 Distributed parameter system

A distributed parameter system (as opposed to a lumped parameter system) is a system whose state space is infinitedimensional. Such systems are therefore also known as infinitedimensional systems. Typical examples are systems described by partial differential equations or by delay differential equations.
Contents
 1 Linear timeinvariant distributed parameter systems
 2 See also
 3 Notes
 4 References
Linear timeinvariant distributed parameter systems
Abstract evolution equations
Discretetime
With U, X and Y Hilbert spaces and ∈ L(X), ∈ L(U, X), ∈ L(X, Y) and ∈ L(U, Y) the following equations determine a discretetime linear timeinvariant 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.
Continuoustime
The continuoustime case is similar to the discretetime 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.
Example: a partial differential equation
The partial differential equation with t > 0 and given by
 w(0,ξ) = w_{0}(ξ),
 w(t,0) = 0,
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 L^{2}(0, 1). 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
Example: a delay differential equation
The delay differential equation
 y(t) = w(t),
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 L^{2}(−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
Transfer functions
As in the finitedimensional case the transfer function is defined through the Laplace transform (continuoustime) or Ztransform (discretetime). Whereas in the finitedimensional case the transfer function is a proper rational function, the infinitedimensionality of the state space leads to irrational functions (which are however still holomorphic).
Discretetime
In discretetime 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 D + Cz(I − zA) ^{− 1}B. An interesting fact is that any function that is holomorphic in zero is the transfer function of some discretetime system.
Continuoustime
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 D + C(sI − A) ^{− 1}B 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^{[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.
Transfer function for the partial differential equation example
Setting the initial condition w_{0} equal to zero and denoting Laplace transforms with respect to t by capital letters we obtain from the partial differential equation given above
 W(s,0) = 0,
This is an inhomogeneous linear differential equation with ξ as the variable, s as a parameter and initial condition zero. The solution is W(s,ξ) = U(s)(1 − e ^{− sξ}) / s. Substituting this in the equation for Y and integrating gives Y(s) = U(s)(e ^{− s} + s − 1) / s^{2} so that the transfer function is (e ^{− s} + s − 1) / s^{2}.
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^{[7]} 1 / (s − 1 − e ^{− s}).
Controllability
In the infinitedimensional case there are several nonequivalent definitions of controllability which for the finitedimensional case collapse to the one usual notion of controllability. The three most important controllability concepts are:
 Exact controllability,
 Approximate controllability,
 Null controllability.
Controllability in discretetime
An important role is played by the maps Φ_{n} which map the set of all U valued sequences into X and are given by . The interpretation is that Φ_{n}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 if the range of Φ_{n} equals X,
 approximately controllable in time n if the range of Φ_{n} is dense in X,
 null controllable in time n if the range of Φ_{n} includes the range of A^{n}.
Controllability in continuoustime
In controllability of continuoustime systems the map Φ_{t} given by plays the role that Φ_{n} plays in discretetime. However, the space of control functions on which this operator acts now influences the definition. The usual choice is L^{2}(0, ∞;U), the space of (equivalence classes of) Uvalued square integrable functions on the interval (0, ∞), but other choices such as L^{1}(0, ∞;U) are possible. The different controllability notions can be defined once the domain of Φ_{t} is chosen. The system is called^{[8]}
 exactly controllable in time t if the range of Φ_{t} equals X,
 approximately controllable in time t if the range of Φ_{t} is dense in X,
 null controllable in time t if the range of Φ_{t} includes the range of e^{At}.
Observability
As in the finitedimensional case, observability is the dual notion of controllability. In the infinitedimensional case there are several different notions of observability which in the finitedimensional case coincide. The three most important ones are:
 Exact observability (also known as continuous observability),
 Approximate observability,
 Final state observability.
Observability in discretetime
An important role is played by the maps Ψ_{n} which map X into the space of all Y valued sequences and are given by (Ψ_{n}x)_{k} = CA^{k}x if k ≤ n and zero if k > n. The interpretation is that Ψ_{n}x is the truncated output with initial condition x and control zero. The system is called
 exactly observable in time n if there exists a k_{n} > 0 such that for all x ∈ X,
 approximately observable in time n if Ψ_{n} is injective,
 final state observable in time n if there exists a k_{n} > 0 such that for all x ∈ X.
Observability in continuoustime
In observability of continuoustime systems the map Ψ_{t} given by (Ψ_{t})(s) = Ce^{As}x for s∈[0,t] and zero for s>t plays the role that Ψ_{n} plays in discretetime. However, the space of functions to which this operator maps now influences the definition. The usual choice is L^{2}(0, ∞, Y), the space of (equivalence classes of) Yvalued square integrable functions on the interval (0,∞), but other choices such as L^{1}(0, ∞, Y) are possible. The different observability notions can be defined once the codomain of Ψ_{t} is chosen. The system is called^{[9]}
 exactly observable in time t if there exists a k_{t} > 0 such that for all x ∈ X,
 approximately observable in time t if Ψ_{t} is injective,
 final state observable in time t if there exists a k_{t} > 0 such that for all x ∈ X.
Duality between controllability and observability
As in the finitedimensional case, controllability and observability are dual concepts (at least when for the domain of Φ and the codomain of Ψ the usual L^{2} choice is made). The correspondence under duality of the different concepts is^{[10]}:
 Exact controllability ↔ Exact observability,
 Approximate controllability ↔ Approximate observability,
 Null controllability ↔ Final state observability.
See also
Notes
 ^ Curtain and Zwart
 ^ Curtain and Zwart Example 2.2.4
 ^ Curtain and Zwart Theorem 2.4.6
 ^ This is the mathematical convention, engineers seem to prefer transfer functions to be holomorphic at infinity; this is achieved by replacing z by 1/z
 ^ Curtain and Zwart Lemma 4.3.6
 ^ Staffans Theorem 4.6.7
 ^ Curtain and Zwart Example 4.3.13
 ^ Tucsnak Definition 11.1.1
 ^ Tucsnak Definition 6.1.1
 ^ Tucsnak Theorem 11.2.1
References
 Curtain, Ruth; Zwart, Hans (1995), An Introduction to InfiniteDimensional Linear Systems theory, Springer
 Tucsnak, Marius; Weiss, George (2009), Observation and Control for Operator Semigroups, Birkhauser
 Staffans, Olof (2005), Wellposed linear systems, Cambridge University Press
 Luo, ZhengHua; Guo, BaoZhu; Morgul, Omer (1999), Stability and Stabilization of Infinite Dimensional Systems with Applications, Springer
 Lasiecka, Irena; Triggiani, Roberto (2000), Control Theory for Partial Differential Equations, Cambridge University Press
 Bensoussan, Alain; Da Prato, Giuseppe; Delfour, Michel; Mitter, Sanjoy (2007), Representation and Control of Infinite Dimensional Systems (second ed.), Birkhauser
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