∈ L(U, Y) the following difference equations determine a discrete-time linear time-invariant system: with
(the input or control) a sequence with values in U and
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,[1] but the inclusion of many other interesting physical examples forces unboundedness of B and C as well.
given by fits into the abstract evolution equation framework described above as follows.
The operator A is defined as It can be shown[2] that A generates a strongly continuous semigroup on X.
The bounded operators B, C and D are defined as The delay differential equation fits into the abstract evolution equation framework described above as follows.
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[3] that A generates a strongly continuous semigroup on X.
The bounded operators B, C and D are defined as As in the finite-dimensional case the transfer function is defined through the Laplace transform (continuous-time) or Z-transform (discrete-time).
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.
[4] 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
If A generates a strongly continuous semigroup and B, C and D are bounded operators, then[5] 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.
[6] 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.
equal to zero and denoting Laplace transforms with respect to t by capital letters we obtain from the partial differential equation given above This is an inhomogeneous linear differential equation with
Proceeding similarly as for the partial differential equation example, the transfer function for the delay equation example is[7]
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.
is the state that is reached by applying the input sequence u when the initial condition is zero.
However, the space of control functions on which this operator acts now influences the definition.
The usual choice is L2(0, ∞;U), the space of (equivalence classes of) U-valued square integrable functions on the interval (0, ∞), but other choices such as L1(0, ∞;U) are possible.
The different controllability notions can be defined once the domain of
The system is called[8] As in the finite-dimensional case, observability is the dual notion of controllability.
is the truncated output with initial condition x and control zero.
However, the space of functions to which this operator maps now influences the definition.
The usual choice is L2(0, ∞, Y), the space of (equivalence classes of) Y-valued square integrable functions on the interval (0,∞), but other choices such as L1(0, ∞, Y) are possible.
The different observability notions can be defined once the co-domain of
The system is called[9] As in the finite-dimensional case, controllability and observability are dual concepts (at least when for the domain of