Task 1 Consider the following first order autonomous nonlinear system. 4(+ 2) *(t) + -x* (t) x(t) = sin(x(t)) 13 Find its equilibrium points (the x points, where i(t) = 0). Hint: try to visualize both sides of the equation and find the intersections of the graphs. There are 3 equilibrium points. If you cannot figure it out quickly, do not waste your time here, move to the next tosk and return if you finished the rest. Task 2 Consider the following first order nonlinear system. (1) i(t) = arctan(u(t) - 4 x(t)) + 3 sinh X(t) Linearize (trim) the system around the (x, u) = (0,0) point and give the differential equation of the corre- sponding linearized first order system. Task 3 The following linear system is given. i(t) + 3 x(t) = 4 u(t) Consider the controllers in the following list up(t) = Kp e(t). upd(t) = Kp e(t) + Kp (t), upi(t) = Kp e(t) + K1 e(T) dr, Kife Seco upID(t) = Kp e(t) + Ki (1) dr + Kp e(t), where e is the error signal. Design a controller for the given system such that 1. the closed loop is stable, 2. the response to a unit step reference satisfies the following constrains the absolute value of the input signal is not more than 6 units, b. the settling time is not more than 3 seconds, c. the steady state error is not more than 0.1 units, d. overshoot is not allowed (if the closed loop system is second order, then must be 1). Give the values of Kp, Ki and Kp (use O if some term is not used) for the chosen controller. Explain why you picked a specific controller to satisfy the constrains. a. Task 1 Consider the following first order autonomous nonlinear system. 4(+ 2) *(t) + -x* (t) x(t) = sin(x(t)) 13 Find its equilibrium points (the x points, where i(t) = 0). Hint: try to visualize both sides of the equation and find the intersections of the graphs. There are 3 equilibrium points. If you cannot figure it out quickly, do not waste your time here, move to the next tosk and return if you finished the rest. Task 2 Consider the following first order nonlinear system. (1) i(t) = arctan(u(t) - 4 x(t)) + 3 sinh X(t) Linearize (trim) the system around the (x, u) = (0,0) point and give the differential equation of the corre- sponding linearized first order system. Task 3 The following linear system is given. i(t) + 3 x(t) = 4 u(t) Consider the controllers in the following list up(t) = Kp e(t). upd(t) = Kp e(t) + Kp (t), upi(t) = Kp e(t) + K1 e(T) dr, Kife Seco upID(t) = Kp e(t) + Ki (1) dr + Kp e(t), where e is the error signal. Design a controller for the given system such that 1. the closed loop is stable, 2. the response to a unit step reference satisfies the following constrains the absolute value of the input signal is not more than 6 units, b. the settling time is not more than 3 seconds, c. the steady state error is not more than 0.1 units, d. overshoot is not allowed (if the closed loop system is second order, then must be 1). Give the values of Kp, Ki and Kp (use O if some term is not used) for the chosen controller. Explain why you picked a specific controller to satisfy the constrains. a