Hankel Operator in Control Theory

In control theory, there are two different, but related, definitions about Hankel Operator, depends on the system the definition for.

For stable and minimum realization system \(G = (A,B,C)\), the Hankel operator is defined by

\[\Gamma_G = P_+ G|_{L_2(-\infty,0]} \]

where \(G|_{L_2(-\infty,0]}\) denotes the restriction of \(G\) to the subspace \(L_2(-\infty,0]\), and \(P_+\) is the operator that projects a signal in \(L_2(-\infty,\infty)\) to \(L_2[0,\infty)\) by truncation. Correspondingly, define the controllability operator \(\Psi_c:L_2(-\infty,0]\to \mathbb{C}^n\) by

\[\Psi_c u= \int_{-\infty}^0 e^{-A\tau}Bu(\tau)d\tau \]

and define the observability operator \(\Psi_o:\mathbb{C}^n \to L_2[0,\infty)\) by

\[\Psi_o x_0 = Ce^{At}x_0,~~t\ge 0. \]

Then it holds that

\[\Gamma_G = \Psi_o \Psi_c \]

An alternative definition of Hankel operator is for unstable system. That is,

\[\bar{\Gamma}_G = P_- G|_{L_2[0,\infty)} \]

where \(P_-\) is the operator that projects a signal in \(L_2(-\infty,\infty)\) to \(L_2(-\infty,0]\) by truncation. The corresponding controllability operator \(\bar{\Psi}_c:L_2[0,\infty)\to \mathbb{C}^n\) is defined by

\[\bar{\Psi}_c u= -\int_{0}^{\infty} e^{-A\tau}Bu(\tau)d\tau \]

and observability operator \(\bar{\Psi}_o:\mathbb{C}^n \to L_2(-\infty,0]\) is defined by

\[\bar{\Psi}_o x_0 = Ce^{At}x_0,~~t\le 0 \]

Then it also holds that

\[\bar{\Gamma}_G = \bar{\Psi}_o \bar{\Psi}_c \]

There is a systemic interpretation for controllability and observability operators. For "stable" definition, \(\Psi_c\) just maps the input \(u \in L_2(-\infty,0]\) supported in the past to \(x(0)\), and \(\Psi_o\) maps \(x(0)\) to the system output \(y(t),t\ge 0\), which no input applied for \(t\ge 0\).

While for "unstable" definition, it is not so intuitive. Note that \(A\) is anti-stable and

\[x(t) = e^{At}x_0 + \int_{0}^{\infty} e^{A(t-\tau)}Bu(\tau)d\tau \]

Then \(e^{-At}x(t) = x_0 + \int_{0}^{\infty} e^{-A\tau}Bu(\tau)d\tau\) and letting \(t \to \infty\) obtains

\[x_0 =- \int_{0}^{\infty} e^{-A\tau}Bu(\tau)d\tau = \bar{\Psi}_c u \]

This can be obtained from another point of view that consider \(x(t)\) as the "initial state" of the system and \(x(0)=x_0\) as the "final state" such that

\[x_0 = x(0) = e^{A(0-t)}x(t) + \int_{t}^{0} e^{A(0-\tau)}Bu(\tau)d\tau \]

For sufficiently large \(t\), \(e^{-At}x(t)\) is small enough such that \(x_0 \approx \int_{t}^{0} e^{-A\tau}Bu(\tau)d\tau\).