Optimal Control Homework 1. Consider uxxxxx+−−==2122132 t0= 0T =1 x 0( ) =12      J = x12T( )+ x22T()+x14+u2()dτt0T∫ Set up (but do not solve) the Hamilton-Jacobi equation for this case, include the boundary conditions. 2. Consider uxx +−= J = x21( )+ x2+ u2( )dτ01∫ Use the Riccati Equation to find the optimal feedback control u*t( ) = F t( )x t( ). 3. Consider uxxxx+==1221 J = x12+ u2( )dτ0∞∫ Use the Algebraic Riccati Equation (ARE) to find u to minimize J. 4. Missile autopilot design problem. The goal of this problem is to design a controller that can satisfy stability, performance, and robustness design goals. Since this is a missile, we want the speed of response to be as fast as possible. The problem will be evaluated based on your explanation as to why the controller you call the “design” is a good one, and how you selected the LQR penalty parameters to achieve the design. The project goal is to design an autopilot (controller) without explicitly using the knowledge of the actuator dynamics that is robust to the actuator specified in part b) below. By not using the actuator dynamics in the design model thestates of the actuator will not have to be measured for feedback in the implementation. Consider at this flight condition a maximum command of 10 degree angle of attack, with an actuator that has a rate limit of 200 deg/s. a) A linear model of a missile's longitudinal dynamics is given by 10ZZMMqqαδαδααδ   = +       Zα= −1.30461sZδ= −0.21421sMα= 47.71091s2Mδ= −104.83461s2 Design a Robust Servo LQR controller to track constant AOA α( ) commands using state feedback. Simulate your design (step response). Analyze your design in the frequency domain. Turn in a step response, Bode plot, Nyquist plot, and singular value frequency responses for ( )ILσ+ and ( )1ILσ−+. Comment on the effect of choosing your LQR penalty parameters on the time domain response and frequency domain plots. b) Add 2nd order actuator dynamics (0.707ζ=, 20nωπ=) to this model and repeat the simulation and frequency domain analysis completed in part 1 using the controller you designed in part 1. Do not recompute the gains, just insert the dynamics for the actuator in the plant input path. Discuss the effect of adding the actuator dynamics to your design. Make sure you list the LQR penalty