The parameter-dependent γ-performance problem is solvable if following LMIs in parameter-dependent symmetric matrices (X, Y) and parameter-dependent quadruple (AK, BK, CK, DK) are feasible.

　

　

 In such case, a gain-scheduled controller of the form (5) is readily obtained with the following two-step scheme [3].

・Solve for N, M, the factorization problem

・Compute for AK, BK, CK

AK = N^-1 (XY + NM^T + AK - X (A - B2DKC2)

BK = N^-1 (BK - XB2DK)

CK = (CK - DKC2Y)M^-1

　

3.3 Regional Pole Constraints

　

 The closed-loop pole constraints are also used in our controller design. The purpose of employing closed-loop pole placement constraints lies in the consideration of some time-domain performance such as settling time, overshoot and hard constraint of implementation [2]. Specifically, a closed-loop pole with large real part leads to fast mode of the controller thus should be avoided.

　

 It is possible to specify an L2 gain bound with regional pole constraints on the closed-loop dynamics of the underlying LTI systems (ρ frozen), thus (7) must be implemented with

　

　

where the data λjk and μkj define the geometry of the region.

　

 To solve the synthesis problem, in general, gridding the parameter space is necessary [10]. However, thanks to the affined dependency of the underwater vehicle system, the synthesis problem will be greatly simplified [3][6], especially in implementation phase.

　

3.4 Implementation Related Problem

　

 The plant's parameter-dependency is mimicked by introducing the affined relationship into the pair (X, Y) and variables (AK(ρ, ρ), BK(ρ), CK(ρ), DK(ρ)),

　

　

where p is the number of scheduling parameter. Intensive study has been done on this kind of approximation in [3]. As pointed out in their study, a controller, which requires time-derivatives of the scheduling variables, is impractical in implementation. In order to get rid of the dependence on ρ(t), either X or Y in (7) is restricted to be fixed.

　

 For the LPV system of this paper, when the parameter ranges in a polytope of R^p with corners {Πi}^Ni=1 (N = 2^p), it can be expressed as the convex combination of the vertexes of the polytope as

　

　

and the system matrix is given by

　

　

 Then the controller can be derived from the values AK (Πi), BK (Πi),・・・ at the corners of the parameter box by