A partially augmented Lagrangian method for low order H-infinity controller synthesis using rational constraints

Abstract

When designing robust controllers, H-in nity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the (augmented) system, the problem becomes nonconvex and it is relatively hard to solve. These problems become very complex, even when the order of the system is low. The approach used in this work is based on formulating the constraint on the maximum order of the controller as a polynomial (or rational) equation. This equality constraint is added to the optimization problem of minimizing an upper bound on the H-in nity norm of the closed loop system subject to linear matrix inequality (LMI) constraints. The problem is then solved by reformulating it as a partially augmented Lagrangian problem where the equality constraint is put into the objective function, but where the LMIs are kept as constraints. The proposed method is evaluated together with two well-known methods from the literature. The results indicate that the proposed method has comparable performance in most cases, especially if the synthesized controller has many parameters, which is the case if the system to be controlled has many input and output signals. Abstract-When designing robust controllers, H-infinity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the (augmented) system, the problem becomes nonconvex and it is relatively hard to solve. These problems become very complex, even when the order of the system is low.The approach used in this work is based on formulating the constraint on the maximum order of the controller as a polynomial (or rational) equation. This equality constraint is added to the optimization problem of minimizing an upper bound on the H-infinity norm of the closed loop system subject to linear matrix inequality (LMI) constraints. The problem is then solved by reformulating it as a partially augmented Lagrangian problem where the equality constraint is put into the objective function, but where the LMIs are kept as constraints.The proposed method is evaluated together with two wellknown methods from the literature. The results indicate that the proposed method has comparable performance in most cases, especially if the synthesized controller has many parameters, which is the case if the system to be controlled has many input and output signals.

Direct Synthesis of Fixed-Order <inline-formula> <tex-math notation="TeX">${\cal H}_{\infty}$</tex-math></inline-formula> Controllers

“…It remains to show that inequality (6) is also satised. Substitute X and Y by their equivalents represented in (16) to obtain,…”

“…A considerable amount of research effort has been expended to implement this approach. These include alternating projections [12], [13], linearization [14], and more recently augmented Lagrangian methods [15], [16]. In [17] a Kronecker Canonical Form (KCF) is utilized to exploit special plant structures which may lead to a drop in the rank of the controller.…”

Direct synthesis of fixed‐order multi‐objective controllers

“…More precisely, the last n x − n k coefficients of the characteristic polynomial of each n x × n x matrix with rank less than or equal to n k must be zero. A series of studies such as References formulate this constraint in an algebraic form and solve the Lagrangian form of the constrained problem. This has led to the development of augmented Lagrangian algorithms which determine sub‐optimal solutions of the fixed‐order synthesis problem.…”

Fixed-order ℋ∞ controller design for MIMO systems via polynomial approach

“…To deal with this non-convexity, usage of an alternating projection method (Grigoriadis and Skelton, 1996), which involves seeking the intersection of convex LMIs and a non-convex rank constraint, and an augmented Lagrangian method (Ankelhed et al, 2011; Fares et al, 2001), which involves relaxing the non-convex constraint, provide reasonable results. In addition to these approaches, nonlinear semi-definite programming can be used to solve this problem (Apkarian et al, 2004).…”