MATLAB CONTROL SYSTEM TOOLBOX 9 Spécifications télécharger pdf (Page 248)

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11 Reliable Computations

11-2

When working with low-order SISO models (less than five states), computers

are usually quite forgiving and insensitive to numerical problems. You

generally won't encounter any numerical difficulties and MATLAB will give

youaccurateanswersregardlessofthemodelor conversionmethodyouchoose.

For high order SISO models and MIMO models, however, the finite-precision

arithmetic of a computer is not so forgiving and you must exercise caution.

In general, to get a numerically accurate answer from a computer, you need:

•A well-conditioned problem

•An algorithm that is numerically stable in finite-precision arithmetic

•A good software implementation of the algorithm

A problem is said to bewell-conditioned ifsmall changes in the data cause only

small corresponding changes in the solution. If small changes in the data have

the potential to induce large changes in the solution, the problem is said to be

ill-conditioned. An algorithm is numerically stable if it does not introduce any

more sensitivity toperturbationthanisalreadyinherentin theproblem. Many

numerical linear algebra algorithms can be shown to be backward stable; i.e.,

the computed solution can be shown to be (near) the exact solution of a slightly

perturbed original problem. The solution of a slightly perturbed original

problem will be close to the true solution if the problem is well-conditioned.

Thus, a stable algorithm cannot be expected to solve an ill-conditionedproblem

any more accurately than the data warrant, but an unstable algorithm can

produce poor solutions even to well-conditioned problems. For further details

and references to the literature see [5].

While most of the tools in the Control System Toolbox use reliable algorithms,

some of the tools do not use stable algorithms and some solve ill-conditioned

problems. These unreliable tools work quite well on some problems (low-order

systems) but can encounter numerical difficulties, often severe, when pushed

on higher-order problems. These tools are provided because:

•They are quite useful for low-order systems, which form the bulk of

real-world engineering problems.

•Many control engineers think in terms of these tools.

•A more reliable alternative tool is usually available in this toolbox.

•They are convenient for pedagogical purposes.

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