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D. Properties of Continuous-Time Riccati Equations


D.5 Duality between Linear Quadratic Regulator and Optimal Linear Filter

The close connections between the optimal filter and the LQR problem can be expressed directly as follows: We consider the problem of estimating a particular linear combination of the states, namely,


\begin{displaymath} z(t)=f^Tx(t) \end{displaymath} (D.5.1)

(The final solution will turn out to be independent of $f$, and thus will hold for the complete state vector.)

Now we will estimate $z(t)$ by using a linear filter of the following form:


\begin{displaymath} \hat{z}(t)=\int_0^t h(t-\tau)^Ty'(\tau)d\tau +g^T\hat{x}_o \end{displaymath} (D.5.2)

where $h(t)$ is the impulse response of the filter and where $\hat{x}_o$ is a given estimate of the initial state. Indeed, we will assume that (22.10.17) holds, that is, that the initial state $x(0)$ satisfies


\begin{displaymath} \ensuremath{{\cal{E}}} {(x(0)-\hat{x}_o)(x(0)-\hat{x}_o)^T}=\ensuremath{\mathbf{P_o}}\end{displaymath} (D.5.3)

We will be interested in designing the filter impulse response, $h(\tau)$, so that $\hat{z}(t)$ is close to $z(t)$ in some sense. (Indeed, the precise sense we will use is a quadratic form.) From (D.5.1) and (D.5.2), we see that


\begin{displaymath} \begin{split} \tilde{z}(t)&= z(t)- \hat{z}(t)\\ &=f^Tx(t)-... ...hbf{C}} x(\tau)+\dot{v}(t)\bigr)d\tau -g^T\hat{x}_o \end{split}\end{displaymath} (D.5.4)

Equation (D.5.4) is somewhat difficult to deal with, because of the cross-product between $h(t-\tau)$ and $x(t)$ in the integral. Hence, we introduce another variable, $\lambda $, by using the following equation

\begin{displaymath} \frac{d\lambda(\tau)}{d\tau}=-\ensuremath{\mathbf{A}} ^T\lambda(\tau)-\ensuremath{\mathbf{C}} ^T u(\tau) \end{displaymath} (D.5.5)


\begin{displaymath} \lambda(t)=-f \end{displaymath} (D.5.6)

where $u(\tau)$ is the reverse time form of $h$:


\begin{displaymath}u(\tau)=h(t-\tau) \end{displaymath} (D.5.7)

Substituting (D.5.5) into (D.5.4) gives

\begin{displaymath} \begin{split} \tilde{z}(t)=&f^Tx(t)+ \int_0^t \left[ \fra... ...&-\int_0^t u(\tau)\dot{v}(\tau)d\tau -g^T\hat{x}_o \end{split}\end{displaymath} (D.5.8)

Using integration by parts, we then obtain


\begin{displaymath} \begin{split} \tilde{z}(t)=&f^Tx(t)+\bigl[\lambda^Tx(\tau) \... ...\tau)-u(\tau)^T \frac{dv(\tau)}{d\tau} \right)d\tau \end{split}\end{displaymath} (D.5.9)

Finally, using (22.10.5) and (D.5.6), we obtain


\begin{displaymath} \begin{split} \tilde{z}(t)&=\lambda(0)^T(x(0)-\hat{x}_o) +... ...{d\tau} \right)d\tau\\ &-(\lambda(0)+g)^T\hat{x}_o \end{split}\end{displaymath} (D.5.10)

Hence, squaring and taking mathematical expectations, we obtain (upon using (D.5.3), (22.10.3) and (22.10.4) ) the following:

\begin{displaymath} \begin{split} \ensuremath{{\cal{E}}}\{\tilde{z}(t)^2\}&=\la... ...\\ &+\parallel(\lambda(0)+g)^T\hat{x}_o\parallel^2 \end{split}\end{displaymath} (D.5.11)

The last term in (D.5.11) is zero if $g=-\lambda(0)$. Thus, we see that the design of the optimal linear filter can be achieved by minimizing


\begin{displaymath} J=\lambda(0)^T\ensuremath{\mathbf{P_o}}\lambda(0)+ \int_0^t\... ...da(\tau)+ u(\tau)^T\ensuremath{\mathbf{R}} u(\tau) \bigr)d\tau \end{displaymath} (D.5.12)

where $\lambda(\tau)$ satisfies the reverse-time equations (D.5.5) and (D.5.6).

We recognize the set of equations formed by (D.5.5), (D.5.6), and (D.5.12) as a standard linear regulator problem, provided that the connections shown in Table D.1 are made.


Regulator Filter   Regulator Filter
$\tau$ $t-\tau$   $t_f$ 0
$\ensuremath{\mathbf{A}} $ $-\ensuremath{\mathbf{A}} ^T$   $\ensuremath{\boldsymbol{\Psi}} $ $\ensuremath{\mathbf{Q}} $
$\ensuremath{\mathbf{B}} $ $ -\ensuremath{\mathbf{C}} ^T $   $ \ensuremath{\boldsymbol{\Phi}} $ $\ensuremath{\mathbf{R}} $
$x$ $\lambda $   $ \ensuremath{\boldsymbol{\Psi}} _f$ $\ensuremath{\mathbf{P_o}} $

Table D.1: Duality in quadratic regulators and filters

Finally, by using the (dual) optimal control results presented earlier, we see that the optimal filter is given by


\begin{displaymath} \hat{z}^o(\tau)=\int_o^tu^o(\tau)^Ty'(\tau)d\tau+g^T\hat{x}_o \end{displaymath} (D.5.13)

where


\begin{displaymath} u^o(\tau)=-\ensuremath{\mathbf{K_f}} (\tau)\lambda(\tau) \end{displaymath} (D.5.14)


\begin{displaymath} \ensuremath{\mathbf{K_f}} (\tau)=\ensuremath{\mathbf{R}} ^{-1}\ensuremath{\mathbf{C}}\ensuremath{\boldsymbol{\Sigma}} (\tau) \end{displaymath} (D.5.15)

and $\ensuremath{\boldsymbol{\Sigma}} (\tau)$ satisfies the dual form of (D.0.1), (22.4.18):


\begin{displaymath} -\frac{d\ensuremath{\boldsymbol{\Sigma}} (t)}{dt}=\ensurema... ...T+ \ensuremath{\mathbf{A}}\ensuremath{\boldsymbol{\Sigma}} (t) \end{displaymath} (D.5.16)


\begin{displaymath} \ensuremath{\boldsymbol{\Sigma}} (0)=\ensuremath{\mathbf{P_o}}\end{displaymath} (D.5.17)

Substituting (D.5.14) into (D.5.5), (D.5.6) we see that


\begin{displaymath} \frac{d\lambda(\tau)}{d\tau}=-\ensuremath{\mathbf{A}} ^T\lam... ...th{\mathbf{C}} ^T\ensuremath{\mathbf{K_f}} (\tau)\lambda(\tau) \end{displaymath} (D.5.18)


\begin{displaymath} \lambda(t)=-f \end{displaymath} (D.5.19)


\begin{displaymath} u^o(\tau)=-\ensuremath{\mathbf{K_f}} (\tau)\lambda(\tau) \end{displaymath} (D.5.20)


\begin{displaymath}g=-\lambda(0) \end{displaymath} (D.5.21)

We see that $u^o(\tau)$ is the output of a linear homogeneous equation. Let $\nu=(t-\tau )$, and define $\ensuremath{\boldsymbol{\Phi}} (\nu)$ as the state transition matrix from $\tau=0$ for the time-varying system having $A-matrix$ equal to $\left[ \ensuremath{\mathbf{A-K}} _f(t-\nu)^T\ensuremath{\mathbf{C}}\right] $. Then


\begin{displaymath}\lambda (\tau ) =-\ensuremath{\boldsymbol{\Phi}} (t-\tau )^Tf \end{displaymath} (D.5.22)


\begin{displaymath}\lambda (0) =-\ensuremath{\boldsymbol{\Phi}} (t)^Tf \nonumber \end{displaymath}  


\begin{displaymath}u^0(\tau ) =\ensuremath{\mathbf{K_f}} (\tau )\ensuremath{\boldsymbol{\Phi}} (t-\tau )^Tf \nonumber \end{displaymath}  

Hence, the optimal filter satisfies


\begin{displaymath}\hat{z}(t) =g^T\hat{x}_o+\int\limits_0^tu^oy'(\tau )d\tau \end{displaymath} (D.5.23)


\begin{displaymath}=-\lambda (0)^T\hat{x}_o+\int\limits_0^tf^T\ensuremath{\bolds... ... )\ensuremath{\mathbf{K_f}} ^T(\tau )y'(\tau )d\tau \nonumber \end{displaymath}  


\begin{displaymath}=f^T\left(\ensuremath{\boldsymbol{\Phi}} (t)\hat{x}_o+\int\li... ...remath{\mathbf{K_f}} ^T(\tau )y'(\tau)d\tau \right) \nonumber \end{displaymath}  


\begin{displaymath}=f^T\hat{x}(t) \nonumber \end{displaymath}  

where


\begin{displaymath}\hat{x}(t)=\ensuremath{\boldsymbol{\Phi}} (t)\hat{x}_o+\int\l... ...i}} (t-\tau )\ensuremath{\mathbf{K_f}} ^T(\tau )y'(\tau )d\tau \end{displaymath} (D.5.24)

We then observe that (D.5.24) is actually the solution of the following state space (optimal filter).

\begin{displaymath} \frac{d\hat{x}(t)}{dt}=\left(\ensuremath{\mathbf{A}} -\ensur... ...hbf{C}}\right)\hat{x}(t)+ \ensuremath{\mathbf{K_f}} ^T(t)y'(t) \end{displaymath} (D.5.25)


\begin{displaymath} \hat{x}(0)=\hat{x}_o \end{displaymath} (D.5.26)


\begin{displaymath} \hat{z}(t)=f^Tx(t) \end{displaymath} (D.5.27)

We see that the final solution depends on $f$ only through (D.5.27). Thus, as predicted, (D.5.25), (D.5.26) can be used to generate an optimal estimate of any linear combination of states.

Of course, the optimal filter (D.5.25) is identical to that given in (22.10.23)

All of the properties of the optimal filter follow by analogy from the (dual) optimal linear regulator. In particular, we observe that (D.5.16) and (D.5.17) are a CTDRE and its boundary condition, respectively. The only difference is that, in the optimal-filter case, this equation has to be solved forward in time. Also, (D.5.16) has an associated CTARE, given by


\begin{displaymath} \ensuremath{\mathbf{Q}} - \ensuremath{\boldsymbol{\Sigma}} ... ...} _{\ensuremath{\boldsymbol{\infty}} } =\ensuremath{\mathbf{0}}\end{displaymath} (D.5.28)

Thus, the existence, uniqueness, and properties of stabilizing solutions for (D.5.16) and (D.5.28) satisfy the same conditions as the corresponding Riccati equations for the optimal regulator.