Modern Control Methods

The design of controllers using classical methods such as root locus and Bode meth- ods is essentially trial and error [89]. The two concepts defining modern control are that: (a) the design is based directly on the state-variable model and (b) perfor- mance specifications are expressed in terms of a mathematically precise performance criterion [90]. All the gains are solved simultaneously and are directly solved from the state space matrix algebra. Modern controllers offer the prospect for improved performance and can, in some cases, adapt to changing conditions. The downside of most modern controller methods is that they require unmeasured states which can be hard to accurately estimate. Whilst many modern control techniques are shown to be closed loop stable and function on real helicopters, improvements in perfor- mance over equivalent classical controllers on the same plant are rarely confirmed or quantified in the literature. Furthermore, they may require accurate system models which, in the case of a highly non-linear helicopter, may be difficult to achieve. It is often stated that the use of PID control requires laborious tuning to take place. Whilst this is true to some extent, the systems identification step required to de- velop a modern controller is no less laborious and opens up many opportunities for numeric error during the controller design stage. I have included a brief overview

20 Related Work

of the application of modern control theory to helicopter control for the sake of completeness.

Many of the modern control techniques are linear methods. As the helicopter is non-linear, linearisation of the system model is necessary in order to design a working controller. Koo and Sastry present approaches to this based on differen- tial flatness [91] and approximate feedback linearisation [92]. In these techniques, approximate models are developed which are minimum phase, whereas the actual plant being modelled is in fact non-minimum phase [93]. Unfortunately their work was conducted in simulation models which ignore the rotor and actuator dynamics, so successful performance on a real system is not guaranteed.

Optimal control is a branch of control theory that aims to find control laws to maximise some optimality criterion. A typical optimality criterion involves a linear quadratic form which serves to simultaneously minimise the control energy and the time it takes the system to reach steady-state. Noting that the model has to be linearised, and is therefore only an approximation of the real plant, it can be argued that the condition of optimality can become somewhat invalid. Furthermore, the choice of weighting factors used in the optimality criterion can become somewhat arbitrary. Optimal controllers for helicopters have been proposed by Morris [94]; Mettler et al [95] and Bogdanov et al [96]. Morris et al tested a Linear Quadratic Gaussian (LQG) controller [97, 98] on a small electric helicopter constrained to moving in only pitch, roll and yaw. Even with this simplified system, the system performance was not good, owing to unmodelled dynamics and the lack of angular rate sensors. Mettler et al developed a successful LQG controller for the lateral- directional dynamics of a small helicopter by ignoring the flapping dynamics and treating the helicopter as a rigid body. Using a notch filter to attenuate the fuselage- rotor mode, the controller was shown to successfully control bank angle in flight, however removal of the notch filter resulted in instability.

Bogdanov et al describe State-Dependant Riccati Equation (SDRE) control [99] of an X-cell 8kg RC helicopter and the Yamaha RMAX helicopter. The approach consists of approximating the non-linear model with a linear set of state update equations at each time step based on the current state. Treating the state equations as linear, the Ricatti equation is then solved and used to calculate optimal control inputs based on a linear quadratic cost function. Bogdanov’s results for trajectory control of an RMAX flying a 183m×183msquare racetrack pattern are good with a position accuracy of 2m and an altitude accuracy of about 1.2m. However, the computational power required to solve the Ricatti equation is demanding, requiring 14ms out of 20ms at 50Hz on a 300MHz Pentium CPU to compute. This level of processing could not be achieved on either of the systems used for this project as it would have required an additional CPU to be added.

One method of dealing with inaccurate systems identification is to apply the principles ofrobust control. In robust control methods, such asH₂andH∞[100], the objective is to ensure sufficient stability remains in all scenarios that could arise from the presence of noise, disturbances and bounded uncertainties in the plant model. A number of researchers have applied robust controllers to small helicopters. Takahashi applied a H₂ controller to a control of hover in a full-size helicopter simulator.

§2.5 Helicopter Control 21

Bendotti and Morris discuss a H_∞ controller for a model helicopter constrained to 3 degrees of freedom only and show that it out performs an LQG controller [101]. Hashimoto et al develop and flight test a third order H∞ velocity controller for a full-scale unmanned helicopter which augments an existing stability augmentation system [102]. Other examples of H∞ techniques applied to helicopter control are La Civita et al [103] ; Shim et al [93] and Yang et al [104–107]. Recent results from test of a H∞ loop shaping controller on a Yamaha R-50 helicopter at CMU have shown tracking performance which is claimed to exceed the performance of any other techniques in the published literature [108]. On the basis of this result, one would expect that H∞ would be a good candidate for future unmanned helicopter control designs. However, this work was only made possible by the presence of a high fidelity simulation model [103], so I have not pursued the approach in this thesis.

A number of schemes for controlling a helicopter using backstepping [109] have appeared in the literature. Backstepping is a recursive technique whereby the com- plete system is built up in cascaded stages, starting with a simple system that can be stabilised using a known Lyapounov function. As each system is stabilised, an integrator is added to the input and then the same process is repeated to design a Lypanov function and feedback law to stabilise the new system. The process continues until the complete system is stabilised. The advantage of backstepping is that it can be applied to nonlinear system without having to make linearising approximations. It also guarantees stability and can be made to be adaptive [110]. Mahoney and Hamel present simulation results for a robust trajectory tracking con- troller using backstepping [111] which can enable a priori bounds on the tracking performance to be set. A very similar approach is taken by Frazzoli et al [112] who also develop a backstepping trajectory control in simulation. The disdavantage of backstepping is that, even for simple systems, the algebra can become very compli- cated and unwieldy. Hence, the currently published applications of backstepping to helicopters tend to make a number of assumptions to simplify the models that are unrealistic [113], such as assuming that the rotor can be manipulated to provided instantaneous control forces and moments. For example the backstepping algorithm by Mahoney et al [114] assumes that the flapping angles are directly measurable and controllable, neither of which is practical. Whilst backstepping does show promise, the complexity is not consistent with my stated aim of keeping the control systems as simple and practical as possible.