Chapter 17 Observer-based controllers

We consider the input-state-output system

\begin{equation} \label {eq:iso:obs} \dot {x}(t)=Ax(t)+Bu(t),\qquad y(t)=Cx(t)+Du(t), \end{equation}

where

\[ A\in \mR ^{n\times n},~ B\in \mR ^{n\times m},~ C\in \mR ^{p\times n},~ D\in \mR ^{p\times m}. \]

  • Definition 17.1. An observer for the input-state-output system (17.1) is an input-state system

    \[ \dot {x}_c(t)=A_cx_c(t)+B_c\bbm {u(t)\\y(t)}, \]

    where \(A_c\in \mR ^{n\times n}\) and \(B_c\in \mR ^{n\times (m+p)}\) such that for all \(x(0)\) and \(x_c(0)\) we have \(\lim _{t\to \infty }x(t)-x_c(t)=0\).

  • Proposition 17.2. Let \(L\in \mR ^{n\times p}\) be such that \(A-LC\) is asymptotically stable. Then

    \[ A_c=A-LC,\qquad B_c=\bbm {B-LD&L}, \]

    gives an observer.

The above proposition is easy to prove. We have

\begin{align*} \dot {x}_c&=(A-LC)x_c+\bbm {B-LD&L}\bbm {u\\Cx+Du}, \\&=(A-LC)x_c+LCx+Bu, \end{align*} so that

\[ \dot {x}-\dot {x}_c=Ax-(A-LC)x_c-LCx=(A-LC)(x-x_c), \]

so that \(z:=x-x_c\) satisfies \(\dot {z}=(A-LC)z\). Since \(A-LC\) is asymptotically stable, we have that \(\lim _{t\to \infty }z(t)=0\) for all \(z(0)\). It follows that \(\lim _{t\to \infty }x(t)-x_c(t)=0\) for all \(x(0)\) and \(x_c(0)\).

  • Remark 17.3. The idea is that the state \(x_c\) of the observer is an estimate of the state \(x\) based only on the measured output \(y\). Of course, by the above definition this is only necessarily a good estimate as \(t\to \infty \). However, by appropriately choosing the output injection \(L\), we can obtain a good estimate for all positive time.

  • Definition 17.4. An observer-based stabilizing controller for (17.1) is an input-state-output system

    \begin{equation} \label {eq:obscontr} \dot {x}_c(t)=A_cx_c(t)+B_c\bbm {u(t)\\y(t)},\qquad u(t)=C_cx_c(t), \end{equation}

    such that the combination of (17.1) and (17.2) is such that for all \(x(0)\) and all \(x_c(0)\) we have \(\lim _{t\to \infty }x(t)=0\) and \(\lim _{t\to \infty }x_c(t)=0\).

  • Proposition 17.5. Let \(L\) be such that \(A-LC\) is asymptotically stable and let \(F\) be such that \(A+BF\) is asymptotically stable, then

    \[ A_c=A-LC,\qquad B_c=\bbm {B-LD&L},\qquad C_c=F, \]

    given an observer-based stabilizing controller.

Also the above proposition is easy to prove. We have

\[ \dot {x}-\dot {x}_c=(A-LC)(x-x_c),\qquad \dot {x}=(A+BF)x-BF(x-x_c), \]

which we can write in matrix form as

\[ \bbm {x-x_c\\x}'=\bbm {A-LC&0\\-BF&A+BF}\bbm {x-x_c\\x}. \]

Because of the structure, the eigenvalues of this matrix are those of \(A-LC\) combined with those of \(A+BF\), so that by asymptotic stability of \(A-LC\) and \(A+BF\) we have \(x(t)-x_c(t)\to 0\) and \(x(t)\to 0\) as \(t\to \infty \) for all initial conditions. By algebra of limits this implies that also \(x_c(t)\to 0\).

  • Remark 17.6. The idea is that \(x_c\) is an approximation of \(x\) and instead of the state feedback \(u=Fx\) we use the feedback \(u=Fx_c\) based on this approximation of \(x\).

  • Remark 17.7. Consider the situation in the output regulation and disturbance rejection measurement feedback problem. Define (we use tildes for the matrices as in this chapter to distinguish them from those in Theorem 11.3)

    \[ \widetilde {A}=\bbm {A&B_1C_e\\0&A_e},\qquad \widetilde {B}=\bbm {B_2\\0},\qquad \widetilde {C}=\bbm {C_2&D_{21}C_e},\qquad \widetilde {D}=0. \]

    The controller in Theorem 11.3 then has the structure of an observer-based stabilizing controller as in Propostion 17.5 for the system

    \[ \bbm {x\\x_e}'=\bbm {A&B_1C_e\\0&A_e}\bbm {x\\x_e}+\bbm {B_2\\0}u,\qquad y=\bbm {C_2&D_{21}C_e}\bbm {x\\x_e}, \]

    where \(F\) has the special structure

    \[ F=\bbm {F_1&V-F_1\Pi }. \]