THE UNIVERSITY OF AUCKLAND Department of Electrical, Computer & Software Engineering ELECTENG 704 – Assignment 2 Instructor: Lily Zhang Due: 21st October, 2021 Student Name: Student ID: This assignment contains 2 questions. Total of points is 100. Distribution of Marks Question Points Score 1 80 2 20 Total: 100 1 ELECTENG 704 Assignment 2 Due: 21st October, 2021 1. In this assignment, we consider the rotary inverted pendulum shown in Figure 1 as the controlled plant. The mechanism includes an arm attached to a motor, and a pendulum attached to the arms tip. The rotation angles are detected by means of encoders installed at the base of the arm and pendulum; these values are fed back to the arm position control in order to maintain the pendulum in the inverted state. The rotary inverted pendulum is a 1-input 2-output system. Figure 1: Rotary inverted pendulum The Lagrangian equations of motion for the arm and pendulum are given as follows: M(q)q¨ + Vm(q, q˙)q˙ +G(q) = Bu where B = [ 1 0 ] , q = [ q1 q2 ] ,M(q) = [ J1 Z1cos(q1 − q2) Z1cos(q1 − q2) J2 ] Vm(q, q˙) = [ µp + µa −µp + Z1sin(q1 − q2)q˙2 −µp − Z1sin(q1 − q2)q˙1 µp ] , G(q) = [−Z2sin(q1) −Z3sin(q2) ] Page 2 of 4 ELECTENG 704 Assignment 2 Due: 21st October, 2021 with J1 = Ja +mas 2 a +mpl 2 a, J2 = Jp +mps 2 p, Z1 = mplasp, Z2 = {masa +mpla}g and Z3 = mpspg The parameter values are, g = 9.81 m/s2,ma = 0.275 kg, la = 0.322, sa = 0.175 m, Ja = 4.52× 10−3 kg m2,mp = 0.065 kg, Jp = 1.36× 10−3 kg m2, sp = 0.196 m,µa = 9.29× 10−3Nms/rad, µp = 5.23× 10−4Nms/rad. q1 denotes the angle between pendulum arm and the vertical line (θ1 in Figure 1) and q2 denotes θ2 in the Figure. (a) (20 points) Verify that the vertical position, q1 = 0, q2 = 0, q˙1 = 0 and q˙2 = 0, is an equilibrium point of the system. Linearize the system about this equilibrium. Verify that the equilibrium is unstable. (b) (20 points) Suppose all the state variables can be measured. Design a LQR controller to stabilise the system about the equilibrium. Check your result in a nonlinear simulation with several initial conditions slightly off the vertical equilibrium. For this stabilising controller, estimate the domain of attraction. HINT: Use small initial conditions, and slightly increase the initial conditions until the overall system becomes unstable. (c) (20 points) Assume that only q1 and q2 can be measured. Design a LQG controller to stabilise the system about the equilibrium. Estimate the domain of attraction and compare that with your LQR design. (d) (10 points) Same assumption as (c) design a LTR controller to stabilise the system about the equilibrium. Estimate the domain of attraction and compare that with your LQG design. (e) (10 points) Based on the model you derived in (a), assume that we have external distur- bance as below: ω(t) =  ω1(t) ω2(t) ω3(t) ω4(t)  =  0.1sin(t) 0.2sin(t) 0.1cos(t) 0.05cos(t)  , 0 ≤ t ≤ 5s, ω(t) = 0, t > 5s where ω1 and ω2 are process noises that affect q˙1 and q˙2, and ω3 and ω4 are measurement noises. Design an H∞ state-feedback controller to stabilise the system about the equi- librium (assume all states are available). HINT: The piece-wise disturbance signal can be described in MatLab function block. Page 3 of 4 ELECTENG 704 Assignment 2 Due: 21st October, 2021 2. Consider a system represented by the following linear continuous-time state equations: x˙ = Ax+Bu where x ∈ Rn is the state vector, u ∈ Rm is the input vector, A and B are known matrices of appropriate dimensions. Design a state-feedback control law u = −Kx using the Lyapunov stability theory, such that the closed loop system is asymptotically stable with the following Lyapunov function: V (t) = xT (t)Px(t), P > 0 (a) (10 points) Based on the Lyapunov stability method, prove that the stability criteria of the closed-loop system is equivalent to the following matrix inequalities: P > 0, Q ≥ 0 ATP + PA− PBK −KTBTP +Q ≤ 0 HINT: To ensure the stability of the CL system, the Lyapunov function should be non- increasing with respect to time. i.e., there exist a positive semi-definite symmetrical matrix Q such that dV (t) dt ≤ −xT (t)Qx(t) (b) (5 points) Notice that the matrix inequality ATP + PA− PBK −KTBTP +Q ≤ 0 is NOT a LINEAR matrix inequality due to the existence of the term PBK. Both P and K are unknown variables. Perform the congruence transformation to transform it into a linear matrix inequality (LMI). (c) (5 points) Solve the LMI problem established in the previous question, regarding the fol- lowing system matrices: A = [−2 −1 1 1 ] , B = [ 1 0 ] and obtain the controller gain K. Verify the stability of the closed-loop system (A−BK). HINT: Use Yalmip or standard LMI solver in MATLAB. Refer to the tutorial document. Page 4 of 4