A SIMPLIFIED ANALYSIS OF NONLINEAR LONGITUDINAL DYNAMICS AND CONCEPTUAL CONTROL SYSTEM DESIGN

Transcription

1 A SIMPLIFIED ANALYSIS OF NONLINEAR LONGITUDINAL DYNAMICS AND CONCEPTUAL CONTROL SYSTEM DESIGN ROBBIE BUNGE 1. Introduction The longitudinal dynamics of fixed-wing aircraft are a case in which classical linearized analysis and control system design methods are not enough. The reasons are two-fold: 1) this is a tightly coupled multi-input, multi-output system, and 2) the transition from one equilibrium condition to another (e.g. steady cruise to steady climb) requires the consideration of nonlinear effects, especially when these the start and end equilibrium conditions are significantly different. Yet, it is possible to develop simple models that capture the main nonlinearities involved. This brief article presents a starting point for the analysis longitudinal dynamics and conceptual design of a longitudinal control law. 2. The nonlinearities The main nonlinear phenomena are: 1) quadratic dependence of aerodynamic forces with airspeed, 2) rotation of aerodynamic forces with flight path angle, 3) quadratic dependence of drag with angle of attack, 4) nonlinear lift curve due to stall, 5) dependence of thrust with airspeed. Of primary importance are the first three sources of nonlinearity. 3. The model We propose a simple flight model, parametrized by the main aircraft design parameters. To do this we do a force balance and moment balance about the CG. (1) Fz = Lcos(γ) mg Dsin(γ) + T sin(θ) = m V z (2) Fx = T cos(θ) Dcos(γ) Lsin(γ) = m V x (3) My = 1 2 ρv 2 [S w c w (C mwac C L w l w /c w + C mq c w 2V o q) C Lh S h l h ] = I y q Adding the kinematic relation: (4) q = θ 1

2 2 ROBBIE BUNGE 4. Trim Analysis Studying the trim relationship are a very important and instructive step in the analysis of longitudinal dynamics. If the airplane is naturally stable (internal dynamics are stable), these will provide insight to what will be the steady state conditions to which the airplane will converge. Thus, if the airplane is stable and there are no external disturbances, we could fly the airplane simply by commanding the proper trim control inputs, and it will eventually converge to the desired flight condition. By setting all derivatives to zero, we obtain the trim conditions Moment balance trim. fg By setting q to zero we obtain: (5) C mwac l w c w C Lw (α) = S hl h S w c w C Lh (α, δ e ) From here it is pretty clear that fixing the elevator deflection maps into a steady state angle of attack. So, steady state elevator position commands steady state angle of attack. In particular, for angles of attack away from stall, there is a linear relation between the angle of attack and the elevator position Vertical force balance trim. To have the airplane trimmed we also require the forces to be balanced. Doing a small angle approximation for both the flight path angle and the pitch angle (which makes sense given these are relatively small angles), we obtain: (6) L(1 D L γ ) = mg }{{} <<1 Given that normally L/D 10 and γ 5deg, these terms can be neglected without loss of first order correctness. Then we get the obvious relationship: lift equals weight. Now if we look at the drag force, we can write it as: (7) L = 1/2ρV 2 SC L (α) = mg Given the previous analysis, we know that the steady state angle of attack is given by the elevator position. Thus, if we solve for the airspeed V, we get: (8) V 2 (δ e ) = 2mg ρsc L (α(δ e )) So, steady state speed is also defined by the steady state elevator position! This makes sense, given that angle of attack is defined by elevator and angle of attack defines lift

3 A SIMPLIFIED ANALYSIS OF NONLINEAR LONGITUDINAL DYNAMICS AND CONCEPTUAL CONTROL SYSTEM DESIGN3 coefficient, which in turn requires a certain dynamic pressure if it is to produce a lift equal to the weight Forward force balance. If we set the derivative to zero, do a small angle approximation and solve for the flight path angle gamma, we get: (9) γ = T D L = T (δ t) mg 1 C L C D (α(δ e )) The flight path angle determines if whether the airplane is in level, climbing or descending flight. Here we can see that the steady state flight path angle is influenced both by the thrust command δ t and the elevator position d e, which already starts to indicate the coupling of control inputs. We can always increase the steady state flight path angle by increasing the thrust. This is not the case for the elevator: if we are at very low speed, such that the angle of attack at which we trim the moment and the vertical weight is above the maximum C L C D point, an increase in elevator will result in descent rather than climb! 5. Flight Dynamics To describe the flight dynamics we propose to have the speed V, flight path angle γ, pitch angle θ and pitch rate q as the state variables. To obtain a set of nonlinear differential equations in terms of these variables, we start by noting that we can express V z and V x from Eqns. (1) and (2), in terms of the speed V and the flight path angle γ, by noting that V z = V sin(γ) and V x = V cos(γ). Thus, (10) (11) If we define F x = (2): V z = V sin(γ) + V cos(γ) γ V γ + V γ V x = V cos(γ) V sin(γ) γ V V γ Fx m and F z = Fz m, we can solve for V and γ using Eqns. (1) and (12) γ = F z F x γ V (1 + γ 2 ) (13) V = Fx + F z γ (1 + γ 2 ) The only thing missing is to describe F x and F z in terms of flight variables V, γ, θ, q and control variables δ e, δ t. We can describe the wing lift coefficient and the tail lift coefficient in a simple manner (neglecting downwash effects on the wing and tail for further simplification of the equations and assuming a symmetrical airfoil for the tail), that allows for easy numerical simulation. These could be included for slightly improved simulations. Note that by definition α = θ γ.

4 4 ROBBIE BUNGE (14) C Lw = 2π(α α Lw=0) (15) C Lh = 2π(α α Lh =0) + C L h δ e For relatively small elevator deflections, thin airfoil theory allows us to estimate C L h by: (16) C Lh = 2(π λ) + 2sin(λ) Where λ = cos 1 (2f 1) and f is the elevator chord fraction (usually f 0.25 ). Finally, we need to describe the thrust force. The simplest approximation is to neglect the dependence with airspeed, and making a linearized approximation about a given trim speed V o. If more data is available the airspeed dependence can be included, which add some more dynamics to the problem. Thus, we have: (17) T = T o (δ to ) + T δ t (δ t δ to ) Vo,δto Putting all these equations together we can go ahead and do some numerical simulations, and conceptually test different control system architectures. 6. Control System Design 6.1. Linearized modes. By putting representative values in the model, and linearizing it about some steady flight condition, as expected we will obtain a fast and highly damped mode (a.k.a. the short period) and a slow and lightly damped mode (a.k.a. the phugoid). Depending on the damping of this last mode, and of the accuracy with which one wants to control the flight, one might want to add additional damping by feeding back a derivative term to some of the control inputs Trim controls + feedback. If we want to attain a given flight condition (e.g. climb), we must first issue the correct trim control inputs. Superimposed (i.e. added) on these we should have feedback terms that improve the damping and the bandwidth, so that the steady state values are reached fast enough and with reasonable overshoot. This might not be easy, since there is always a trade off between speed of response, damping (or, even worse, stability) and control effort (both in magnitude and frequency).

5 A SIMPLIFIED ANALYSIS OF NONLINEAR LONGITUDINAL DYNAMICS AND CONCEPTUAL CONTROL SYSTEM DESIGN Integral terms. It is also an option to include some integral feedback terms, which has the property of learning the trim control inputs for us, but at the expense of retarding the response and adding some overshoot, and also the risk of making it unstable! For this reason, integral terms should be include very consciously and always include anti-windup elements, to cap the potential damage it can cause Tracking altitude. To track altitude we should note that: (18) ḣ = V sin(γ) Thus, if for example we have inner loops on γ and V, we can wrap a loop around these, which produces a desired ḣd, based on the altitude we want to track. 7. Conclusion In order to fly correctly we must first understand the trim curves, and experimentally obtain these trim values. In addition we might need to add some feedback terms to reject disturbances and speed up response, which might include integral terms to learn any discrepancies in the trim values.