7065MAA - Nonlinear Control and Estimation in Engineering Assignment

Assignment Task

1. Linearisation and Equilibrium points

a) Explain the differences (at least 5 points) between linear and nonlinear systems and give an example for each point of differences.

b) Consider a system represented by a differential equation

x(t) + x(t) + 4 x(t) − x 3 (t) = 0, t > 0

• Explain what is commonly understood by the “stability of a nonlinear system”.
• Express the system in state-space formx = f(x), where x= [x1 x2] T = [x x] T . Then find the equilibrium points (equilibria) of the
• Using the linearisation method, determine the stability of the If it is stable, be precise with the type of stability that the origin possesses.

c) Of the equilibria obtained in (1b),

• Find the type of each equilibrium and determine the stability of each of
• Sketch the phase portrait of each equilibrium in one (Note: please do not use MATLAB or any other computer aid to sketch it)

2. Lyapunov Stability Theory 

Given a nonlinear system:

_x1 = x₁ − 2 x _{1 x 2} + u x ₂

_x2 = −x ₂ + 2 x ²

a) Using

v(x, x ) = 1 x ² + 1 (x − 1) ² , x ≥ 1

as a Lyapunov function for the system, determine the stability of the system for u = 0 andx ₂ ≥ 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:

_{x 1} = −x ³ +x ₂

_x2 = x ₃

_{x 3} = u

design a backstepping controller to stabilise the system.

4. Nonlinear Controller Design (Feedback Linearisation)

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)q + C(q, q)q + g(q) = r

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

( ) C ,

M(q) = >θ ₁ + 2 θ ₃ cos(q ₂ ) θ₂ + θ ₃ cos(q ₂ )

_θ2 +θ ₃ cos q _2 θ 2

c(q, q) = >−θ₃ sin(q ₂ )q ₂ −θ ₃ sin(q ₂ )q ₁ − θ ₃ sinq ₂ )q _2c

_θ3 sin(q ₂ )q ₁ 0

( ) C,

g(q) = >θ _% g cos(q₁ ) + θ _{& g} cos(q ₁ + q ₂ )

_{θ & g} cos q ₁ + q ₂

where

_{θ 1} = (m ₁ + m ₂ )l² + m _2 l² , θ ₂ = m _{2 l} ² , θ ₃ = m_{2 l1 l 2} , θ _% = (m₁ + m ₂ )l ₁ , θ _& = m _2 l2 

Express the system in state-space form x = f(x) + u(x, r), where x = [x1 x2 x3 x%] t = [q1 q2 q _{1 q 2} ] 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) = 0.5 sin(4t)

_q2d(t) = 0.5 cos(4t)

for joint positions. Using the system parameters m₁ = 500gr, m₂ = 400gr, l₁ = 300mm, l₂ = 200mm, and g = 9.81 m/s², illustrate the behaviour of the system for 10 seconds by Matlab.

• Plot q₁,q_1d with respect to time on the same figure and comment on your observations
• Plot position and velocity errors with respect to Comment on what are the restrictions to increasing the static control gain in practical application.

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