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control and analysis of systems with hard constraints

 This area of research is dedicated to the design of robust linear feedback controllers for MIMO and LTI plants, subject to hard constraints on the plant input (control variable). Most applications in control engineering involve hard constraints on the plant input (eg a flow rate in a valve); unfortunately, there are only few design procedures which deal with such constraints. In practical usage, this has led to ad-hoc methods (or, in the worst case, wait-and-see-what-happens methods), which have little theoretical justification.
The approach used is a saturation avoiding approach, i.e. the control signal is forced to be smaller in amplitude than a prescribed value, in all operating conditions. The advantage is that we remain in a linear setting and do not need any further assumptions on the plant to control, in contrast to the schemes that incorporate a saturation nonlinearity (e.g. the works on switching gain controllers on nested ellipsoids etc.).

To solve the problem of the constrained plant input as mentioned above, the amplitude of the external (here: reference) signal has to be bounded, too. Earlier works of Dourdoumas and Reichel showed, that the average load of the control variable is better (and moreover some kind of optimisation is possible), when the first derivative of the reference signal is bounded (in amplitude) as well. A central problem within this approach is the determination of the maximum output of a LTI-system (under these prerequisites), which is solved via Reichel's "Balkenverfahren", that constructs the optimal (i.e. the maximum output amplitude achieving input signal). An improvement of this algorithm is possible via nonlinear optimisation. In many cases, however, the optimal solution can be determined by linear programming techniques. This problem is discussed in the Master's thesis and following works:

Having this "computational" tool at hand, the actual design problem can be attacked. Here, our aim is a direct consideration of these constraints. Our first approach employs the well-known design of robust controllers (in particular, H Loop Shaping) as looks at proper and systematic adaption of the weights, in order to meet prescribed hard bounds on the control signal. This method, however, leads very well to robust controllers, but does not guarantee the hard bound on the control signal when actually facing this model uncertainty. Parts of the thesis deal with this problem, as well as some recent papers:

This lack of robustness has been attacked in two ways: one can either check the bounds on the control signal when having parameter variations a-posteriori, or one could try to incorporate these uncertainties directly in the design process. The second method, however, is in a rather developing stage. These two approaches can be found in: Another topic within this framework is the design of optimal controllers, optimal for instance in the sense that they produce an as small as possible error signal. This problem turns out to be somewhat related to assessing the general solvability of the design problem: what bounds on the control signal can be dealt with with what kind of external signals. Note, that this is the turn-around to the discussion of which initial states can be brought to the origin with what "size" of control signal. This is discussed in the Master's thesis and in some recent works: Future research directions include for example the incorporation of rate constraints of the control signal (interesting for flight control) and the examination of mixed hard and soft (2-norm) bounds for different signals of a control system.


I collected the software used for all the works in a package called cc - Constraint Control . Version 1.0 appeared in September 1999 and is for use with Matlab 5.3 and 6.0. Contact me if you are interested in using the software.

Funding and Cooperations

This work used to be supported by the German research council, but obviously that stopped (at least for me), when I moved to Sweden. People, closely involved in these activities are the control groups at Paderborn University (F. Gausch), Graz Technical University (N. Dourdoumas, A. Hofer) and Politecnico di Torino (M. Canale).

Research Seminars

[excluding conference presentations]
Last updated Mon Sep 2 19:25:43 2002 by Wolfgang Reinelt . Legal Disclaimer