Bruce A. Wade

Nonlinear PID Control with Partial State Knowledge: Damping without Derivatives

Nonlinear PID (NPID) control is implemented by allowing the controller gains to vary as a function of system state. NPID control has been previously described and implemented, and recently a constructive Lyapunov stability proof has been given. The controllers arising with the constructive Lyapunov method will in general depend on knowledge of the full state vector. In the present work, NPID controllers that operate without knowledge of some state variables are demonstrated. A general but conservative design method is presented with an experimental demonstration. For a special case, complete necessary and sufficient conditions are established; for this case, simulation of a robotic force control application demonstrates well-damped control with no requirement for a force-rate signal. The extension to cases of partial state knowledge is important for NPID control, which is most practical when some state variables—particularly rate variables—are poorly known, confounding full-state feedback or other high-damping linear control designs. Extension of NPID control to MIMO systems and computed torque control is also shown.

Stability of Phase-Based Gain Modulation with Designer-Chosen Switch Functions

Phase-Based Gain-Modulation (PBGM) control is realized by modulating controller gains in response to the phase of the system state or tracking error. PBGM controllers have been applied to robotic hands, parallel manipulators and flexible mechanisms to give increased damping, reduced tracking error and friction compensation. A novel method is presented to establish Lyapunov stability for PBGM control. Prior PBGM stability results incorporated a constraint which limited the range of provably stable systems. The present result removes this constraint, establishing Lyapunov stability for a substantially broader class of systems. Additionally, the new approach decouples the selection of the Lyapunov function from the controller design, permitting the controls designer to independently specify a switch function which determines the application of gain modulation. The present results are applied to analyze PBGM control of the Sarcos dextrous manipulator, illuminating the stability properties of control experiments previously reported in the literature. Numerical methods for design calculations are also presented.

A survey of the Kreiss matrix theorem for power bounded families of matrices and its extensions

We survey results related to the Kreiss Matrix Theorem, especially examining extensions of this theorem to Banach space and Hilbert space. The survey includes recent and established results together with proofs of many of the interesting facts concerning the Kreiss Matrix Theorem.

Recent advances in computational and applied mathematics in science and engineering

We are honoured to bring you this special issue dedicated to recent advances in computational and applied mathematics in science and engineering. These articles comprise a diverse collection of the...

A real distinct poles Exponential Time Differencing scheme for reaction-diffusion systems

A second order Exponential Time Differencing (ETD) method for reaction-diffusion systems which uses a real distinct poles discretization method for the underlying matrix exponentials is developed. The method is established to be stable and second order convergent. It is demonstrated to be robust for problems involving non-smooth initial and boundary conditions and steep solution gradients. We discuss several advantages over competing second order ETD schemes.

AN EXTENSION OF THE KREISS MATRIX THEOREM

A new condition is shown to be equivalent to the other conditions of the Kreiss Matrix Theorem for power bounded families of matrices. This new condition is important for the application of the theory of pseudodifference operators to stability estimates for variable coefficient finite difference equations. As an example of the usefulness of this new condition, it is used to prove stability of the leapfrog scheme for hyperbolic equations with variable coefficients.

On the periodic solutions of a rigid dumbbell satellite in a circular orbit

The aim of the present paper is to provide sufficient conditions for the existence of periodic solutions of the perturbed attitude dynamics of a rigid dumbbell satellite in a circular orbit.

Numerical solution of a long-term average control problem for singular stochastic processes

This paper analyzes numerically a long-term average stochastic control problem involving a controlled diffusion on a bounded region. The solution technique takes advantage of an infinite-dimensional linear programming formulation for the problem which relates the stationary measures to the generators of the diffusion. The restriction of the diffusion to an interval is accomplished through reflection at one end point and a jump operator acting singularly in time at the other end point. Different approximations of the linear program are obtained using finite differences for the differential operators (a Markov chain approximation to the diffusion) and using a finite element method to approximate the stationary density. The numerical results are compared with each other and with dynamic programming.

Nonlinear PID control with partial state knowledge: design by quadratic programming

Nonlinear PID (NPID) control is implemented by allowing the controller gains to vary as a function of system state. NPID controllers will in general depend on knowledge of the full state vector. In this work, NPID controllers which operate without knowledge of some state variables are demonstrated. A general but conservative design method is presented with an experimental demonstration. For a special case, complete necessary and sufficient conditions are established.

Cesàro means and the Kreiss matrix theorem

Abstract We present some new equivalent conditions in the Kreiss matrix theorem involving higher order Cesaro means. These are interesting for the issue of treating families of matrices with no bound on the dimensions.