the inverse of the control effectiveness matrix in order to obtain actuator commands to drive that error to zero. As with any system, there are dynamics that are not of interest or are not directly controllable. For an aircraft with sufficient control degrees of freedom, it is possible to select a set of controlled variables, such as rotational velocities, and design a controller that in the closed loop would keep rotational degrees of freedom independent of the translational axes. Hence, provided that the translational dynamics follow certain assumptions, e.g., they are stable, we are ensured that the desired dynamics are not corrupted by the translational degrees of freedom and the system is stable by design. In the case of a flexible vehicle, the nature of internal system dynamics changes. The translational and rotational degrees of freedom can still be separated and rotation can be controlled without interference from the former, but now the dynamics in the frequency range beyond the controller bandwidth must be considered. These dynamics become internal dynamics as well as influence the controlled dynamics and the internal dynamics of the translational degrees of freedom. Specifically, assume that there is interest in controlling the pitch rate of the aircraft as well as the behavior of the first few fuselage flexible modes that come close to the pilot operating bandwidth. Two things make this problem very different from a typical one. First, in typical cases the actuator dynamics are fast enough to be negligible within the frequency range of interest. Second, the higher frequency dynamics are sufficiently far enough away from both the controller bandwidth and the actuator dynamic frequency and can be neglected. If these dynamics are close enough to the actuator dynamic frequencies such that their effect cannot be ignored, then they are dealt with on separate basis employing notch filters. However, in the problem under consideration neither case holds. In this problem, there are several actuators for controlling the vehicle with different dynamic capabilities. In addition, the higher frequency dynamics are very close to the dynamics being controlled and hence cannot be discounted or controlled with notch filters. As already mentioned in Chapter 3 during the discussion of the open loop dynamics, the proximity of flexible mode
dynamics to the flight dynamics as well as close clustering of a number of flexible modes suggest a need for an integrated flight/SMC law for optimum aircraft performance.
In fact, further modifications are required for successful implementation of an integrated flight/SMC controller using dynamic inversion. The modifications to the
methodology and the motivation behind them are discussed in detail below. A dynamic inversion controller designed in a typical manner, i.e., dynamic cancellation in the frequency range of interest, and then applied to the entire system that includes higher frequency modes and actuator dynamics behaves nothing like the design. Consider only the system dynamics for which the controller has been designed and then add the actuator dynamics. The resulting system immediately reflects the destabilizing effect of this addition on the closed loop system. The dynamics that are destabilized are the flexible modes, which are half as fast as the slowest actuator, i.e., ~12 rad/sec. This is reflected in Figure 5.3 presented in a later part of the chapter. If the actuator dynamics are sped up to about 6 to 7 times the speed of the primary fuselage mode under control then the original controller functions as per design. Hence, somehow the original controller must be modified in order to effectively deal with actuator dynamics as well as higher frequency dynamics that are also destabilized by the presence of actuator dynamics. (Recall from Chapter 3 Figures 3.7 and 3.8 the effect the presence of actuator dynamics had on open loop response.) One way of doing this is to make sure that the controller bandwidth is as small as the performance requirements allow and have the controller roll-off as quickly as possible past that bandwidth frequency.
Consider again the control variables of interest, [qma,qps −qma]. The difference between the desired and the actual dynamics can be compared at two different frequency intervals. The first frequency interval upper bound is restricted to the highest frequency of the system model used for controller. In this range, the error between desired and actual is, in fact, something that the controller is designed to drive to zero. The second interval contains the frequencies higher than those of the design model. In this case, the error is in reality just the actual system response in that frequency range. If this were fed back to the controller, then the controller would attempt to react to this error for which it was not designed, move the actuator in response to these higher frequencies and hence start destabilizing the system in both frequency intervals. One solution would be to impose a low-pass filter on the error fed into the control effectiveness matrix in order to minimize the impact of the higher frequency dynamics and thus let the controller deal
would be placed in the inversion loop of the controller preceding the feed into the control effectiveness matrix.
Further consideration must be given to the question of direct flexible mode control
within the frequency range of the design model. Applying standard dynamic inversion to
this portion of the problem has several drawbacks. First, canceling dynamics close to the jω-axis is not very prudent in case there are model mismatches and exact cancellation does not occur. More importantly, simply canceling flexible mode dynamics only means that they are not observed at the pilot station, it does nothing to improve their response to turbulence or other excitation. So instead of cancellation, the proper objective for
flexible modes is to improve their dynamic response by increasing their damping ratio and design the flight control portion of the compensator such that it minimizes the excitation of flexible modes in the first place. These dual objectives, in addition to discussion in the paragraph above, lead to the development of the novel dynamic inversion technique that is the subject of this research.
The linear controller portion can also be viewed as specifying the desired dynamics of the variable being controlled that the inversion part of the controller will match where dynamic inversion is perfect. With this in mind, the linear controller portion started as a PI compensator. The PI controller, also known as a lag compensator, is essentially a low- pass filter. The attenuation characteristic of the PI compensator is useful and permits an increase in loop gain. The lag part of the phase-shift characteristic is detrimental to system performance (destabilizing effect) but must be tolerated. The proportional part increases the gain of the system at low frequencies for better performance response while the integral part rolls off the controller for better response to high frequency dynamics and disturbance attenuation. That is a classical reason and approach to pitch rate control, which is made even more important here by the fact of close proximity of flexible modes to the pilot bandwidth. The case in the other loop, elimination of elastic mode excitation at the pilot station, produces the best response with only a proportional controller. The destabilizing effect of the lag on the phase at high frequency plays a decisive role.
The details of the novel dynamic inversion controller design based on the principals discussed in this section are described below, followed by analysis of the actual