Chapter 9, Linear Control Video Solutions, Structural Motion Engineering | Numerade (2024)

Consider Eq. (8.75). Integrating this cquation between $t_j$ and $t_{j+1}$ leads to
$$
\mathbf{X}_{j+1}-\mathbf{X}_j=\int_{t_j}^{t_{j+1}}\left(\mathbf{A X}+\mathbf{B}_f \mathbf{F}+\mathbf{B}_g a_g+\mathbf{B}_p p\right) d t
$$

Suppose the integrand is assumed to vary linearly over the time interval, and the coefficient matrices $\mathbf{A}, \mathbf{B}_f, \ldots$, are constant.
1. Derive the expression for corresponding to these conditions. Compare this result with Eq. (8.78). Comment on the nature of the error.
2. Specialize 1. for negative linear feedback, and compare with Eq. (8.83).
3. Specialize 2. for no time delay and free vibration response. Compare with Eq. (8.87). Define the stability requirement for this approximation.

Refer to Eq. (9.36). Will the LQR algorithm ever produce an unstable system?

Consider Eq. (9.44). Let
$$
\begin{aligned}
\mathbf{D} & =\mathbf{Q}+\mathbf{K}_f^T \mathbf{R} \mathbf{K}_f \\
\mathbf{S}_n & =\sum_{j=0}^n \mathbf{C}^j{ }^T \mathbf{D} \mathbf{C}^j
\end{aligned}
$$

Noting the identity,
$$
\mathbf{C}^T \mathbf{S}_n \mathbf{C}-\mathbf{S}_n=-\mathbf{S}+\mathbf{C}^{n+1} \cdot T \mathbf{D C}^{n+1}
$$
and the limit condition,
$$
\mathbf{C}^j \rightarrow 0 \text { as } j \rightarrow \infty
$$
derive Eq. (9.45).

Refer to Figs. 9.3 and 9.4 of Example 9.1. Suppose the time ratio $\Delta t / T$ is determined by the external loading, and is equal to 0.1 . Suggest a value for $\bar{q}_v$ such that $\xi_a$ is close to 0.2 when $\xi=0.05$.

Consider the following system:
$$
\begin{aligned}
m & =1,000 \mathrm{~kg} \\
k & =60,000 \mathrm{~N} / \mathrm{m} \\
c & =1,000 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}
\end{aligned}
$$

Suppose $\Delta t=0.02 \mathrm{~s}$. Select the parameters for discrete time feedback control such that the effective damping ratio is equal to 0.2 . Use Figs, 9.3 and 9.4 of Example 9.1 to obtain an initial estimate, and the function dare in MATLAB to refine the estimate. Note that the solution tends toward the continuous time feedback case as $\Delta t / T$ approaches 0 .

Refer to Example 9.2. Compare the expressions for $k_d$ and $k_v$ corresponding to $q_d=0$ with the continuous time Riccati solution defined by Eq. (9.37). Use the discrete time Riccati solution for $\xi=0.02$ listed in Example 9.1 to compare the values of $q_v$ required to produce $\left(k_v / 2 \omega m\right)=0.2$ for two time increments, $\Delta t / T=0.02$ and 0.1 .

Rework Problem 9.12 using the finite interval discrete time algebraic Riccati equation. Note that the weighting factors for the finite interval formulation are different from the corresponding weighting factors for the discrete time algebraic Riccati equation.

Consider the 5DOF systems shown in Table P9.13a.

1. Suppose control forces are applied at all five nodes. Determine the modal coordinate weighting parameters such that the equivalent damping ratio corresponding to continuous feedback is equal to 0.15 for the first mode. Assume uniform weighting.
2. Suppose self-equilibrating sets of control forces are applied on all five elements and the weighting is applied to the element deformation time rates. Determine the weights such that the first mode damping ratio is 0.15 . Assume uniform weighting.
3. Apply the Northridge earthquake to the models obtained in 1. and 2. Compare
(a) the internode displacement profiles
(b) the peak power
(c) the peak value of the control forces

Consider a 10DOF shear beam with constant mass, element stiffness, and element damping. Take $m=10,000 \mathrm{~kg}$.
1. Determine the stiffness and damping constants such that the properties for the first mode are
$$
\begin{aligned}
& \text { Period }=1 \mathrm{~s} \\
& \text { Damping ratio }=0.02
\end{aligned}
$$
2. Select an active control force scheme which provides a damping ratio of 0.2 for the first mode.
3. Apply the Kobe ground acceleration to the system defined in 2. Examine the responses of the first three modes. Generate both the time histories and the Fourier components.
4. If the design objective is to have uniform peak element shear deformation throughout the system, what design modifications would you suggest? Illustrate
your strategy for the case where the target value of the relative internodal displacement is $0.0125 \mathrm{~m}$.

Consider an undamped SDOF system with acceleration-based negative linear feedback. The governing equation for free vibration response allowing for time delay is
$$
m \ddot{u}+k u+k_m \ddot{u}(t-\tau)=0
$$
1. Express the hom*ogeneous solution as $u=e^{\lambda r}$. Derive the following expression for $\lambda$.
$$
\lambda^2\left(m+e^{-\lambda \tau} k_m\right)+k=0
$$
2. Substitute for $e^{-\lambda \tau}$ using the Pade approximation, Eq. (8.52), in (P17.1) and the following notation:
$$
\begin{aligned}
k_m & =\alpha m \\
\tau & =\frac{1}{\omega} \bar{\tau}=\frac{T}{2 \pi} \bar{\tau}
\end{aligned}
$$

Show that (P17.1) expands to
$$
\bar{\lambda}^3(1-\alpha) \frac{\bar{\tau}}{2}+\bar{\lambda}^2(1+\alpha)+\bar{\lambda} \frac{\bar{\tau}}{2}+1=0
$$
3. Take $\alpha=0.05$. Solve (P17.2) for a set of values of $\bar{\tau}$ ranging from 0 to $\pi$. Plot $\lambda_I$ vs $\lambda_R$. Discuss whether instability is possible as the delay increases.
4. Following the approach described in Sect. 8.2, express $\lambda$ as
$$
\lambda=i \Theta
$$
and substitute for $\bar{\lambda}$ in (P17.1). Verify that the solution for $\Theta$ and the maximum allowable $\bar{\tau}$ is
$$
\begin{aligned}
\Theta^2 & =\frac{1}{1-\alpha} \\
\bar{\tau}_{\max } & =\frac{\pi}{\Theta}=(1-\alpha)^{1 / 2} \pi \\
\tau_{\max } & =\frac{T}{2}(1-\alpha)^{1 / 2}
\end{aligned}
$$

Compare this result with that obtained with the Pade approximation.