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.
Contents
- Discrete time
- Continuous time
- Example a partial differential equation
- Example a delay differential equation
- Transfer functions
- Transfer function for the partial differential equation example
- Transfer function for the delay differential equation example
- Controllability
- Controllability in discrete time
- Controllability in continuous time
- Observability
- Observability in discrete time
- Observability in continuous time
- Duality between controllability and observability
- References
Discrete-time
With U, X and Y Hilbert spaces and
with
Continuous-time
The continuous-time case is similar to the discrete-time case but now one considers differential equations instead of difference equations:
An added complication now however is that to include interesting physical examples such as partial differential equations and delay differential equations into this abstract framework, one is forced to consider unbounded operators. Usually A is assumed to generate a strongly continuous semigroup on the state space X. Assuming B, C and D to be bounded operators then already allows for the inclusion of many interesting physical examples, but the inclusion of many other interesting physical examples forces unboundedness of B and C as well.
Example: a partial differential equation
The partial differential equation with
fits into the abstract evolution equation framework described above as follows. The input space U and the output space Y are both chosen to be the set of complex numbers. The state space X is chosen to be L2(0, 1). The operator A is defined as
It can be shown that A generates a strongly continuous semigroup on X. The bounded operators B, C and D are defined as
Example: a delay differential equation
The delay differential equation
fits into the abstract evolution equation framework described above as follows. The input space U and the output space Y are both chosen to be the set of complex numbers. The state space X is chosen to be the product of the complex numbers with L2(−τ, 0). The operator A is defined as
It can be shown that A generates a strongly continuous semigroup on X. The bounded operators B, C and D are defined as
Transfer functions
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).
Discrete-time
In discrete-time the transfer function is given in terms of the state space parameters by
Continuous-time
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
Transfer function for the partial differential equation example
Setting the initial condition
This is an inhomogeneous linear differential equation with
Transfer function for the delay differential equation example
Proceeding similarly as for the partial differential equation example, the transfer function for the delay equation example is
Controllability
In the infinite-dimensional case there are several non-equivalent definitions of controllability which for the finite-dimensional case collapse to the one usual notion of controllability. The three most important controllability concepts are:
Controllability in discrete-time
An important role is played by the maps
Controllability in continuous-time
In controllability of continuous-time systems the map
Observability
As in the finite-dimensional case, observability is the dual notion of controllability. In the infinite-dimensional case there are several different notions of observability which in the finite-dimensional case coincide. The three most important ones are:
Observability in discrete-time
An important role is played by the maps
Observability in continuous-time
In observability of continuous-time systems the map
Duality between controllability and observability
As in the finite-dimensional case, controllability and observability are dual concepts (at least when for the domain of