7065MAA - Faculty of Engineering Environment and Computing Assignment

Assignment Task

Questions

1. Linearisation and Equilibrium points

1. Consider a system represented by a differential equation

x(t) + x(t) + 4x(t) - X³(t) = 0, t>0.

2. Explain what is commonly understood by the “stability of a nonlinear system”

3. Express the system in state-space form x = f(x), where x(t) + 4x(t) - x³. Then find the equilibrium points (equilibria) of the system.

4. Using the linearization method, determine the stability of the origin. If it is stable, be precise with the type of stability that the origin possesses.

5. Find the type of each equilibrium and determine the stability of each of them.

6. Sketch the phase portrait of each equilibrium in one plot. (Note: please do not use MATLAB or any other computer aid to sketch it) 

2. Lyapunov Stability Theory

Given a nonlinear system:

X_{1 = X}1 _2X1 X2 _{+ uX2}

x₂ = -x₂+2x²

a. Using

V(x₁, x₂) = 1/2 x₁² + 1/2 (x₂ −1)², X² ≥1

As a Lyapunov function for the system, determine the stability of the system for x = 0 and x₂ ≥ 1.

b. Find all the equilibria of the system for u = 0, and determine the type and stability of each equilibrium.

c. Relate your finding about the stability of the equilibria in (b) with the stability of the system in (a).

3. Nonlinear Controller Design (Backstepping)

Given a nonlinear system

X1 = −x²₁ + x₂

x₂ = X₃

x₃ = u

Design a backstepping controller to stabilise the system.

4. Nonlinear Controller Design

Figure 1 shows a structure of a two-link planar manipulator system. The links are cascaded in a serial fashion and are actuated by individual motors.

Equation of motion of the two-link planar manipulator is described by.

M(q)ä + C(q,q)q+g(q) = t

Where q and q are 2- dimensional vectors of generalized coordinates representing joint position and velocity respectively, m(q) is a symmetric inertia matrix. The c(q, q)q accounts for centrifugal and Coriolis forces. g(q) denotes gravity forces.

a. A two-link planar robot manipulator model’s system matrices are given as.

M(q) = [0₁, +20₃ cos(q2) 0₂ +0₃ cos(q₂) 0₂ +0₃ cos(q₂)

C(q, q) =( -0₃, sin(q2)q₂ - 03 sin(q₂)q_{1 +} (q) = [0q₉ - 03 02+ 03 cos(92)] 3cos (q₂)], q [q] 0² sin(2)91 – 03 sin(92)42]

g (q) = 0₄ g cos(q1)+0₅ g cos(q₁ +9₂) 0₅ g cos(q1 +q2)+92) ],

Where

0₁ = (m₁+m₂)l₁² + m₂l₂², 0₂ = m₂l₂², 03 = m²l₁l², 04 = (m₁ + m₂)4₁, 0₅ m₂ l₂. Express the system in state-space form x = f(x) + u(x, t),

where x = [x₁ X₂ X₃ X₄]^T = [q₁ q₂ q₁ q₂]^T

b. Using the feedback linearisation technique to design a controller for the two-link planar robot manipulator to ensure asymptotic stability.

c. Assign appropriate values to the static gain matrix (k) of your controller to track the reference signals.

q₁d(t) = 0.5Sin(4t)

q₂d(t) = 0.5cos(4t)

For joint positions. Using the system parameters m₁ = 500gr,m₂ = 400gr,l_1" = 300mm, l₂ = 200mm, and g = 9.81 m/s₂, illustrate the behaviour of the system for 10 seconds by Matlab.

Plot q₁", q₁d"' with respect to time on the same figure and comment on your observations ii. Plot position and velocity errors with respect to time. Comment on what are the restrictions to increasing the static control gain in practical application

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