Reference no: EM132983628

Question 1:

The ball and beam system (see Figure 1) has been used frequently in testing different control algorithms.

Figure 1 Figures 2 and 3 illustrate the model of the system.

Figure 2

Figure 3

(1). Define x₁ = x, x₂ = x·, write down the nonlinear state space model of the ball and beam system. Then assume the beam angle (input) is small, linearise this system and get the linear continuous time state space model of the system.

(2). Using either hand calculation or MATLAB, perform the following tasks for the linear state space model obtained in (1),

(2A) Analyze the system stability, controllability and observability.

(2B) Design the state feedback control law such that the poles of closed loop system are -2 and -3.

(2C) Design the state feedback LQR control law for

(2D) Design the state observer such that the poles are -1 and -2.

(2E) Design the Ka1man-Bucy filter with

(3). Analyze the performance of the closed-loop systems:

Compare the LQR performance index for the closed-loop system using state feedback (2B) and the closed-loop system using state feedback in (2C). Use the Q and R given in (2C). Compute the value of the performance index of each system for three different initial state values

(4). Assume we want to control the ball position to -0.2 instead of 0, design the state feedback set point control law with the poles of the closed-loop to be -2 and -3.

Question 2:

Examine the state space model of the inverted pendulum (Question 1 in Assignment 1), write MATLAB code to

(1) Design a state observer such that the poles of the observer are -1,-1.5,-2,-2.5.

(2) Design state feedback control law u=-Kx such that the poles of the closed-loop system are -2, -3, -4, -6 (as in Assignment 1 Question 1). Describe the state-space model of the closed-loop system including the observer and feedback control.

(3) Simulate the state trajectory of the closed-loop for any given initial states.

Question 3:

Consider system

choose V(x) = x12 + 2x22 and use Lyapunov stability theorem to demonstrate that the equilibrium state of the system is stable.

Question 4:

Consider a mobile robot following a circular trajectory. The mobile robot position at time t is denoted as (x(t), y(t)) and its orientation at time t is denoted as θ(t), the velocity and turn rate are denoted as v(t) and ω(t) respectively. The nonlinear motion model can be simplified as

x^· = v cosθ, y^· = v sinθ, θ^·· = ω

Suppose the desired trajectory (reference trajectory) is a circular trajectory

x_r(t) = x_o + R sin(ω_rt), y_r (t) = y_o - R cos(ω_rt), θ_r(t) = ω_rt

where (x_o, y_o) is the centre of the circle, and R is the radius of the circle. Then the reference trajectory satisfies

x^·_r = V_r cos θ_r, y^·_r = V_rsin θr, θ^·r = ω_r,

where v_r = Rω_r.

Consider the difference between the actual robot trajectory and the desired robot trajectory in the robot local frame. That is, define new state as

Then the tracking error model is

x^·_e = ωy_e - v + v_rcosθ_e

y^·_e = -ωx_e + v_rsinθ_e

θ^·e = ω_r - ω

Define the new control as

v_e = v - v_rcosθ_e, ω_e = ω-ω_r.

(1) Derive the nonlinear state space model of the system with state (x_e, y_e, θe) and control (y_e, ω_e).

(2) Assuming Xe, ye, θe, ve, ωe are all close to zero, linearise the system at the operating point (xe, ye, θe, ve, ωe) = (0, 0, 0, 0, 0) to derive the state-space model of the linearised control system.

(3) Let v_r = 0.3, ωr = 0.2. Design state feedback control law for the lin¬earised system such that the poles of the closed loop system are -1,-2,-3.

(4) Let v_r = 0.3, ωr = 0.2. Design the state feedback LQR control for the linearised system with

Question 5:
Use MATLAB simulator to test the controller design using the model developed in Question 4 (set the desired velocity as 0.3 and the desired turn rate as 0.2). A template of the MATLAB simulator is provided which is used to simulate a robot moving along a straight line. You are required to
(1) change the simulator to be the circular trajectory situation and complete the open-loop (simply fix the velocity and turn rate) and closed-loop control design (using pole placement and LQR on the linearised model as in Question 4) and integrate it with the MATLAB simulator,
(2) compare the performance of the closed-loop systems using different controllers with that of the open-loop system, and discuss the results obtained.

Run Ass2_49329_Simulation_template.m for the simulator. Choose open-loop or closed-loop in line 9-10:
closed_loop=0; % open-loop closed_loop=1; % closed-loop

You can change the simulation time period in line 12:
time_period = 100; % simulation time period

For the closed-loop design, you need to modify the file: control_velocity_turnrate_49329.m

Attachment:- Control of Mechatronic Systems.rar