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ELECTENG 704 ASSIGNMENT #1 Date of Submission: 2020/9/22

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 = q q1 2 ; M(q) = Z1cos(Jq11 – q2) Z1cos(Jq21 – q2)

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ELECTENG 704 ASSIGNMENT #1 Date of Submission: 2020/9/22

Vm

(q; q_) = -µp – Zµ1psin +(µqa1 – q2) _ q1 -µp + Z1sin µp(q1 – q2) _ q2 ; G(q) = – -Z Z2 3sin sin( (q q1 2) )

with

J1 = Ja + mas2 a + mpla2; J2 = Jp + mps2 p; Z1 = mplasp; Z2 = fmasa + mplagg 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:

(a) (15 points) Verify that the vertical position, q1 = 0, q2 = 0, _ q1 = 0 and _ q2 = 0, is an

equilibrium. Linearise the system about this equilibrium. Verify that the equilibrium is

unstable.

(b) (15 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 different initial conditions.

(c) (15 points) Now 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) (15 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) (15 points) Same assumption as (c) design an H1 controller to stabilise the system about

the equilibrium. Estimate the domain of attraction and compare that with your LQG

design.

2. Consider a system represented by the following linear continuous-time state equations:

x_ = Ax + Bu

where x 2 Rn is the state vector, u 2 Rm is the input vector, A and B are are known matrices

of appropriate dimensions. The objective is to determine the control input u which minimises

the following performance index (LQR Problem):

J = Z01 xTQx + uTRu dt;

where Q 2 Rn×n is real symmetric positive semi-definite matrix and R 2 Rp×p is a real positivedefinite matrix. The optimal control input which minimises J is given by,

u(t) = Kx(t); K = R-1BTP

where the matrix P is obtained by solving the is obtained by solving the following Riccati

equation.

ATP + PA + PBR-1BTP + Q < 0; P > 0; R > 0

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ELECTENG 704 ASSIGNMENT #1 Date of Submission: 2020/9/22

(a) (15 points) Using Schur compliment show that above inequalities can be represented using

following linear matrix inequalities. HINT: The Riccati equation, in contrast to Lyapunov

equations, is a nonlinear equation of P. This is because the quadratic term PBR-1BTP

appears in the inequality. This can be simplified using Schur compliment.

P > 0; Q > 0; R > 0 and

ATP +BTPA P + Q PB -R < 0:

(b) (10 points) For following A and B matrices find the optimal control gain K. HINT: Use

MATLAB LMI Solver or YALMIP toolbox with any solver.

A = -12 1 -1 ; B = 1 0

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