply found, and examples of the algorithm applied to some simple systems are included.     

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     

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.     

A note on overshoot estimation in pole placements

In this note we show that for a given controllable pair (A, B) and any λ>0, a gain matrix K can be chosen so that the transition matrix e~((A+BK)t) of the system =(A+BK)x decays at the exponential rate e~(-λt) and the overshoot of the transition matrix can be bounded by Mλ~L for some constants M and L that are independent of λ. As a consequence, for any h>0, a gain matrix K can be chosen so that the magnitude of the transition matrix e~((A+BK)t) can be reduced by1/2 (or by any given portion) over [0, h]. An interesting application of the result is in the stabilization of switched linear systems with any given switching rate (see [1]).