Optimal projection equations

The reduced-order LQG control problem is almost identical to the conventional full-order LQG control problem. Let $\backslash hat\{\backslash mathbf\{x\}\}\_r(t)$ represent the state of the reduced-order LQG controller. Then the only difference is that the state dimension $n\_r=dim(\backslash hat\{\backslash mathbf\{x\}\}\_r(t))$ of the LQG controller is a-priori fixed to be smaller than $n=dim(\{\backslash mathbf\{x\}\}(t))$, the state dimension of the controlled system.

The reduced-order LQG controller is represented by the following equations:

: $\backslash dot\{\backslash hat\{\backslash mathbf\{x\}\}\}\_r(t)\; =\; A\_r(t)\backslash hat\{\backslash mathbf\{x\}\}\_r(t)\; +\; B\_r(t)\{\backslash mathbf\{u\}\}(t)+K\_r(t)\; \backslash left(\; \{\backslash mathbf\{y\}\}(t)-C\_r(t)\backslash hat\{\backslash mathbf\{x\}\}\_r(t)\; \backslash right),\backslash hat\{\backslash mathbf\{x\}\}\_r(0)=\{\backslash mathbf\{x\}\}\_r(0),$

: $\{\backslash mathbf\{u\}\}(t)=\; -L\_r(t)\; \backslash hat\{\backslash mathbf\{x\}\}\_r(t).$

These equations are deliberately stated in a format that equals that of the conventional full-order LQG controller. For the reduced-order LQG control problem it is convenient to rewrite them as

: $\backslash dot\{\backslash hat\{\backslash mathbf\{x\}\}\}\_r(t)\; =\; F\_r(t)\backslash hat\{\backslash mathbf\{x\}\}\_r(t)\; +\; K\_r(t)\; \{\backslash mathbf\{y\}\}(t),\backslash hat\{\backslash mathbf\{x\}\}\_r(0)=\{\backslash mathbf\{x\}\}\_r(0),$

: $\{\backslash mathbf\{u\}\}(t)=\; -L\_r(t)\; \backslash hat\{\backslash mathbf\{x\}\}\_r(t),$

where

: $F\_r(t)=A\_r(t)-B\_r(t)L\_r(t)-K\_r(t)C\_r(t).$

The matrices $F\_r(t),\; K\_r(t),\; L\_r(t)$ and $\{\backslash mathbf\{x\}\}\_r(0)$ of the reduced-order LQG controller are determined by the so-called optimal projection equations (OPE).

The square optimal projection matrix $\backslash tau(t)$ with dimension $n$ is central to the OPE. The rank of this matrix is almost everywhere equal to $n\_r.$ The associated projection is an oblique projection: $\backslash tau^2(t)=\backslash tau(t).$ The OPE constitute four matrix differential equations. The first two equations listed below are generalizations of the matrix Riccati differential equations associated to the conventional full-order LQG controller. In these equations $\backslash tau\_\backslash perp(t)$ denotes $I\_n-\backslash tau(t)$ where $I\_n$ is the identity matrix of dimension $n$.

: $$

\begin{align}

\dot{P}(t) = {} & A(t)P(t)+P(t)A'(t)-P(t)C'(t)W^{-1}(t) C(t)P(t)+V(t) \\[6pt]

& {} +\tau_\perp (t)P(t)C'(t)W^{-1}(t) C(t)P(t)\tau'_\perp (t), \\[6pt]

P(0)= {} & E \left({\mathbf{x}}(0){\mathbf{x}}'(0) \right), \\[6pt]

& {} -\dot{S}(t) = A'(t)S(t)+S(t)A(t)-S(t)B(t)R^{-1}(t)B'(t)S(t)+Q(t) \\[6pt]

& {} + \tau'_\perp (t)S(t)B(t)R^{-1}(t)B'(t)S(t) \tau_\perp (t),

\end{align}

: $S(T)\; =\; F.$

If the dimension of the LQG controller is not reduced, that is if $n=n\_r$, then $\backslash tau(t)=I\_n,\; \backslash tau\_\backslash perp(t)=0$ and the two equations above become the uncoupled matrix Riccati differential equations associated to the conventional full-order LQG controller. If separable. The oblique projection $\backslash tau(t)$ is determined from two additional matrix differential equations which involve rank conditions. Together with the previous two matrix differential equations these are the OPE. To state the additional two matrix differential equations it is convenient to introduce the following two matrices:

: $\backslash Psi\_1(t)=(A(t)-B(t)R^\{-1\}(t)B\text{'}(t)S(t))\backslash hat\{P\}(t)+\backslash hat\{P\}(t)$

(A(t)-B(t)R^{-1}(t)B'(t)S(t))'

:::: $\{\}+P(t)C\text{'}(t)W^\{-1\}(t)C(t)P(t),$

: $\backslash Psi\_2(t)=(A(t)-P(t)C\text{'}(t)W^\{-1\}(t)$

C(t))'\hat{S}(t)+\hat{S}(t)(A(t)-P(t)C'(t)W^{-1}(t)C(t))

:::: $\{\}+S(t)B(t)R^\{-1\}(t)B\text{'}(t)S(t).$

Then the two additional matrix differential equations that complete the OPE are as follows:

: $\backslash dot\{\backslash hat\{P\}\}(t)=1/2\; \backslash left(\backslash tau(t)\backslash Psi\_1(t)+\backslash Psi\_1(t)\backslash tau\text{'}(t)\; \backslash right),\backslash hat\{P\}(0)=E(\{\backslash mathbf\{x\}\}(0))E(\{\backslash mathbf\{x\}\}(0))\text{'},\; \backslash operatorname\{rank\}(\backslash hat\{P\}(t))=n\_r$ almost everywhere,

: $-\backslash dot\{\backslash hat\{S\}\}(t)=1/2\; \backslash left(\backslash tau\text{'}(t)\backslash Psi\_2(t)+\backslash Psi\_2(t)\backslash tau(t)\; \backslash right),\backslash hat\{S\}(T)=0,\; \backslash operatorname\{rank\}(\backslash hat\{S\}(t))=n\_r$ almost everywhere,

with

: $\backslash tau(t)=\; \backslash hat\{P\}(t)\; \backslash hat\{S\}(t)\; \backslash left(\; \backslash hat\{P\}(t)\; \backslash hat\{S\}(t)\; \backslash right)^*.$

Here * denotes the group generalized inverse or Drazin inverse that is unique and given by

: $A^*=A(A^3)^+A.$

where + denotes the Moore–Penrose pseudoinverse.

The matrices $P(t),S(t),\backslash hat\{P\}(t),\backslash hat\{S\}(t)$ must all be nonnegative symmetric. Then they constitute a solution of the OPE that determines the reduced-order LQG controller matrices $F\_r(t),\; K\_r(t),\; L\_r(t)$ and $\{\backslash mathbf\{x\}\}\_r(0)$:

: $F\_r(t)=H(t)\backslash left(\; A(t)-P(t)C\text{'}(t)W^\{-1\}(t)$

C(t)-B(t)R^{-1}(t)B'(t)S(t) \right)G(t)+\dot{H}(t)G'(t),

: $K\_r(t)=H(t)P(t)C\text{'}(t)W^\{-1\}(t),$

: $L\_r(t)=R^\{-1\}(t)B\text{'}(t)S(t)G\text{'}(t),$

: $\{\backslash mathbf\{x\}\}\_r(0)=H(0)E(\{\backslash mathbf\{x\}\}(0)).$

In the equations above the matrices $G(t),H(t)$ are two matrices with the following properties:

: $G\text{'}(t)H(t)=\backslash tau(t),G(t)H\text{'}(t)=I\_\{n\_r\}$ almost everywhere.

They can be obtained from a projective factorization of $\backslash hat\{P\}(t)\backslash hat\{S\}(t)$.

The OPE can be stated in many different ways that are all equivalent. To identify the equivalent representations the following identities are especially useful:

: $\backslash tau(t)\backslash hat\{P\}(t)=\backslash hat\{P\}(t)\backslash tau\text{'}(t)=\backslash hat\{P\}(t),\; \backslash tau\text{'}(t)\backslash hat\{S\}(t)=\backslash hat\{S\}(t)\backslash tau(t)=\backslash hat\{S\}(t)$

Using these identities one may for instance rewrite the first two of the optimal projection equations as follows:

: $\backslash dot\{P\}(t)\; =\; A(t)P(t)+P(t)A\text{'}(t)-P(t)C\text{'}(t)W^\{-1\}(t)C(t)P(t)+V(t)+\backslash tau\_\backslash perp(t)\backslash Psi\_1(t)\backslash tau\text{'}\_\backslash perp\; (t),$

: $P(0)=\; E\; \backslash left(\{\backslash mathbf\{x\}\}(0)\{\backslash mathbf\{x\}\}\text{'}(0)\; \backslash right),$

: $-\backslash dot\{S\}(t)\; =\; A\text{'}(t)S(t)+S(t)A(t)-S(t)B(t)R^\{-1\}(t)B\text{'}(t)S(t)+Q(t)+\backslash tau\text{'}\_\backslash perp\backslash Psi\_2(t)\backslash tau\_\backslash perp(t),$

: $S(T)\; =\; F.$

This representation is both relatively simple and suitable for numerical computations.

If all the matrices in the reduced-order LQG problem formulation are time-invariant and if the horizon $T$ tends to infinity, the optimal reduced-order LQG controller becomes time-invariant and so do the OPE. In that case the derivatives on the left hand side of the OPE are zero.

Similar to the continuous-time case, in the discrete-time case the difference with the conventional discrete-time full-order LQG problem is the a-priori fixed reduced-order discrete-time OPE it is convenient to introduce the following two matrices:

: $\backslash Psi^1\_i=\backslash left(A\_i-B\_i(B\text{'}\_iS\_\{i+1\}B\_i+R\_i)^\{-1\}B\text{'}\_iS\_\{i+1\}A\_i)\backslash right)\backslash hat\{P\}\_i$

\left(A_i-B_i(B'_iS_{i+1}B_i+R_i)^{-1}B'_iS_{i+1}A_i)\right)'

:::: $\{\}+A\_iP\_iC\text{'}\_i(C\_iP\_iC\text{'}\_i+W\_i)^\{-1\}C\_iP\_iA\text{'}\_i$

: $\backslash Psi^2\_\{i+1\}=\backslash left(A\_i-A\_iP\_iC\text{'}\_i(C\_iP\_\{i\}C\text{'}\_i+W\_i)^\{-1\}C\_i\backslash right)\text{'}\backslash hat\{S\}\_\{i+1\}$

\left(A_i-A_iP_iC'_i(C_iP_{i}C'_i+W_i)^{-1}C_i\right)

:::: $\{\}+A\text{'}\_iS\_\{i+1\}B\_i(B\text{'}\_iS\_\{i+1\}B\_i+R\_i)^\{-1\}B\text{'}\_iS\_\{i+1\}A\_i$

Then the discrete-time OPE is

: $P\_\{i+1\}\; =\; A\_i\; \backslash left(\; P\_i\; -\; P\_i\; C\text{'}\_i\; \backslash left(\; C\_i\; P\_i\; C\text{'}\_i+W\_i\; \backslash right)^\{-1\}\; C\_i\; P\_i\; \backslash right)\; A\text{'}\_i+V\_i+\backslash tau\_\{\backslash perp\; i+1\}\backslash Psi^1\_i\; \backslash tau\text{'}\_\{\backslash perp\; i+1\},\; P\_0=E\; \backslash left(\; \{\backslash mathbf\{x\}\}\_0\{\backslash mathbf\{x\text{'}\}\}\_0\; \backslash right)$.

: $S\_i\; =\; A\text{'}\_i\; \backslash left(\; S\_\{i+1\}\; -\; S\_\{i+1\}B\_i\; \backslash left(\; B\text{'}\_iS\_\{i+1\}B\_i+R\_i\; \backslash right)^\{-1\}\; B\text{'}\_i\; S\_\{i+1\}\; \backslash right)\; A\_i+Q\_i+\backslash tau\text{'}\_\{\backslash perp\; i\}\backslash Psi^2\_\{i+1\}\; \backslash tau\_\{\backslash perp\; i\},\; S\_N=F$.

:$\backslash hat\{P\}\_\{i+1\}=1/2(\backslash tau\_\{i+1\}\backslash Psi\_i^1+\backslash Psi\_i^1\backslash tau\text{'}\_\{i+1\}),\backslash hat\{P\}\_0=E(\{\backslash mathbf\{x\}\}(0))\; E(\{\backslash mathbf\{x\}\}(0))\text{'},\; \backslash operatorname\{rank\}(\backslash hat\{P\}\_i)=n\_r$ almost everywhere,

:$\backslash hat\{S\}\_\{i\}=1/2(\backslash tau\text{'}\_i\; \backslash Psi\_\{i+1\}^2+\backslash Psi\_\{i+1\}^2\backslash tau\_i),\backslash hat\{S\}\_N=0,\; \backslash operatorname\{rank\}(\backslash hat\{S\}\_i)=n\_r$ almost everywhere.

The oblique projection matrix is given by

: $\backslash tau\_i=\backslash hat\{P\}\_i\backslash hat\{S\}\_i\; \backslash left(\backslash hat\{P\}\_i\backslash hat\{S\}\_i\; \backslash right)^*.$

The nonnegative symmetric matrices $P\_i,S\_i,\backslash hat\{P\}\_i,\backslash hat\{S\}\_i$ that solve the discrete-time OPE determine the reduced-order LQG controller matrices $F\_i^r,\; K\_i^r,\; L\_i^r$ and $\{\backslash mathbf\{x\}\}\_0^r$:

: $F\_i^r=H\_\{i+1\}\backslash left(\; A\_i-P\_i\; C\text{'}\_i\; \backslash left(\; C\_i\; P\_i\; C\text{'}\_i+W\_i\; \backslash right)^\{-1\}C\_i-B\_i\backslash left(\; B\text{'}\_iS\_\{i+1\}B\_i+R\_i\; \backslash right)^\{-1\}\; B\text{'}\_i\; S\_\{i+1\}\backslash right)G\text{'}\_i,$

: $K\_i^r=H\_\{i+1\}P\_i\; C\text{'}\_i\; \backslash left(\; C\_i\; P\_i\; C\text{'}\_i+W\_i\; \backslash right)^\{-1\},$

: $L\_i^r=\backslash left(\; B\text{'}\_iS\_\{i+1\}B\_i+R\_i\; \backslash right)^\{-1\}\; B\text{'}\_i\; S\_\{i+1\}G\text{'}\_i,$

: $\{\backslash mathbf\{x\}\}\_0^r=H\_0E(\{\backslash mathbf\{x\}\}\_0).$

In the equations above the matrices $G\_i,H\_i$ are two matrices with the following properties:

: $G\text{'}\_iH\_i=\backslash tau\_i,\; G\_iH\text{'}\_i=I\_\{n\_r\}$ almost everywhere.

They can be obtained from a projective factorization of $\backslash hat\{P\}\_i\backslash hat\{S\}\_i$. To identify equivalent representations of the discrete-time OPE the following identities are especially useful:

: $\backslash tau\_i\backslash hat\{P\}\_i=\backslash hat\{P\}\_i\backslash tau\text{'}\_i=\backslash hat\{P\}\_i,\; \backslash tau\text{'}\_i\backslash hat\{S\}\_i=\backslash hat\{S\}\_i\backslash tau\_i=\backslash hat\{S\}\_i$

As in the continuous-time case if all the matrices in the problem formulation are time-invariant and if the horizon $N$ tends to infinity the reduced-order LQG controller becomes time-invariant. Then the discrete-time OPE converge to a steady state solution that determines the time-invariant reduced-order LQG controller.

The discrete-time OPE apply also to discrete-time systems with variable state, input and output dimensions (discrete-time systems with time-varying dimensions). Such systems arise in the case of digital controller design if the sampling occurs asynchronously.

Category:Optimal control

Category:Control theory

Category:Stochastic control