Question

We consider the equations of motion of an airplane in a vertical plane. If the units are scaled appropriately (forward speed equal to one for instance), then these equations are approximately $$ \begin{aligned} &\dot{\gamma}=\sin \alpha, \\ &\ddot{\theta}=-(\alpha-u), \\ &\dot{h}=\sin \gamma, \end{aligned} $$ where \- \(h\) is the height of the airplane with respect to a certain reference height; \- \(\gamma=\theta-\alpha\) is the flight angle; \- \(\theta\) is the angle between the reference axis of the airplane and the horizontal; \- \(u\) is the rudder control. One must design an automatic pilot to keep \(h\) constant (and equal to zero) in the presence of all kinds of perturbations such as vertical gusts. \- Linearize the equations of motion and write them as a set of first order differential equations. \- Show that the designer who proposes a feedback of the form \(u=k h\), where \(k\) is a suitably chosen constant, cannot be successful. \- Prove that a feedback of the form \(u=k_{1} h+k_{2} \theta\), with suitably chosen constants \(k_{1}\) and \(k_{2}\), 'does the job', i.e. the resulting closed-loop system is asymptotically stable.

Short answer

Using \( u = k_1 h + k_2 \theta \) with suitable constants stabilizes the system. Feedback \( u = k h \) fails to keep \( h \) constant.

Step-by-step solution

Linearize the Equations of Motion

To linearize the equations around the equilibrium point where the height \( h = 0 \), the angles \( \gamma = 0 \), and \( \theta = 0 \), assume that the small perturbations are \( \Delta h \), \( \Delta \gamma \), and \( \Delta \theta \). The small angle approximations \( \sin \alpha \approx \alpha \) and \( \sin \gamma \approx \gamma \) can be used.The linearized system becomes:\[\begin{aligned}&\dot{\Delta \gamma}=\alpha, \&\ddot{\Delta \theta}=-(\alpha-u), \&\dot{\Delta h}=\gamma.\end{aligned}\]

Convert to First Order Differential Equations

To convert the second-order differential equation \( \ddot{\Delta \theta}=-(\alpha-u) \) into a system of first-order equations, we introduce new variables:Let \( \Delta \theta_1 = \Delta \theta \) and \( \Delta \theta_2 = \dot{\Delta \theta} \). Therefore, \( \dot{\Delta \theta}_1 = \Delta \theta_2 \) and \( \dot{\Delta \theta}_2 = -(\alpha - u) \).The complete first order system becomes:\[\begin{aligned}&\dot{\Delta \gamma}=\alpha, \&\dot{\Delta \theta}_1=\Delta \theta_2, \&\dot{\Delta \theta}_2=-(\alpha-u), \&\dot{\Delta h}=\gamma.\end{aligned}\]

Analyze Feedback Control \(u = k h\)

Substitute \( u = k \Delta h \) into the system:\[\begin{aligned}&\dot{\Delta \gamma}=\alpha, \&\dot{\Delta \theta}_1=\Delta \theta_2, \&\dot{\Delta \theta}_2=-(\alpha - k \Delta h), \&\dot{\Delta h}=\Delta \gamma.\end{aligned}\]Notice that \( \Delta \gamma \) still explicitly depends on \( \alpha \), which indicates that the perturbation \( \alpha \) cannot be canceled by the control \( u = k \Delta h \), thus the height cannot be held constant effectively.

Analyze Feedback Control \(u = k_1 h + k_2 \theta\)

Now, substitute \( u = k_1 \Delta h + k_2 \Delta \theta \) into the system:\[\begin{aligned}&\dot{\Delta \gamma}=\alpha, \&\dot{\Delta \theta}_1=\Delta \theta_2, \&\dot{\Delta \theta}_2=-(\alpha - k_1 \Delta h - k_2 \Delta \theta), \&\dot{\Delta h}=\Delta \gamma.\end{aligned}\]For the closed-loop system to be asymptotically stable, the eigenvalues of the system matrix should have negative real parts. By choosing appropriate values for \( k_1 \) and \( k_2 \), such as ensuring the characteristic polynomial of the system matrix has roots with negative real parts, the system becomes stable, thus "doing the job."

Key concepts

Linearization of Differential Equations

The process of linearization is vital in simplifying complex systems. It takes non-linear equations and approximates them with linear equations. This is particularly useful at an equilibrium point where the variables are small, allowing for simpler analysis and solutions. In the case of the airplane's equations of motion, the system was linearized around the equilibrium point where the height, flight angle, and orientation angle are all zero.

This linearization uses the small angle approximations: \( \sin \alpha \approx \alpha \) and \( \sin \gamma \approx \gamma \). These approximations simplify the equations significantly, transforming terms involving trigonometric functions into simple linear expressions of the angles.

This transformation is crucial because it allows us to apply linear control techniques, which are generally easier and more robust compared to dealing with non-linear controls. By considering the first-order approximations, the control systems can be constructed more intuitively, paving the way for analyzing feedback mechanisms effectively.

Asymptotic Stability

Asymptotic stability refers to a system's ability to return to equilibrium after experiencing a disturbance. This characteristic is crucial in control systems, ensuring that any perturbation, such as a sudden gust of wind in the case of an airplane, does not lead to instability. When analyzing the feedback control for the airplane, we aim for asymptotic stability.

One way to ensure asymptotic stability is to examine the eigenvalues of the system matrix derived from our linearized control equations. Specifically, all the eigenvalues should have negative real parts. This property ensures that any deviation from the equilibrium state diminishes over time, driving the system back to its stable state.

In practical terms, for the airplane’s system, this means that carefully selecting feedback constants \( k_1 \) and \( k_2 \) can "do the job" by ensuring all disturbances are mitigated over time, leading the closed-loop system to stabilize.

Feedback Control Design

Feedback control design involves creating strategies to adjust inputs to a system to achieve desired outputs. In our airplane problem, the goal is to design a control mechanism that maintains constant height despite disturbances.

Initially, a simple feedback of the form \( u = k h \) was suggested, where \( k \) is a constant. However, this approach proved insufficient because it failed to account for variations introduced by \( \alpha \), the perturbation angle. Consequently, the system could not achieve the desired height stabilization.

A more effective strategy incorporates not only the height \( h \) but also the orientation angle \( \theta \). By implementing a feedback control \( u = k_1 h + k_2 \theta \), it introduces an adaptive response that can fine-tune the control inputs based on both variables, h and \( \theta \). This dual-variable approach allows the system to react more dynamically to changes and ensures that the control system remains robust and asymptotically stable.