User`s guide

10 Reliable Computations
10-2
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
you accurate answers regardlessofthe modelorconversion method you choose.
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 be well-conditioned if small 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
moresensitivity to perturbation than is already inherent in the problem. 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 tosolve an ill-conditioned problem
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.