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System with an infinite dimensional state space

In control theory, a distributed parameter system (as opposed to a lumped parameter system) is a system whose state space is infinite dimensional. Such systems are therefore also known as infinite dimensional systems. Typical examples are systems described by partial differential equations or by delay differential equations.

As in the finite dimensional case the transfer function is defined through the Laplace transform (continuous time) or Z transform (discrete time). Whereas in the finite dimensional case the transfer function is a proper rational function, the infinite dimensionality of the state space leads to irrational functions (which are however still holomorphic).

In discrete time the transfer function is given in terms of the state space parameters by and it is holomorphic in a disc centered at the origin. In case 1/z belongs to the resolvent set of A (which is the case on a possibly smaller disc centered at the origin) the transfer function equals An interesting fact is that any function that is holomorphic in zero is the transfer function of some discrete time system.

If A generates a strongly continuous semigroup and B, C and D are bounded operators, then the transfer function is given in terms of the state space parameters by for s with real part larger than the exponential growth bound of the semigroup generated by A. In more general situations this formula as it stands may not even make sense, but an appropriate generalization of this formula still holds. To obtain an easy expression for the transfer function it is often better to take the Laplace transform in the given differential equation than to use the state space formulas as illustrated below on the examples given above.

Proceeding similarly as for the partial differential equation example, the transfer function for the delay equation example is .