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The analysis and control of linear systems with varying parameters, called linear parameter-varying (LPV) systems, have become a major research area since the beginning of the nineties. This stems from the fact that LPV systems naturally arise when modeling manufacturing systems (wafer stages, machine tools, cranes), chemical processes such as liquid-gas separation, electro-hydraulic systems and active suspension systems, automotive engines, etc. (for an example we refer to [1]). In addition, in most of these applications the dynamics are inherently infinite-dimensional, characterized by a spatio-temporal distribution of state variables. A distribution in space gives rise to partial differential equations, while a distribution of state variables in time (to take into account latency, heredity, memory, etc.) involves functional differential equations. Recently new optimization based methods have been developed to design a controller with a pre-scribed structure or an order that is much smaller than the system order[2; 3]. Although the resulting optimization problems are non-convex and may even be non-smooth, these methods are promising for the control of infinite-dimensional systems, where any finite-dimensional controller is by definition a low order controller. This is illustrated by new results in the area of time-delay systems (a particular class of infinite-dimensional systems), on which OPTEC has gained key expertise (an overview can be found in Part II of the monograph [4]). The current methods for the analysis and control of LPV systems on the other hand are founded on linear matrix inequalities [5], giving rise to convex optimization problems. However, these methods are conservative and enforcing a fixed controller structure is difficult. In the two classes of methods, i.e. non-smooth optimization based and LMI based, incorporating model uncertainty is challenging. Moreover, the analysis and control of linear systems that have both infinite dimension and varying parameters is currently an open problem.
The aim of the research is to develop optimization based methods for the analysis and control of linear infinite-dimensional systems with time-varying parameters. These methods should deal with uncertainty on the system model and must be capable of designing controllers with a pre-scribedstructure or order that optimize stability, robustness and performance measures in aH2 -H1 setting.
The research topics at OPTEC are:
- Extending the two classes of analysis and control design methods. (i) Extend the nonsmooth optimization based methods to design fixed structure/order controller to general linear infinite-dimensional systems, hereby accounting for uncertainty on the system model. (ii) Extend the LMI based control design methods for LPV system to handle model uncertainty and a prescribed controller structure.
- Tackling the analysis and control of linear infinite-dimensional systems with time-varying parameters. To this end two complementary directions will be explored, and possibly combined: (i) an optimization based approach for a direct computation of the controller parameters in the infinite-dimensional closed-loop system, relying on state-of-the art numerical methods for large-scale systems and non-convex optimization techniques, possibly combined with interpolation techniques. (ii) an approach based on parameterized model reduction, where a reduced model for the infinite-dimensional system is obtained that explicitly depends on the original parameters, followed by a control design for the reduced model. In both cases the design problems give rise to large-scale but structured linear algebra problems, in particular large-scale matrix equations and inequalities, which need to be efficiently solved and embedded in an optimization framework. In order to handle a fast time-variation of the parameters an additional challenge consists of appropriately combining the inherently frequency domain techniques (both the current fixed-structure design techniques [2] and approaches for parameterized model order reduction [6] are eigenvalue / frequency domain based) with timedomain characterizations.
- Developing a flexible software environment for system modeling and controller synthesis froman interconnection point of view. This viewpoint allows us to define both the (sub)systems, controller structures and their interconnections in a natural way. Mathematically, it reflects in a model description involving both differential and algebraic equations.
