Distributed parameter system

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 Banach spaces and $A\in L(X)$, $B\in L(U,X)$, $C\in L(X,Y)$ and $D\in L(U,Y)$ the following equations determine a discrete-time linear time-invariant system:

$x(k+1)=Ax(k)+Bu(k)$

$y(k)=Cx(n)+Du(n)$

with x (the state) a sequence with values in X, u (the input or control) a sequence with values in U and y (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:

${\dot {x}}(t)=Ax(t)+Bu(t)$

$y(t)=Cx(t)+Du(t)$.

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 $t>0$ and $\xi \in [0,1]$ given by

${\frac {d}{dt}}w(t,\xi )=-{\frac {d}{d\xi }}w(t,\xi )+u(t),$

$w(0,\xi )=w_{0}(\xi ),$

$w(t,0)=0,$

$y(t)=\int _{0}^{1}w(t,\xi )\,d\xi ,$

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

$Ax=-x',~~~D(A)=\left\{x\in X:w{\text{ absolutely continuous }},x'\in L^{2}([0,1]){\text{ and }}x(0)=0\right\}.$

It can be shown that A generates a strongly continuous semigroup on X. The bounded operators B, C and D are defined as

$Bu=u,~~~Cx=\int _{0}^{1}x(\xi )\,d\xi ,~~~D=0.$

Example: a delay differential equation

The delay differential equation

${\dot {w}}(t)=w(t)+w(t-\tau )+u(t),$

$y(t)=w(t),$

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 $L^{2}([-\tau ,0])$. The operator 'A' is defined as

$A{\begin{pmatrix}r\\f\end{pmatrix}}={\begin{pmatrix}r+f(-\tau )\\f'\end{pmatrix}},~~~D(A)=\left\{{\begin{pmatrix}r\\f\end{pmatrix}}\in X:f{\text{ absolutely continuous }},f'\in L^{2}([-\tau ,0]){\text{ and }}r=f(0)\right\}.$

It can be shown that A generates a strongly continuous semigroup on X. The bounded operators B, C and D are defined as

$Bu={\begin{pmatrix}u\\0\end{pmatrix}},~~~C{\begin{pmatrix}r\\f\end{pmatrix}}=r,~~~D=0.$

See also

• Control theory
• State space

References

• Curtain, Ruth; Zwart, Hans (1995), An Introduction to Infinite-Dimensional Linear Systems theory, Springer
• Tucsnak, Marius; Weiss, George (2009), Observation and Control for Operator Semigroups, Birkhauser
• Staffans, Olof (2005), Well-posed linear systems, Cambridge University Press
• Luo, Zheng-Hua; Guo, Bao-Zhu; 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