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2) Consider the mathematical model of the system given by $(ML^2 + mr^2)\ddot{\theta} + 2mr\dot{r}\dot{\theta} - (mr + ML)g \cos\theta = \tau$ $m\ddot{r} - m\dot{\theta}^2r + mg(1 - \sin\theta) = 0$ where $\theta$ and $r$ are angular position of rod and position of mass $m$ respectively. a) Find the state-space model of the system if $\theta = x_1$, $\dot{\theta} = \dot{x_1} = x_2$, $r = x_3$, $\dot{r} = \dot{x_3} = x_4$ b) Linearize this nonlinear system by using Taylor series expansion.

          2) Consider the mathematical model of the system given by
$(ML^2 + mr^2)\ddot{\theta} + 2mr\dot{r}\dot{\theta} - (mr + ML)g \cos\theta = \tau$
$m\ddot{r} - m\dot{\theta}^2r + mg(1 - \sin\theta) = 0$
where $\theta$ and $r$ are angular position of rod and position of mass $m$ respectively.
a) Find the state-space model of the system if
$\theta = x_1$, $\dot{\theta} = \dot{x_1} = x_2$, $r = x_3$, $\dot{r} = \dot{x_3} = x_4$
b) Linearize this nonlinear system by using Taylor series expansion.

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2) Consider the mathematical model of the system given by
(ML^2 + mr^2)θ̈ + 2mrṙθ̇ - (mr + ML)g cosθ = τ
mr̈ - mθ̇^2r + mg(1 - sinθ) = 0
where θ and r are angular position of rod and position of mass m respectively.
a) Find the state-space model of the system if
θ = x1, θ̇ = ẋ1̇ = x2, r = x3, ṙ = ẋ3̇ = x4
b) Linearize this nonlinear system by using Taylor series expansion.

Added by Timothy L.

University Physics with Modern Physics

Hugh D. Young 14th Edition

Instant Answer

Solved by Expert Adi S

08/08/2023

Step 1

The state-space representation is given by: \[ \dot{x} = Ax + Bu \] \[ y = Cx + Du \] where x is the state vector, u is the input vector, y is the output vector, A is the state matrix, B is the input matrix, C is the output matrix, and D is the feedforward matrix. Show more…

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Consider the mathematical model of the system given by (ML^(2)+mr^(2)) heta ^(¨)+2mrr^(˙) heta -(mr+ML)gcos heta = au mr^(¨)-m heta ^(˙)^(2)r+mg(1-sin heta )=0 where heta and r are angular position of rod and position of mass m respectively. a) Find the state-space model of the system if heta =x_(1), heta ^(˙)=x_(1)^(˙)=x_(2),r=x_(3),r^(˙)=x_(3)^(˙)=x_(4) b) Linearize this nonlinear system by using Taylor series expansion. 2) Consider the mathematical model of the system given by (ML2 +mr2)0+2mrr0 - (mr + ML)g cos 0 = T mr - m02r + mg(1 - sin 0) = 0 where O and r are angular position of rod and position of mass m respectively a) Find the state-space model of the system if 0=x1 0=x1=X2 r =x3, =x3=X4 b) Linearize this nonlinear system by using Taylor series expansion

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