Suboptimal adaptive control system for flight uality improvement Andrzej Tomczyk Department of Avionics and Control, Faculty of Mechanical Engineering and Aeronautics Rzeszów University of Technology, W. Pola 2, 35-959 Rzeszów, Poland ATomczyk@prz.edu.pl; http://www.prz.edu.pl/~atomczyk Keywords: Flight control system, adaptive control, suboptimal control Abstract In this paper the suboptimal algorithm of adaptive control system is presented, which is specially adjusted for automatic flight control systems of general aviation and commuter aircraft, and unmanned aircraft (UMA) that conduct flights in atmospheric turbulence. At first, the method could be applied for correction these changes in flight dynamics parameters, which cannot be compensated with the aid of an open adaptation loop. At the same time, full identification of aircraft model in real time is not reuired. This method is based on the estimation of most typical parameters of the aircraft mathematical model, which are most closely related to parameters, which are unmeasurable during flight, like aircraft real mass and position of center of gravity. The structure of an adaptation algorithm of aircraft flight control laws is based on the expert knowledge in the field of flight dynamics and is the result of optimization calculations. The examples which show attaining better flight comfort of the PZL M20,,Mewa" general aviation aircraft and uality improvement of the UMA,,Vector" pitch angle automatic control, have been presented. A. Tomczyk Suboptimal adaptive control system... 1
Introduction An aircraft is an object whose dynamic properties have wide scope of change that is also dependent on changes of flight conditions and airframe configuration. Obtaining reuired flight control uality reuires adjustment of an autopilot's control laws to changing properties of an object. Applied methods of adaptive control can be divided into three groups (Aström and Wittenmark, 1989; Sastry and Bodson, 1989): parametric adaptation methods which are based on knowledge of an object's model, and on chosen parameters' measurements describing its state (open system adaptation), methods based on algorithms of controlled object's (process) mathematical model identification in real time (on-line), methods of tuning controller which minimise chosen criteria of flight control uality in real time. Adaptation methods mentioned above are characterized by various levels of controlled object's necessary knowledge, disturbance sensitivity, amount of calculations in real time, control uality, etc. Practically it means that the method of adaptation should be chosen according to a given control task. In this paper, the method of an autopilot's control laws synthesis has been presented. It combines chosen features of adaptation steering methods mentioned, and, moreover, some elements typical for expert systems. The basic features of this method declare the following steps of control laws synthesis: 1. Choosing the enhance coefficients of autopilot for basic state of flight and aircraft configuration, 2. Choosing the algorithm of identification the parameters which describe the aircraft's properties during flight state under analysis, 3. Choosing the method of flight control uality evaluating and calculating of A. Tomczyk Suboptimal adaptive control system... 2
enhance coefficients optimal values in the range of unmeasuerable parameters of flight dynamics, 4. Choosing the algorithm for control laws suboptimal adaptation, 5. Evaluation of the results using non-linear computer simulation which takes into consideration aircraft's and flight system's real properties, as well as limits on steering signals value (saturation effect), discretisation and uantisation of measurements, actuator inertia, backlash, etc. 6. Hardware implementation of adaptive aircraft's suboptimal control laws and flight tests. The design method aims to obtain the simplest possible structure of an on-board control system. Optimization calculations are conducted during the step of autopilot design; during flight, only basic arithmetic operations are conducted. The method of behavior shown above leads to suboptimal control synthesis, which is close to optimal control in the sense of assumed control uality coefficient minimization. The structure of control system The methodology of choosing the autopilot's adaptive control laws will be shown on an example of pitch attitude angle stabilization. Figure 1 shows the system's block scheme used for numerical calculations in the MATRIX x package. Aircraft is described by the linear differential state euations (block 3) with three measurable output signals: pitch rate, pitch attitude increment (theta) and normal acceleration a z (az). An additional output of an aircraft model is the hinge moment of elevator. This moment is not directly measured, but its value influences real angle of elevator displacement. The hinge moment (Hinge_M) is calculated for real actuator properties modeling, which is illustrated in Figure 2. An aircraft is A. Tomczyk Suboptimal adaptive control system... 3
forced by the atmospheric turbulence (block 2) described by the Dryden model (Houbolt, Steiner and Pratt, 1964). Signals U g and W g are horizontal and vertical gust velocity, respectively. Elevator is turned by the angle E (deltae) with the help of the actuator (block 5) with non-linear parameters. The full model of the actuator is shown in Figure 2. The model contains the main real properties of electric motor and gear-box, as inertia (block 94), backlash (block 91), rate and position saturation (blocks 96, 32). The safe clutch characteristics are taken into consideration (block 20), and influence of the automatic trim servo is included (block 90). Integrator (block 98) with non-linear feedback loop (block 97) and threshold module (block 1) describe the logic control of the automatic elevator trim. The adaptive autopilot (AP_ADA_2, block 11) minimizes the differences between real (theta) and desired (theta_d) pitch angle according to control laws: u k K kor k1 k2 d (1) where: u k - corrected control signal (u_k), d - tracking error of pitch angle (theta-theta_d), K=[k 1, k 2 ] - matrix of autopilot gains described for basic flight state, K kor - correction coefficient (kor), calculated in adaptation algorithm. The structure of an autopilot is shown on Fig. 3. The upper part actively controls the feedback loop while lower part deals with the algorithm of correction coefficient (kor) value calculation. take in figure 1 and 2 A. Tomczyk Suboptimal adaptive control system... 4
The suboptimal algorithm of autopilot enhance gains adaptation The parametric adaptation of control laws based on measurable features (airspeed, altitude) does not allow to take into consideration the influence of factors which cannot be measured during the flight, like real mass of an aircraft or center of gravity position, and changes in aerodynamics performance caused by ice, for example. Methods based on on-line identification of controlled object's mathematical model and self-tuning methods reuire significant amount of calculations by an on-board steering computer (Hammond and D'Azzo, 1991). The method proposed in the present paper is based on approximate estimation of aircraft's dynamic properties by on-line calculating of parametric sensitivity coefficient (index), which describes the influence of elevator displacement on pitch rate (Tomczyk, 1997; Tomczyk, 1999). The measure of sensitivity index S() may be the magnitude of transfer function M(): 1 S( ) ; M Gs ; Gs ; and M (2) s j M ( ) s s E s s E where: - pitch rate, E - elevator displacement. The value of magnitude of M( ) is not measuring during a flight and it's a function of input signal E freuency, which is dependent on disturbance freuency. If an aircraft is affected by random disturbances caused by atmospheric turbulence, statistical measurements are needed for practical calculations. In order to simplify the calculations in real time, absolute values of pitch rate speed and elevator deflection, smoothed with a low-pass filter, were used for calculating the sensitivity index: E 1 1 u (3) T s 1 T s 1 s s; E s E s u where: E - statistical estimator of a signal (t), if (t)= (t), A. Tomczyk Suboptimal adaptive control system... 5
E u - statistical estimator of signal E t, if E t t, E T - time constants of low-pass filters used for averaging the signals., T u In the time domain, statistical estimates of signals (t) and E, E and E u, will be calculated as filtered absolute values of the signals: E t F t; E t F t; (4) u u E where F and F u are operations of low-pass filters. Time constants of filters T and T u have been selected in such a way that oscillations of pitch rate resulting from disturbance effects are averaged and, simultaneously, adeuate speed of autopilot's gain coefficients adaptation is assured. Finally, the estimate actual value S(t) of the sensitivity index S() can be calculated: Eu ( t) S( t) (5) E ( t) Practical realisation of calculations is presented on Fig. 3, where also a high-pass filter (block 30) was used in order to separate a constant component of elevator deflection (deltae). Blocks 12 and 21 represent low-pass filters, which form pitch rate (E ) and elevator deflection (E u ) estimators (formula (3)). take in figure 3 The main part of the design is the tuning function description for correction coefficient K kor (euation (1)). There is no theoretical basis for choosing the tuning function. The decision should be based upon expert opinion in the fields of flight dynamics, onboard flight control systems and designing experience. The tuning function can be written in a general form: K kor =f K (S, P) (6) A. Tomczyk Suboptimal adaptive control system... 6
where: f K expert-defined estimate function, S=S(t) actual value of sensitivity index, P matrix of parameters. In this paper, the following tuning function is proposed: K S( t) p 1 min max kor( t ) 1 ; K K kor K (7) 1 p2 where: p 1, p 2 - parameters calculated from the data set {K kor, S(t)} by least-meansuares estimation. Analytical synthesis of autopilot control laws reuires establishing a definition of a uality index in order to optimize its parameters. The form of uality performance index should result from substantial evidence. In this design method, for the purpose of evaluating an aircraft control uality during flight in atmospheric turbulence, the C* criterion concept has been used (Tobie, Elliot and Malcolm, 1966; Stevens and Levis, 1992): C t Vco n zp g (8) where: n zp - the incremental acceleration in g's at the pilot station, V co - crossover speed, assumed V co =400ft/s, g - gravity. The value of the C * criterion describes well the flight comfort from the pilot point of view and level of flight precision for UMA. So, for measuring control uality, the performance index (Fig. 1, block 13) was used: I T C T 0 t 2 dt (9) where: T-T 0 - time interval assumed for evaluation. A. Tomczyk Suboptimal adaptive control system... 7
As a step towards calculating, the minimum value of performance index (9), the vector of autopilot gain matrix K (Fig. 3, block 90) for chosen basic flight state should be calculated. This value will be corrected according to estimated dynamic state of controlled aircraft. Calculations done for the set of unmeasuerable during flight parameter's forecasted ranges of change allow obtaining optimal values of correction coefficient K kor (used in formula (1)) and to calculate estimators E and E u (formula (3 or 4)). Block 20 on Figure 3 (ADAPTATION) secures euation (7), while next element (block 32) with feedback loop is a low-pass filter which smoothes short-term fluctuations of correction coefficient. Block 22 is a multiplying element, which introduces corrections into main loop of control. Proving stability of adaptive control system described by euations (1) and (5) directly is rather difficult. Indirect proof of correct controlling is based on the following assumptions: for each value of correction coefficient in range K kor = {K min, K max }, the control system is stable, rate of change of K kor coefficient is small, which is assured by appropriate selection of low-pass filters' time constants T and T u, the final evaluation of the closed-loop non-linear control system stability can be based on representative simulation calculations (so-called technical stability). Numerical example The calculations have been conducted with the help of software package MATRIX X for five flight conditions of PZL M20 "Mewa" aircraft (Polish version of Piper Seneca II; Bociek, Dolega and Tomczyk, 1992) and for four flight conditions of unmanned aircraft A. Tomczyk Suboptimal adaptive control system... 8
,,Vector" (Gruszecki [ed.], 2002), presented in Table 1. Versions "A" and "F" are base flight condition (minimum mass, medium e.g. position), which will serve as a comparison for autopilot's enhances coefficients correction. Calculations were done for the same realization of moderate atmospheric turbulence, at the sea level. The example of the UMA,,Vector" flight parameters time-history is presented in Figure 4. take in figure 4, table 1 and figure 5 Figure 5 presents the correction coefficient's self-tuning process for the case of discrete changes in,,vector" aircraft's properties (changes in mass or/and center of gravity position). Assumed parameters of the adaptation module filters, the steady-state values of correction coefficient are obtained after about 150 seconds; this time constant is acceptable in real flight. The best adaptation results have been obtained for low-pass filters' time constants T = T u = 15 sec. The effects of using the enhance coefficient adaptation may be evaluated on the basis of comparison the performance index values (formula (9) for T 0 =150 seconds and T=300 seconds) in case of constant enhance coefficients (I C ) and adaptive autopilot (I A ). The relative effect of the suboptimal adaptive control system using can be described by the coefficient e: e I I I C A (10) C The results of numerical calculations are summarized in Table 1 and Figure 6. take in figure 6 The relative decrease in control performance uality e[%] depends on flight state and characteristics of disturbances, for example, on turbulence scale length in Dryden's model. A drop of a few percentage points means lower average level of random load factor of plane construction and higher level of flight comfort. This effect is obtained with minimal complications in control algorithm being conducted by the autopilot-computer. A. Tomczyk Suboptimal adaptive control system... 9
Closing remarks The method of adaptive on-board flight control system synthesis, presented in this paper, is accommodated for designing relatively simple and cheap digital autopilots for general-aviation and unmanned flying vehicles as well. Calculated control is not optimal (from the theoretical point of view), because in search for minimal value of performance index (9), a simplified method of aircraft's dynamic properties evaluation is used. Simplifications lead to the suboptimal system synthesis, which practical properties will depend in high degree on knowledge of an expert who can define the tuning function. An important stage of design process is the solution verification by the means of computer simulation, taking into consideration real properties of aircraft and its control system. Further improvements in performance uality can be obtained by including the real measured data from the flight tests. Relatively simple adaptation algorithms based only on low- and high-pass filters allow easy implementation in control system's microprocessor computers. Adaptation mechanism described may be also employed in simple autopilots without advanced systems of measurements and data processing. References Aström, K.J., Wittenmark, B. (1989), "Adaptive Control", Addison-Wesley Publishing Company. Bociek, S., Dołęga, B.,Tomczyk, A. (1992), "Synthesis of the Microprocessor Digital Autopilot", Systems Science, vol.18, No 4, Wrocław, pp. 99-115. Gruszecki J. [ed.] (2002), Unmanned Air Vehicles; Control and Navigation Systems, A. Tomczyk Suboptimal adaptive control system... 10
Rzeszów University of Technology Press, ISSN 83-7199-221-1 (in Polish). Hammond, D., D'Azzo, J.J. (1991), "Parameter Adaptive Multivariable Flight Controller Using a Full Autoregressive Moving Average (ARMA) Model and Recursive Least Suares (RLS) Estimation", AIAA 91-0420, 29th Aerospace Sciences Meeting, Reno, Nevada. Houbolt, J.C., Steiner, R., Pratt, K.G (1964), "Dynamic Response of Airplanes to Atmospheric Turbulence Including Flight Data on Input and Response", NASA Technical Report, R-199. Sastry, S., Bodson, M. (1989), "Adaptive Control -Stability, Convergence, and Robustness", Prentice Hall, Inc. Stevens, B.L., Levis, F.L. (1992), "Aircraft Control and Simulation", J.Wiley & Sons, Inc. ISBN 0-471-61397-5. Tobie, H.N., Elliot, E.M, Malcolm, L.G. (1966), "A New Longitudinal Handling Qualities Criterion", Proc. National Aerospace Electronics Conference, Dayton, Ohio. Tomczyk A. (1997), "A Simple Method of Compensating the Unmeasurable Influence on Automatic Flight Control Quality", AIAA Paper 97-3635, AIAA Guidance, Navigation, and Control Conference, New Orleans, LA, USA, 11-13.08.1997, Part 3, 1316-1321 Tomczyk A. (1999), Digital Flight Control Systems, Rzeszów University of Technology Press, Rzeszów, ISBN 83-7199-093-6 (in Polish). A. Tomczyk Suboptimal adaptive control system... 11
FIGURES and TABLE Figure 1. Block-diagram for computer simulation (MATRIX X ) Figure 2. Model of elevator servo-actuator(matrix X ) A. Tomczyk Suboptimal adaptive control system... 12
Figure 3. Block-scheme of an analyzed adaptive algorithm for autopilot (MATRIX X ) Figure 4. Atmospheric turbulence influence on the longitudinal flight parameters of PZL M20 Mewa aircraft, flight condition D : [deg/s] pitch rate, theta [deg] pitch angle, az [m/s 2 ] vertical acceleration, Ug [m/s] horizontal gust velocity, Wg [m/s] vertical gust velocity, kor [-] correction coefficient, Cstar [-] C * criterion value, I control performance index value (formula (9)) for T 0 =50 sek A. Tomczyk Suboptimal adaptive control system... 13
Figure 5. Autopilot s correction coefficient (kor=k kor ) self-tuning process for UMA Vector aircraft; F basic flight condition, G-K different flight conditions defined in Table 1 Table 1. Results of the numerical calculations Aircraft PZL M-20 "Mewa", V=62.8 m/s UMA "Vector", V=41.7 m/s Flight conditions Parameters A B C D E F G H K Mass [kg] 1400 1400 1400 2040 2040 300 300 300 380 c.g. position [m] 0.25 0.198 0.292 0.292 0.198 0.21 0.17 0.24 0.24 Cost function I C 10.66 9.16 13.71 11.26 8.73 16.33 14.95 18.87 17.03 K kor (steady state) 1.01 1.12 0.91 0.67 1.23 0.99 1.09 0.82 0.78 Cost function I A 10.67 8.82 12.90 10.81 8.56 16.32 14.58 17.53 16.47 e=(i C I A )/I C [%] - 3.7 5.9 4.0 1.9-2.5 7.1 3.3 A. Tomczyk Suboptimal adaptive control system... 14
Figure 6. Effect of the suboptimal adaptive control system using. F the base flight conditions, IC= I C performance index for non-adaptive flight control system, IA= I A performance index for suboptimal adaptive flight control system, e [%] relative effect of suboptimal adaptive flight control system application 20 15 10 5 IC IA e [%] 0 F G H K A. Tomczyk Suboptimal adaptive control system... 15