The Solution of the Equations of Motion
5.7 STATE SPACE MODEL AUGMENTATION
5.7.4 Addition of engine dynamics
Provided that the thrust producing devices can be modelled by a linear transfer func-tion then, in general, it can be integrated into the aircraft state descripfunc-tion. This then enables the combined engine and airframe dynamics to be modelled by the overall system response transfer functions. A very simple engine thrust model is described by equation (2.34), with transfer function:
τ(s)
ε(s) = kτ
(1+ sTτ) (5.130)
where τ(t) is the thrust perturbation in response to a perturbation in throttle lever angle ε(t). The transfer function equation (5.130) may be rearranged thus
sτ(s)= kτ
Tτ
ε(s)− 1 Tτ
τ(s) (5.131)
and this is the Laplace transform, assuming zero initial conditions, of the following time domain equation:
˙τ(t) = kτ
Tτε(t)− 1
Tττ(t) (5.132)
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132 Flight Dynamics Principles
The longitudinal state equation (4.67) may be augmented to include the engine dynamics described by equation (5.132) which, after some rearrangement, may be written:
⎡
⎢⎢
⎢⎢
⎣
˙u˙w
˙q
˙θ
˙τ
⎤
⎥⎥
⎥⎥
⎦=
⎡
⎢⎢
⎢⎢
⎣
xu xw xq xθ xτ zu zw zq zθ zτ mu mw mq mθ mτ
0 0 1 0 0
0 0 0 0 −1/Tτ
⎤
⎥⎥
⎥⎥
⎦
⎡
⎢⎢
⎢⎢
⎣ u w q θ τ
⎤
⎥⎥
⎥⎥
⎦+
⎡
⎢⎢
⎢⎢
⎣
xη 0 zη 0
mη 0
0 0
0 kτ/Tτ
⎤
⎥⎥
⎥⎥
⎦
η ε
(5.133)
Thus the longitudinal state equation has been augmented to include thrust as an additional state and the second input variable is now throttle lever angle ε. The output equation (4.68) remains unchanged except that the C matrix is increased in order to the (5× 5) identity matrix I in order to provide the additional output variable corresponding to the extra state variable τ.
The procedure described above in which a transfer function model of engine dynam-ics is converted to a form suitable for augmenting the state equation is known as system realisation. More generally, relatively complex higher order transfer functions can be realised as state equations although the procedure for so doing is rather more involved than that illustrated here for a particularly simple example. The mathemat-ical methods required are described in most books on modern control theory. The advantage and power of this relatively straightforward procedure is very considerable since it literally enables the state equation describing a very complex system, such as an aircraft with advanced flight controls, to be built by repeated augmentation. The state descriptions of the various system components are simply added to the matrix state equation until the overall system dynamics are fully represented. Typically this might mean, for example, that the basic longitudinal or lateral (4× 4) airframe state matrix might be augmented to a much higher order of perhaps (12× 12) or more, depending on the complexity of the engine model, control system, surface actuators and so on. However, whatever the result the equations are easily solved using the tools described above.
Example 5.8
To illustrate the procedure for augmenting an aeroplane state model, let the longitudi-nal model for the Lockheed F-104 Starfighter of Example 5.2 be augmented to include height h and flight path angle γ and to replace normal velocity w with incidence α.
The longitudinal state equation expressed in terms of concise derivatives is given by equation (5.103) and this is modified in accordance with equation (5.121) to replace normal velocity w with incidence α,
⎡
⎢⎢
⎣
˙u˙α
˙q˙θ
⎤
⎥⎥
⎦ =
⎡
⎢⎢
⎣
−0.0352 32.6342 0 −32.2
−7.016E − 04 −0.4400 1 0
1.198E− 04 −4.6829 −0.4498 0
0 0 1 0
⎤
⎥⎥
⎦
⎡
⎢⎢
⎣ u α q θ
⎤
⎥⎥
⎦ +
⎡
⎢⎢
⎣ 0
−0.0725
−4.6580 0
⎤
⎥⎥
⎦η
(5.134)
Equation (5.134) is now augmented by the addition of equation (5.116), the height equation expressed in terms of incidence α and pitch attitude θ:
˙h = V0(θ− α) = 305θ − 305α (5.135)
whence the augmented state equation is written:
⎡
⎢⎢
⎢⎢
⎣
˙u˙α
˙q˙θ
˙h
⎤
⎥⎥
⎥⎥
⎦=
⎡
⎢⎢
⎢⎢
⎣
−0.0352 32.6342 0 −32.2 0
−7.016 × 10−4 −0.4400 1 0 0
1.198× 10−4 −4.6829 −0.4498 0 0
0 0 1 0 0
0 −305 0 305 0
⎤
⎥⎥
⎥⎥
⎦
⎡
⎢⎢
⎢⎢
⎣ u α q θ h
⎤
⎥⎥
⎥⎥
⎦+
⎡
⎢⎢
⎢⎢
⎣ 0
−0.0725
−4.6580 0 0
⎤
⎥⎥
⎥⎥
⎦η
(5.136) The corresponding output equation is augmented to included flight path angle γ as given by equation (5.127) and is then written:
⎡
⎢⎢
⎢⎢
⎢⎢
⎣ u α q θ h γ
⎤
⎥⎥
⎥⎥
⎥⎥
⎦
=
⎡
⎢⎢
⎢⎢
⎢⎢
⎣
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1
0 −1 0 1 0
⎤
⎥⎥
⎥⎥
⎥⎥
⎦
⎡
⎢⎢
⎢⎢
⎣ u α q θ h
⎤
⎥⎥
⎥⎥
⎦ (5.137)
This, of course, assumes the direct matrix D to be zero as discussed above. Equations (5.136) and (5.137) together provide the complete state description of the Lockheed F-104 as required. Solving these equations with the aid of Program CC results in the six transfer functions describing the response to elevator;
(i) The common denominator polynomial (the characteristic polynomial) is given by
Δ(s)= s(s2+ 0.033s + 0.022)(s2+ 0.892s + 4.883) (5.138) (ii) The numerator polynomials are given by
Nηu(s)= −2.367s(s − 4.215)(s + 5.519) ft/s/rad
Nηα(s)= −0.073s(s + 64.675)(s2+ 0.035s + 0.023) rad/rad Nηq(s)= −4.658s2(s+ 0.134)(s + 0.269) rad/s/rad
(5.139) Nηθ(s)= −4.658s(s + 0.134)(s + 0.269) rad/rad
Nηh(s)= 22.121(s + 0.036)(s − 4.636)(s + 5.085) ft/rad Nηγ(s)= 0.073s(s + 0.036)(s − 4.636)(s + 5.085) rad/rad
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134 Flight Dynamics Principles
Note that the additional zero pole in the denominator is due to the increase in order of the state equation from four to five and represents the height integration. This is easily interpreted since an elevator step input will cause the aeroplane to climb or descend steadily after the transient has died away when the response becomes similar to that of a simple integrator. Note also that the denominator zero cancels with a zero in all numerator polynomials except that describing the height response. Thus the response transfer functions describing the basic aircraft motion variables u, α, q and θ are identically the same as those obtained from the basic fourth order state equations. The reason for the similarity between the height and flight path angle response numerators becomes obvious if the expression for the height equation (5.135) is compared with the expression for flight path angle, equation (5.127).
REFERENCES
Auslander, D.M., Takahashi, Y. and Rabins, M.J. 1974: Introducing Systems and Control.
McGraw Hill Kogakusha Ltd, Tokyo.
Barnett, S. 1975: Introduction to Mathematical Control Theory. Clarendon Press, Oxford.
Duncan, W.J. 1959: The Principles of the Control and Stability of Aircraft. Cambridge University Press, Cambridge.
Goult, R.J., Hoskins, R.F., Milner, J.A. and Pratt, M.J. 1974: Computational Methods in Linear Algebra. Stanley Thornes (Publishers) Ltd., London.
Heffley, R.K. and Jewell, W.F. 1972: Aircraft Handling Qualities Data. NASA Contractor Report, NASA CR-2144, National Aeronautics and Space Administration, Washington D.C. 20546.
Owens, D.H. 1981: Multivariable and Optimal Systems. Academic Press, London.
Shinners, S.M. 1980: Modern Control System Theory and Application. Addison-Wesley Publishing Co, Reading, Massachusetts.
Teper, G.L. 1969: Aircraft Stability and Control Data. Systems Technology, Inc., STI Technical Report 176-1. NASA Contractor Report, National Aeronautics and Space Administration, Washington D.C. 20546.
PROBLEMS
1. The free response x(t) of a linear second order system after release from an initial displacement A is given by
x(t)=1 2Ae−ωζt
1+ ζ
(ζ2− 1
e−ωt
√ζ2−1+
1− ζ
(ζ2− 1
eωt
√ζ2−1
where ω is the undamped natural frequency and ζ is the damping ratio:
(i) With the aid of sketches show the possible forms of the motion as ζ varies from zero to a value greater than 1.
(ii) How is the motion dependent on the sign of ζ?
(iii) How do the time response shapes relate to the solution of the equations of motion of an aircraft?
(iv) Define the damped natural frequency and explain how it depends on
damping ratio ζ. (CU 1982)
2. For an aircraft in steady rectilinear flight describe flight path angle, incidence and attitude and show how they are related. (CU 1986) 3. Write down the Laplace transform of the longitudinal small perturbation equa-tions of motion of an aircraft for the special case when the phugoid motion is suppressed. It may be assumed that the equations are referred to wind axes and that the influence of the derivativesZ◦q,Z◦˙wandM◦ ˙wis negligible. State all other assumptions made:
(i) By the application of Cramer’s rule obtain algebraic expressions for the pitch rate response and incidence angle response to elevator transfer functions.
(ii) Derivative data for the Republic Thunderchief F-105B aircraft flying at an altitude of 35,000 ft and a speed of 518 kt are,
Z◦w
m = −0.4 1/s M◦w
Iy = −0.0082 1/ft s M◦q
Iy = −0.485 1/s M◦η
Iy = −12.03 1/s2 Z◦η
m = −65.19 ft/s2
Evaluate the transfer functions for this aircraft and calculate values for the longitudinal short period mode frequency and damping.
(iii) Sketch the pitch rate response to a 1◦step of elevator angle and indicate the significant features of the response. (CU 1990) 4. The roll response to aileron control of the Douglas DC-8 airliner in an approach
flight condition is given by the following transfer function:
φ(s)
ξ(s) = −0.726(s2+ 0.421s + 0.889) (s− 0.013)(s + 1.121)(s2+ 0.22s + 0.99)
Realise the transfer function in terms of its partial fractions and by calculating the inverse Laplace transform, obtain an expression for the roll time history in response to a unit aileron impulse. State all assumptions.
5. Describe the methods by which the normal acceleration response to elevator transfer function may be calculated. Using the Republic Thunderchief F-105B model given in Question 3 calculate the transfer function az(s)/η(s):
(i) With the aid of MATLAB, Program CC or similar software tools, obtain a normal acceleration time history response plot for a unit elevator step input. Choose a time scale of about 10 s.
(ii) Calculate the inverse Laplace transform of az(s)/η(s) for a unit step eleva-tor input. Plot the time hiseleva-tory given by the response function and compare with that obtained in 5(i).
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136 Flight Dynamics Principles
6. The lateral–directional equations of motion for the Boeing B-747 cruising at Mach 0.8 at 40,000 ft are given by Heffley and Jewell (1972) as follows:
⎡
⎢⎢
⎢⎢
⎢⎣
(s+ 0.0558) | −(62.074s+ 32.1)
774 | (771.51s− 2.576)
− − − − − − − | − − − − − − −− | − − − − − − −774
3.05 | s(s+ 0.465) | −0.388
− − − − − − − | − − − − − − −− | − − − − − − −
−0.598 | 0.0318s | (s+ 0.115)
⎤
⎥⎥
⎥⎥
⎥⎦
⎡
⎢⎢
⎢⎣ β p s r
⎤
⎥⎥
⎥⎦
=
⎡
⎢⎢
⎢⎢
⎣
0 | 0.00729
− − − −| − − − − 0.143 | 0.153
− − − −| − − − − 0.00775 | −0.475
⎤
⎥⎥
⎥⎥
⎦
ξ ζ
where s is the Laplace operator and all angles are in radians. Using Cramer’s rule, calculate all of the response transfer functions and factorize the numerators and common denominator. What are the stability modes characteristics at this flight condition?
7. The longitudinal equations of motion as given by Heffley and Jewell (1972) are
⎡
⎢⎢
⎢⎢
⎣
(1− X˙u)s− X∗u | −X˙ws− X∗w | (−Xq+ We)s+ g cos θe
− − − − − − − − − | − − − − − − − − − | − − − − − − − − −
−Z˙us− Z∗u | (1− Z˙w)s− Zw | (−Zq− Ue)s+ g sin θe
− − − − − − − − − | − − − − − − − − − | − − − − − − − − −
−M˙us− M∗u | −(M˙ws+ Mw) | s2− Mqs
⎤
⎥⎥
⎥⎥
⎦
⎡
⎣u w θ
⎤
⎦
=
⎡
⎣Xη
Zη
Mη
⎤
⎦ η q = sθ
˙h = −w cos θe+ u sin θe+ (Uecos θe+ Wesin θe) az = sw − Ueq+)
g sin θe
*θ
Note that the derivatives are in an American notation and represent the mass or inertia divided dimensional derivatives as appropriate. The * symbol on the speed dependent derivatives indicates that they include thrust effects as well as the usual aerodynamic characteristics. All other symbols have their usual meanings.
Rearrange these equations into the state space format:
M˙x(t) = Ax(t)+ Bu(t) y(t)= Cx(t) + Du(t)
with state vector x= [u w q θ h], input vector u = η and output vector y= [u w q θ h az]. State all assumptions made.
8. Longitudinal data for the Douglas A-4D Skyhawk flying at Mach 1.0 at 15,000 ft are given in Teper (1969) as follows:
Trim pitch attitude 0.4◦
Speed of sound at 15,000 ft 1058 ft/s
Xw − 0.0251 1/s M˙w −0.000683 1/ft
Xu − 0.1343 1/s Mq −2.455 1/s
Zw − 1.892 1/s Xη −15.289 ft/rad/s2
Zu − 0.0487 1/s Zη −94.606 ft/rad/s2
Mw − 0.1072 Mη −31.773 1/s2
Mu 0.00263 1/ft s
Using the state space model derived in Problem 7, obtain the state equations for the Skyhawk in the following format:
˙x(t) = Ax(t) + Bu(t) y(t)= Cx(t) + Du(t)
Using MATLAB or Program CC, solve the state equations to obtain the response transfer functions for all output variables. What are the longitudinal stability characteristics of the Skyhawk at this flight condition?
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