# wcnorm

Worst-case norm of uncertain matrix

## Syntax

```maxnorm = wcnorm(m)
[maxnorm,wcu] = wcnorm(m)
[maxnorm,wcu] = wcnorm(m,opts)
[maxnorm,wcu,info] = wcnorm(m)
[maxnorm,wcu,info] = wcnorm(m,opts)
```

## Description

The norm of an uncertain matrix generally depends on the values of its uncertain elements. Determining the maximum norm over all allowable values of the uncertain elements is referred to as a worst-case norm analysis. The maximum norm is called the worst-case norm.

As with other uncertain-system analysis tools, only bounds on the worst-case norm are computed. The exact value of the worst-case norm is guaranteed to lie between these upper and lower bounds.

### Basic syntax

Suppose `mat` is a `umat` or a `uss` with M uncertain elements. The results of

```[maxnorm,maxnormunc] = wcnorm(mat) ```

`maxnorm` is a structure with the following fields.

Field

Description

`LowerBound`

Lower bound on worst-case norm, positive scalar.

`UpperBound`

Upper bound on worst-case norm, positive scalar.

`maxnormunc` is a structure that includes values of uncertain elements and maximizes the matrix norm. There are M field names, which are the names of uncertain elements of `mat`. The value of each field is the corresponding value of the uncertain element, such that when jointly combined, lead to the norm value in `maxnorm.LowerBound`. The following command shows the norm:

``` norm(usubs(mat,maxnormunc)) ```

### Basic syntax with third output argument

A third output argument provides information about sensitivities of the worst-case norm to the uncertain elements ranges.

```[maxnorm,maxnormunc,info] = wcgain(mat) ```

The third output argument `info` is a structure with the following fields:

Field

Description

`Sensitivity`

A `struct` with M fields. Fieldnames are names of uncertain elements of `sys`. Field values are positive numbers, each entry indicating the local sensitivity of the worst-case norm in `maxnorm.LowerBound` to all of the individual uncertain elements uncertainty ranges. For instance, a value of 25 indicates that if the uncertainty range is increased by 8%, then the worst-case norm should increase by about 2%. If the `Sensitivity` property of the `wcgainOptions` object is `'off'`, the values are `NaN`.

`ArrayIndex`

1-by-1 scalar matrix with the value of 1. In more complicated situations (described later) the value of this field depends on the input data.

## Examples

You can construct an uncertain matrix and compute the worst-case norm of the matrix, as well as its inverse. Your objective is to accurately estimate the worst-case, or the largest, value of the condition number of the matrix.

```a=ureal('a',5,'Range',[4 6]); b=ureal('b',2,'Range',[1 3]); b=ureal('b',3,'Range',[2 10]); c=ureal('c',9,'Range',[8 11]); d=ureal('d',1,'Range',[0 2]); M = [a b;c d]; Mi = inv(M); [maxnormM] = wcnorm(M) maxnormM = LowerBound: 14.7199 UpperBound: 14.7327 [maxnormMi] = wcnorm(Mi) maxnormMi = LowerBound: 2.5963 UpperBound: 2.5979 ```

The condition number of `M` must be less than the product of the two upper bounds for all values of the uncertain elements making up `M`. Conversely, the largest value of `M` condition number must be at least equal to the condition number of the nominal value of `M`. Compute these crude bounds on the worst-case value of the condition number.

```condUpperBound = maxnormM.UpperBound*maxnormMi.UpperBound; condLowerBound = cond(M.NominalValue); [condLowerBound condUpperBound] ans = 5.0757 38.2743 ```

How can you get a more accurate estimate? Recall that the condition number of an `nxm` matrix `M` can be expressed as an optimization, where a free norm-bounded matrix Δ tries to align the gains of `M` and `M`1

If `M` is itself uncertain, then the worst-case condition number involves further maximization over the possible values of `M`. Therefore, you can compute the worst-case condition number of an uncertain matrix by using a `ucomplexm` uncertain element, and then by using `wcnorm` to carry out the maximization.

Create a 2-by-2 `ucomplexm` object, with nominal value equal to zero.

```Delta = ucomplexm('Delta',zeros(2,2)); ```

The range of values represented by `Delta` includes 2-by-2 matrices with the maximum singular value less than or equal to 1.

You can create the expression involving `M, Delta` and `inv(M)`.

```H = M*Delta*Mi; ```

Finally, consider the stopping criteria and call `wcnorm`. One stopping criteria for `wcnorm(H)` is based on the norm of the nominal value of `H`. During the computation, if `wcnorm` determines that the worst-case norm is at least

`ABadThreshold+MBadThreshold*norm(N.NominalValue)`

then the calculation is terminated. In our case, since `H.NominalValue` equals 0, the stopping criteria is governed by `ABadThreshold`. The default value of `ABadThreshold` is 5. To keep `wcnorm` from prematurely stopping, set `ABadThreshold` to 38 (based on our crude upper bound above).

```opt = wcgopt('ABadThreshold',38); [maxKappa,wcu,info] = wcnorm(H,opt); maxKappa maxKappa = LowerBound: 26.9629 UpperBound: 27.9926 ```

You can verify that `wcu` makes the condition number as large as `maxKappa.LowerBound`.

```cond(usubs(M,wcu)) ans = 26.9629 ```

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