Research

        Accurate descriptions of complex dynamical systems often require highly sophisticated mathematical modeling to capture the finer details in the behaviour of the systems. This is especially true when models are built from first principles where the interactions between the subsystems that make up the whole system may be quite complicated. One ready example that comes to mind is a Power Network (Power Grid) over a large region. In the control parlance, this means the resulting realization has a very high state order. Our primary interest in modeling such systems is to perform analysis to understand and predict the behaviour in order to build suitable controllers to achieve certain performance objectives.

        However, in many situations, modeling introduces unnecessary complexity that makes it difficult to analyze or predict the system's response (we fail in our objectives right at the outset!). For example, one may want to model a motor's behaviour by a third order differential equation to take into account fast dynamics like vibrations/chatter at high frequencies. But, it may be sufficient to approximate the dynamics of interest by a single-pole system for, say, a low frequency application. While there may be ways to perform accurate analysis on highly complex systems, our computing resources are not sophisticated enough to do so at present. Moreover, control design carried out using well-known methods like H2/H-infinity Synthesis, when performed even with the current computing capabilites, often result in controllers with orders equal to or comparable to the original order of the system. Even such superior standards of computing technology and performance cannot guarantee the design of the simplest, most efficient controllers for large-scale systems as it turns out to be an NP-hard problem. Such controllers are not only complicated for analysis, simulation or validation but also makes them practically expensive and in some occasions, unrealizable. Simply put, existing algorithms, softwares and processors are just not good enough for solving this problem. 

        There is also another practical aspect to system modeling. On most occasions, one specific model, however complex it is, can never represent all possible dynamics exhibited by the physical system. Hence, controllers designed using a single-model representation may not work well in practice. This problem is overcome by introducing uncertainty into the model. One induces perturbations on the nominal system to include a range of possible models around the nominal one. Then, a controller is designed that works for all possible plant models in this range. But again, this uncertainty comes with a price tag.   The controller we come up with may be more than what we need (in fact, the cost of the controller goes up steadily as the uncertainty increases as one would expect). In other words, the controller obtained may still be complex and unworkable. Hence, it would be very useful to obtain simplifications in the uncertain model too (may be even get rid of some needless uncertainty descriptions). In short, there exists a great practical necessity to approximate the system models with simpler, more tractable ones with pre-determinable error on the approximation. This calls for systematic numerical methods for Model Reduction of Systems.

        Starting in the late 70s, several procedures have been developed to reduce the model order with apriori error bounds. The early methods were concerned with reducing the well-known class of Linear Time Invariant (LTI) Systems with no uncertainty in the models (the simplest case!). One of the popular methods that held much promise was the Balanced Truncation Algorithm(BTA). It comes under the broader category of Gramian-based approaches and has an appealing symmetry to it. With the advent of the information age, several software tools were developed to implement these algorithms. For example, the Control System Toolbox, Robust Control Toolbox and Mu-synthesis Toolbox, that were developed in the 90s, each have their own set of routines to compute lower order approximations of systems. BTA underwent significant modifications and refinements that made the implementations more efficient and robust to finite-precision numerical errors. At the time of writing this, there are several excellent solvers  available as freewares on the web for this purpose. However, as it suggests itself, their application is restricted to a very small class of systems - LTI with no uncertainty in the models. This renders them not as useful as physical systems are almost always not LTI and as mentioned earlier, the models should have uncertainty incorporated to effectively capture the behaviour of the system.

        The theory and generalization of BTA to Multi-dimensional and Uncertain systems came in place in the 90s. This led to the development of the Linear Matrix Inequalities (LMI) as a basis for solving reduction problems. The LMI-based approach, as it is called, has one of its origins as a relaxation of the well-known Lyapunov Equations to inequality constaints. Alternatives to the classical Ricatti-based control synthesis methods were also formulated. Inevitably, such a  framework served as the right platform for optimization using matrix variables and matrix inequality constraints. The optimization problems that are formulated using the LMI machinery may also be called Semi-Definite Programs (SDPs). One way to think about LMIs and SDPs is as generalizations of the well-known Linear Programming (LP) to the matrix case. Karmarkar's seminal paper on solving LPs published in the mid-90s was the starting point in enthusing interest in solving SDPs. Active research led to the extension of Karmarkar's Algorithm to the semi-definite case - now, called the Interior Point Methods. Much of the earlier work is credited to Nesterov and Nemirovsky in this area. The algorithms they developed compute the global optimum of a convex Semi-Definite program. More importantly, they are solvable in polynomial-time. Such computational tractability of control-based optimization problems aroused tremendous research interest and activity in the control arena since a lot of control problems that looked unsolvable could now be re-formulated as LMIs and the Interior-Point methods can now be successfully applied. For the past decade or so, LMIs and SDPs have been and still are, very hot research topics!

       One needs to observe that model reduction and control synthesis are closely tied.  Naturally, the LMI-based approaches found their applicability in this field as well.  Our problem is,  say we are given a multi-dimensional/uncertain system realization that has a high state order and complex uncertainty description. How does one go about obtaining a lower-order approximation to this model with an apriori upper error bound ?  This research endeavor and MRedTool attempt to give the solution to this problem. As well, the algorithms implemented in the toolbox may also be used to reduce calssical multiD realizations and spatially distributed control models.
 
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