Engg. tutorials

Engg. tutorials

5.14 A pendulum is attached with a rod as shown in figure. It can be described by the equation of motions $\ddot{\psi}(t)-\sigma^2 \psi (t) - w^2[\psi (t) +\theta(t)]=u(t)$; $\ddot{\theta(t)}+w^2_0\theta(t) +\epsilon\ddot{\psi}(t)=0$, where $\sigma^2=3g/2l$, $w^2=3mg/Ml$, $\epsilon=l/a$, $w^2_0=g/a$, $u=Q/Ml$. $Q$ is the torque applied at the point A.

a) Write the state space equation where $x_1=\psi$, $x_2=\dot{\psi}, x_3=\theta, x_4=\dot{\theta}$. Assume, we measure time in unit of $1/w^2$ and $u(.)$ in the unit of $w^2_0$ (so you may take $1/w_0\rightarrow1$, and $u/w^2_0 \rightarrow u$).

b) Find the open loop pole positions.

c) Design a state feedback controller so that the mass $m$ will start swinging with a time constant of about $T/2$, where the period of oscillation of the hanging mass $(T)=2\pi\sqrt(a/g)$. [Hints: Check placing poles at $-1/3 \pm 3j/2$ and $-3/2 \pm j/2$ will satisfy the design criteria.

a) Write the state space equation where $x_1=\psi$, $x_2=\dot{\psi}, x_3=\theta, x_4=\dot{\theta}$. Assume, we measure time in unit of $1/w^2$ and $u(.)$ in the unit of $w^2_0$ (so you may take $1/w_0\rightarrow1$, and $u/w^2_0 \rightarrow u$).

b) Find the open loop pole positions.

c) Design a state feedback controller so that the mass $m$ will start swinging with a time constant of about $T/2$, where the period of oscillation of the hanging mass $(T)=2\pi\sqrt(a/g)$. [Hints: Check placing poles at $-1/3 \pm 3j/2$ and $-3/2 \pm j/2$ will satisfy the design criteria.

Fig 5.4 Pendulum on rod

5.15 A helicopter near hover can be described by the equations:
$\begin{pmatrix}\dot{x_1} \\ \dot{x_2} \\ \dot{x_3} \\ \end{pmatrix}=\begin{pmatrix}-0.02 & -1.4 & 9.8 \\-0.01 & -0.4 & 0 \\ 0 & 1.0 & 0\end{pmatrix}\begin{pmatrix}x_1 \\ x_2 \\ x_3 \\ \end{pmatrix} + \begin{pmatrix} 9.8 \\ 6.3 \\ 0 \end{pmatrix}u$;
where $x_1$ is horizontal velocity, $x_2$ is pitch rate, $x_3$ is pitch angle, $u$ is rotor tilt angle (For a helicopter the control I/P is rotor tilt angle). a) Find out open loop poles. b) Design a state feedback such that the poles move to -2, $-1\pm j$.

5.16 Multivariable integral control: My lectures on pole placement do not consider any reference or set point to be followed by the system. In other word, we considered the states go to zero point as time tends to infinite. But in practice that may not be the case. In many cases, you may have reference which needs to be followed. Let us consider a system whose structure is as shown in the figure below. We describe the system with state space as:
$\begin{pmatrix}\dot{x_1} \\ \dot{x_2} \end{pmatrix}=\begin{pmatrix}-3 & 2 \\ 4 & 5 \end{pmatrix}\begin{pmatrix}x_1 \\ x_2 \end{pmatrix} + \begin{pmatrix} 1 \\ 0 \end{pmatrix}u + \begin{pmatrix} 1 \\ 0 \end{pmatrix}w$; $y= \begin{pmatrix} 0 & 1 \end{pmatrix}x$. Please note that open loop eigenvalue without any state feedback is at 0, -1, -7 positions. Now calculate the feedback gains such that the eigenvalues shift to -2, -1, -7 position.

Hints: Read Vegte pp370.

Hints: Read Vegte pp370.

Fig 5.5 State feedback with Integral control

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