# Robust Quadratic Optimal Control for Discrete-Time Linear Systems with Non-Stochastic Noises
## 1 Introduction  
Linear quadratic (LQ) optimal control problems with indeterminacy factors have been widely studied in theoretical and practical perspectives. In most cases, these indeterminacy factors are modeled as stochastic processes. Many achievements in the study of stochastic control have been made, such as minimum variance control [1], Linear-quadratic-Gaussian (LQG) [2; 3; 4] and probability density function (PDF) control [5]. Actually, due to lack of sufficient information and data, the statistical properties of the uncertainties in some special systems are very difficult to obtain. Taking the mobile system as an example, because the terrain conditions cannot be reused, the stochastic model can not be modeled with the insufficient data [6]. Moreover, there are several uncertainties which are not essentially stochastic. Non-stochastic uncertainties as an alternative which only consider the range of uncertain information have attracted the interests of scholars. [PERSON] proposed a recursive robust linear quadratic regulator based on the robust weighted least-squares method for discrete-time systems with norm-bounded parametric uncertainties [7]. [PERSON] investigated the robust quadratic optimal control based on robust optimization for the systems in the presence of ellipsoidal bounded disturbance [8]. [PERSON] studied the robust optimal control with adjustable uncertainty sets which treated the uncertainty sets as additional decision variables [9]. However, the methods above were with the assumption that the system states were observed completely. Actually, the states in the real systems are usually not measurable completely and accurately. Set-membership filtering can provide an estimated feasible set of true states for systems with non-stochastic noises, and so far, many approaches have been proposed to obtain the solutions [10; 11; 12; 13]. Based on the set-membership estimation theory, [PERSON] the output tracking control problem for systems with ellipsoidal uncertainties in system parameters and noise, and an adaptive robust tracking control algorithm was proposed [14]. However, the performance index was given with \(l_{1}\) norm of the output deviation over the finite time horizon, whose transient performance is poorer than the linear quadratic optimal control. [PERSON] studied the linear quadratic control problems with norm-bounded uncertainties [15]. However, for the imperfect state information case, the state estimation error set was assumed in fixed bounded set which neglected the learning feature of the state estimation. [PERSON] investigated the output feedback robust predictive control with ellipsoidal estimation error sets [16; 17]. However, the methods updated the estimation state error set by solving one convex optimal problem each time. Motivated by the above discussions, the LQ control problem with non-stochastic noises is considered in this paper. Specifically, the non-stochastic process and measurement noises are confined to ellipsoidal bounded sets. Ellipsoidal bounded sets as more realistic and flexible non-stochastic uncertain sets, on the one hand, are of good tractability and well-motivated from a stochastic setting, where they naturally occur when a normal distribution is cut off at a certain level of probability. On the one hand, they can describe kinds of bounded uncertainties as well, such as 2-norm bounded (by selecting the identity matrix as the size matrix for the ellipsoid set) [18]. In addition, compared with the traditional stochastic assumption, the bound of the noises can be obtained more easily. As the noises are bounded, the traditional Kalman filtering and Dynamic Bellman Equation are not applicable the proposed control problem. For the LQ control methods with ellipsoidal bounded noises, most of them considered the states to be completely observed and for the output feedback control methods, most of them need to solve one complex optimization problem at each time to update the estimation state error set.