8.9: Chapter References

References

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Solutions to Practice Problems

1. The number of degrees of freedom is b) 3.
2. The answer is b) False. From the middle equation of \ref{8.30}, we can see that when the mini-Segway pendulum angle \(θ_p\) is not equal to zero, there is a term \({\dot{\phi }}^2\) in the equation, affecting the linear motion in \(x\) direction.
3. The answer is a) True. From the bottom equation of \ref{8.30}, we can see that when the mini-Segway pendulum control inputs \(u_{aL}=u_{aR}\), the forcing term of the differential equation for \(\phi\) is zero. As a result, \(\dot{\phi }\) will be damped out or goes to zero.
4. The answer is b) False. From the top and middle equations of \ref{8.30}, we can see that mini-Segway linear motion in \(x\) direction and angular pendulum motion \(θ_p\) are highly coupled. This is the main reason we can use the cart linear motion to stabilize the mini-Segway pendulum.
5. The answer is discrete-time PID is preferred for implementation to a microcontroller. The main reason is that implementing a discrete-time PID controller into a microcontroller will significantly reduce the computational load.
6. The answer is c) 6\(f_h.\) Note that if a) or b) were selected, certain physical system dynamics could be filtered out, leading to slow derivative action for the PID controller. The rule is to filter out as much as noise but keep the system dynamics in the error signal \(e\left(t\right).\)

1. The answer is a), b) and d). Note that for case c), if the integral is enabled, the unsaturated PID control output will continuously increase, causing integration windup.
2. The answer is a).
3. The answer is b) 5.
4. The statement is true. However, for the actual physical system, these two dynamics (cart turning \(\dot{\phi }\) and cart-speed/pendulum-position \({\dot{x}}_{cart}\)/\(θ_p\)) are coupled with a quadratic term. During the process of linearization to obtain the state-space model, the coupling relation is eliminated due to the quadratic coupling relation.
5. The created mode is not controllable. This is mainly due to the fact that any controller designed cannot change/move the integration pole location.
6. The control gain related to the integration error state \(x_{ce}\) is not affected by the \(ε\) value. For example, when the \(ε\) value is adjusted from \(0.0001\) to \(0.001\), the control gain associated with the integration error state \(x_{ce}\) changed from \(0.4880\) to \(0.4877\) with a slight reduction of \(0.0003\) (about 0.0615%). This is very small compared with a ten-time increment of the \(ε\) value. As a result, \(ε\) value does not need to be fine-tuned.

\[A^{-1}_p\left(e^{A_pT_s}-I\right)=T_s+\dfrac{A_pT^2_s}{2!}+\dfrac{A^2_pT^3_s}{3!}+\dfrac{A^3_pT^4_s}{4!}+\cdots\]

Therefore, even though \(A_p\) is not invertible, term \(A^{-1}_p\left(e^{A_pT_s}-I\right)\) can be calculated (it exists). But the inverse of \(A_p\) does not exist because it is singular.

1. The answer is b) because there is at least one pole located at the origin of the complex plan, which makes the system matrix not invertible.
2. The answer is a).
3. The answer is a); see Figure 8.19.
4. The answer is c); see Figure 8.20.
5. The answer is b); see the Simulink nonlinear model shown in Figure 8.8.
6. True. See the discussion before Figure 8.19.

The simulation results are plotted below. Due to increased gains on both cart speed and pendulum position, we can see that the initial oscillation magnitude is reduced, whereas the cart speed tracking performance remains almost unchanged.

1. The answers are b) and c).
2. The answer is c), the I2C bus link.
3. The answer is that all the factors listed from a) to c) could result in the gain difference.
4. The answer is: Yes, a correction factor is needed. The factor should be used to multiply to the designed model-based controller gain, and the factor is \(k_c=22.17\).
5. This modification can be achieved by replacing the dual-PID control logics with the state feedback one similar to the LQTI controller in Figure 8.26. (Delete the integration block and set both the cart-turning displacement and speed to zero).
6. The answer is c); \(γ_c\) is between 0.8 and 1.0.