# University of Twente Student Theses

## Finding the nearest positive-real system of lower order

Swart, D.F.H.
(2018)
*Finding the nearest positive-real system of lower order.*

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Abstract: | Many physical situations can be modeled as an linear time-invariant (LTI) system. These models can become very complex, consisting of many parameters, making the order of the model (the size of the state vector) very large. Models of high order can on one hand be very realistic, but on the other hand very heavy, which makes simulation expensive. This motivates us to find accurate lower-order approximations, where they should still be a realistic image of the modelled process. Another aspect of realistic models is the positive realness of a system. An LTI system that is postive-real (PR), which is equivalent to passivity in the case of LTI-systems, loses energy when there is no input, something that happens in nature as well. This brings us to the idea that if we approximate a physical system, the approximation should at least be positive-real to follow nature's law. In this report, we will start by showing how to find the nearest positive-real system to a given non-PR one. This is being done for descriptor systems (system described by Ex_ = Ax + Bu and y = Cx+Du). We will state the problem of finding the nearest PR-system and reformulate this problem to an equivalent problem, which has a simple convex set of solutions. Then we will formulate an algorithm to find the nearest PR-system. The second part of this report is about model reduction. We first describe truncation and residualization for standard input/state/output LTI systems (E = I). Since truncation and residualization strongly depend on the initial realization, we present a way to balance the system in order to improve the model reduction techniques. This results in balanced truncation and balanced residualization, and we will give an upper bound for the error between the original and the reduced system. Before we can combine the two previous parts to finding the nearest positive-real system of reduced order, we have to generalize model reduction for standard systems to descriptor systems. In order to do so, we will generalize classical resulsts, such as Lyapunov equations, Controllability and Observability Gramians and balanced realizations. We will conclude this part with an algorithm that gives a balanced truncation for descriptor systems. Next follows a chapter about the implementation in MATLAB. That chapter will cover the most important MATLAB functions. Finally, the results will be presented via a numerical example. We will discuss the results, argue about the combination of finding the nearest PR system and model reduction and conclude with future research topics. |

Item Type: | Essay (Master) |

Faculty: | EEMCS: Electrical Engineering, Mathematics and Computer Science |

Subject: | 31 mathematics |

Programme: | Applied Mathematics MSc (60348) |

Link to this item: | https://purl.utwente.nl/essays/77219 |

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