S.T. Glad

Model structure identifiability and persistence of excitation

An algorithm procedure for determining identifiability of nonlinear systems is presented. It also gives conditions for the control signal to be persistently exciting. The algorithm is based on differential algebraic concepts.<<ETX>>

PARAMETRIZATION OF NONLINEAR MODEL STRUCTURES AS LINEAR REGRESSIONS

Abstract The parameterization of a set of dynamical models is one of the most important issues in system identification. An especially favorable class of parametrizations is formed by so called linear regressions, where the unknown parameters enter linearly in expressions formed by measured inputs and outputs. We consider nonlinear systems that can be described by differential polynomials, i.e. polynomials in several variables and their derivatives. Using results from differential algebra, we show in this contribution that, generically, all parametrizations of such nonlinear systems, can be transformed to linear regressions. The regularity condition that is required for this can be interpreted as a global identifiability condition. The method of proof gives an algorithm, that can transform the original equations to get the linear regression relationship explicitly, in a finite number of steps. This algorithm also gives an excitation condition for the system.

Implementing Ritt's Algorithm of Differential Algebra

Abstract Ritt’s algorithm for computation of characteristic sets has a central role in several constructive calculations in differential algebra. The paper presents the algorithm and discusses some possible modifications of it.

Estimation of the primary current in a saturated transformer

It is pointed out that, when a current in a power system is measured using a transformer, the measurement can be distorted by transformer saturation. The authors examine two methods of estimating the primary current from the secondary in the presence of saturation. The first is based on an extended Kalman filter, while the second is based on an off-line Gauss-Newton method.<<ETX>>

Observability and nonlinear dead beat observers

It is shown that conditions that guarantee observability for continuous time nonlinear systems also imply observability for discrete measurements. This is used to show the existence of nonlinear dead beat observers.

Backstepping control of a rigid body

A method for backstepping control of rigid body motion is proposed. The control variables are torques and the force along the axis of motion. The proposed control law and Lyapunov function guarantee asymptotic stability from all initial values except one singular point.

Identifiability with Constraints

It is shown that identifiability of very general nonlinear models, described by polynomial implicit differential equations, inequations and inequalities, can be decided algorithmically. An algorithm based on differential algebra and real algebra can perform the calculations.

Nonlinear input output relations and identifiability

The identifiability of general systems, described by characteristic sets, is considered. It is shown that there is a natural canonical parametrization of input-output descriptions which gives global identifiability.<<ETX>>

On the gain margin of nonlinear and optimal regulators

The robustness of non-linear regulators for nonlinear systems with respect to variations in gain is investigated. It is shown that there exist regulators that produce asymptotically stable closed loop systems but do not tolerate any variation in gain without instability. However, if the linearized closed loop system is also asymptotically stable, then there is always some gain margin. For a wide class of optimal regulators it is shown that the gain margin is infinite with respect to increases in gain and that decreases down to 0.5 can be tolerated. The robustness properties of linear quadratic control laws are thus generalized.

Input output representations from characteristic sets

The authors present an algorithm for computation of input-output descriptions. Characteristic sets can be considered as generalization of state space descriptions. Input-output descriptions can also be regarded as characteristic sets under a different ranking. This forms the basis of the differential algebraic algorithm for computation of input-output relations. For polynomial state-space descriptions the calculations are easier than the general computations of characteristic sets.<<ETX>>