Stability Under Constraint
Both civil and military helicopters are required to operate in confined spaces, often in conditions of poor visibility and in the presence of disturbed atmospheric conditions. To assist the pilot with flight path (guidance) and attitude (stabilization) control, some helicopters are fitted with automatic stability and control augmentation systems (SCAS) that, through a control law, feedback a combination of errors in aircraft states to the rotor controls. The same effect can be achieved by the pilot, and depending on the level of SCAS sophistication and the task the share of the workload falling on the pilot can vary from low to very high. The combination of aircraft, SCAS and pilot, coupled together into a single dynamic system, can exhibit stability characteristics profoundly different than the natural behaviour discussed in detail in Chapter 4. An obvious aim of the SCAS and the pilot is to improve stability and task performance, and in most situations the control strategy to achieve this is conceptually straightforward – proportional control to cancel primary errors, rate control to quicken the response and integral action to cancel steady-state errors. In some situations, however, the natural control option does not always lead to improved stability and response, e. g., tight control of one constituent motion can drive another unstable. It is of interest to be able to predict such behaviour and to understand the physical mechanisms at work. A potential barrier to physical insight in such situations, however, is the increased dimension of the problem. The sketch of the Lynx SCAS in Chapter 3 (see Fig. 3.36) highlights the complexity of a relatively simple automatic system. Integrating the SCAS with the aircraft will lead to a dynamic system of much higher order than that of the aircraft itself; the pilot behaviour will be even more complex and the scope for deriving further understanding of the dynamic behaviour diminishes, as the complexity and order of the integrated system model increases.
One solution to this dilemma was first discussed by Neumark in Ref. 5.1, who identified that it was possible to imagine control so strong that one or more of the motion variables could actually be held at equilibrium or some other prescribed values. The behaviour of the remaining, unconstrained, variables would then be described by a reduced order dynamical system with dimension even less than the order of the natural aircraft. This concept of constrained flight has considerable appeal for the analysis of motion under strong pilot or SCAS control because of the potential for deriving physical understanding from tractable, low-order analytic solutions. Neumark’s attention was drawn to solving the problem of speed stability for fixed-wing aircraft operating below minimum drag speed; by applying strong control of flight path with elevator, the pilot effectively drives the aircraft into speed instability. In a later report, Pinsker (Ref. 5.2) demonstrated how, through strong control of roll attitude on fixed-wing aircraft with relatively high values of aileron-yaw, a pilot could drive the effective directional stiffness negative, leading to nose-slice departure characteristics. Clearly, a helicopter pilot has only four controls to cope with 6 DoFs. If the operational situation demands that the pilot constrains some motions more tightly than others, then there is always a question over the stability of the unconstrained motion. If strong control is required to maintain a level attitude for example, then flight path accuracy may suffer and vice versa. Should strong control of some variables lead to a destabilizing of others, then the pilot should soon recognize this and subsequently share his workload between constraining the primary DoFs and compensating for residual motions of the weakly constrained DoFs. This apparent loss of stability can be described as a pilot-induced oscillation (PIO). However, this form of PIO can be insidious in two respects: first, where the unconstrained motion departs slowly, making it difficult for the pilot to identify the departure until well developed, and second, where a rapid loss of stability occurs with only a small increase in pilot gain. With helicopters, the relatively loose coupling of the rotor and fuselage (compared with the wing and fuselage of a fixed-wing aircraft) can exacerbate the problem of constrained flight to the extent that coupled rotor/fuselage motions can occur which have a limited effect on the aircraft flight path, yet cause significant attitude excursions.
In the following analyses, we shall deal with strong attitude control and strong flight path control separately. We shall make extensive use of the theory of weakly coupled systems (Ref. 5.3), which was used to investigate motion under constraint in Ref. 5.4. The theory is described in Appendix 4A and has already been utilized in Chapter 4 in the derivation of approximations for a helicopter’s natural stability characteristics. The reader is referred to these sections of the book and the references for further elucidation. The method is ideally suited to the analysis of strongly controlled aircraft, when the dynamic motions tend to split into two types – those under control and those not – with the latter tending to form into new modes with stability characteristics quite different from those of the uncontrolled aircraft. In control theory terms, these modes become the zeros of the closed-loop system in the limit of infinite gain.