Abstract: This paper presents the design of an optimal control strategy for a 2 degree of freedom standard laboratory system - Ball and Beam. A linear quadratic regulator (LQR) is designed and implemented with an objective to control … 3.3. Output Feedback. 2 optimal control problems, including the linear quadratic regulator (LQR) in Sec. Eating your cake and having it in optimal control problems. Optimal Control • a dynamical system is described as where maps a state , a control (the action) , and a disturbance , to the next state , starting from an initial state . of Lyapunov in the time-domain control of nonlinear systems. 6.1 Finite-horizon LQR problem In this chapter we will focus on the special case when the system dynamics are linear and the cost is quadratic. The next [Kalman 1960a] discussed the optimal control of systems, providing the design equations for the linear quadratic regulator (LQR). The third paper [Kalman 1960b] discussed optimal filtering and estimation Next, linear quadratic Gaussian (LQG) control … n Optimal Control for Linear Dynamical Systems and Quadratic Cost (aka LQ setting, or LQR setting) n Very special case: can solve continuous state-space optimal control problem exactly and only requires performing linear algebra operations n Running time: O(H n3) Note 1: Great reference [optional] Anderson and Moore, … 37 Example: Open-Loop Stable and Unstable Second-Order System Response to Initial Condition Stable Eigenvalues = –0.5000 + 3.9686i –0.5000 - 3.9686i Unstable Eigenvalues = +0.2500 + 3.9922i x t+1 = f t (x t,u t,w t) f t x t ∈ R d u t ∈ R k w t x t+1 ∈ R d x 0 • The objective is to find the control policy which minimizes the long term cost, : same as the optimal finite horizon LQR control, T −1 steps before the horizon N • a constant state feedback • state feedback gain converges to infinite horizon optimal … These problems are chosen because of their simplic-ity, ubiquitous application, well-defined quadratic cost-functions, and the existence of known optimal solutions. Optimal Control: LQR. A simple feedback control scheme is to use the outputs to compute the control inputs according to the Proportional (P) feedback law u Ky v where v(t) is the new external control … This depends upon how in-depth you’d like to understand the concepts. While this additional structure certainly makes the optimal control problem more tractable, our goal is not merely to specialize our earlier results to this simpler setting. in general, optimal T-step ahead LQR control is ut = KTxt, KT = −(R+BTPTB)−1BTPTA where P1 = Q, Pi+1 = Q+A TP iA−ATPiB(R+BTPiB)−1BTPiA i.e. MATLAB function: lqr Optimal control gain matrix Optimal control t f!" 3. Optimal Control • maps a state , a control (the action) , and a disturbance , to the next state , starting from an initial state . 2. Matrix A is the system or plant matrix, B is the control input matrix, C is the output or measurement matrix, and D is the direct feed matrix. The system is an open loop and nonlinear system, which is inherently unstable. • The objective is to find the control policy which minimizes the long term cost, where is the time horizon (which can be … 3.2 and Kalman filters in Sec.