Note: Sections 4.1 and 4.2 are taken from Aerospace Measurements and Instrumentations (ASE 369K) lab report Behavior of Second Order Systems [18]. These sections discuss second order system theory and have been included because the controller set-up is a second order system, which can be modeled as an RLC circuit.
An RLC circuit driven by numerous frequencies of a sinusoidal function can be modeled to represent a second-order system. The total response of an RLC circuit to a sinusoidal function input can be characterized as the sum of the transient and steady state responses of the system. RLC circuit theory as well as second order system is discussed extensively in this section.
For an RLC circuit model, the configuration shown in Figure 15 can represent the process dynamics of the system.
Figure 15: Diagram of a typical RLC circuit model [17]
The components of the RLC circuit are the source voltage v(t), a resistor R, an inductor L, and a capacitor C. The circuit can be represented as a differential equation of the form
(4.1)
Next, an equation is derived for the voltage across the capacitor, which is used to determine the current flowing in the circuit. Consequently, Eq. (4.1) can also be represented as Eq. (4.4)
(4.2)
(4.3)
(4.4)
With this result, the current in the RLC circuit can be found once Eq. (4.4) has been solved for the voltage across the capacitor [17].
The expression “second order system” simply means that the power of the highest derivative in the general equation describing the system will be two. If the number of derivatives in a second order system is n = 2, a second order system will have integration constants. The general equation for a second order system can be written as
(4.5)
where on the right hand side of the equation is known as the forcing function. For an RLC circuit, the general equation for a second order system will take the form
(4.6)
Where
The form of the general second-order system equation in Eq. (4.6) will have both a transient and a steady state solution. The sum of the transient and steady state solutions is the total solution of a second order system [17].
The steady state response of a system, also known as the particular solution, will be directly related to the input function (forcing function) going into the system. With the steady state response constrained by the forcing function, the assumed solution for the steady state response will be
(4.7)
Assuming a solution of this form, Eq. (4.7) is then substituted into Eq. (4.6):
(4.8)
The constant, is then solved by simple integration.
The transient solution of a second order system, also known as the homogeneous solution, is dependant on the design of the system or the system parameters. Again, considering the general equation for second order systems, Eq. (4.6), the transient solution can by found by adding the steady state response to the left hand side of the equation and setting the right hand side of the equation equal to zero. This step yields
(4.9)
Because the derivatives of the solution to Eq. (4.9) must have the same form as the actual solution, an exponential function is proposed as the solution to Eq. (4.9):
(4.10)
However, it is clearly seen from Eq. (4.10) that the value of (damping) will affect the response of the second order system. The three cases which the value of may attain are given as (> 1), (= 1), and (< 1). The value of can directly affect that rate at which any oscillations in a system will decay as well as the lag (phase angle) of the response to the corresponding input. Figures 16 and 17 illustrate how the value of the damping factor may impact the rate of decay and lag of a response [17].
Figure16: Second Order System Time Response [17]
Figure 17: Second Order System Phase Response [17]
Now that a formal discussion of how second order systems work has been presented, Active Wing Engineering may present their results from the control systems analysis with confidence that the reader will be able to understand what is being discussed. As previously mentioned, Active Wing Engineering determined the closed loop transfer function of the process dynamics to be
With this transfer function, it was possible to determine how the actuation system would respond to a step input from the controller. Using the step function from the Matlab computer program, the following plot was attained.
Figure18: Step response plot of control system
From the step response plot, many important results can be extrapolated. Figure 18 is repeated below, but this time data labels have been placed on key areas of the plot.
Figure19: Step response plot of control system with data labels
As can be seen from the figure above, the rise time of the control system is favorably small (approximately .0793 sec). This means that the control system will come online almost immediately after an input command is given. Hydraulic actuation systems would typically not have this quick of a response time because they exhibit some compression in the hydraulic fluid before pressure waves are allowed to propagate.
The next important result to discuss is the peak amplitude of the control system, which is labeled on Figure 19. It is important to notice that for a step input, the system responds at a peak amplitude of A = 1.59. This is approximately a 60% overshoot of the desired result. This value is significant because damage could be caused when driving the actuation system because the system will initially respond 1.6 times the original input. On the positive side, however, the control system is quickly able to compensate for this overshoot. It is seen that the effects of the initial overshoot are settled after a period of about 1.78 seconds.
Perhaps the most important result of all is the steady-state error from the response plot. As can be seen, the final value of the control system from the step input is 1. This means that the model shows no steady-state DC voltage bias. In the special case of ideal actuation system components, this would also imply a zero steady-state frequency bias. However, it is highly doubtful that any realistic actuation system could be constructed without having any inefficiencies or frictional losses present in the system. In a more realistic sense, the actuation system would likely still exhibit a zero steady-state DC voltage bias, however it would also exhibit some degree of frequency bias dependant on the frictional losses and damping present in the system.
Referring back to Figure11 of the electric motor, it can be seen that the rotational motion of the motor must be converted into linear motion to move the bell-crank of the actuation system. This is done with the use of slider cranks. On the following page, Figure 11 has been re-printed to show how the slider crank will be employed.
Although the electric motor will be operating at a set frequency, the frequency of the flap oscillations will be determined by the size of the gears used to tie to the connecting rod. Also, the location of where the connecting rod is tied to the gear will affect the oscillating frequency of the flaps. Using the definition of frequency
Active Wing Engineering can gear down the motor so that the actuation system will operate at the desired frequency of about 20 Hz.
The task of gearing and constructing the actuation system has been reserved for the second half of the semester. From the primitive design from Figure 11 Active Wing Engineering has determined that the process dynamics of the control system will have to be remodeled because of the added complexity of the system. Additionally, the group suspects that the addition of gears and a slider crank assembly will introduce a steady-state error to the control system. However, with careful planning the group believes that the actuation system can be constructed with minimal errors.
