Active Control of Wind-Tunnel Model Aeroelastic Response Using Neural Networks

Active Control of Wind-Tunnel Model

Aeroelastic Response

Using Neural Networks

Robert C . Scott*

NASA Langley Research Center, Hampton, VA 23681

Under a joint research and development effort conducted by the National Aeronautics and Space Administration and The Boeing Company (formerly McDonnell Douglas) three neural-network based control systems were developed and tested. The control systems were experimentally evaluated using a transonic wind-tunnel model in the Langley Transonic Dy- namics Tunnel. One system used a neural network to schedule flutter suppression control laws, another employed a neural network in a predictive control scheme, and the third employed a neural network in an inverse model control scheme. All three of these control schemes successfully suppressed flutter to or near the limits of the testing apparatus, and represent the first experimental applications of neural networks to flutter suppression. This paper will summarize the findings of this project.

Keywords: neural network, adaptive control, aeroelasticity, flutter suppression effort conducted by the National Aeronautics and Space Introduction Administration, Langley Research 'Center and The Boe- CTIVE control of aeroelastic phenomena will be- ing Company (formerly McDonnell Douglas) under a

A come more prevalent on future flight vehicles. One

Memorandum of Agreement. The goal of the program of the key issues in gaining acceptance of such systems was to develop and demonstrate neural-network based is reliability. Systems that can reliably adapt to sensor adaptive control systems using the Benchmark Active and actuator failures or plant changes will improve sys- Controls Technology (BACT) wind-tunnel model. The tem reliability. Since neural networks can be trained to ANCAR program consisted of two phases. Phase I was model dynamic systems, their use has been suggested the development and demonstration of a neural net- for adaptive control and many studies proposing a va- work gain scheduled flutter suppression system. Under riety of control system architectures have been studied.

Phase 11, two adaptive neural-network based control sys- Some common types of algorithms include model refer- tems were to be developed and demonstrated. These ence control where the neural network is used to model systems used predictive control and inverse model con- the plant and inverse control where the neural network is trol methodologies. This paper will summarize all the used to model the inverse of the plant. Studies to inves- accomplishments of the ANCAR project; however, the tigate the aeronautical applications of neural networks majority of the paper will focus on the neural-network have been much more limited. References 1, 2, and 3 based inverse model system which has not previously describe analytical studies where neural networks were been reported.

applied to flight controls in either fixed wing or rotorcraft applications. These studies have focused on utilizing The work presented in this paper was an exploratory neural networks to achieve improvements in trajectory study. Additional research is required to determine the tracking. Studies to investigate the application of neural merits of using neural networks for aeroelastic control.

networks to controlling aeroelastic response (flutter and In addition to the neural-network based control systems, gust load alleviation, for example) have, however, been there were several other control systems tested using the even more limited. The purpose of the present research BACT wind-tunnel model. These control systems in- is to investigate the use of neural networks for controlling clude those described in references 4 and 5. The control aeroelastic response.

systems described in these papers were designed using The work described in this paper is part of the Adap- classical or modern methods. While the performance of tive Neural Control of Aeroelastic Response (ANCAR) these control systems was not directly compared with project. ANCAR was a joint research and development the neural-network based control systems, it was appar- ent that they were significantly more robust than either *ResearchEngineer, Aeroelasticity Branch, Structures and Ma- the neural predictive or inverse model control systems.

terials Competency.

The neural gain scheduled system is based on classical control law design and its performance was qualitatively similar to the systems in references 4 and 5.

Apparatus Transonic Dynamics Tunnel Wind-tunnel testing was conducted in the NASA Lan- gley Transonic Dynamics Tunnel (TDT).' The TDT is a single-return variable-density transonic wind tunnel.

The slotted test section is 16 ft by 16 ft square with cropped corners. The speed and pressure are indepen- dently controllable over a range of Mach number from 0.0 to 1.2 (unblocked), and a range of stagnation pressures from near zero to one atmosphere. Either air or a heavy gas can be used as the test medium. The heavy gas used in these wind-tunnel tests was R-12, but the TDT has since been modified to use R134a as the heavy gas. The Fig. 1 BACT wind-tunnel model.

TDT is also equipped with quick-opening bypass valves which can be activated to rapidly reduce test-section dy- Input namic pressure and Mach number when flutter occurs. Neuron Vector The combinations of large scale, high speed, high density, variable pressure, and the bypass-valve system make the T D T ideally suited for aeroelastic testing.

Wind-Tunnel Model The BACT wind-tunnel model is a rigid, rectangu- +bi lar wing with an NACA 0012 airfoil section. It is equipped with a trailing-edge (TE) control surface and R 1 fNonlinea+ ai=tanh(Ep.w-+bi) upper- and lower-surface spoilers, all independently con- j=1 J 1 1 trollable. The model is attached to a flexible mount system, the Pitch And Plunge Apparatus (PAPA), that Fig. 2 Neural network computational element or allows both pitch and plunge degrees-of-freedom. An neuron.

image of the model mounted in the TDT and a sepa- rate image showing only the model and mount system summary of only the neural networks applied in this pa- are shown in figure 1. The model is extensively instru- per.

mented with pressure transducers and accelerometers to The name neural network comes from the fact that the measure surface pressures and model dynamic response, networks emulate the structure of the brain. In biological and the mount system is instrumented with strain gauges nervous systems, the output of one neuron is connected to measure normal force and pitching moment. Parame- to many other neurons. It is these connections that de- ters which could be varied during the test include Mach termine the function of the network. In practice, neural number, dynamic pressure, model angle-of-attack, and networks are composed of many computational elements control surface deflection. Reference 7 contains a more or neurons operating in parallel. The general structure detailed description of this wind-tunnel model.

of an individual neuron is shown in figure 2. The func- tion of each neuron is to sum the weighted inputs (wij) Neural Networks and the bias ( b j ) and process this sum through a trans- Neural networks have been studied for many years in fer function (f). The transfer functions considered in a variety of fields. These fields include speech and image this study will be limited to the two shown in figure 2: recognition, credit and insurance policy evaluation, and a linear transfer function or a sigmoid transfer function (tanh). The use of a tanh or tan-sigmoid transfer func- trajectory control to name a few. They have also been tion allows the modeling of nonlinear effects.

studied extensively for use in controlling dynamic sys- tems. The MATLAB Neural Network Toolbox Manual8 Two or more neurons can be used to form a layer and provides an excellent overview of neural network appli- one or more layers forms a neural network. An example cations with many references. While numerous control of the network structure used in this paper is shown in system architectures and network types have been in- figure 3. This network consists of an input vector and vestigated, this section of the paper will provide a brief two layers of neurons, a hidden layer and an output layer.

SPIE PAPER 3991-30, MARCH 2000 sure varied from 75 to 250 psf. These models were used to design a fixed-gain control law and to design a series of control laws optimized to minimize accelerometer out- put for each combination of M and q. A neural network was used to schedule the series of 56 optimized control laws.

Fixed-Gain Control Law The fixed-gain feedback control law was designed to stabilize and minimize the wing response over the entire Input output Vector Hidden Layer Layer range of the state-space models. A washout filter was included in the compensation to eliminate any drift due Neural network with one hidden layer.

Fig. 3 to bias errors in the accelerometers. Root-locus pole and zero placement methods were used to design the fixed- Typically the output layer uses linear transfer functions gain robust feedback control law given below.

and the hidden layer uses linear or tan-sigmoid transfer functions. This network structure is the simplest form of

s ( s 2 + 12s + 520)

multilayer feedforward network. More hidden layers can

s4 + 27s3 + 491s2 + 4515s + 13050

be used, but no more than one hidden layer was used in this paper. Given sufficient neurons on the hidden layer, Neural-Network Scheduled Control Laws this type of network can approximate most functions ar- For the neural-network scheduled control laws, the bitrarily well.

control law described above was tailored for the pole-zero Neural networks must be trained prior to use as pre- dynamics of each state-space model. Fifty-six custom de- dictive models. Training is a common term in the field signs were generated for the various M and q condition of neural networks and simply refers to the process by state-space models. All of the resulting control laws had which the network weights and biases are selected. The the same order numerator (3 zeros) and denominator (4 training process begins by selecting or acquiring train- poles) as the robust fixed-gain control law. The general ing data, a set of input and output data for the plant or form of the these control laws is given below: function to be approximated with the network. During

the training process the weights and biases are adjusted 53 + 0282 + a1s

k until the error between the training data output and the

s4 + b3s3 + bzs2 + bls + bo

network output is minimized. There are several meth- The neural network, shown in the lower portion of fig- ods or algorithms for adjusting the network weights and ure 4, was trained using backpropagation to output the biases. A common training method, and the one used two numerator coefficients, the four denominator coef- in this paper, is backpropagation. Backpropogation is ficients, and the overall gain as functions of M and q.

a gradient descent optimization algorithm that can be The control law parameters used to train the neural net- used on multilayer networks as long as the neurons have work were in the continuous domain, rather than the differentiable transfer functions.

discrete domain. This was required because continuous- domain coefficients vary smoothly as a function of M Gain Scheduler and q and do not require the high numerical precision of This section of the paper will summarize the imple- discrete-domain control law coefficients. The neural net- mentation and findings of the neural network gain sched- work outputs were transformed into the discrete domain uled flutter suppression system described in reference 9.

before being transferred to the digital controller.

The objective of this system was to use a neural net- Experimental Results work to schedule flutter suppression control laws. The approach taken in this study was to use a series of state- The control system architecture that was implemented space models of the BACT wind-tunnel model to design in the 1995 BACT wind-tunnel test is shown in figure 4.

classical single input single output (SISO) flutter sup- The trained neural network and Tustin (continuous-to- pression control laws. The state-space models were gen- discrete) transformation was implemented on a Macin- erated using the Integrated Structures, Aerodynamics, tosh computer. As M and q were varied, the Macintosh and Controls (ISAC) code." The state-space models all computer transferred control laws to the real time sys- used the same structural and aerodynamic models with tem. The Active Digital Controller (ADC)ll was used as Mach number (M) and dynamic pressure (4) being var- the real-time digital control system operating at a rate of ied. In all, fifty six state-space models were used where 200 Hz. The fixed-gain control law was also implemented Mach number varied from 0.3 to 0.9 and dynamic pres- on the ADC.

SPIE PAPER 3991-30. MARCH 2000 TE Accelerometer TE Control Signal Surface Signal < > Discrete State-Space Control Law Dynamic 150

4 -

Sun Real Time Digital Pressure,

t

0.50 0.60 0.70 0.80 0.90 1.00 Mach Number I 0 I I Dynamic

-

,045 Pressure I

I -

.040 I I

-

,035 I

.. 0 00

I

-

TE Accel. .030 0 0 0

L - - - - - - - - - - - - - - - - - - - A

RMS, g's .0*5

-

om^^ 0 0

Fig. 4 Neural network gain scheduling system ar-

0 0 o o

-

.020 chitecture.

e o

-

.015 The control systems were tested along four constant

-

.010 total-pressure lines which define test paths within the

-

.005 TDT. These H-lines and the BACT model open-loop I Each H-line is flutter boundary are shown in figure 5.

TE Accelerometer Signal, y(n) BACT model u is the T E control surface command and y is the TE accelerometer signal.

3. Evaluate the performance of the trial input value TE Control Surface Signal according to a performance index based on the cost _ _ _ _ _ _ - - - of regulation/tracking error and the required control I I I input power.

I I I - u(n: 4. Repeat steps 2 and 3 with an optimization scheme until the termination criteria (desired performance or iteration limit) is achieved.

, -\.. ' I J \

\ 5. Output the trial control value selected by the op- uln-ol timization process to the plant. This is where the switch in figure 7 is moved from the down position to the up position so that the TE control surface command can be sent to the physical model. After Fig. 7 Neural predictive control system architecture the signal is sent the switch is returned to the down position for the next control cycle.

a linear single layer network or neural-network plant model. As depicted in this figure, the NPC system im- 6. If on-line learning is engaged, update the plant plemented in this study used a linear plant model to model using a set of input/output data and an ap- capture the input/output dynamics of the BACT model.

propriate training algorithm.

This predictive plant model is then utilized in an on-line optimization scheme to select the optimal control val- 7. Repeat the entire process for each control cycle.

ues for each control cycle. This system used a sample rate of 100 Hz, and each cycle had a duration of 0.01 The inputs to the plant model consist of time-delayed seconds. The plant model can initially be trained by samples of the plant inputs, u ( n ) , and outputs, y(n), as exciting the actuators, measuring the sensor response, shown in figure 7. For nonlinear control applications, a and training the plant model. This model can then be multi-layer neural network architecture with backpropa- updated on-line to handle with changing conditions and gation training can be used. For linear plants, a linear time-varying plant dynamics. The plant model training autoregressive moving average (ARMA) model is used.

portion of this system is performed in parallel with the The NPC architecture was implemented using a Pentium closed-loop portion and is depicted by the error feedback 60 MHz PC with a plug-in Alacron neural accelerator to the plant model in figure 7.

board and analog input/output boards. The Pentium The method implemented for flutter suppression in- host CPU is responsible for all NPC computations and volves minimizing a cost function. The cost function, data transfer to the input/output and neural accelera- or performance index, is typically a quadratic function tor boards. The neural accelerator board performs the of the regulation/tracking error and required control in- on-line model adaptation which occurs in parallel with put power. consequently, NPC is an optimal controller the host CPU control loop computations. Due to several in the same sense that the Linear Quadratic Regulator limitations of this hardware and software, only the linear (LQR) is optimal. The advantage of NPC over LQR lies ARMA plant models were considered in this study.

in its capability to be easily extended to nonlinear sys- Simulation Results tems and to explicitly account for plant constraints in real time.

The simulation studies were performed to design the A step-by-step description of the NPC algorithm is appropriate plant model architecture, evaluate the ro- given below: bustness of the controller, and validate the real time control software. Successful adaptation and control was 1. Generate a reference trajectory, yd(n), which rep- demonstrated across the range of simulated wind tunnel conditions, with the NPC system running at a rate of resents the desired value of the future plant output (yd(n) = O for flutter suppression). 100 Hz. One representative simulation time history is il- lustrated in figure 8 for an open-loop unstable condition, 2. Predict the future plant output, y p ( n + l ) , using the M=0.75 and q=175 psf. Starting with an untrained net- plant model. This prediction is based on the current work, a white noise excitation signal was sent to the T E and past values of the plant input and output, u ( n ) control surface for four seconds and the accelerometer and y(n), and a new trialinput value, u k ( n ) . For the response recorded providing 400 data points for neural S P I E PAPER 3991-30, MARCH 2000 200 0

NN I Control & I Control &I Control & I Control

NN, I Excitation I &NN Learning I Excitaztion I

I I Learning I ILeaming

190 1 " 7

0.5 TE Accel.

Signal, g's 0 Dynamic -0.5 Pressure, 150

I I I -1.0 - I

I I

Control 4ctivated I I

I I I II I I -1.5 I II

+ Flutter Points

0 2 4 6 8 10 12 14 16 18 20 Time, sec. 0 0 Neural Predictive Control NPC simulation time history. Fig. 8

0.60 0.70 0.80 0.90 1 .oo

Mach Number network learning. The learning then occurred during NPC wind-tunnel data points.

Fig. 9 the next 2.7 seconds, allowing control to be activated at about 6.7 seconds. As shown in the figure, from t=3.9 to Inverse Model Control t=6.7, the wing response grew steadily due to open-loop This section of the paper will provide a detailed de- flutter until the controller was initiated for flutter sup- scription of the application of neural networks and in- pression. Once the system was activated, learning and verse modelling to the control of aeroelastic response.

control occur simultaneously, allowing model updates to occur every 6.7 seconds. The length of this time interval Architecture is determined by the speed of the processors, the control The inverse modelling control architecture used in this cycle rate, and the amount of data needed for accurate study borrows elements from several proposed control plant modeling.

schemes. This section of the paper will describe the relevant aspects of these systems and the system ar- Experimental Results chitecture implemented here. Inverse modelling control has generally been applied to trajectory tracking ap- The NPC system was tested at several M and q condi- plications. References 13, 14, and 15 describe several tions along two H-lines. Testing began at the lower end approaches to inverse modelling control and introduce of the H-line, in the open-loop stable region, with plant several elements applicable to the present problem. In model learning activated. Dynamic pressure and Mach general, the key assumption in inverse model control is number were then gradually increased to conditions well that an unknown plant can be made to track an input beyond the open-loop flutter boundary. The continuous command signal when this signal is applied to a con- adaption of the plant model, which was one goal of the troller whose transfer function approximates the inverse project, was not reliable enough to use for long periods of the plant's transfer function. An adaptive control of time. Periodically, the plant model generated by the system can be created by applying an adaptive inverse on-line learning algorithm was not accurate enough and modelling process to the plant. A simple form of such resulted in an unstable control system. Additional anal- a system is depicted in figure 10. The upper portion ysis is necessary to isolate the root of the problem, but of this figure shows the training procedure. Here the it may be related to the level of random noise used to arrow passing through the network box indicates the excite the wing dynamics, the amount of data used for feedback of the error for training of the network using training, or tuning of the performance index used by the back propagation. The lower portion of figure 10 shows NPC system.

the implementation of the inverse model. In practice After several unsuccessful attempts to operate in a the adaptive inverse model will be continuously updated.

fully adaptive mode, it was decided to activate learning Thus, no direct feedback is used, except that the plant only when a new model was required. Thus, for each output is monitored and utilized to adapt the parameters H-line the plant model was generated at the low end of the controller.

of the H-line at an open-loop stable condition, and then The type of system just described is only suitable for learning was turned off, thereby freezing the plant model minimum phase systems. The introduction of a time parameters. This arrangement was used to successfully delay on the plant input used in the inverse modelling suppress flutter along two H-lines as shown in figure 9.

process allows one to obtain an approximate delayed in- 6 OF 1 2

Input I 4 I Response I

A I _ - Prefilter

Y

a) Inverse modelling process.

Desired Response Response Network Plant a) Data acquisition and controller training.

b) Control process.

TE Control Fig. 10 Generic inverse model control block dia- gram.

verse model to both minimum and non-minimum phase plants. Reference 14 also suggests using a prefilter on the plant input signal. Both the time delay and prefilter con- cepts will be employed in this study to generate inverse plant models. As the aeroelastic control application in- vestigated here is not a trajectory following problem, b) Controller implementation.

the actual implementation of the inverse model in the control system will be different than that proposed by Fig. 11 Inverse model control system architecture.

Widrow.13-15 The implementation of the inverse model (A) was an integer number of time steps. The prefilter in a closed-loop system is discussed next.

was implemented using the MATLAB FILTFILT func- The inverse model can take numerous forms including tion. This function digitally filters the data in both the the one used here, a neural network. The implemen- forward and reverse directions. The result has zero phase tation of the neural-network inverse model used in this distortion with a magnitude modified by the square of study is similar to that proposed in reference 16. Unlike the filter’s magnitude response. The prefilter is tailored many other studies, reference 16 suggests a relatively to have a peak magnitude near the frequency of the simple implementation where the neural network inputs physical phenomenon to be controlled. For flutter sup- and outputs are connected directly to the plant like a pression, the flutter frequency is used.

standard feedback controller. The present implementa- The lower portion of figure 11 shows the implementa- tion will use this approach. However, unlike reference tion of a trained network. Here the network is inserted 16 which used perceptron neural networks trained us- in the control loop with the the BACT model’s T E con- ing genetic algorithms, the present application will use trol surface command as the output of the network. This a two-layer feed-forward neural network trained using control system also was implemented on the ADC” sam- back-propagation.

pling at 200 Hz. The gain, IC, on the output signal could The present implementation of the inverse modelling be varied on-line.

approach is shown in figure 11. The upper part of the fig- The following steps summarize the operation of this ure shows the acquisition of training data and the off-line system.

network training procedure. The input to the network is one of the BACT model’s accelerometer signals. As 1. Send excitation signal to the BACT model recording indicated on the figure, the input to the network is made both the excitation (TE control surface command) up of the current and delayed values of this signal. The and the model response (accelerometer signal).

number of time-delayed inputs used can be varied. Also, as discussed earlier, the model input signal used to train 2. Train the network using delayed and filtered excita- the network is delayed and filtered prior to used. The tion signal and BACT accelerometer response time data acquisition was performed using the ADCll sam- history.

pling at 200 Hz. The network training was performed off-line using the MATLAB software. The time delay 3. Implement trained network on the real-time system.

7 OF 1 2 SPIE PAPER 3991-30. MARCH 2000 model to be used for either accelerometer signal (trail- ing edge or leading edge) or either control surface (TE control surface of upper spoiler) without changing the model. For instance, if the gains on the upper spoiler control signal and the gain on the leading edge ac- celerometer signal are both zero, the resulting system would be SISO utilizing the T E control surface and the T E accelerometer. This was the configuration considered in this investigation. In addition, excitation generators and their associated gains were introduced for obtaining training data and for studying the effects of noise and turbulence.

Several parameters that could also be varied but are not shown in figure 12, include the number of hidden LSS &In layers on the neural network and the number of time- delayed network inputs. The hidden layer could be either Fig. 12 Inverse model control system simulation linear or nonlinear.

block diagram.

Parametric Variations Prior to implementation of this system, many parame- This section of the paper will present results from sev- ters were explored using the simulation model discussed eral parametric variations using the simulation model in the next section.

described above. First, a few words about how these parametric studies were performed. The primary objec- Simulation Model tive of this system is flutter suppression and the per- The simulation model developed by W a s ~ a k ~ ~ ~ ' ~ was formance of the control system will be evaluated at a used in this study. The model was formed by combin- dynamic pressure where the model is open-loop unsta- ing the equations of motion for the BACT wind-tunnel ble, 175 psf. The data for training the network must be model with actuator models and a model of wind-tunnel obtained initially at a dynamic pressure were the model turbulence. Wherever possible, the numerical model pa- is open-loop stable. The initial training was obtained rameters were determined experimentally. The mass and from an open-loop simulation using a dynamic pressure inertia parameters were obtained by measuring the mass, of 133 psf where a pseudo-random noise (PPN) or lin- stiffness, and damping properties of the BACT flexible ear sign sweep (LSS) input was used to drive the T E mount. The static aerodynamic parameters were deter- control surface. Both excitation signals had frequency mined from experimental data when the BACT model content between 0 and 12 Hz. The prefilter for the flutter was mounted to a five-degree-of-freedom balance.' The suppression applications was selected to have peak mag- dynamic derivatives were obtained computationally us- nitude of unity in the vicinity of the flutter frequency of ing 1SAC.l' The numerical values for the static and the BACT model. The prefilter used for flutter suppres- dynamic stability and control derivatives are only valid sion control laws is shown below.

at a single Mach number of 0.77; however, the dynamic -431. IS pressure for the model could be changed. The analyt- '

ical flutter boundary for this simulation model occurs s3 + 1 9 . 6 ~ ~ + 811.6s + 2529.8

at a dynamic pressure of 150.8 psf. Figure 12 shows It has a peak magnitude of unity at 4 Hz and significant the version of the simulation model used in this study.

washout below and rolloff above this target frequency.

The BACT equations of motion and actuator models In all the parametric results presented, the RMS val- developed by Waszak are contained in the appropriately ues of the T E accelerometer and the T E control sur- labeled blocks. The turbulence model in the upper left face will be plotted against the parameter being varied.

portion of the model is based on a Dryden spectrum Lower is better and values off the scale of the plot indi- with parameters tuned to match power spectrum data cate that the system is not stable. The value of A, the obtained in the TDT. The other elements of the block number of time delays used in training the network, is diagram were added for this investigation.

a very important parameter and will often be used as The primary changes to the model developed by the independent variable in these plots. The nominal Waszak included the addition of the neural-network con- simulation parameters are as follows trol blocks and numerous other gain blocks. The control Network Controller: system studies in this investigation were SISO so the gain blocks were included to allow the same SIMULINKl' Hidden Nodes = 6 8 OF 1 2 SPIE PAPER 3991-30, MARCH 2000 Accel RMS, 0.4

kJ7, 0 With Prefilter

g s 0.2

2o 15 L

TE RMS, Degrees lo I I I I 0 5 10 15 20 ' 5 0 35 40 45 50 55 Number of time delays, A 0 5 10 15 20 25 30 35 40 45 50 55 Number of time delays, A Fig. 13 Comparison of linear inverse model control system response with and without the use of the pre- Linear network inverse model controller per- Fig. 14 filter.

formance for varying A and turbulence gain.

Time Delayed Inputs = 25 A = 25 Training Data Simulation:

, I I

q = 133.0 psf Turbulence Gain = 3.0 Simulation Time = 30 seconds T E Control Surface Excitation = PPN 0 With Prefilter Controller Evaluation Simulation: TE RMS, q =175.0 psf Degrees l o Turbulence Gain = 3.0 Simulation Time = 30 seconds 0 5 10 15 20 25 30 35 40 45 50 55 Control Law Gain, IC = 2.0 Number of time delays, A The first issue to be explored is the use of the pre- Fig. 15 Comparison of nonlinear inverse model con- filter. Figure 13 shows a comparison of linear controller trol system response with and without the use of the performance where the linear network controllers were prefilter.

trained with and without the prefilter.

The compar- ison shows that a controller that will suppress flutter The use of a nonlinear network (sigmoid transfer func- can be obtained without using the prefilter, but it will tion on the hidden layer) is now considered. Figure 15 have poor robustness properties. shows a comparison between nonlinear networks trained Typically, an accept- able flutter suppression control system will have control with and without a prefilter as a function of A . An im- portant feature of the nonlinear result is that while the surface RMS values significantly below unity. Without the prefilter, only one value of A achieves acceptable system may become unstable, the control command is control system performance. limited by the saturation of the sigmoid transfer func- tions. Otherwise, the linear and nonlinear network con- The next parametric variation considered a linear net- trollers achieve similar levels of performance. To further work where A and the turbulence gain were varied. The explore the use of nonlinear network controllers, the tur- turbulence gain was varied for the open-loop simulations bulence gain and prefilter were varied. Figure 16 shows where the training data was acquired. A gain of three a comparison of the performance of nonlinear network was used to evaluate the closed-loop performance. This controllers where the training data was obtained using data is shown in figure 14. From these data, it is ap- various values of the turbulence gain. As with the lin- parent that a time delay between 25 and 35 achieves the ear case, moderate levels of turbulence were found to be best performance. It can also be observed that moder- desirable.

ate levels of turbulence improve the performance of the network controllers.

The next parameter to be considered in this study 0 Linear, PPN 0 Nonlinear, PPN 0 Linear, LSS A Nonlinear, LSS

I I

n .

U TE RMS, 0 5 10 15 20 25 30 35 40 45 50 55 Degrees Number of time delays, A Nonlinear network inverse model controller Fig. 16 performance for varying A and turbulence gain.

Number of time delays, A 1 .o Linear and nonlinear inverse model con- Fig. 18 0.8 trollers trained using PPN and LSS excitation.

TE Accel. 0.6 RMS, g’s o.4 This completes the discussion on the inverse model control parametric studies. Note that not all param- 0.2 eters were varied. Future studies should consider the use of the leading accelerometer sensor and the use of 0 Linear, q=175 psf the spoiler. The amplitude of the control surface excita- 0 Nonlinear, q=175 psf tion was not varied, and sensor noise was not considered 0 Linear, q=195 psf A Nonlinear, q=195 psf at all. The only control systems considered were SISO,

TE RMS, I

Degrees l o yet the simulation model can be modified to use both leading-edge and trailing-edge accelerometers as input signals to the controller. Finally, a truly adaptive sys- 0 10 20 30 40 50 60 tem needs to continually re-acquire data and train new Time Delayed Network Inputs inverse model controllers as the plant changes. This was Fig. 17 Inverse model controller performance as a not considered in the present study.

function of the number of delayed network inputs.

Experimental Results A=25.

This section of the paper describes experimental re- was the number of delayed inputs to use on the network sults for application of inverse modelling control to the controller. Figure 17 shows a comparison of performance BACT model during the 1996 wind-tunnel test. The of a linear and nonlinear network where the number of networks used during this wind-tunnel test all had 25 time-delayed inputs was varied from 0 to 60. The value of time-delayed inputs to the network and 6 nodes on the A in these analyses was 25. The systems were evaluated hidden layer. These controllers were SISO with the in- at two dynamic pressures, 175 psf and 195 psf. At 175 put coming from the T E accelerometer and the output psf dynamic pressure, the performance of the system was being sent to the T E control surface. Unless otherwise insensitive to the number of time delays, but at 195 psf noted, the prefilter used is the same as that described in dynamic pressure, a minimum value of 15 time delays the preceding subsection. Due to time constraints only was required.

a limited number of parameter variations were explored experiment ally.

The final parameter considered is the excitation type used to obtain the training data. So far only a PPN Two inverse model flutter suppression control systems excitation has been used. A comparison of PPN results were demonstrated. Figure 19 shows a portion of the with those obtained using a LSS excitation is presented TDT operating envelope with the BACT open-loop flut- in figure 18 for both linear and nonlinear controllers.

ter boundary superimposed. The two inverse model Both types of excitations yield controllers with similar control systems are designated A and B. System A was performance, but the PPN excitation appears to have a trained using data acquired at conditions well below slight advantage over the LSS excitation.

the open-loop flutter boundary (M=0.65, q=133 psf).