Loose eigenstructure assignment via rank-one LMI approach with application to transient response shaping in H_∞ control

Abstract

This article proposes a new regional eigenstructure assignment via rank-one LMI approach. A gain parameter condition for the regional eigenvalue/eigenstructure assignment is newly derived. This assignment condition is easily combined with H _∞ design by means of enhanced LMI characterisation. In the present approach, the desired assignment of closed-loop eigenvalues (i.e. poles) are not previously fixed but constrained into individual assignment regions to bring us more design freedom than classical exact assignment. The regional assignment discussed in this article never falls into an ordinary LMI root clustering because the union of assignment regions is disjoint in general. Each closed-loop eigenvector is also constrained into individual alignment cone. To show the practical use of the extra design freedom brought by regional assignment, a transient response shaping in H _∞ design is also discussed. A useful design algorithm based on linearisation algorithm is proposed.

Keywords:

• eigenstructure assignment
• linear matrix inequality
• rank constraint
• transient response shaping
• H _∞ control
• root clustering

Notes

Notes

1. The absolute misalignment angle between complex vectors η and ξ is defined as θ ≔ cos⁻¹(|⟨η, ξ⟩|/‖η‖‖ξ‖) ∈ [0, π/2].

2. Note that (26–32) are complex LMIs (both coefficient matrices and decision variables are complex-valued). A complex LMI problem is solved via an equivalent real LMI problem. A complex LMI F(x) > 0 (≥ 0) is feasible, iff a real LMI is feasible. However, some solvers (e.g. SeDuMi by Sturm 1999) can treat complex LMI problems directly.