Institute of Technology of Cambodia Modern Control Systems (I4-GEE-EA) IT Chivorn
TD3 – State Space Representation
Problem 1. Express the state space equation of following transfer function
𝑌 (𝑠) 𝐾 𝐾 𝑁𝑠
𝐻(𝑠) = =𝐾 + +
𝑈(𝑠) 𝑠 𝑠+𝑁
Where, 𝐾 , 𝐾 , 𝐾 , 𝑁 are constants
Problem 2. Given a process described by a sinusoid movement. This process may be e.g. a
disturbance, like waves and temperatures. Let
𝑦(𝑡) = sin(𝜔𝑡 + 𝜙)
A sinusoid disturbance is common in many processes. This example is therefore of importance when
modeling disturbances. Write a state space description of the sinusoid process.
Problem 3. Consider a system described be the following couple of differential equations
𝑧 ̈ − 4𝑣̈ + 2𝑒 ̇ + 𝑣 = −𝑢 ̇ + 2𝑢
𝑒 ̇ − 2𝑣̇ + 𝑧 = 𝑢
𝑣̈ + 𝑒 = 𝑢 + 𝑢
Where 𝑢 and 𝑢 are defined as control inputs and 𝑒 and 𝑣 are defined as measurement or output.
Obtain the state space model of system.
Problem 4. The nonlinear dynamic equation for a pendulum is given by
𝑚𝑙𝜃 ̈ = −𝑚𝑔 sin 𝜃 − 𝑘𝑙𝜃 ̇
Where 𝑙 the length of the pendulum is, 𝑚 is the mass of the bob, and 𝜃 is the angle subtended by the
rod and the vertical axis through the pivot point,
a) Choose appropriate state variables and write down the state equations.
b) Find all equilibria of the system.
c) Linearize the system around the equilibrium points, and determine whether the system
equilibria are stable or not.
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Institute of Technology of Cambodia Modern Control Systems (I4-GEE-EA) IT Chivorn
Problem 5. A synchronous generator connected to an infinite bus can be modeled by
𝑀𝛿 ̈ = 𝑃 − 𝐷𝛿 ̇ − 𝜂 𝐸 sin 𝛿
𝜏 𝐸 ̇ = −𝜂 𝐸 + 𝜂 cos 𝛿 + 𝐸
where 𝛿 is the angle in radians, 𝐸 is voltage, 𝑃 is mechanical input power, 𝐸 is field voltage (input),
𝐷 is damping coefficient, 𝑀 is inertial coefficient, 𝜏 is a time constant, and 𝜂 , 𝜂 , and 𝜂 are constant
parameters.
a) Using 𝛿, 𝛿 ̇ , and 𝐸 as state variables, find the state equation.
b) Linearize the system around 𝛿 = 1, 𝐸 = 2 under control input 𝑃 = 0, 𝐸 =4
𝑀 = 1, 𝐷 = 0.1, 𝜏 = 0.2, 𝜂 = 0.5, 𝜂 = 1, 𝜂 = 2
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