Question

State equation for (a)NLS, (b)LS)

Find out a state space representation for the following control system:

y³ + yy² + ysiny = (1+y)u

Answer 1

In this case, the state equation for the given control system is: 3x₁² * x₁' + x₁'*x₂ + x₁ + x₁*cos(x₁)*x₁' - x₁'u - (1 + x₁)u' = 0

Step-by-step explanation:

The given equation is a nonlinear control system, and we need to find its state space representation. The state space representation consists of state equations and output equations.

The state equation represents the dynamics of the system, while the output equation relates the states to the outputs.

To find the state equation, we need to introduce state variables. Let's define:

x₁ = y

x₂ = y²

Now, we can rewrite the given equation in terms of these state variables:

x₁³ + x₁*x₂ + x₁*sin(x₁) = (1 + x₁)u

To obtain the state equation, we take the derivative of the state variables with respect to time:

dx₁/dt = dy/dt = x₁' (since x₁ = y)

dx₂/dt = d(y²)/dt = 2y * dy/dt = 2x₁ * x₁'

Using these derivatives, we can rewrite the equation:

x₁³ + x₁*x₂ + x₁*sin(x₁) = (1 + x₁)u

as:

x₁³ + x₁*x₂ + x₁*sin(x₁) = (1 + x₁)u

Differentiating both sides with respect to time, we get:

3x₁² * x₁' + x₁'*x₂ + x₁ + x₁*cos(x₁)*x₁' = x₁'u + (1 + x₁)u'

Rearranging the equation, we obtain the state equation:

3x₁² * x₁' + x₁'*x₂ + x₁ + x₁*cos(x1)*x₁' - x₁'u - (1 + x₁)u' = 0

So, the state equation for the given control system is:

3x₁² * x₁' + x₁'*x₂ + x₁ + x₁*cos(x₁)*x₁' - x₁'u - (1 + x₁)u' = 0