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# EE 350 Control Systems Assignments

## Assignment 4: State space equation and it's solution

4.4 Consider a unit mass subjected to an inverse square low force field. It is assumed that the unit mass (may be a satellite) can thrust in radial and in the tangential direction with thrust $u_{1}$ and $u_{2}$ respectively. Assume $X=\begin{pmatrix}r & \dot{r} & \theta &\dot{\theta} \end{pmatrix}^T$. Write the state space equation. Observe that the state space equation is nonlinear. Linearize the state space equation across the nominal point $X=\begin{pmatrix}r_{0} & 0 & \theta_{0} & w_{0} \end{pmatrix}$ and $u_{1}(t)=0$ ,$u_{2}(t)=0$

Fig 4.3

4.5 Consider an inverted pendulum problem. It is desired to keep the pendulum in upright position by moving the cart in x axis. To start with, we have model to the system. Assume state vector as $X=\begin{pmatrix} \theta & \dot{\theta} & x &\dot{x}\end{pmatrix}^T, where$ M=2kg $,$m=0.1kg, $l=0.5m$. Write the state space equation. Assume $\theta$ and $\dot{\theta}$ are small i.e $sin \theta=\theta$, $cos\theta=1$, $\theta\dot{\theta}^2=0$. Write the state space in linear form. Comment about the stability of the of the system.

Fig 4.4 Inverted Pendulum

4.6 State space equation of a system is given by $\dot{x}_{1}=x_{1}^2 +x_{2}^2+x_{2}cosx_{1}$; and $\dot{x}_{2}=(1+x_{1})x_{1} +(1+x_{2})x_{2}+x_{1}sinx_{2}$. Linearize the system about the point $x^T=[0\hspace{.5cm} 0]$.

4.7 Proof the following properties of the state transition matrix
1. $\phi(0)=I$
2. $\phi(t)=\phi(-t)$
3. $\phi(t_{1}+t_{2})=\phi(t_{2})\phi(t_{1})$
4. $[\phi(t)]^n=\phi(nt)$
5. $\phi(t_{2}-t_{1})\phi(t_{1}-t_{0})=\phi(t_{1}-t_{0})\phi(t_{2}-t_{1})$

4.8 For a system, $\dot{x}_{1}=5x_{1}-2x_{2};$ $\dot{x}_{2}=4x_{1}-x_{2}$; solve the state space equation.

4.9 Determine the TF of the systems given by
1. $A=\begin{pmatrix}0 & 1 \\ -1 & -2 \end{pmatrix}$; $B=\begin{pmatrix}0 \\ 1 \end{pmatrix}$; $C=\begin{pmatrix} -3 & 3 \end{pmatrix}; D=0$
2. Let $A=\begin{pmatrix}0 & 1 \\ -2 & -3\end{pmatrix}$; $B=\begin{pmatrix}0 \\ 1\end{pmatrix}$; $C=\begin{pmatrix}-1 & -5 \end{pmatrix}; D=1$