# Robust Quadratic Optimal Control for Discrete-Time Linear Systems with Non-Stochastic Noises
## 2 Problem Formulation  
Consider the following uncertain discrete-time linear dynamic system,  
\[x(k+1) = A(k)x(k)+B(k)u(k)+w(k),k=0,1,\ldots,N-1, \tag{1}\] \[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]  
where \(x(k)\in R^{n}\) is the state vector, \(u(k)\in R^{m}\) is the control vector, \(y(k)\in R^{p}\) is the measure output vector. \(w(k)\in R^{n}\) and \(v(k)\in R^{p}\) are the process noise and observation noise respectively, which are confined to specified ellipsoidal sets,  
\[w(k)\in\Omega(0,W(k)),\]  
\[v(k)\in\Omega(0,V(k)),\]where \(W(k)\) and \(V(k)\) represent the bounds of noises \(w(k)\) and \(v(k)\) respectively. The initial state \(x(0)\) takes value in the following ellipsoidal set,  
\[x(0)\in\Omega(\hat{x}(0),P(0)).\]  
Considering the quadratic cost function,  
\[J_{o}=\sum_{k=0}^{N}x^{T}(k)Q(k)x(k)+\sum_{k=0}^{N-1}u^{T}(k)R(k)u(k), \tag{3}\]  
where matrices \(Q(k)\geq 0\) and \(R(k)>0\) are non-negative definite and positive definite symmetric matrices of appropriate dimensions, respectively. Our target of this work is to find a feedback control series \(\{u(k)\}_{k=0}^{N-1}\) such that the quadratic cost function \(J_{o}\) is minimized when the noises \(w(k)\) and \(v(k)\) belong to bounded ellipsoidal uncertain sets. Namely,  
\[\min_{u(0),\cdots,u(N-1)}J_{o} \tag{4}\] \[s.t.\;\;x(k+1) =A(k)x(k)+B(k)u(k)+w(k)\] \[y(k) =C(k)x(k)+v(k)\] \[w(k) \in\Omega(0,W(k))\] \[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]0  
Due to the noises \(w(k)\) and \(v(k)\) being bounded in this paper, the dynamics Bellman equation cannot be applied to solve the multi-step quadratic optimal control problem. In order to effectively address the proposed optimal control problem, we attempt to find a feasible approximate optimal solution. Because the value of \(x^{T}(0)Q(0)x(0)\) is not influenced by the control law \(u(k)\), so we simplified the cost function to \(J=\sum_{k=0}^{N-1}[x(k+1)Q(k+1)x(k+1)+u^{T}(k)R(k)u(k)]\). To ensure the optimality of \(J\) for all the possible values in the noises sets \(\Omega(0,W(k))\) and \(\Omega(0,V(k))\), the minimization of the worst case of \(J\) is considered. To realize the approximate optimality of the multi-step quadratic cost function \(J\), we optimize each one-step quadratic cost function \(J(k)(k=0,1,\ldots,N-1)\) which is defined as follows,  
\[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]1  
Then, the multi-step quadratic global optimal control problem is converted to multiple one-step quadratic local optimal control problems which can provide a feasible approximate optimal solution. Each one-step quadratic optimal control problem is as follows,  
\[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]2 \[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]3 \[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]4  
In optimization problem (6), the performance index \(J(k)\) is a function with respect to the system states which are not measurable completely. As a result, the state estimation error directly affects the feedback control performance. If we can yield more accurate true state feasible set, the maximum value of \(J(k)\) can be more accurate. Then by minimizing the maximum value of \(J(k)\), optimal control law \(u(k)\) can be obtained as well. The following lemmas are used in the designing of the optimal control law. **Lemma 1**. : \(S\)_-procedure [19] Let \(\widetilde{F}_{0},\cdots,\widetilde{F}_{p}\) be quadratic functions of the variable \(\zeta\in R^{m}\),_  
\[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]5_where_ \(T_{i}=T_{i}^{T}\)_,_ \(\phi_{i}\) _and_ \(\phi_{i}\) _are with appropriate dimensions. The following condition on_ \(\widetilde{F}_{0}\)_,_ \(\cdots\)_,_ \(\widetilde{F}_{p}\)_,_ \[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]6 _holds if there exist_ \[y(k) = C(k)x(k)+v(k),k=1,2,\ldots,N, \tag{2}\]7 _such that_ \[\left[ \begin{array}{cc}T_{0}&\phi_{0}\\ \phi_{0}^{T}&\phi_{0}\end{array} \right]-\sum_{i=1}^{p}\tau_{i}\left[ \begin{array} []{cc}T_{i}&\phi_{i}\\ \phi_{i}^{T}&\phi_{i}\end{array}