{Equilibrium solutions of systems of parameterized ordinary differential equations \dot x = f(x, α) , x ∈ Rn , α∈ Rm can be characterized by their parametric distance to manifolds of critical solutions at which the behavior of the system changes qualitatively. Critical points of interest are bifurcation points and points at which state variable constraints or output constraints are violated. We use normal vectors on manifolds of critical points to measure the distance between these manifolds and equilibrium solutions as suggested in I. Dobson [J. Nonlinear Sci., 3:307-327, 1993], where systems of equations to calculate normal vectors on codimension-1 bifurcations were presented. We present a scheme to derive systems of equations to calculate normal vectors on manifolds of critical points which (i) generalizes to bifurcations of arbitrary codimension, (ii) can be applied to state variable constraints and output constraints, (iii) implies that the normal vector defining system of equations is of size c1 n+ c2 m+ c3 , ci∈ R , i.e., no bilinear terms nm or higher-order terms occur, (iv) reduces the number of equations for normal vectors on Hopf bifurcation manifolds compared to previous work, and (v) simplifies the proof of regularity of the normal vector system. As an application of this scheme, we present systems of equations for normal vectors to manifolds of output/state variable constraints, to manifolds of saddle-node, Hopf, cusp, and isola bifurcations, and we give illustrative examples of their use in engineering applications.}

Normal Vectors on Manifolds of Critical Points for Parametric Robustness of Equilibrium Solutions of ODE Systems

Starting from plysical laws a four-dimensional nonlinear model for mecano-hydraulic servomechanisms is deduced. The stability of its equilibria is analysed using a theorem of Lyapunov and Malkin to handle the critical case due to the presence of zero in the spectrum of the matrix of the linear part around equilibria. Stability diagrams are drawn and simulation results are presented through phase diagrams.

Stability of Equilibria in a Four-dimensional Nonlinear Model of a Hydraulic Servomechanism

The problem of operating or designing a system with robust stability with respect to many parameters can be viewed as a geometric problem in multidimensional parameter space of finding the position of nominal parameters λ0 relative to hypersurfaces at which stability is lost in a bifurcation. The position of λ0 relative to these hypersurfaces may be quantified by numerically computing the bifurcations in various directions in parameter space and the bifurcations closest to λ0. The sensitivity of the distances to these bifurcations yield hyperplane approximations to the hypersurfaces and optimal changes in parameters to improve the stability robustness. Methods for saddle-node, Hopf, transcritical, pitchfork, cusp, and isola bifurcation instabilities and constraints are outlined. These methods take full account of system nonlinearity and are practical in high dimensional parameter spaces. Applications to the design and operation of electric power systems, satellites, hydraulics and chemical process control are summarized.

Distance to Bifurcation in Multidimensional Parameter Space: Margin Sensitivity and Closest Bifurcations

Optimization of a pressure control valve for high power automatic transmission considering stability

On Modelling, System Identification and Control of Servo-Systems with a Flexible Load

A critical evaluation of current hydraulic servo system analysis methods indicates a need for alternative methods better able to quantify robust stability One promising method recently developed for analysing large-scale power systems determines stability robustness in a high-dimensional parameter space by computing the distance to the ‘closest’ Hopf bifurcation (which corresponds to the birth of a limit cycle oscillation) In this paper a procedure is developed for applying closest Hopf bifurcation theory in the design and analysis of robustly stable hydraulic servo systems The procedure addresses practical implementation issues such as the impact of an inhomogeneous parameter space and the choice of a metric that yields a meaningful quantitative measure of stability robustness Results from the new procedure applied to a common position control system compare favourably with published describing function results and new simulation results Additionally, the new procedure is easier to apply and

G G Kremer

Papers

A bifurcation-based procedure for designing and analysing robustly stable non-linear hydraulic servo systems: