UFSS Vehicle: Pitch-Angle Control Representation We return to the Unmanned Free-Swimming Submersible (UFSS) vehicle introduced in the case studies in Chapter 5 (Johnson, 1980). We will represent in state space the pitchangle control system that is used for depth control. PROBLEM: Consider the block

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UFSS Vehicle: Pitch-Angle Control Representation We return to the Unmanned Free-Swimming Submersible (UFSS) vehicle introduced in the case studies in Chapter 5 (Johnson, 1980). We will represent in state space the pitchangle control system that is used for depth control.

PROBLEM: Consider the block diagram of the pitch control loop of the UFSS vehicle shown on the back endpapers. The pitch angle, θ, is controlled by a commanded pitch angle, [latex] heta _{e}[/latex], which along with pitch angle and pitch-rate feedback determines the elevator deflection, [latex]\delta _{e}[/latex], which acts through the vehicle dynamics to determine the pitch angle. Let [latex]K_{1} = K_{2}=1[/latex] and do the following:

a. Draw the signal-flow graph for each subsystem, making sure that pitch angle, pitch rate, and elevator deflection are represented as state variables. Then interconnect the subsystems.
b. Use the signal-flow graph obtained in a to represent the pitch control loop in state space.

Accepted Answer

a. The vehicle dynamics are split into two transfer functions, from which the signalflow graph is drawn. Figure 5.36 shows the division along with the elevator actuator. Each block is drawn in phase-variable form to meet the requirement that particular system variables be state variables. This result is shown in Figure 5.37(a). The feedback paths are then added to complete the signal-flow graph, which is shown in Figure 5.37(b).

b. By inspection, the derivatives of state variables [latex] x_{1} [/latex] through [latex] x_{4} [/latex] are written as

[latex] \dot{x}_{1} = [/latex] [latex] x_{2} [/latex]

[latex] \dot{x}_{2} = -0.0169x_{1}-0.226x_{2}+0.435x_{3}-1.23x_{3} -0.125x_{4} [/latex]

[latex] \dot{x}_{3} = [/latex] [latex] -1.23x_{3} -0.125x_{4} [/latex]

[latex] \dot{x}_{4} = [/latex] [latex] 2x_{1} +2x_{2} [/latex] [latex] -2x_{4}-2 heta _{c} [/latex]

Finally, the output [latex]y = x_{1} [/latex].

In vector-matrix form the state and output equations are

[latex]\dot{x}=\left[\begin{matrix} 0 & 1 & 0 & 0 \ -0.0169 & -0.226 & -0.795 & -0.125 \ 0 & 0 & -1.23 & -0.125 \ 2 & 2 & 0 & -2 \end{matrix} ight] x+\left[\begin{matrix} 0 \ 0 \ 0 \ -2 \end{matrix} ight] heta _{c}[/latex]

[latex] y= \left[\begin{matrix} 1 & 0 & 0 & 0 \end{matrix} ight] x [/latex]

CHALLENGE: We now give you a problem to test your knowledge of this chapter’s objectives. The UFSS vehicle steers via the heading control system shown in Figure 5.38 and repeated on the back endpapers. A heading command is the input. The input and feedback from the submersible’s heading and yaw rate are used to generate a rudder command that steers the submersible (Johnson, 1980). Let [latex]K_{1} = K_{2}=1[/latex]and do the following:

a. Draw the signal-flow graph for each subsystem, making sure that heading angle, yaw rate, and rudder deflection are represented as state variables. Then interconnect the subsystems.
b. Use the signal-flow graph obtained in a to represent the heading control loop in state space.
c. Use MATLAB to represent the closed-loop UFSS heading control
system in state space in controller canonical form.

Question 5.CS.2: UFSS Vehicle: Pitch-Angle Control Representation We return t...

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