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.

Linear time-invariant distributed-parameter systems

Abstract evolution equations

Discrete-time

With U, X and Y Hilbert spaces and ∈L(X), ∈L(U,X), ∈L(X,Y) and ∈L(U,Y) the following difference equations determine a discrete-time linear time-invariant system:

with (the state) a sequence with values in X, (the input or control) a sequence with values in U and (the output) a sequence with values in Y.

Continuous-time

The continuous-time case is similar to the discrete-time case but now one considers differential equations instead of difference equations:

1. ,

2. .

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

given by

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

and it is holomorphic in a disc centered at the origin. In case 1/z belongs to the resolvent set of A. An interesting fact is that any function that is holomorphic in zero is the transfer function of some discrete-time system.

Continuous-time

If A generates a strongly continuous semigroup and B, C and D are bounded operators, then 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.

Transfer function for the partial differential equation example

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 as the variable, s as a parameter and initial condition. Substituting this in the equation for Y.

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:

• Exact controllability,

• Approximate controllability,

• Null controllability.

Controllability in discrete-time

which map the set of all U is the state that is reached by applying the input sequence u when the initial condition is zero. The system is called

• exactly controllable in time n equals X,

• approximately controllable in time n is dense in X,

• null controllable in time n includes the range of An.

Controllability in continuous-time

plays in discrete-time. 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 is chosen. The system is called

• exactly controllable in time t equals X,

• approximately controllable in time t is dense in X,

• null controllable in time t.

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:

• Exact observability (also known as continuous observability),

• Approximate observability,

• Final state observability.

Observability in discrete-time

which map X into the space of all Y if k≤n and zero if k>n is the truncated output with initial condition x and control zero. The system is called

• exactly observable in time n if there exists a kn> for all x∈X,

• approximately observable in time n is injective,

• final state observable in time n if there exists a kn> for all x∈X.

Observability in continuous-time

for s[0,t] and zero for s>t plays in discrete-time. 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 is chosen. The system is called

• exactly observable in time t if there exists a kt> for all x∈X,

• approximately observable in time t is injective,

• final state observable in time t if there exists a kt> for all x∈X.

Duality between controllability and observability

the usual L2 choice is made). The correspondence under duality of the different concepts is:

• Exact controllability ↔ Exact observability,

• Approximate controllability ↔ Approximate observability,

• Null controllability ↔ Final state observability.

See also

• Control theory

• State space (controls)

Notes

References