On general ℋ₂ control: from frequency to time domain

Abstract

This article presents a general formula for discrete-time ℋ₂ control. It works with the regular and singular case of ℋ₂ control, i.e. in the case of possibly non-left-invertible matrices G ₁₂ and non-right-invertible matrices G ₂₁, with possible unit circle invariant zeros. In the generic case, it can be simplified and adapted to work with the plant transfer matrix directly, without invoking the matrices of the parameterisation of stabilising controllers. A further result of this article is the presented necessary and sufficient conditions for state-space ℋ₂ control, under only stabilisability and detectability assumptions. If the conditions are satisfied, an observer-based ℋ₂ controller is constructed. The corresponding numerical algorithm consists of solving two discrete-time algebraic Riccati systems (DARSs) and two eigen-problems.

Keywords:

• parameterisation of stabilising controllers
• partial fraction expansion
• spectral factorisation
• inner and coinner matrices

Notes

Notes

1. A polynomial matrix is generalised column reduced if its column-leading coefficient matrix has rank equal to the normal rank of the polynomial matrix. Analogously, we define the notion of generalised row-reduced polynomial matrix.

2. Instead of DARSs, one can solve linear matrix inequalities (LMIs), with the same result (Saberi et al. 1995, Section 6.4). Actually, the existence in Saberi et al. (1995) is formulated for LMIs.

3. We can obtain matrices X ₁, Ā₁₁, , and F ₁ as follows: Let where is an orthogonal matrix, Ā₁₁ is a matrix with eigenvalues on the unit circle only and Ā₂₂ is a stable matrix. Partition compatibly with (F.1): Then . We choose matrix so that the eigenvalues of matrix are arbitrary stable, and .

4. We can obtain matrices X ₂, , , and F ₂ as follows: Let where is an orthogonal matrix, is a matrix with eigenvalues on the unit circle only and is a stable matrix. Partition, compatibly with (F.2) Then . We choose matrix so that the eigenvalues of matrix are arbitrary stable, and .

5. Actually, by DARS (4.1) it follows more generally that there are no unobservable modes on the unit circle of (A, N ₁₂ K ₁) and uncontrollable modes on the unit circle of (A, K ₂ N ₂₁).