Peter Hippe

Brief Systematic closed-loop design in the presence of input saturations

This contribution distinguishes between two conceptually different windup phenomenons, namely controller and plant windup. Even in the absence of dynamic controller elements, saturating nonlinearities can trigger undesired oscillations. This effect is obviously due to inappropriate plant states, and is therefore called plant windup. Based on the frequency response of the linear part, a design aid for plant windup prevention is presented. It facilitates the search for suitable closed-loop dynamics, or for an appropriate additional dynamic network. State and disturbance observers add dynamic elements to the compensator, and in the presence of very slow or unstable controller modes, the well-known reset windup or integral windup occurs. This effect, the controller windup, is related to mismatched controller states. By feeding the limited plant input signal into the observer it is completely removed, so that the remaining effects of input saturation are solely attributable to plant windup. Various known windup prevention techniques can be explained within the presented framework.

Brief Windup prevention for unstable systems

Presented is a windup prevention method for linear, time-invariant, exponentially unstable systems with saturating input nonlinearity such that, given a linear nominal controller and disturbances of known maximum amplitudes, arbitrary reference input sequences do not cause closed loop instability.

Optimal reduced-order estimators in the frequency domain: the discrete-time case

The discrete-time stationary Kalman filter problem is solved in the z-domain for nth-order systems with >m outputs, where 0 ≤ >k ≤ >m measurements are noise-free. The design equations are very similar to the continuous-time cage, and they can be solved by spectral factorization, giving the polynomial matrix D¯∼( z) which parametrizes the reduced-order optimal estimator in the frequency domain. By solving a single linear equation the equivalent time-domain representation for the optimal filter can also be obtained. A simple example demonstrates the filter design in the frequency domain.

Parameterization of reduced-order compensator in the frequency domain

After a brief survey of the relations between time-and frequency-domain representations of the observer-based linear state feedback loop, the parameterization of the compensator is discussed. It is shown that the free parameters specified by the state feedback matrix K and the output error feedback matrix L in the time domain appear as coefficients in the polynomial matrices [Dtilde](s) and (s), characterizing the closed-loop dynamics in the frequency domain. These polynomial matrices can either be chosen arbitrarily (pole-placement problem) or they may be derived as solutions from the corresponding optimal control or the optimal estimation problems, respectively. Starting from the polynomial matrices [Dtilde](s) and (s), the doubly coprime fractional representations of the system and of the compensator transfer matrices can be computed directly without recurrence to the time domain results. From these representations, the right and left coprime compensator matrix fraction descriptions can easily be evalu...

Environ analysis of linear compartmental systems: The dynamic, time-invariant case

Abstract This paper gives the extension of the known static input-output analysis for linear time-invariant compartmental systems to the case of time-dependent input functions. The environ analysis partitioning compartment storages and flows into components associated with particular system inputs or outputs therefore yields time-dependent results. By this extension the powerful tool of environ analysis as introduced by Matis and Patten (1979) can also be used when the flow pattern within an ecosystem is not constant but varies with time, as, for example, when subject to diurnal, seasonal or weather-related influences. The compartmental system is investigated from two different points of view. In one, the flows are expressed as fractions of the recipient compartment, and in the other as fractions of the donor compartment. The output environs can be defined using the linear time-invariant differential equations of the ecosystem, whereas for the input environ partitions a modified production matrix has to be formed. Examples for stepwise input functions are used to clarify the methods, and an extension to harmonic input functions, using Laplace transforms, is given at the end of the paper.

Parametric compensator design

A parametrization of dynamical output-feedback compensators that will assign a complete set of distinct eigenvalues to a given state-space system is presented. The main result is a compact parametric expression for the gain matrices of an output-feedback compensator, of dynamical order I > n − (m + r), explicitly characterized by two sets of free parameters: the set of n + l closed-loop system eigenvalues and a set of N = mr − n + (m + r − 1)/ effective free parameters for n-state, r-input, m-output systems. These latter N effective free parameters are determined by a sequential design procedure. The principal benefits of the explicit characterization of a parametric class of dynamical output-feedback compensators lie in the ability to directly accommodate various different design criteria.

Comments on ‘A regulating and tracking PI(D) controller’

Eitelberg (1987) presents a modification of the classical PID controller which amounts to a two-degree-of-freedom structure. Beside the fact that this modification is already known from the literature, its potentials are not fully exploited by Eitelberg. Based on a closed-loop interpretation of the approach, a systematic tuning procedure and an additional extension for PI controllers are presented.

Design of pole placing controllers for stable and unstable systems with pure time delay

ABSTRACT Based on improved low order dead time approximants or starting from an overall approximation of linear time invariant systems with pure time delay, pole placing controller design is applied. This procedure yields excellent results for stable and unstable systems. The main advantage of this approach is the use of well-known state feedback control methods which can be used to shape the closed-loop response in a desired manner. As long as the approximations are of sufficient quality, the closed loop transients of model and original loop coincide within small error limits.

A nonminimal representation of reduced order observers

Abstract A new nonminimal representation of reduced-order observers of order n − k in the time and in the frequency domain is introduced. This reduced-order observer is based on a full-order plant model where the order reduction to n − k results from an infinite gain feedback of k outputs. This new representation is crucial for establishing the links between time and frequency domain design methods incorporating reduced-order observers. The main result are the doubly coprime fractional representations of a transfer matrix related to an observer of order n − k .

Strictly doubly coprime factorizations and all stabilizing compensators related to reduced-order observers

Abstract Doubly coprime factorizations (DCF) play an important role in the design of compensators for lumped linear time invariant systems. They are defined as stable proper rational matrices, and they not only constitute a starting point for the fractional approach to all stabilizing compensators, but they are also closely related to the frequency domain design of (optimal) state feedback and estimation, and of the resulting observer-based compensators. When defining DCFs related to reduced-order observers, some rational matrices become improper. This can be prevented by the introduction of artificial (stable) dynamics—the identity elements—which cancel in the system and in the compensator transfer matrices. Such pole-zero cancellations, however, are contrary to the usual notion of coprimeness. It is shown here that these additional dynamics, which have no meaning in an observer-based compensator scheme are superfluous, and that the parametrization of all stabilizing compensators is feasible on the basis of possibly nonproper factorizations not containing identity elements. For such factorizations, the new definition of strictly doubly coprime factorizations (SDCF) is proposed.