One of the approaches of controller design is to split the process in two steps:

1. Find the family of all controllers that make the plant P(s) stable
2. From the family found in the step above choose the controller that best matches the performance and robustness criteria

We take as reference the following model of a feedback control system.

P(s) is the transfer function of the plant, C(s) is our controller, F(s) is a feedback compensator, r(s) is the reference signal, y(s) is the system output, d(s) and n(s) are external noises or disturbances. We're obviously talking about Laplace Transforms.

C(s) and F(s) are assumed to be proper while P(s) is assumed to be strictly proper.

We can write down the following set of relations between the system inputs and the block inputs:

With the assumptions made just above we ensure that the system is well defined: 1 + P(s)C(s)F(s) is not null and all the transfer functions are proper.

Such a system is said to be internally stable if all the transfer functions in the relations above (thus from the system inputs to the block inputs) are stable.

We now want to find the generic family of controllers C(s) which ensure the internal stability of the system. For this purpose we make the assumption that the systems involved are S.I.S.O. and that F(s) = I = 1. We obtain then the following set of simplified relations:

We define R as the set of all the real, rational, proper and stable transfer functions. Such a set is closed by the operation of addition and multiplication: if P(s) belongs to R then P(s)X(s)+Y(s) still belongs to R.

We shall first examine the easy case of P(s) known being stable (thus belonging to R). The set of all the controllers C(s) for that the system is internally stable can be expressed by:

For any Q(s) in R!

To show that for any given Q(s) in R, the controller C(s)=Q(s)/(1 - P(s)Q(s)) internally stabilizes the system we simply need to substitute its expression in the system relations shown above. We obviously obtain:

Which yelds:

Since P(s) and Q(s) are in R and R is closed under the addition and multiplication then all the transfer functions in the relation above are in R (thus stable).

Conversely, assuming that a given C(s) internally stabilizes the system we shall prove that it can be expressed as Q(s)/(1 - P(s)Q(s)) for some Q(s) in R.

Such a Q(s) is found by taking the transfer function from the reference (r) to the input of the second block (x2).

This function is stable by hypotesis and obviously yelds:

To provide a solution for the general case of P(s) being unstable we need to define the coprime factorization of a transfer function.

Two functions N(s) and M(s) are said to be coprime if there exist X(s) and Y(s), elements of R that

The formula above is a Diophantine Equation and its solutions are being studied since centuries.

Such definition of coprimeness is a generalization of the integer coprimeness over the commutative ring R.

The coprime factorization of a (rational) transfer function G(s) is defined as

It can be proved that a sufficient condition for N(s) and M(s) to be coprime (as defined above) is that the functions have no common zeros in the right half closed complex plane. This makes finding M(s) and N(s) easy but doesn't help in finding X(s) and Y(s). To find these two we use the (extended) Euclidean Algorithm.

The set of all the controllers C(s) for that the system is internally stable can be expressed by:

where N(s),M(s),X(s) and Y(s) are the elements of a coprime factorization of the plant transfer function P(s)

Preamble.

Let

be a coprime factorization of the controller C(s).

By substituting the two factorizations in the system relations defined earlier we obtain

That can be transformed to

and then to

which clearly shows that the system is internally stable if and only if

is stable and thus lies in R

Assumed that we can now prove that given a coprime factorization of P(s) and a Q(s) e R the controller of the form

makes the system internally stable.

Let

We can then write

which means that Nc(s) and Mc(s) are elements of a coprime factorization of C(s). It's also clear that

and thus the system is internally stable because of the statement shown in the preamble.

Conversely we should show that if C(s) makes the system internally stable then there exists a Q(s) in R for that the controller can be written as

This part of the proof is still trivial but quite long thus we choose to omit it to avoid annoying the reader.

Please note that if P(s) is known stable then finding the coprime factorization is trivial: N(s) = P(s), M(s) = 1, X(s) = 0 and Y(s) = 1. The set of stabilizing controllers reduces then to the formula defined in the first paragraphs.

Note also that all the feedback path functions are affine expressions in Q(s). In particular the sensivity function S(s) and the complementar sensivity T(s) can be expressed as