Anders Helmersson

Model reduction using LMIs

This paper treats the problem of approximating an nth order continuous system by a system of order k<n. Optimal solutions minimizing the H/sub /spl infin// norm of the approximation error exist for k=0, n-1. This paper presents an iterative two-step LMI method for improving the H/sub /spl infin// model error compared to Hankel norm reduction.<<ETX>>

Data driven local coordinates for multivariable linear systems and their application to system identification

In this paper we introduce a new parametrization for state-space systems: data driven local coordinates (DDLC). The parametrization is obtained by restricting the full state-space parametrization, where all matrix entries are considered to be free, to an affine plane containing a given nominal state-space realization. This affine plane is chosen to be perpendicular to the tangent space to the manifold of observationally equivalent state-space systems at the nominal realization. The application of the parametrization to prediction error identification is exemplified. Simulations indicate that the proposed parametrization has numerical advantages as compared to e.g. the more commonly used observable canonical form.

System identification using an over-parametrized model class-improving the optimization algorithm

The use of an over-parametrized state-space model for system identification has some clear advantages: A single model structure covers the entire class of multivariable systems up to a given order. The over-parametrization also leads to the possibility to choose a numerically stable parametrization. During the parametric optimization the gradient calculations constitute the main computational part of the algorithm. Consequently using more than the minimal number of parameters required slows down the algorithm. However, we show that for any chosen (over)-parametrization it is possible to reduce the gradient calculations to the minimal amount by constructing the parameter subspace which is orthonormal to the tangent space of the manifold representing equivalent models.

An IQC-based stability criterion for systems with slowly varying parameters

An integral quadratic constraints (IQC) is introduced for stability analysis of linear systems with slowly varying parameters. The parameters are assumed to be bounded and with bounded derivatives. Other types of uncertainties can be included in the problem. The new criterion yields less conservative bounds than previously proposed criteria.

State-space parametrizations of multivariable linear systems using tridiagonal matrix forms

Tridiagonal parametrizations of linear state-space models are proposed for multivariable system identification. The parametrizations are surjective, i.e. all systems up to a given order can be described. The parametrization is based on the fact that any real square matrix is similar to a real tridiagonal form as well as a compact tridiagonal form. These parametrizations has significantly fewer parameters compared to a full parametrization of the state-space matrices.

IQC Synthesis based on Inertia Constraints

Abstract Integral quadratic constraints (IQCs) can be used for proving stability of systems with uncertainties and nonlinearities. Similarly, IQCs can also be used for controller synthesis. Necessary and sufficient conditions for the existence of such a controller is derived. These conditions include linear matrix inequalities (LMIs) and matrix inertia specifying the number of negative eigenvalues of a matrix. In general, these conditions are non-convex. Connections to bilinear matrix inequalities and LMIs with rank constraints are also given.

Parameter dependent Lyapunov functions based on linear fractional transformation

Abstract A parameter dependent Lyapunov function is introduced for stability analysis of linear systems with slowly varying parameters. The Lyapunov function is a quadratic form using a linear fractional transformation (LFT) of the parameters. The parameters are assumed to be bounded and with bounded derivatives. Other types of uncertainties can be included in the problem. The new criterion yields less conservative bounds than previously proposed criteria.

A dynamic minimal parametrization of multivariable linear systems and its applications to optimization and system identification

Abstract A minimal state-space parametrization which is dynamically or adaptively changed during the optimization process is proposed. The basis is an overparametrized model structure which can describe all systems of given dimensions. At each step in the iterative optimization, a local minimal parametrization is constructed. This reduces the search space to the minimal dimension. The construction of this local parametrization can be done efficiently and in such a way that the numerical operations in each optimization iteration becomes well conditioned.

μ synthesis and LFT gain scheduling with real uncertainties

This paper presents the solution to the mixed µ synthesis problem, and how to design gain scheduling controllers with linear fractional transformations (LFTs). The system is assumed to have a param ...

A Quasi-Newton Interior Point Method for Low Order H-Infinity Controller Synthesis

This technical note proposes a method for low order H-infinity synthesis where the constraint on the order of the controller is formulated as a rational equation. The resulting nonconvex optimization problem is then solved by applying a quasi-Newton primal-dual interior point method. The proposed method is evaluated together with a well-known method from the literature. The results indicate that the proposed method has comparable performance and speed.