Torkel Glad

On global identifiability for arbitrary model parametrizations

It is a fundamental problem of identification to be able—even before the data have been analyzed—to decide if all the free parameters of a model structure can be uniquely recovered from data. This is the issue of global identifiability. In this contribution we show how global identifiability for an arbitrary model structure (basically with analytic non-linearities) can be analyzed using concepts and algorithms from differential algebra. It is shown how the question of global structural identifiability is reduced to the question of whether the given model structure can be rearranged as a linear regression. An explicit algorithm to test this is also given. Furthermore, the question of ‘persistent excitation’ for the input can also be tested explicitly is a similar fashion. The algorithms involved are very well suited for implementation in computer algebra. One such implementation is also described.

An Algebraic Approach to Linear and Nonlinear Control

The analysis and design of control systems has been greatly influenced by the mathematical tools being used. Maxwell introduced linear differential equations in the 1860’s. Nyquist, Bode and others started the systematic use of tranfer functions, utilizing complex analysis in the 1930’s. Kalman brought forward state space analysis around 1960. For nonlinear systems, differential geometric concepts have been of great value recently. We will argue here that algebraic methods can be very useful for both linear and nonlinear systems. To give some motivation we will begin by looking at a few examples.

On Diffusion Driven Oscillations in Coupled Dynamical Systems

The paper deals with the problem of destabilization of diffusively coupled identical systems. It is shown that globally asymptotically stable systems being diffusively coupled, may exhibit oscillat ...

A multiplier method with automatic limitation of penalty growth

This paper presents a multiplier method for solving optimization problems with equality and inequality constraints. The method realizes all the good features that were foreseen by R. Fletcher for this type of algorithm in the past, but which suffers from none of the drawbacks of the earlier attempts.

On parameter and state estimation for linear differential-algebraic equations

The current demand for more complex models has initiated a shift away from state-space models towards models described by differential-algebraic equations (DAEs). These models arise as the natural product of object-oriented modeling languages, such as Modelica. However, the mathematics of DAEs is somewhat more involved than the standard state-space theory. The aim of this work is to present a well-posed description of a linear stochastic differential-algebraic equation and more importantly explain how well-posed estimation problems can be formed. We will consider both the system identification problem and the state estimation problem. Besides providing the necessary theory we will also explain how the procedures can be implemented by means of efficient numerical methods.

Nonlinear regulators and Ritt’s remainder algorithm

Ritt’s algorithm can be used to compute a controller for a nonlinear system, so that the closed loop dynamics agrees with a specified differential polynomial. A necessary condition for a practical controller is that the system is minimum phase.

A Method for State and Control Constrained Linear Quadratic Control Problems

Abstract A method for linearly constrained linear quadratic control problems is presented. The method is based on ideas from nonlinear programming and quadratic programming. It is shown how the regularity conditions of mathematical programming can be translated into controllability conditions and how the solution of the quadratic program is carried out in the standard Riccati equation. The use of the method for general nonlinear control problems is briefly indicated.

Computing the controllability function for nonlinear descriptor systems

The computation of the controllability function for nonlinear descriptor systems is considered. Three different methods are derived. The first method is based on the necessary conditions for optimality from the Hamilton-Jacobi-Bellman theory for descriptor systems. The second method uses completion of squares to find the solution. The third method gives a series expansion solution, which with a finite number of terms can serve as an approximate solution

Velocity Estimation from Irregular, Noisy Position Measurements

Abstract Velocity estimatio based on position measurements obtained at irregular time instants is considered. A common case could be to estimate angular velocity from a sequence of light-pulses, registered when holes in a disc mounted on the axis pass through light source. The problem is formulated as a state vector estimation problem and a filter for this formulation is compared to conventional solutions. It is found that the more elaborate solution may give a considerable improvement in performance.

MODEL REDUCTION OF NONLINEAR DIFFERENTIAL-ALGEBRAIC EQUATIONS

In this work, a computational method to compute balanced realizations for nonlinear differential-algebraic equation systems is derived. The work is a generalization of an earlier work for nonlinear control-affine systems, and is based on analysis of the controllability and observability functions.