My research interests are
- Functional analysis
- Control theory
- Partial differential equations
- Complex analysis
I am especially interested in the intersection of the above research domains.
Main research projects
Model Reduction
It is often desirable to replace an accurate but complex model for a physical system by a perhaps slightly less accurate but simpler model. The process to extract the simpler model from the more complex one is called model reduction.
My research on model reduction mainly focuses on various types of balancing methods where the original complex model is assumed to be given by a partial differential equation. Analysis of error-bounds is of particular interest.
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- My earliest work in this area (Opmeer and Curtain 2004, Opmeer 2007) generalized LQG-balancing (including error-bounds in the gap metric) to infinite-dimensional systems.
- Later work considered the behavior of the known error-bound for Lyapunov-balanced model reduction. In (Opmeer 2008) this was done for ultra-differentiable control systems (which model certain damped vibrating systems). A more detailed analysis of the error (meaning the decay rate of the Hankel singular values) was carried out in (Opmeer 2010) for analytic control systems.
- My PhD student Chris Guiver investigates model reduction for dissipative systems. A manuscript on the finite-dimensional case has been submitted and the infinite-dimensional case is currently under study.
State space methods in the factorization approach to controller design
One of the main problems in control theory is to find a (robustly) stabilizing controller for a given system. It is often desirable to have state space formulas for such a stabilizing controller.
Such state space formulas can be obtained from the solution of the linear quadratic optimal control problem and involve the solutions of algebraic Riccati equations. As intermediate results state space formulas for Bezout factors and for the solution of the Nehari problem have to be obtained.
In my research into these problems the objective is to use minimal assumptions so as to cover as wide a class of (partial differential equation) systems as possible.
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- Work on this topic started for me with (Curtain and Opmeer 2005), where state space formulas for a solution of the suboptimal Nehari problem were obtained under very moderate stability assumptions. These were used in (Curtain and Opmeer 2006) to obtain state space formulas for Bezout factors under very moderate stabilizability assumptions. State space formulas for the resulting dynamic robustly stabilizing controller were subsequently obtained in articles by my co-author Ruth Curtain. This was all for the continuous-time case. The corresponding discrete-time results are in (Curtain and Opmeer 2009) for the Nehari problem and in a submitted manuscript for the Bezout factors and dynamic robustly stabilizing controllers.
- Together with Olof Staffans, I have looked at these issues in a slightly different light. The above results are proven under moderate stabilizability assumptions, but nonetheless assume that the state space chosen is 'appropriate'. In applications this amounts to choosing the correct spaces on which to study the partial differential equation. In the approach that Olof Staffans and I take, the choice of state space becomes part of the solution of the problem rather than something that has to be a priori determined appropriately. This led us to fundamentally reconsider the classical linear quadratic optimal control problem. We have done this first for discrete-time systems. The first article in the series is (Opmeer and Staffans 2008), where we treat the linear quadratic optimal control problem on the positive time axis, right factorizations and unbounded solutions of the control Riccati equation. In the second article in the series (Opmeer and Staffans 2010) we treat the linear quadratic optimal control problem on the negative time axis, left factorizations and unbounded solutions of the filter Riccati equation. The third article in the series has been submitted and deals with the linear quadratic optimal control problem on the doubly infinite time axis, strongly coprime factorizations and stabilizing dynamic controllers. The corresponding continuous-time results are currently under investigation.
Other research projects
- Distributional control systems (more information)
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Usually control systems that are L^p well-posed are studied. Several interesting examples (such as the heat equation with Dirichlet control and Neumann observation) are however not L^p well-posed. To cover such examples in an abstract framework, I introduced distributional control systems. The frequency domain theory appears in (Opmeer 2005) and the time domain theory in (Opmeer 2006).
- Non-dissipative feedback for beams (more information)
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In (Guiver and Opmeer 2010) my PhD student Chris Guiver and I show that a non-dissipative boundary feedback that is known to stabilize an Euler-Bernoulli beam model is actually destabilizing for Rayleigh and Timoshenko beam models.