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Hankel minimum degree approximation (MDA) without balancing
GRED = hankelmr(G) GRED = hankelmr(G,order) [GRED,redinfo] = hankelmr(G,key1,value1,...) [GRED,redinfo] = hankelmr(G,order,key1,value1,...)
hankelmr 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 Hankel singular values indicate the respective state energy of the system. Hence, reduced order can be directly determined by examining the system Hankel SV's, σι.
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]:
$${\Vert G-Gred\Vert}_{\infty}\le 2{\displaystyle \sum _{k+1}^{n}{\sigma}_{i}}$$
Note It seems this method is similar to the additive model reduction routines balancmr and schurmr, but actually it can produce more reliable reduced order model when the desired reduced model has nearly controllable and/or observable states (has Hankel singular values close to machine accuracy). hankelmr will then select an optimal reduced system to satisfy the error bound criterion regardless the order one might naively select at the beginning. |
This table describes input arguments for hankelmr.
A batch run of a serial of different reduced order models can be generated by specifying order = x:y, or a vector of 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'.
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.
G can be stable or unstable, continuous or discrete.
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:
rng(1234,'twister'); G = rss(30,5,4); [g1, redinfo1] = hankelmr(G); % display Hankel SV plot % and prompt for order (try 15:20) [g2, redinfo2] = hankelmr(G,20); [g3, redinfo3] = hankelmr(G,[10:2:18]); [g4, redinfo4] = hankelmr(G,'MaxError',[0.01, 0.05]); for i = 1:4 figure(i); eval(['sigma(G,g' num2str(i) ');']); end
Singular Value Bode Plot of G (30-state, 5 outputs, 4 inputs) shows a singular value Bode plot of a random system G with 20 states, 5 output and 4 inputs. The error system between G and its Zeroth order Hankel MDA has it infinity norm equals to an all pass function, as shown in All-Pass Error System Between G and Zeroth Order G Anticausal.
The Zeroth order Hankel MDA and its error system sigma plot are obtained via commands
[g0,redinfo0] = hankelmr(G,0); sigma(G-redinfo0.Ganticausal)
This interesting all-pass property is unique in Hankel MDA model reduction.
Singular Value Bode Plot of G (30-state, 5 outputs, 4 inputs)
All-Pass Error System Between G and Zeroth Order G Anticausal
[1] Glover, K., "All Optimal Hankel Norm Approximation of Linear Multivariable Systems, and Their L_{∝}-error Bounds," Int. J. Control, vol. 39, no. 6, pp. 1145-1193, 1984.
[2] Safonov, M.G., R.Y. Chiang, and D.J.N. Limebeer, "Optimal Hankel Model Reduction for Nonminimal Systems," IEEE Trans. on Automat. Contr., vol. 35, no. 4, April 1990, pp. 496-502.