Question

Open Loop Analysis For Aircraft Pitch Control

With Pitch angle input of 0.15 radians

Open Loop Analysis Using Simulink Procedure: Plot a step response of the open loop system Determine the overshoot, settling time, rise time, peak time and a steady state value of the output. Comment the system's characteristics. Actuator Aircraft -I0 s + 10 -10(s 1s0.01) (s2 + 2s + 2)(s2 + 0.02s + 0.0101)

Answer 1

Below is the Simulink model for the open-loop control system given in the question.

0.7424 -10 S+ 10 Actuator Y(s) U(s) Aircraft theta dot d theta dot

By running the simulation multiple numbers of times, it is observed that the output reaches a steady state value somewhere around 650-700seconds. So, the simulation is run for 1000 seconds.

Below are the block parameters for the Actuator and aircraft blocks.

Block Parameters: Actuator Transfer Fcn The numerator coefficient can be a vector or matrix expression. The number of rows in the numerator coefficient. You should specify the coefficients in descending order of powers of s Parameters Numerator coefficients: 1-10] Denominator coefficients: 1 101 Absolute tolerance: auto State Name: (e.g., 'position') OKCancel Hlp Apply

aBlock Parameters: Aircraft Transfer Fcn The numerator coefficient can be a vector or matrix expression. The denominator coefficient must be a vector. The output width equals thee number of rows in the numerator coefficient. You should specify the coefficients in descending order of powers of s Parameters Numerator coefficients: -10*conv([1 1],[1 0.01]) Denominator coefficients: conv([1 2 2],[1 0.02 0.0101]) Absolute tolerance: auto State Name: (e.g., 'position') OK Cancel Help Apply

The block parameters for the block theta_dot_d for pitch angle input of 0.15 radians is

| Block Parameters: theta-dot,d Step Parameters Step time: C) Initial value: C) Final value: 0.15 Sample time: C) Interpret vector parameters as 1-D Enable zero-crossing detection OK Cancel Help Apply

Since the simulation is run for a very long duration, we need to set the configuration parameters appropriately to view the desired results.

Configuration Parameters: untitled/Configuration (Active) Q Search Solver Data Import/Export Optimization Diagnostics Hardware Implementation Model Referencing Simulation Target Code Generation Coverage HDL Code Generation Simulation time Start time: 0.0 Stop time: 1000 Solver options Type: Variable-step Solve ode45 (Dormand-Prince) ▼ Additional parameters Max step size: 10 (-3) Min step size: auto Initial step size: auto Number of consecutive min steps: 1 Relative tolerance:1e-3 Absolute tolerance: auto Shape preservation: Disable All Zero-crossing options Zero-crossing control: Use local settings Time tolerance Number of consecutive zero crossings: 1000 ▼ Algorithm: Nonadaptive 10 128 eps Signal threshold: auto Tasking and sample time options Automatically handle rate transition for data transfer OK Cancel Help Apply

Below is the output on the scope window for the 1000 second duration.

File Tools View Simulation Help 100 200 300 400 500 600 700 800 900 1000 Sample based T-1000.000

The steady state value is found to be at 0.7424 on the display block of the Simulink model.

0.7424 -10 S + 10 Actuator Y(s) theta dot d Aircraft theta dot

In order to get the accurate results, we need to zoom in to the data of the scope and check the values by using the 'Cursor Measurement' option on the scope window.

Rise Time (t_r)

It is the time taken for the output to rise from 10% to 90% of the steady-state value.

In order to properly measure the rise time, we will zoom into five to six seconds and measure the times when the output is 10% and 90% of the steady state value and obtain the difference. From the below figure, we can see that the rise time is around 860 milliseconds.

File Tools View Simulation Help Cursor Measurements Settings Measurements Time 0.420 1.280 Value 7.39Be-02 6.691e-01 ΔΙ 860.000 ms AY 5.951e-01 1.163 Hz 691.945 (ks) 1/Δ AY .2 Ready Sample based T-1000.000

Settling Time (t_s)

It is the time by which the output settles within the 2% envelope of the steady-state value. In our case, the 2% envelope is 0.7276 to 0.7572. From the below figure, we can see that the settling time is around 617 seconds.

File Tools View Simulation Help ▼ Cursor Measurements Settings Time Value 7.398e-02 7.277e-01 6.537e-01 617.001 AT 1.622 mHz AYIAT 1.060 (ks) 100 200 300 500 600 700 800 900 1000 Sample based T-1000.000

Peak time (t_p) and Overshoot (OS)

The first peak occurs within the first 25 to 30 seconds. So, we will zoom into that duration (0 to 30 seconds). Peak time is the time by when the output reaches the maximum output for a given input and will trace back to the steady state value. Overshoot is the amount by which the peak differs from the steady-state value.

We will measure the time at which the output reaches the peak and the output after 1000 seconds. The difference between the outputs at these times gives the overshoot. From the below figure we can see that the peak occurs at a peak time of 15.808 seconds and the peak overshoots the steady-state value by 6.379radians

File Tools View Simulation Help FCursor Measurements Settings Measurements Value 7.121e+00 7.424e-01 15.808 1.016 mHz (/ks) -2 4 10 15 20 25 Ready Sample based T-1000.000