Robust Control Toolbox™

Balanced model truncation via square root method

Syntax

• GRED = balancmr(G)
  GRED = balancmr(G,order)
  [GRED,redinfo] = balancmr(G,key1,value1,...)
  [GRED,redinfo] = balancmr(G,order,key1,value1,...)
  

Description

balancmr returns a reduced order model GRED of G and a struct array redinfo containing the error bound of the reduced model and Hankel singular values of the original system.

The error bound is computed based on Hankel singular values of G. For a stable system these values indicate the respective state energy of the system. Hence, reduced order can be directly determined by examining the system Hankel SV's, .

With only one input argument G, the function will show a Hankel singular value plot of the original model and prompt for model order number to reduce.

This method guarantees an error bound on the infinity norm of the additive error || G-GRED || for well-conditioned model reduced problems [1]:

This table describes input arguments for balancmr.

Argument	Description
G	LTI model to be reduced (without any other inputs will plot its Hankel singular values and prompt for reduced order)
ORDER	(Optional) an integer for the desired order of the reduced model, or optionally a vector packed with desired orders for batch runs

A batch run of a serial of different reduced order models can be generated by specifying order = x:y, or a vector of positive integers. By default, all the anti-stable part of a system is kept, because from control stability point of view, getting rid of unstable state(s) is dangerous to model a system.

'MaxError' can be specified in the same fashion as an alternative for 'Order'. In this case, reduced order will be determined when the sum of the tails of the Hankel sv's reaches the 'MaxError'.

This table lists the input arguments 'key' and its 'value'.

Argument	Value	Description
'MaxError'	Real number or vector of different errors	Reduce to achieve H error. When present, 'MaxError'overides ORDER input.
'Weights'	{Wout,Win} cell array	Optimal 1-by-2 cell array of LTI weights Wout (output) and Win (input). Defaults are both identity. Weights must be invertible.
'Display'	'on' or 'off'	Display Hankel singular plots (default 'off').
'Order'	Integer, vector or cell array	Order of reduced model. Use only if not specified as 2nd argument.

Weights on the original model input and/or output can make the model reduction algorithm focus on some frequency range of interests. But weights have to be stable, minimum phase and invertible.

This table describes output arguments.

Argument	Description
GRED	LTI reduced order model. Becomes multidimensional array when input is a serial of different model order array
REDINFO	A STRUCT array with three fields: REDINFO.ErrorBound (bound on || G-GRED ||) REDINFO.StabSV (Hankel SV of stable part of G) REDINFO.UnstabSV (Hankel SV of unstable part of G)

G can be stable or unstable, continuous or discrete.

Algorithm

Given a state space (A,B,C,D) of a system and k, the desired reduced order, the following steps will produce a similarity transformation to truncate the original state space system to the k^th order reduced model.

1. Find the SVD of the controllability and observability grammians

                                         P = U_p p VpT

                                        Q = U_q q VqT

2. Find the square root of the grammians (left/right eigenvectors)

                                        L_p = U_p p1/2

                                                    Lo = Uq q1/2

3. Find the SVD of (L_o^TL_p)

                                      L_o^T L_p = U VT

4. Then the left and right transformation for the final k^th order reduced model is

                                     S_L,BIG = L_o U(:,1:k) (1;k,1:k))-1/2

                                     S_R,BIG = L_p V(:,1:k) (1;k,1:k))-1/2

5. Finally,

The proof of the square root balance truncation algorithm can be found in [2].

Example

Given a continuous or discrete, stable or unstable system, G, the following commands can get a set of reduced order models based on your selections:

• rand('state',1234); randn('state',5678);G = rss(30,5,4);
  [g1, redinfo1] = balancmr(G); % display Hankel SV plot
                                % and prompt for order (try 15:20)
  [g2, redinfo2] = balancmr(G,20); 
  [g3, redinfo3] = balancmr(G,[10:2:18]);
  [g4, redinfo4] = balancmr(G,'MaxError',[0.01, 0.05]);
  rand('state',12345); randn('state',6789);
  wt1 = rss(6,5,5); wt1.d = eye(5)*2;
  wt2 = rss(6,4,4); wt2.d = 2*eye(4);
  [g5, redinfo5] = balancmr(G, [10:2:18], 'weight',{wt1,wt2});
  for i = 1:5
      figure(i); eval(['sigma(G,g' num2str(i) ');']);
  end
  

Reference

[1]   Glover, K., "All Optimal Hankel Norm Approximation of Linear Multivariable Systems, and Their Lµ - error Bounds," Int. J. Control, Vol. 39, No. 6, 1984, p. 1145-1193

[2]  Safonov, M.G., and R.Y. Chiang, "A Schur Method for Balanced Model Reduction," IEEE Trans. on Automat. Contr., Vol. 34, No. 7, July 1989, p. 729-733

See Also
reduce      Top level model reduction function

schurmr     Balanced model truncation via Schur method

hankelmr    Hankel minimum degree approximation

bstmr       Balanced stochastic model truncation via Schur method

ncfmr       Balanced model truncation for normalized coprime factors

hankelsv    Hankel singular values

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