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Incontrol theory, adistributed-parameter system (as opposed to alumped-parameter system) is asystem whosestate space is infinite-dimensional. Such systems are therefore also known as infinite-dimensional systems. Typical examples are systems described bypartial differential equations or bydelay differential equations.

WithU,X andYHilbert spaces and${\displaystyle A}$ ∈ L(X),${\displaystyle B}$ ∈ L(U, X),${\displaystyle C}$ ∈ L(X, Y) and${\displaystyle D}$ ∈ L(U, Y) the followingdifference equations determine a discrete-timelinear time-invariant system:

${\displaystyle x(k+1)=Ax(k)+Bu(k)}$

${\displaystyle y(k)=Cx(k)+Du(k)}$

with${\displaystyle x}$ (the state) a sequence with values inX,${\displaystyle u}$ (the input or control) a sequence with values inU and${\displaystyle y}$ (the output) a sequence with values inY.

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

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

${\displaystyle y(t)=Cx(t)+Du(t)}$.

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 considerunbounded operators. UsuallyA is assumed to generate astrongly continuous semigroup on the state spaceX. AssumingB,C andD 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 ofB andC as well.

The partial differential equation with${\displaystyle t>0}$ and${\displaystyle \xi \in [0,1]}$ given by

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}w(t,\xi )=-\frac{\mathrm{\partial }}{\mathrm{\partial }\xi }w(t,\xi )+u(t),}$

${\displaystyle w(0,\xi )=w_0(\xi ),}$

${\displaystyle w(t,0)=0,}$

${\displaystyle y(t)={\int }_0^{1}w(t,\xi )d\xi ,}$

fits into the abstract evolution equation framework described above as follows. The input spaceU and the output spaceY are both chosen to be the set of complex numbers. The state spaceX is chosen to beL²(0, 1). The operatorA is defined as

${\displaystyle Ax=-x^{\prime }}$,${\displaystyle D(A)=\left\{x\in X:x\text{ absolutely continuous },x^{\prime }\in L^2(0,1),x(0)=0\right\}.}$

It can be shown^[2] thatA generates a strongly continuoussemigroup onX. The bounded operatorsB,C andD are defined as

${\displaystyle Bu=u,Cx={\int }_0^{1}x(\xi )d\xi ,D=0.}$

The delay differential equation

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

${\displaystyle y(t)=w(t),}$

fits into the abstract evolution equation framework described above as follows. The input spaceU and the output spaceY are both chosen to be the set of complex numbers. The state spaceX is chosen to be the product of the complex numbers withL²(−τ, 0). The operatorA is defined as

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

It can be shown^[3] thatA generates a strongly continuous semigroup on X. The bounded operatorsB,C andD are defined as

${\displaystyle Bu=\left(\begin{array}{c}u\\ 0\end{array}\right),C\left(\begin{array}{c}r\\ f\end{array}\right)=r,D=0.}$

As in the finite-dimensional case thetransfer function is defined through theLaplace transform (continuous-time) orZ-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 stillholomorphic).

In discrete-time the transfer function is given in terms of the state-space parameters by${\displaystyle D+\sum _{k=0}^{\mathrm{\infty }}C{A}^{k}B{z}^k}$ and it is holomorphic in a disc centered at the origin.^[4] In case 1/z belongs to theresolvent set ofA (which is the case on a possibly smaller disc centered at the origin) the transfer function equals${\displaystyle D+Cz(I-zA{)}^{-1}B}$. An interesting fact is that any function that is holomorphic in zero is the transfer function of some discrete-time system.

IfA generates a strongly continuous semigroup andB,C andD are bounded operators, then^[5] the transfer function is given in terms of the state space parameters by${\displaystyle D+C(sI-A{)}^{-1}B}$ fors with real part larger than the exponential growth bound of the semigroup generated byA. 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.

Setting the initial condition${\displaystyle w_0}$ equal to zero and denoting Laplace transforms with respect tot by capital letters we obtain from the partial differential equation given above

${\displaystyle sW(s,\xi )=-\frac{d}{d\xi }W(s,\xi )+U(s),}$

${\displaystyle W(s,0)=0,}$

${\displaystyle Y(s)={\int }_0^{1}W(s,\xi )d\xi .}$

This is an inhomogeneouslinear differential equation with${\displaystyle \xi }$ as the variable,s as a parameter andinitial condition zero. The solution is${\displaystyle W(s,\xi )=U(s)(1-e^{-s\xi })/s}$. Substituting this in the equation forY and integrating gives${\displaystyle Y(s)=U(s)(e^{-s}+s-1)/s^2}$ so that the transfer function is${\displaystyle (e^{-s}+s-1)/s^2}$.

Proceeding similarly as for the partial differential equation example, the transfer function for the delay equation example is^[7]${\displaystyle 1/(s-1-e^{-s})}$.

In the infinite-dimensional case there are several non-equivalent definitions ofcontrollability 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.

An important role is played by the maps${\displaystyle {\mathrm{\Phi }}_n}$ which map the set of allU valued sequences into X and are given by${\displaystyle {\mathrm{\Phi }}_{n}u=\sum _{k=0}^n{A}^{k}B{u}_k}$. The interpretation is that${\displaystyle {\mathrm{\Phi }}_{n}u}$ is the state that is reached by applying the input sequenceu when the initial condition is zero. The system is called

• exactly controllable in timen if the range of${\displaystyle {\mathrm{\Phi }}_n}$ equalsX,
• approximately controllable in timen if the range of${\displaystyle {\mathrm{\Phi }}_n}$ is dense inX,
• null controllable in timen if the range of${\displaystyle {\mathrm{\Phi }}_n}$ includes the range ofAⁿ.

In controllability of continuous-time systems the map${\displaystyle {\mathrm{\Phi }}_t}$ given by${\displaystyle {\int }_0^t{\mathrm{e}}^{As}Bu(s)ds}$ plays the role that${\displaystyle {\mathrm{\Phi }}_n}$ plays in discrete-time. However, the space of control functions on which this operator acts now influences the definition. The usual choice isL²(0, ∞;U), the space of (equivalence classes of)U-valued square integrable functions on the interval (0, ∞), but other choices such asL¹(0, ∞;U) are possible. The different controllability notions can be defined once the domain of${\displaystyle {\mathrm{\Phi }}_t}$ is chosen. The system is called^[8]

• exactly controllable in timet if the range of${\displaystyle {\mathrm{\Phi }}_t}$ equalsX,
• approximately controllable in timet if the range of${\displaystyle {\mathrm{\Phi }}_t}$ is dense inX,
• null controllable in timet if the range of${\displaystyle {\mathrm{\Phi }}_t}$ includes the range of${\displaystyle {\mathrm{e}}^{At}}$.

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.

An important role is played by the maps${\displaystyle {\mathrm{\Psi }}_n}$ which mapX into the space of allY valued sequences and are given by${\displaystyle ({\mathrm{\Psi }}_{n}x{)}_k=C{A}^{k}x}$ ifk ≤ n and zero ifk > n. The interpretation is that${\displaystyle {\mathrm{\Psi }}_{n}x}$ is the truncated output with initial conditionx and control zero. The system is called

• exactly observable in timen if there exists ak_n > 0 such that${\displaystyle \Vert {\mathrm{\Psi }}_{n}x\Vert \ge k_n\Vert x\Vert }$ for allx ∈ X,
• approximately observable in timen if${\displaystyle {\mathrm{\Psi }}_n}$ isinjective,
• final state observable in timen if there exists ak_n > 0 such that${\displaystyle \Vert {\mathrm{\Psi }}_{n}x\Vert \ge k_n\Vert A^{n}x\Vert }$ for allx ∈ X.

In observability of continuous-time systems the map${\displaystyle {\mathrm{\Psi }}_t}$ given by${\displaystyle ({\mathrm{\Psi }}_t)(s)=C{\mathrm{e}}^{As}x}$ fors∈[0,t] and zero fors>t plays the role that${\displaystyle {\mathrm{\Psi }}_n}$ plays in discrete-time. However, the space of functions to which this operator maps now influences the definition. The usual choice isL²(0, ∞, Y), the space of (equivalence classes of)Y-valued square integrable functions on the interval(0,∞), but other choices such asL¹(0, ∞, Y) are possible. The different observability notions can be defined once the co-domain of${\displaystyle {\mathrm{\Psi }}_t}$ is chosen. The system is called^[9]

• exactly observable in timet if there exists ak_t > 0 such that${\displaystyle \Vert {\mathrm{\Psi }}_{t}x\Vert \ge k_t\Vert x\Vert }$ for allx ∈ X,
• approximately observable in timet if${\displaystyle {\mathrm{\Psi }}_t}$ isinjective,
• final state observable in timet if there exists ak_t > 0 such that${\displaystyle \Vert {\mathrm{\Psi }}_{t}x\Vert \ge k_t\Vert {\mathrm{e}}^{At}x\Vert }$ for allx ∈ X.

As in the finite-dimensional case, controllability and observability are dual concepts (at least when for the domain of${\displaystyle \mathrm{\Phi }}$ and the co-domain of${\displaystyle \mathrm{\Psi }}$ the usualL² 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.

• Control theory
• State space (controls)

• 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

• Control theory

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