The Scientific World Journal

Volume 2015 (2015), Article ID 523914, 10 pages

http://dx.doi.org/10.1155/2015/523914

## Helicopter Control Energy Reduction Using Moving Horizontal Tail

College of Aviation, Erciyes University, 38039 Kayseri, Turkey

Received 7 November 2014; Revised 14 February 2015; Accepted 1 April 2015

Academic Editor: Zheng Zheng

Copyright © 2015 Tugrul Oktay and Firat Sal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Helicopter moving horizontal tail (i.e., MHT) strategy is applied in order to save helicopter flight control system (i.e., FCS) energy. For this intention complex, physics-based, control-oriented nonlinear helicopter models are used. Equations of MHT are integrated into these models and they are together linearized around straight level flight condition. A specific variance constrained control strategy, namely, output variance constrained Control (i.e., OVC) is utilized for helicopter FCS. Control energy savings due to this MHT idea with respect to a conventional helicopter are calculated. Parameters of helicopter FCS and dimensions of MHT are simultaneously optimized using a stochastic optimization method, namely, simultaneous perturbation stochastic approximation (i.e., SPSA). In order to observe improvement in behaviors of classical controls closed loop analyses are done.

#### 1. Introduction

Traditionally in order to control helicopters, collective and cyclic (i.e., longitudinal and lateral) rotor blade pitches are used. Presently almost all of the helicopters employ a swashplate mechanism and pitch links (it consists of two circular plates and a ball bearing arrangement separating them; see [1] for more details) to transmit two cyclic and collective pitch commands to the blade root. However, this mechanism is heavy and complex and also causes important drag during high flight speeds. Throughout history some other control methods have been considered in order to avoid these drawbacks and also for some other reasons such as reduction of control energy and redundancy in case of failures. Some of these alternatives are using trailing edge flaps (TEFs) with (see [2–4]) and without (see [5–7]) classical swashplate mechanism, passive (see [8–10]) and active (see [11–13]) helicopter morphing, and MHT (see [14–18]). For example, in [2] TEFs were integrated into blades for the case of a failure of the pitch link making the blade free float in pitch. By this method catastrophic results of a pitch link failure were corrected. In [5] TEFs were replaced with a conventional swashplate mechanism. Via eliminating swashplate mechanism and using TEFs, important reduction in weight, drag, and cost and also improvement in rotor performance were obtained. Moreover, in [8] passive morphing was used in order to reduce helicopter FCS energy. In that study many blade parameters (e.g., blade length and blade chord length) were simultaneously optimized with helicopter FCS parameters in order to save FCS energy. Substantial reduction in helicopter FCS energy was obtained using passive morphing idea. In [11] active morphing was used to save helicopter FCS energy. In that study actively morphing parameters were blade chord length, blade length, blade twist, and main rotor angular speed. The main difference between this and previous study (i.e., passive morphing) was that for the active case the helicopter design parameters are able to change (except helicopter FCS) during flight, but in prescribed interval. Using active morphing idea significant reduction in helicopter FCS energy was obtained.

MHT idea was firstly studied in [14] in 1953. In this study differential control of each side of a canted horizontal tail was permitted. In [15] collective control of horizontal tail was mechanically achieved. In this study it was claimed that a fixed horizontal tail is advantageous in order to improve longitudinal stability of helicopters in forward flight, but it is not enough during gliding and climbing flights. More recently in [16] a moveable horizontal tail to give the desired attitude at different flight speed for UH-60 was designed. Recently in [17, 18] a moveable horizontal tail was designed in order to reduce TEF deflection for swashplateless helicopters since the stroke capacity of existing smart material actuators is not enough for the required TEFs inputs. Using MHT TEFs deflections were relieved.

Numerous helicopter FCS design methods have been studied throughout the years, in historical sequence classical pole placement techniques (see [19, 20]), simple feedback control methods (see [21, 22]), and modern control approaches depend on linear matrix algebra such as linear quadratic regulator (LQR) and linear quadratic Gaussian (LQG) techniques (see [23, 24]), control synthesis (see [25, 26]), and model predictive control (MPC) (see [27, 28]). In this paper, a modern constrained control method, namely, OVC, is chosen for the design of helicopter FCS. OVC has many advantages with respect to the other control strategies existing in the literature. First, these controllers are modified LQG controllers and they benefit from Kalman filters as state estimators. Second, variance constrained controllers apply second-order information (i.e., state covariance matrix; see [29, 30] for details) and this kind of information is very beneficial during multivariable control system design because all stabilizing controllers are parameterized in relation to the physically meaningful state covariance matrix. Last, for large and strongly coupled multi-input, multi-output (MIMO) systems as in air vehicles control and especially in this paper, variance constrained control methods give guarantees on the transient behavior of independent variables by enforcing upper limits on the variance of these variables.

Variance constrained controllers have been used for many aerospace vehicles (e.g., helicopters, see [8, 11, 31–36]; tiltrotor aircraft, see [37]; Hubble space telescope, see [38]; tensegrity structures, see [39]) in last thirty years. For instance, in [32] variance constrained controllers were used for helicopter FCS during maneuvers, specifically level banked turn and helical turn. In that paper, performance of them was also considered during failures of some helicopter sensors. Reasonable consequences (meaning that variance constraints on outputs/inputs were satisfied and also closed loop systems were exponentially stabilized) were found in terms of helicopter FCS. Robustness of the closed loop systems (obtained via integration of linearized helicopter model and FCS) with respect to some modeling uncertainties (i.e., variation of flight conditions and all helicopter inertial parameters) was also studied and it was found that these controllers have stability robustness with respect to modeling uncertainties.

In this paper, MHT is for the first time simultaneously designed with helicopter FCS. For this purpose, a specific variance constrained controller OVC is also used for the first time for FCS. It is important to note that when MHT is integrated with classical helicopter, the number of controls increases. This causes an important result. The number of trim unknowns increases with additional MHT controls. Nevertheless, there are no additional trim equations. Therefore, in order to solve the resulting nonlinear trim equations, a useful optimization algorithm is required. For its solution, a stochastic optimization method specifically simultaneous perturbation stochastic approximation (i.e., SPSA) (see [40, 41] for brief description of SPSA) is for the first time applied for the simultaneous trimming and FCS design problem since it is computationally cheap and effective during solving constrained optimization problems when it is impossible to compute derivatives such as gradients and Hessians, analytically as in the situation herein. This paper first presents helicopter models used for simultaneous MHT and FCS design. Second, MHT is illustrated and motions of it are described. Then, definition of applied FCS (i.e., OVC) is given briefly. After that, trimming the system (i.e., the one obtained via integration of helicopter, MHT, and FCS) via simultaneous trimming and FCS design idea is explained. Then, the specific optimization method, namely, SPSA, applied in order to trim the system is summarized. Finally, this simultaneous design idea is applied for Puma SA 330 helicopter and closed loop responses of classical helicopter and helicopter with MHT are compared.

#### 2. Helicopter Model

The modeling approach of used helicopter models in this paper is presented in detail in [31, 42]. The essential modeling assumptions are given next. First of all, multibody system approach was used to include all helicopter components: fuselage, horizontal tail, tail rotor hub and shaft, landing gear, and fully articulated main rotor with 4 rigid blades with blade flapping and lagging hinges. Secondly, a static inflow formulation (i.e., Pitt-Peters formulation) was applied for helicopter main rotor downwash. Thirdly, linear incompressible aerodynamics was used for the main rotor blades, but an analytical formulation was applied for the modeling of fuselage.

The modeling procedure requires using physics principles and because of the assumptions described in the previous paragraph it directly led to helicopter dynamic models that consisted of finite sets of ordinary differential equations (ODEs). This mathematical structure is fairly beneficial for control system design since it assists the direct use of modern control theory, which relies on state space representations of the system’s dynamics, easily obtained from ODEs.

The modeling methodology summarized above was applied in Maple and it led to a nonlinear helicopter model in implicit form:where , , and . Here and are nonlinear state and control vectors, respectively, and represents the linear space of -dimensional real vectors, where “” can be 28, 25, or 4. It should be noted that the inconsistency between the size of (28) and the size of (25) is due to the three static downwash equations. The 28 nonlinear equations in (1) are categorized as follows: 9 fuselage equations, 8 blade flapping and 8 blade lead-lagging equations, and 3 static main rotor downwash equations. The helicopter models obtained have too many terms, making its use in fast computation impractical. For that reason, a systematic model simplification technique, named ordering scheme, was applied to reduce the number of terms in the nonlinear ODEs. The ordering scheme iteratively deletes terms from an equation depending on their relative magnitude with respect to the other terms in that equation. Each term’s magnitude is guessed depends on expected values that the state and control variables can take during helicopter flight (see [31, 42]). It is significant to note that the ordering scheme does not change the number or type of equations generated using physics principles; it just shortens the equations by retaining the dominant terms.

The model found after using the ordering scheme is still reasonably complex (i.e., with a total of 28 nonlinear equations). In this paper, for FCS design the nominal trajectories considered are straight level flights. When the straight level flight conditions were applied for the nonlinear equations of motion, 17 trim equations were found (i.e., equations were deleted). These equations were solved using MATLAB for different straight level flight speeds. After trimming, the model was linearized using Maple, yielding continuous linear time-invariant (LTI) systems:Here and are the perturbed state and perturbed control vectors. Matrices and are of size and . The state vector consists of 9 fuselage states, 8 blade flapping states, and 8 blade lead-lagging states. The control vector includes 3 main rotor controls (collective, , longitudinal cyclic, , and lateral cyclic, , blade pitch angles) and 1 tail rotor control (collective, ).

Puma SA 330 helicopter (see [31, 43]) was used to validate the models used in this paper. These models are leading to acceptable agreement on trim values, flight dynamics modes, and qualitatively similar flapping and lead-lagging mode behavior (see [43]). In Table 1 and Figure 1 some validation results show how the models correctly capture the dynamics of Puma SA 330 helicopter (see [31] for more validation data). For instance, most of the flight dynamics modes (linearized system eigenvalues) of the models for hover and straight level flights (i.e., 40 kts and 80 kts) match well the results reported in [43]. The mode displaying the largest discrepancy is the 4th mode (it is important to note that this is due to modeling discrepancy between the models used and [43]); nevertheless, the qualitative behavior is similar (they are both exponentially stable modes).