Robust Control Toolbox™

Compute Hankel singular values for stable/unstable or continuous/discrete system

Syntax

• hankelsv(G)
  hankelsv(G,ErrorType,style) 
  [sv_stab,sv_unstab]=hankelsv(G,ErrorType,style) 
  

Description

[sv_stab,sv_unstab]=hankelsv(G,ErrorType,style) returns a column vector SV_STAB containing the Hankel singular values of the stable part of G and SV_UNSTAB of anti-stable part (if it exists). The Hankel SV's of anti-stable part ss(a,b,c,d) is computed internally via ss(-a,-b,c,d). Discrete model is converted to continuous one via the bilinear transform.

hankelsv(G) with no output arguments draws a bar graph of the Hankel singular values such as the following:

This table describes optional input arguments for hankelsvd.

Argument	Value	Description
ERRORTYPE	'add' 'mult' 'ncf'	Regular Hankel SV's of G Hankel SV's of phase matrix Hankel SV's of coprime factors
STYLE	'abs' 'log'	Absolute value logarithm scale

Algorithm

For ErrorType = 'add', hankelsv implements the numerically robust square root method to compute the Hankel singular values [1]. Its algorithm goes as follows:

Given a stable model G, with controllability and observability grammians P and Q, compute the SVD of P and Q:

• [Up,Sp,Vp] = svd(P);
  [Uq,Sq,Vq] = svd(Q);
  

Then form the square roots of the grammians:

• Lr = Up*diag(sqrt(diag(Sp)));
  Lo = Uq*diag(sqrt(diag(Sq)));
  

The Hankel singular values are simply:

• H = svd(Lo'*Lr);
  

This method not only takes the advantages of robust SVD algorithm, but also ensure the computations stay well within the "square root" of the machine accuracy.

For ErrorType = 'mult', hankelsv computes the Hankel singular value of the phase matrix of G [2].

For ErrorType = 'ncf', hankelsv computes the Hankel singular value of the normalized coprime factor pair of the model [3].

Reference

[1]  Safonov, M.G., and R.Y. Chiang, "A Schur Method for Balanced Model Reduction," IEEE Trans. on Automat. Contr., vol. AC-2, no. 7, July 1989, pp. 729-733.

[2]  Safonov, M.G., and R.Y. Chiang, "Model Reduction for Robust Control: A Schur Relative Error Method," International J. of Adaptive Control and Signal Processing, Vol. 2, pp. 259-272, 1988.

[3]  Vidyasagar, M., Control System Synthesis - A Factorization Approach. London: The MIT Press, 1985.

See Also
reduce      Top level model reduction routines

balancmr    Balanced truncation via square-root method

schurmr     Balanced truncation via Schur method

bstmr       Balanced stochastic truncation via Schur method

ncfmr       Balanced truncation for normalized coprime factors

hankelmr    Hankel minimum degree approximation

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