ncfmr

Model reduction from normalized coprime factorization

Syntax

[Gred,info] = ncfmr(G,ord)

[~,info] = ncfmr(G)

Gred = ncfmr(G,ord,info)

ncfmr(G)

Description

ncfmr computes a reduced-order approximation of a model by truncating modes in a coprime factorization of the full-order model. This method is related to the balanced truncation method of balred, but it is particularly well-suited to controller order reduction. For a stabilizing controller, the reduced controller is also stabilizing as long as the approximation error is smaller than the robustness margin computed by ncfmargin.

[Gred,info] = ncfmr(G,ord) computes a reduced-order approximation of the dynamic system model G. Specify the desired reduction order as ord. If ord is a vector, then Gred is an array of approximations of the corresponding order. The structure info contains information about the computation such as bounds on the approximation error.

[~,info] = ncfmr(G) computes the coprime factorization of G given by [M,N] such that G = M\N (see lncf), the Hankel singular values of the factorization, and the error bounds. You can use this information to determine the target reduction order programmatically based on desired fidelity or robust stability considerations. Then, use the syntax Gred = ncfmr(G,ord,info) to compute the reduced-order model.

Gred = ncfmr(G,ord,info) computes the reduced-order approximation using the normalized coprime factorization and Hankel singular values that you provide in info. Obtain info using the previous syntax, [~,info] = ncfmr(G). Providing a previously computed info to ncfmr allows you to perform model reduction without having to recompute the factorization and Hankel singular values. This syntax is therefore particularly useful when performance is a concern.

ncfmr(G) plots the Hankel singular values and bounds on the approximation error corresponding to each order. Examine the plot to determine a reduced order based on desired fidelity or robust stability considerations. You can then use Gred = ncfmr(G,ord) to compute the reduced-order model.

Examples

Reduce Model Order

ncfmr computes Hankel singular values and approximation errors to help you select a suitable target reduction order. One way to do so is to examine a plot of these values. Load the 30-state plant model G.

load("ncfmrModel.mat","G")
size(G)

State-space model with 2 outputs, 3 inputs, and 30 states.

Call ncfmr without an output argument. The function generates a Hankel singular value plot, which shows the relative energy contributions of each state in the coprime factorization of G, arranged in decreasing order by energy. The plot also shows the upper bound on the error between the original and reduced-order models that you obtain by truncating the states at that point. Examine this plot to choose the target order. For instance, for a maximum error of 0.01, you can reduce the model to 13th order.

ncfmr(G)

Call ncfmr again with an output argument and using order = 13. Doing so computes the reduced model Gred. Examine the singular values of G and of the difference between G and Gred. The difference is very small across all frequencies, showing that the reduced-order model is a good approximation of the full-order model.

Gred = ncfmr(G,13);
sigma(G,G-Gred)
legend("G","G-Gred")

Reduce Controller Order While Preserving Stability and Robustness

When you use ncfmr to reduce a plant G or controller K for which the closed-loop response feedback(G*K,eye(n)) is stable, the resulting closed-loop response is also stable as long as the approximation error of the reduced model does not exceed the robustness margin computed by ncfmargin. To see this benefit of ncfmr, load a plant G and design a controller for it. For this example, use ncfsyn to design the controller.

load ncfmrStability.mat G
size(G)

State-space model with 1 outputs, 1 inputs, and 3 states.

% shaping weights
s = tf('s');
W1 = 3.35*tf([1 20.89],[1 0]);
W2 = 1;
% controller
[K,~,~,Kinfo] = ncfsyn(G,W1,W2); 
size(K)

State-space model with 1 outputs, 1 inputs, and 5 states.

ncfsyn designs a controller by optimizing the ncfmargin robustness margin using a plant shaped by weighting functions W1 and W2 (see ncfsyn). To analyze margins with ncfmargin and reduce controller order with ncfmr, work with the shaped plant Gs and the controller Ks designed for it.

Gs = Kinfo.Gs;
Ks = Kinfo.Ks;

Use ncfmargin to find the robustness margin of the system with the full-order controller. ncfsyn assumes a positive feedback loop while ncfmargin assumes negative feedback, so reverse the sign of the controller for this computation.

emax = ncfmargin(Gs,-Ks)

emax = 0.1956

As long as the approximation error of the reduced-order controller does not exceed emax, stability of the closed-loop system is preserved. Suppose that you can tolerate up to a 50% reduction in this margin in exchange for the computational benefit of a lower order controller. To select the reduced order, first compute the errors associated with each target order. ncfmr returns these values in the ErrorBound field of the info argument. Then find the index of the last entry in info.ErrorBound that exceeds the target error of emax/2.

[~,info] = ncfmr(Ks);
r = find(info.ErrorBound>emax/2,1,'last')

r = 3

Thus, you can approximate the original controller by only three states without too much loss of stability. To avoid recomputing the Hankel singular values of Ks, use info as an input argument to ncfmr.

Ksr = ncfmr(Ks,r,info);
size(Ksr)

State-space model with 1 outputs, 1 inputs, and 3 states.

The reduced-order controller yields a very similar stability margin to the original controller.

ncfmargin(Gs,-Ksr)

ans = 0.1949

Reducing the controller order further leads to additional reduction in the stability margin. Reducing too far can lead to loss of closed-loop stability. For instance, try reducing to first order.

Ksru = ncfmr(Ks,1,info);
ncfmargin(Gs,-Ksru)

ans = 0

Thus, for further analysis or implementation, use the third-order controller. To do so, convert Ksr, the reduced controller for Gs, into Kr, the reduced controller for G.

Kr = W1*Ksr*W2;

To confirm that this controller is satisfactory, compare the closed-loop response to the response with the full-order controller. Again, reverse the sign of the controller to account for ncfsyn assuming positive feedback.

CL = feedback(-G*K,1);
CLr = feedback(-G*Kr,1);
step(CL,CLr)
legend

The large overshoot in this case is due to instability of the original plant G.

Compute Multiple Reduced-Order Models

ncfmr can compute multiple reduced-order models at once and return them in a model array. This can be useful, for example, when you want to test a controller design with multiple approximations to choose the one that yields the best balance between accuracy and computational efficiency. To compute multiple models, provide a vector of target reduction orders instead of a single value for order.

Load the 30-state plant model G. Compute five approximations of orders 11−15.

load("ncfmrModel.mat","G")
orders = 11:15;
Gred = ncfmr(G,orders);
size(Gred)

5x1 array of state-space models.
Each model has 2 outputs, 3 inputs, and between 11 and 15 states.

Gred is an array of reduced-order state-space (ss) models. You can use the SamplingGrid property of ss to associate each entry in the array with its corresponding model order.

Gred.SamplingGrid = struct('order',orders);

Assigning SamplingGrid can be useful for keeping track of the entries in a model array. For instance, if you plot the frequency response of Gred in a MATLAB® figure, clicking one of the resulting responses creates a tooltip that includes information drawn from SamplingGrid.

Input Arguments

G — Model to reduce

dynamic system model

Model to reduce, specified as a dynamic system model such as a state-space (ss) model. G can be stable or unstable. If G is a generalized state-space model with uncertain or tunable control design blocks, then the function uses the nominal or current value of those elements. sys cannot be an frd model or a model with time delays.

ord — Reduction order

positive integer | vector of positive integers

Reduction order, specified as a positive integer or a vector of positive integers. If ord is a scalar, ncfmr returns the model Gred of that order. If ord is a vector, then Gred is an array of models reduced to the corresponding orders.

To determine ord, you can use one of two methods:

• Use the syntax ncfmr(G) to obtain a plot of Hankel singular values and bounds on approximation errors at each order. Examine the plot to choose a reduction order with a tolerable approximation error. For an example, see Reduce Model Order.

• Use the syntax [~,info] = ncfmr(G) to obtain the info structure. Programmatically examine the approximation error bounds in info.ErrorBounds to choose a reduction order. For an example, see Reduce Controller Order While Preserving Stability and Robustness.

If G has unstable states, then ord must be at least the number of unstable states.

Output Arguments

Gred — Reduced-order model

ss model

Reduced-order model, returned as a state-space (ss) model. If ord is a scalar, then Gred is a single model of order ord. If ord is a vector, then Gred is an array of ss models of corresponding orders.

info — Information about model-reduction calculation

structure

Information about model-reduction calculation, returned as a structure with the following fields.

• GL — Left normalized coprime factorization of G, returned as a state-space (ss) model. This factorization is given by GL = lncf(G). For more information, see lncf.

• HSV — Hankel singular values of GL, returned as a vector whose length is the number of states in G. These values indicate the relative energy contribution of each state. You can choose a target reduction order by examining these values and choosing a number of states after which the energy contribution drops off significantly.

• ErrorBound — Upper bound on approximation errors, returned as a vector. The approximation error is given by $${\Vert \left[M_r,N_r\right]-\left[M,N\right]\Vert }_{\infty }$$, where [M,N] = lncf(G) and [M_r,N_r] = lncf(Gred). (For more information about these expressions, see lncf.) Each entry info.ErrorBound(j) is the maximum approximation error associated with reducing to j states. Thus, for instance, if you want an approximation error of no more than 0.01, examine info.ErrorBound to find the index of the first entry that is less than 0.01. Use that index as ord.

Tips

• You can use ncfmr to reduce the plant G or controller K while preserving closed-loop stability of the following SISO or MIMO feedback loop.

  

  Stability of this loop is preserved as long as the approximation error of the reduced plant is smaller than the robustness margin for this loop given by ncfmargin(G,K).

  For controllers computed with ncfsyn, reducing the controller K_s that ncfsyn computes for the shaped controller G_s is preferable. Both K_s and G_s are returned by ncfsyn in the info output argument. You can then compute K_r, the reduced controller for the original plant G, from K_r = W₁K_srW₂, where W₁ and W₂ are the shaping weights used with ncfsyn. For an example, see Reduce Controller Order While Preserving Stability and Robustness.

  For controllers obtained by other techniques, reduction with ncfmr also preserves stability if the error does not exceed the ncfmargin margin. However, such reduction can partially remove integral action and introduce steady-state tracking errors. Therefore, removing any integrator terms from the controller before reduction with ncfmr and replacing them in the reduced controller is recommended.

Algorithms

ncfmr performs the following steps to reduce the input model G to the desired order k.

1. Find the left normalized coprime factorization [M,N] of G, where G = M\N (see lncf).

2. Obtain the kth-order approximation [M_r,N_r] of [M,N], using balanced model truncation with absolute error control (see balred).

3. Set Gred = M_r\N_r.

Version History

Introduced before R2006a

R2021b: Syntax and output change

The behavior of ncfmr changed in R2021b to bring it in line with balred and to allow fully programmatic selection of the reduced-model order. As a result, some syntaxes and output arguments have changed, and some options are no longer recommended.

Prior to R2021b, calling ncfmr(G) without a specified reduction order invoked an interactive workflow that prompted you to enter the desired order. As of R2021b, this syntax displays a plot but no longer prompts for the desired order. Instead, call ncfmr again using the ord input argument to specify the desired order. The following table further describes this change in the interactive workflow.

Interactive Workflow Before R2021b	Interactive Workflow in R2021b
Enter Gred = ncfmr(G). The software produces a Hankel singular-value plot and prompts you to enter the desired reduction order. Examine the plot to determine the desired order based on the Hankel singular values. Enter the desired order. The software computes the reduced-order model Gred.	Enter ncfmr(G) without an output argument. The software produces a Hankel singular-value plot that also shows the approximation error of truncation at each mode. Examine the plot to determine the desired order based on the Hankel singular values or the approximation error. Call ncfmr again by entering Gred = ncfmr(G,ord). The software computes the reduced-order model Gred.

The fields of the info output argument have also changed in R2021b. The following table summarizes these changes.

info Fields Before R2021b	info Fields in R2021b
info.GL — Left normalized coprime factorization info.GR — Right normalized coprime factorization info.hsv — Hankel singular values	info.GL — Left normalized coprime factorization info.HSV — Hankel singular values of info.GL info.ErrorBound — Bound on approximation error

Additionally, the 'MaxError' and 'Display' options are not recommended. Instead:

• To display a plot of the Hankel singular values of a dynamic system model G, call nfcmr(G) without an output argument. This command generates a singular-value plot that also displays the approximation error of truncation at each mode.

• To select a model order based on the approximation error, examine the plot. Or, call [~,info] = ncfmr(G) and use the information in info.ErrorBound.