# Robust Quadratic Optimal Control for Discrete-Time Linear Systems with Non-Stochastic Noises
## 3 Robust Optimal Control Strategy
### Robust Control Strategy  
According to the definition \(\eta(k)\) and the dynamics system state Equation (1), we have  
\[x(k+1) = A(k)x(k)+B(k)u(k)+w(k) \tag{14}\] \[= A(k)\hat{x}(k)+A(k)\eta(k)+B(k)u(k)+w(k).\]  
Here, let  
\[G(k)=\left[ \begin{array}{cc}I&A(k)\end{array} \right],\]  
\[\rho(k)=\left[ \begin{array}{cc}w(k)\\ \eta(k)\end{array} \right],\]  
then Equation (14) can be rewritten as follows,  
\[x(k+1)=A(k)\hat{x}(k)+B(k)u(k)+G(k)\rho(k). \tag{15}\]  
Substituting Equation (15) into Equation (5), it has  
\[J(k) = \hat{x}^{T}(k)\widetilde{A}\hat{x}(k)+2b^{T}u(k)+u^{T}(k) \widetilde{B}u(k) \tag{16}\] \[+2c^{T}\rho(k)+2u^{T}(k)D\rho(k)+\rho^{T}(k)\widetilde{G}\rho(k),\]  
where  
\[\widetilde{A}=A^{T}(k)\mathcal{Q}(k+1)A(k),\]  
\[b=B^{T}(k)Q^{T}(k+1)A(k)\hat{x}(k),\]  
\[\widetilde{B}=B^{T}(k)Q(k+1)B(k)+R(k),\]  
\[c=G^{T}(k)Q^{T}(k+1)A(k)\hat{x}(k),\]  
\[D=B^{T}(k)Q(k+1)G(k),\]\[\widetilde{G}=G^{T}(k)Q(k+1)G(k).\]  
It can be seen from (16) that \(J(k)\) contains two uncertainties, one is the process noise and the other is the estimation state error. To realize the control optimality, how to deal with the uncertainties effectively is important. In this subsection, we introduce two uncertainties into the constrains of optimization problem, based on the lemmas outlined in Section 2, the uncertain problem (6) can be converted into one certain semidefinite programming (SDP) problem by using the robust optimization technique. The main result is given by the following theorem. Under the bounded ellipsoid uncertain noises, the problem (6) can be solved by the following certain semi-definite programming (SDP) problem,  
\[\min\lambda \tag{17}\] \[J(k) = \hat{x}^{T}(k)\widetilde{A}\hat{x}(k)+2b^{T}u(k)+u^{T}(k) \widetilde{B}u(k) \tag{16}\]0 \[\left[ \begin{array}{ccc}I&z&F\\ z^{T}&\lambda-\sum_{i=1}^{z}\tau_{i}&-h^{T}\\ F^{T}&-h&-\widetilde{G}+\sum_{i=1}^{2}\tau_{i}M_{i}(k)+F^{T}F\\ \tau_{i}\geq 0,\end{array} \right]\geq 0\]  
where \(\lambda\), \(\tau_{1}\), \(\tau_{2}\) and \(z\) are the decision variables, and  
\[F=\widetilde{B}^{-1/2}D,h=c-D^{T}\widetilde{B}^{-1}b,\]  
\[M_{1}(k)=\left[ \begin{array}{cc}W^{-1}(k)&0\\ 0&0\end{array} \right],\]  
\[M_{2}(k)=\left[ \begin{array}{cc}0&0\\ 0&P^{-1}(k)\end{array} \right],\]  
the control law is  
\[u(k)=-\widetilde{B}^{-1}b+\widetilde{B}^{-1/2}z.\]  
For the system performance index in Equation (16), because \(Q(k+1)\geq 0\) and \(R(k)>0\), therefore \(B^{T}(k)Q(k+1)B(k)\geq 0\) and \(\widetilde{B}>0\), respectively. Meanwhile, because \(Q(k+1)\) and \(R(k)\) are symmetric matrices, so \(\widetilde{B}\) is symmetric and positive definite matrix. Therefore, it can yield that \(\widetilde{B}^{-1/2}=(\widetilde{B}^{-1/2})^{T}\). Let \(u(k)=\widetilde{B}^{-1/2}z-\widetilde{B}^{-1}b\), then substituting it into Equation (16), it has  
\[J(k) = \tilde{x}^{T}(k)\widetilde{A}(k)-b^{T}\widetilde{B}^{-1}b+\rho^ {T}(k)\widetilde{G}\rho(k)\] \[+z^{T}z+2(c-D^{T}\widetilde{B}^{-1}b)^{T}\rho(k)\] \[+2z^{T}(\widetilde{B}^{-1/2}D)\rho(k).\]  
At stage \(k\), the first two parts of \(J(k)\) can be considered as a known constant. Let  
\[F=\widetilde{B}^{-1/2}D,h=c-D^{T}\widetilde{B}^{-1}b,\]  
the optimal problem of \(J(k)\) can be converted to be that of \(\widetilde{J}(k)\),  
\[\widetilde{J}(k)=z^{T}z+2h^{T}\rho(k)+2z^{T}F\rho(k)+\rho^{T}(k)\widetilde{G} \rho(k).\]  
Therefore, the problem (6) is equivalent to the following problem,  
\[\min_{u(k)\ \varpi(k)\in\Omega(0,W(k)),\eta(k)\in\Omega(0,P(k))}\widetilde{J}(k). \tag{18}\]By introducing an auxiliary variable \(\lambda\), problem (18) can be written as follows,  
\[\min \lambda \tag{19}\] \[s.t. \bar{f}(k)\leq\lambda\] \[w^{T}(k)W^{-1}(k)w(k)\leq 1\] \[J(k) = \tilde{x}^{T}(k)\widetilde{A}(k)-b^{T}\widetilde{B}^{-1}b+\rho^ {T}(k)\widetilde{G}\rho(k)\]0  
the constrain in (19)  
\[J(k) = \tilde{x}^{T}(k)\widetilde{A}(k)-b^{T}\widetilde{B}^{-1}b+\rho^ {T}(k)\widetilde{G}\rho(k)\]1  
can be rewritten in the following form  
\[\left[ \begin{array}{c}w(k)\\ \eta(k)\end{array} \right]^{T}\left[