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Dynamic Systems and Control Theory
Published in Mohammad Monir Uddin, Computational Methods for Approximation of Large-Scale Dynamical Systems, 2019
Two special matrices, namely controllability Gramian and observability Gramian, play important roles in the control theory. Among many applications, system Gramians they are extremely useful in the model reduction of large-scale dynamical systems. The Gramian-based MOR methods are in general based on the principle of the systemobservability Gramiancontrollability Gramian and observability Gramian.controllability Gramian In the following subsections, we briefly introduce the Gramians including their properties.
HSH-norm optimal MOR for the MIMO linear time-invariant systems on the Stiefel manifold
Published in International Journal of Control, 2022
Besides the norm, the norm of LTI systems is also differentiable. The norm is defined as the Frobenius norm of the Hankel operator, and it can be computed by using the controllability Gramian and the observability Gramian (Fernando & Nicholson, 1982). The controllability Gramian is used to provide the controllability information of the system. Similarly, the observability information of the system can be obtained by studying the observability Gramian. The norm is first proposed in Hanzon (1992). Schelfhout (1996) develops the first-order necessary conditions for the norm optimal MOR problem and gives a few MOR methods. Schelfhout (1996) shows that the norm can be considered as a special case of the time-weighted norm, and it is invariant under the bilinear transform. For more properties of the norm, one can refer to Hanzon (1992) and Schelfhout (1996).
Minimal Realized Power Systems for Load Frequency Control Using Optimal Theory Based PID Controller
Published in IETE Journal of Research, 2022
Anju G. Pillai, Elizabeth Rita Samuel
The controllability and observability Gramians of (1) and (2) are solutions of dual Lyapunov equations and Instead of computing X and Y, compute and of low rank and Then calculate a singular value decomposition of the form Now, the balancing transformations are and where we can compute reduced system as The controllability Gramian gives an idea about how much energy is required for the change of a system from its initial state to a new state using finite non-impulsive inputs [14]. While the observability Gramian is an indication of the accuracy of the estimation of initial state vector from the output signal without any input signal. The realized system shown in (1) and (2) are considered to be balanced one, if the controllability and observability Gramians are equal and diagonal. Thus it is concluded that each and every state of the balanced realization is equally controllable and observable.