Hankel singular values

The original linear prediction and state-space methods are known in the nuclear magnetic resonance literature as LPSVD and Hankel singular value decomposition (HSVD), respectively, and many variants of them exist. Not only do these methods model the data, but also the fitted model parameters relate directly to actual physical parameters, thus making modelling and quantification a one-step process. The analysis is carried out in the time domain, although it is usually more convenient to display the results in the frequency domain by Fourier transformation of the fitted function. 

P and Q possess real non-negative eigenvalues. Large eigenvalues indicate good controllability and observability, respectively, while very small or zero eigenvalues correspond to non-controllable and non-observable states, respectively. Every linear time-invariant system ((5.3), (5.4)) can be transformed into its balanced realisation [4]. For collocated actuators and sensors P equals Q, with the Hankel singular values... 

Control Structure Selection for Unstable Processes Using Hankel Singular Value... 

Control Structure Selection for open loop unstable processes is the main theme of this paper. Hankel singular value has been used as a controllability measure for input-output selection. This method ensures feedback stability of the process with minimal control effort as well as it provides a quantitative justification for the controllability. Simulation results with Tennesse-Eastman test-bed problem justify the proposed theory. 

Francis (1986) argues that for technical reasons it is assumed that Gy has no poles on the imaginary axis thus Gy belongs to RL , but not RIf°. In that case, the minimum value of over all stabilizing Ks equals the reciprocal of minimum Hankel singular value... 

To observe the effect of individual inputs and outputs on the process, only one input (or output) has been taken at a time along with all the outputs (or inputs) for analysis. Hankel singular values are calculated and shown in the Fig. 3 which clearly shows dominance of liquid levels in stripper and the separator (outputs 12 15 ) and reactor cooling water flow (input 10) over the other inputs and outputs. From engineering understanding it can be concluded that outputs can be controlled with stripper and separator liquid flow and reactor cooling water flow can be used to control the reactor temperature. Further analysis using a Branch Bound method has proven this selection correct (Saha Cao, 2002). 

This work investigates the use of reduced order models of reactive absorption processes. Orthogonal collocation (OC), finite difference (FD) and orthogonal collocation on finite elements (OCFE) are compared. All three methods are able to accurately describe the steady state behaviour, but they predict different dynamics. In particular, the OC dynamic models show large unrealistic oscillations. Balanced truncation, residualization and optimal Hankel singular value approximation are applied to linearized models. Results show that a combination of OCFE, linearization and balanced - residualization is efficient in terms of model size and accuracy. 

In this chapter several model reduction techniques will be discussed. The first method is based on firequency response matching, other methods make use oficonversion ofi the model structure to a state space model and subsequently truncating the states that have a minimum impact on the input-output relationship. The main indicator used fior this purpose is the so-called Hankel singular value. In addition, the model structure is converted to a balanced realization, afiter which the reduction techniques can be applied. Several examples are given on how to apply the dififierent methods. 

The following steps should now be followed in MATLAB to calculate the Gramians, transformation matrix, Hankel singular values and balanced system matrices ... 

Step 5 the Hankel singular values crcan now be computed from sigma = sqrt(S) ... 

File modred25.m is used to compare the various model reduction techniques and show their results. First the system is balanced and the Hankel singular values or Gramian diagonal elements are calculated for a 25th order state space model. Figure 26.5 shows the result. 

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