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Rational Approximation in Hankel Norm
Published in Phillip A. Regalia, Adaptive IIR Filtering in Signal Processing and Control, 2018
Suppose that σN is negligible in comparison with the other singular values. Then (4.7) shows that the last state component xN(1) receives negligible energy from the input. Similarly, (4.8) shows that xN(1) in turn makes a negligible contribution to the output as well. Thus if σN is small, the final state xN(·) contributes little to the input-output behavior of the system. Truncating this final state should then have little influence on the system. This leads to a model reduction method which consists of truncating away those states of a balanced realization corresponding to the smallest Hankel singular values. This method was proposed by Moore [8] for continuous time systems, and examined later by Glover [5], who established an a priori bound concerning the approximation error incurred from this procedure.
Empirical Gramian-based spatial basis functions for model reduction of nonlinear distributed parameter systems
Published in Mathematical and Computer Modelling of Dynamical Systems, 2018
Mian Jiang, Jigang Wu, Wenan Zhang, Xuejun Li
where and denote the th Hankel singular value [35]. For the controllability Gramian, the Hankel singular values correspond to the amount of energy that has to be put into the system in order to move the corresponding states. The singular values of the observability Gramians refer to the energy generated by the corresponding states. If the linear system (4) is balanced, the state with the largest singular value is the one most affected by control moves and the output is the most affected by a change in this state. Therefore, the states corresponding to the largest singular values influence the input–output behaviours the most. The transformed Gramians are then obtained from:
Mechanistic model-based control of biodiesel production processes: a review of needs and scopes
Published in Chemical Engineering Communications, 2023
Rasheed O. Kelani, Zainal Ahmad, Dipesh Patle
The issue of screening for the best control structure was investigated by Cao and Saha (2005). An enhanced and better performing algorithm called "Branch and Bound (BAB)" method was developed to screen for the best control structure. BAB is often used to screen the best control structure. The design of a control structure involves the process of combining or connecting inputs and outputs. Thus, the method shows how global ranking can be obtained for all inputs and outputs pairing. Hankel singular value (HSV) was used to discriminate control structures for stabilization. A literature search revealed that this method had not been applied to the biodiesel production process.
On model order priors for Bayesian identification of SISO linear systems
Published in International Journal of Control, 2019
Patricio E. Valenzuela, Thomas B. Schön, Cristian R. Rojas
Since the Hankel singular values are closely related to the order of a model, they have been widely employed in model order reduction (Green & Limebeer, 2012). A Hankel singular value close to zero indicates that the order of the model can be reduced by one with almost negligible effect on the model properties (e.g. its bandwidth, step response, DC gain, etc.). Thus, the specification of a prior over the Hankel singular values of Gθ reflects prior knowledge (or belief) about the order of the system (2).