Arbitrary pole placement by state feedback with minimum gain

Abstract

We consider the classic problem of pole placement by state feedback. We offer an eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing gain matrix that can deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. This parametric formula is then exploited to introduce an unconstrained nonlinear optimisation algorithm to obtain a gain matrix that delivers the desired pole placement with minimum gain.

Arbitrary pole placement with the extended Kautsky-Nichols-van Dooren parametric form with minimum gain

“…For a suitable real or complex m×n parameter matrix K, we obtain the eigenvector matrix X(K) and gain matrix F(K) by building the Jordan chains starting from the selection of eigenvectors from the kernel of certain matrix pencils, and thus avoid the need for matrix inversions, or the solution of Sylvester matrix equations. Thus the results of this paper neatly parallel the achievements of [11] in providing a new parametric form to achieve pole placement with arbitrary multiplicities, while employing an m × n-dimensional parameter matrix. The second aim of the paper is to employ this novel parametric form to seek the solution to the minimum gain exact pole placement problem (MGEPP), which involves solving the EPP problem and also obtaining the feedback matrix F that has the smallest gain, which is of as a measure of the control amplitude or energy required.…”

“…This result ties up with the method in [11]. Example 4.3: Now we study the example in [14] with n = 9 and m = 4 in which a gain matrix is sought to place all the closed-loop poles at λ = −0.55.…”

“…Recently the general problem of assigning any desired set of poles with any desired multiplicities with minimum Frobenius gain has been considered in [15]. Finally, we demonstrate the performance of the resulting algorithm by considering an example involving the assignment of deadbeat modes, and compare the performance against the methods of [11], [14], [15]. We see that the methods introduced in this paper are able to deliver the desired eigenstructure with equivalent or smaller gain than these alternative methods.…”

“…Where methods [1], [4], [5] all employ parameter matrices of dimension m × n, the parameter matrices in [6] and [7] have dimension n × n. In our recent papers [8]- [9], we gave a novel parametric form for X and F based on the famous pole placement algorithm of Moore [10]. This parameterisation employed parameter matrices of dimension m × n, but required Λ to be diagonal, and hence also assumes the closed-loop eigenvalues have multiplicities of at most m. Very recently in our papers [11]- [12] we generalized this parametric form to accommodate arbitrary multiplicities; the method was based on the pole placement method of Klein and Moore [13]. The principal merit of this approach was to obtain a parameterisation that combines the generality of [6] and [7] with the computational efficiency that comes from an m × n dimensional parameter matrix.…”

“…In this section, we compare the algorithm presented in this paper with the methods given in [11], [14], [15]. Example 4.1: We consider Example 1 in [15], and seek to design a deadbeat controller, which can be achieved with one Jordan mini-block of dimension two, and two blocks of dimension 1.…”