Parameterized LMIs for robust and state feedback control of continuous‐time polytopic systems

Abstract

Summary This paper presents new extended linear matrix inequality (LMI) characterizations for the synthesis of robust H∞ and H2 state feedback controllers for continuous‐time linear time‐invariant systems with polytopic uncertainty. Based on a suitable change of variables and the Elimination Lemma, the proposed robust control design techniques are stated as extended LMI conditions parameterized in terms of 2 scalar parameters. One parameter is shown to belong to a bounded domain, thus limiting the scalar search domain. For the other parameter, a bounded subset is provided from numerical experiments. The benefits of the methodology are illustrated through numerical simulations performed on an uncertain model borrowed from the literature. It is shown that the proposed LMI relaxations can provide less conservative results with fewer scalar searches than some existing methods in the literature.

New iterative linear matrix inequality based procedure for H2 and H∞ state feedback control of continuous‐time polytopic systems

“…5 2]. It has been suggested in Rodrigues et al, 21 that in order to avoid the large computational burden caused by an exhaustive search procedure on scalar parameters, the search range of scalar variables is constrained to the following values (total of 12 searches): ∈ {10 −1 , 10 0 } and ∈ {−0.9, −0.54, −0.18, 0.18, 0.54, 0.9}. For the other parameterized LMI based methods, the scalar is limited to thirteen logarithmically spaced values (total of 13 searches): r ∈ {10 −6 , 10 −5 , … , 10 0 , 10 1 , … 10 6 }.…”

“…The following result, which is a simple extension of standard well-known results in the literature, gives parameter-dependent BMI equivalent conditions for RDP. Lemma 1 (See the work of Geromel and Korogui, 27 Altun, 15 and Rodrigues et al 21 ). Consider the closed-loop system in (1).…”

“…• Note that P( ) in (5) is required to be a general function of for (2)-(4) to be necessary and sufficient. For a practical implementation, we follow the standard practice 12,18,21,26,27 and restrict the Lyapunov function to be affine in the parameters:…”

“…Among the current methods, Altun 15 gives the minimum upper bound on the As shown in the second column in Table 3, the H ∞ -norm upper bound provided by Rodrigues et al 21 Table 3 also gives the associated numerical complexity of the existing methods and Algorithm 1 for H ∞ control. For each LMI test, in contrast to the current approaches, the greater number of scalar variables and LMI rows in Corollary 1 is due to the additional matrix variables V ij .…”

“…The first row in Table 4 demonstrates that ROC18 21,28 provides the best performance of among the existing approaches in the literature; their robust stability conditions are always feasible for all 0 ≤ d ≤ 14.29. However, when d is slightly larger than 14.29 all the above methods give infeasibility for finding a stabilizing state feedback gain.…”