Adaptive control of nonlinear aeroelastic systems

ITALIAN ASSOCIATION OF AERONAUTICS AND ASTRONAUTICS XXII CONFERENCE NAPOLI, 9-12 SEPTEMBER 2013

Adaptive control of nonlinear aeroelastic systems through recurrent neural networks A. Mannarino1,∗ and P. Mantegazza1 1

Politecnico di Milano, Dipartimento di Scienze e Tecnologie Aerospaziali, Via La Masa 34, 20156, Milano ∗

Corresponding author, email: [email protected] Abstract

Aeroelastic systems have the peculiarity of changing their stability behaviour with flight conditions. In such a view, it is difficult to design a control law able to perform efficiently in very different circumstances. Moreover, control laws are tipically designed on linearized, low-fidelity models, and this fact introduces the need of a scheduling tuned on a wide range of flight conditions. Obviously such a design process can be quite a burden, because of the high number of simulations and flight tests required for its tuning. In this work recurrent neural networks are employed with the aim of suppressing the auto-excited oscillations of aeroelastic systems beyond the stability limit, identified by the flutter speed. The control law adopted is tailored on linear low-fidelity models through various numerical simulations, then applied, without changing the pre-computed parameters, to a higher-fidelity representation of the same system, including nonlinear, unsteady aerodynamics. The training of such a neural network is kept active during all the simulations, so to achieve the required level of adaptiveness. The effectiveness of such a control system is verified on the BACT benchmark case, evaluating its robustness properties against changing parameters and flight conditions.

Keywords: Recurrent Neural Networks, Nonlinear Aeroservoelasticity, Adaptive Control, Validation and Verification

1

INTRODUCTION

The improvement of aircraft performances through active control systems is a well established research and industrial topic. As different sources state [1, 2, 3], the next generation of flight control systems will utilize adaptive and non-deterministic techniques to provide more stable and maneuvrable aircrafts. The study of the interaction of structure compliance, aerodynamic loads and control laws involves many diverse disciplines and deals with problems such as wing flutter, buffeting, divergence, control surface effectiveness and reversal, buzz and gust load [4]. Until the last decade, the lack of efficient computational tools has restricted the study of aeroservoelastic systems to linear subsonic and supersonic 1

flight regimes [5]. The progress in Computational Fluid Dynamics (CFD) has permitted to compute accurately unsteady transonic aerodynamic loads for viscous and inviscid flows in the time domain. The adoption of CFD-based estimates of aerodynamic loads is also becoming graduately a standard in industry [6], thanks to the increasing computational power and efficient numerical algorithms available. The use of Euler equations allowed studies of transonic flows with strong shock waves, where nonlinear phenomena, such as weak divergence and limit cycle oscillation can occour. The full control of these dangerous events is of utmost importance in avoiding unacceptable vibrations, self-induced oscillations, ride-quality deterioration and fatigue failures [4]. Different approaches to the control of mechanical and aeroelastic systems can be found in the literature, e.g: classical LQG design [7], robust H∞ framework [8], input limiting [9, 10], immersion and invariance approach [11] and neural networks, both static and recurrent [12, 13, 14]. Since aeroelastic systems change their stability properties in relation to the flight condition, an efficient control law should be able to perform its functions over the whole flight envelope without losing its efficiency. To achieve such a results, two different approaches can be considered: control law scheduling and adaptive control. While the first approach has proved to be very robust and it is currently widely employed in service [15], the latter has not seen any operational deployment yet. The reason is that the certification process for adaptive flight control software for operational use has not yet been decided [1]. Actually, there are two possible ways to realize the control. In one, a learning algorithm is used to compute flight control inputs to augment the controls produced by a non-adaptive flight controller, while the other approach uses a system identification algorithm to compute gain parameters used by the flight controller. A notational diagram of an adaptive control system is shown in figure 1.

Figure 1: Adaptive controller block diagram In this work a neural controller is designed using a recurrent network characterized by on-line learning. The proposed method can be classified within the latter approach mentioned, since the controller is composed by two networks, where as one is used to identify the acceleration of some particular points of the dynamic system, the information so obtained being as input to the second network which acts like a controller. The present work has several objectives: modeling the strong fluid-structure interactions in transonic regime, prove the efficacy of a recurrent neural network-based controller, designing it over a linear, low-order representation of an aeroelastic system, verifying the resulting control law with a high fidelity simulation. Eventually, possible standards for the validation and verification of the results obtained by an adaptive controller will be defined.

2

2

NEURAL NETWORKS

A Neural Network (NN) is a massively parallel distributed process made up of simple processing units (the neurons) that have a natural capability to store the knowledge accumulated through experience, making it available for later uses. Knowledge is acquired through a learning process, and it is stored in the synaptic connections linking the neurons. A general network is composed by an input layer, where the input data is stored and passed to the computational units, represented by the neurons, which receive a linear combination of the input, where the coefficients of the combination are called synaptic weights of the network, and use this input as the argument of a nonlinear function, usually a squashing function, in this work assumed to be the hyperbolic tangent, located in the hidden layer. Because of the universal approximation theorem, a neural network with one hidden layer is sufficient for approximating a nonlinear function with arbitrary accuracy, ensuring that the number of neurons is adequate [16]. In order to represent a dynamic process, the network should be able to learn its evolution path from the input-output data pairs only. The memory is included through a time delay applied to the hidden neurons, defining thus the state of the network. Basically, a network state is an hidden neuron which is connected to an other hidden neuron. Such a network is defined as Recurrent (RNN). In this case the input layer is composed by the set of the input data, function of the time now, and the set of the delayed output of the hidden neurons. A graphical representation of such a recurrent network is given in figure 2. An RNN presents many advantages over static networks, where recurrent connections are absent: smaller dimensions, reduced computational cost and a faster learning: all characteristics that are very important for real-time control problems [13, 14, 16].

Figure 2: Example of recurrent neural network A mathematical description of such a model can be given in the following compact form:  x

n+1

= φ Wa xn + Wb un

y

n

= W c xn





(1)

where xn ∈ Rnx is the network state at time tn , u ∈ Rm is the network input, yn ∈ Rp is the network output, Φ : Rnx −→ Rnx is the set of activation functions, Wa ∈ Rnx ×nx , Wb ∈ Rnx ×m and Wc ∈ Rp×nx are the matrices containing the network synaptic weights; nx , m and p being respectively the state, input and output space dimensions. In practice, nx represents the number of neurons involved in the network. In general, the network state xn has only an internal meaning: no physical interpretation can be associated to 3

it. However, assuming the output matrix Wc with a fixed structure, able to extract the first p elements of the state, we know that such state elements are actually the physical output of the model, gaining some physical insight in the process [16].

3

CONTROL METHODOLOGY

As discussed before, the proposed controller is composed by two networks, the identifier and the control sections. Let us start with the first.

3.1

Identifier network (ID-RNN)

Let us assume to have available a model of the process which is able to give in output some signal, in this case represented by the acceleration of some points on the dynamic system. These quantities are the known, measured, quantities of the model, indicated by ynM . Subjected to the same input of the physical model, the ID-RNN should be able to minimize the error between its output and the measured output. Recalling equation 1, in the case of ID-RNN the input unID is composed by the measured accelerations, expressed in [g] units and the actual control input, measured in [rad], while the output vector ynID is represented by the estimated accelerations, measured in [g] too. The training of such a network can be interpreted as an optimization algorithm which minimizes the following cost function: 1 en = ynID − ynM (2) EnID = eT n en 2 In order to fulfill some of the certification gaps described in [1], in this work the network training is tackled in two different ways, one for the low-order model, the other for the verification model. Regarding the first, an Extended Kalman Filter is coupled to the RNN. Collecting all the optimization variables in a unique vector θID = 



vec W

a,ID

T



, vec W

b,ID

T T

, being vec the operator which orders the columns of a

matrix in a unique column vector, the updating of its value is performed by: 

ID θn+1 = θnID + Gn ynID − ynM



(3)

The matrix Gn is called Kalman gain and it is computed by a recursive solution of the Riccati equations. For the related details we refer the reader to [16]. In the latter case instead, the update of the weights value is obtained through the simpler gradient technique: ID θn+1 = θnID − η ID

ID ∂EnID ID ID T c,ID ∂xn c,ID ID = θ − η e W = θnID − η ID eT Λn n n nW ID ID ∂θn ∂θn

(4)

As can be noted, the variation of weights goes in the opposite direction of the gradient of the identification error. The parameter η is called learning rate and it is the fundamental parameter governing the ID-RNN training velocity. The learning rate value is limited by stability considerations: too-high values may destabilize the training algorithm, causing the divergence of synaptic weights. On the other hand, a low learning rate would limit the control system capability to quickly adapt itself to system variations. Unfortunately, there is no general theory for predicting the η limiting values for ID-RNN, and so its best value must be tuned through design simulations. 4

Note that the EKF does not require any learning parameters, therefore requiring even less expertise in its utilization with respect to the gradient descent algorithm. However a redundant procedure has been chosen in order to approach a more reliable control system. We expect some sort of difference in the results that will be obtained with the two approaches: of course part of this difference will be due to the different training methodologies, while the rest will be represented by the different models of the plant, respectively linear in the low fidelity and nonlinear in the high fidelity simulations. Even with these transient differences, we require that the stabilization task is accomplished by both methods. The gradient of the quadratic identification error, required by both methods, is computed according to the Real-Time Recurrent Learning (RTRL) algorithm [16]. Note that ΛID n is a dynamic variable, like xn , therefore a difference equation should be derived for updating its value at each time step. Since the structure of Eq. 1 is known, such a difference equation can be derived analitically, resulting in the following general form: 





ID ID xnID , unID ΛID xnID , unID ΛID n +U n+1 = H

3.2



(5)

Controller network (CO-RNN)

Recalling equation 1, in the case of CO-RNN the input unCO is composed by the identified accelerations and the actual control input, meanwhile the output vector ynCO is represented by the new value of the control input. The training of such a network can be interpreted as an optimization algorithm which minimizes the following cost function: 1 1 T EnCO = eT ρ ynCO ynCO en = ynID (6) n en + 2 2 The parameter ρ is defined as the control penalization parameter and is used to limit the control effort in a way similar to what is done in a classical LQG controller. In a unstable system, such as the one under investigation, all zeros in the right half of the complex plane may be transformed into unstable poles of the controller by a straight system inversion. In this case, the penalty term is necessary to avoid the divergence of the control [13, 14]. The same redundant approach to network training employed in the ID-RNN case is used here. However in this case care must be taken to consider that the dynamics of the ID-RNN and CO-RNN are coupled. The structure of the resulting control system is represented in figure 3. As a matter of fact, the adaptivity of the control system is achieved by keeping the networks training always active, so that the control adapts itself to system variations by modifying the networks weights appropriately.

CO-RNN

ID-RNN

Figure 3: RNN controller block diagram

4

AERODYNAMIC MODELING

In the verification phase, the aerodynamic sub-system is modelled by the Finite-Volume (FV) aerodynamic solver AeroFoam, developed at Dipartimento di Scienze e Tecnologie 5

Aerospaziali of Politecnico di Milano, Italy. It kicked-off back in 2008, as an ambitious academic experiment, which had the challenging target of filling the empty space left in Open-FOAM, which had no density-based compressible solver available. So AeroFoam is a density-based compressible RANS (Reynolds Averaged Navier Stokes) solver which provides also Euler equations modelling options, as selected in this work. Among the innovative contributions, it is worthwhile to remark: the dedicated aeroelastic interface scheme based on Moving Least Squares (MLS) interpolation strategy, providing all the functionalities necessary to link the structural sub-system to the aerodynamics, and the dedicated hierarchical mesh deformation tool for dealing with moving boundary problems in Arbitrary Lagrangian Eulerian (ALE) formulation. Convergence analyses and capabilities in aeroelastic applications can be found in [17].

5

NUMERICAL RESULTS

The Benchmark Active Controls Technology (BACT) project is part of NASA Langley Research Center Benchmark Models Program for studying transonic aeroservoelastic phenomena. The BACT wind-tunnel model was developed to collect high quality unsteady aerodynamic data (pressures and loads) near transonic flutter conditions and demonstrate the potential of designing and implementing active control systems for flutter suppression using flaps and spoilers [18]. The BACT wind-tunnel model has been used in this work as a reference test case, since a wide database of experimental and numerical results is available. It is expected that nonlinear transonic effects and aerodynamic time delays will have an important role in the dynamic behavior of the system, and CFD will be employed to tackle the high fidelity aeroelastic problem. The BACT wind-tunnel model is a rigid rectangular wing with a NACA 0012 airfoil section. It is equipped with a trailing-edge control surface and upper and lower-surface spoilers that can be controlled independently by means of hydraulic actuators. The wind-tunnel model is instrumented with pressure transducers, accelerometers, control surface position sensors, and hydraulic pressure transducers. The model and its structural idealisation are shown in figure 4.

Figure 4: BACT physical and idealised structural model The wing is mounted to a device called the Pitch and Plunge Apparatus (PAPA) which is designed to permit motion in principally two modes, rotation (or pitch) and vertical translation (or plunge). The combination of the BACT wing section and PAPA mount (which will be referred to as BACT system) was precisely tuned to flutter within the operating range of the Transonic Dynamics Tunnel (TDT) at NASA Langley Research Center. The BACT system has a dynamic behavior very similar to the classical 2 Degrees Of Freedom (DOFs) typical wing section, primarily differing from it because of the complexity of the aerodynamic behavior and the presence of additional structural modes. Moreover, finite span and low aspect ratio of the BACT wing introduce significant three-dimensional flow 6

effects. Higher frequency structural degrees of freedom are associated with the PAPA mount and the fact that the wing section is not truly rigid. The control surfaces also introduce complexities not typically reflected in the classical 2 DOFs system. The mass and inertia of the control surfaces and potential flexibility in their support structures introduce inertial coupling effects. The finite span of the control surfaces and their close proximity to each other also introduce significant aerodynamic effects. All these issues influence the development of the equations of motion but, as will be seen, do not force the abandonment of the 2 DOFs system structure.

5.1

Design on low fidelity model

For the design of active control systems it is convenient to adopt an approximated and efficient numerical model able to provide satisfactory results within the frequency range of interest, rather than choosing an over-detailed and expensive numerical model. To this end, the linear model proposed in [18], with a quasi-steady approximation of the aerodynamic loads is employed. The need to include more sophisticated aerodynamic modelling options such as unsteady kernel approximations or time accurate CFD is albeit mitigated by the fact that the flutter reduced frequency for the BACT wing is relatively low, apωc ≈ 0.05. The aerodynamic derivatives have been estimated through proximately k = 2V∞ wind tunnel testing, with exception of the unsteady ones, obtained from computational aerodynamic analysis. Including also the actuator dynamics, as derived in [19], the model so obtained can be written in the following form, where the subscript SE represent the servoelastic part of the model, meanwhile the subscript A represent its aerodynamic part: 

¨ + CSE q˙ + KSE q = MSE q

c 2 V∞

2

¨+ MA q

c CA q˙ + KA q + Bs βC 2 V∞

(7)

where c is the aerodynamic chord, V∞ the flow asymptotic velocity and the matrices are detailed as:     h   θ q=   β    z



MSE

m  Shθ = Shβ 0

Shθ Jθθ Iθβ 0

Shβ Iθβ Jββ 0

 0 0  0 1

CSE

 chh  0 =  0 0

 0   0  MA = q∞ 2S Bs =    0 c 2 kact ω0act   −CL,α −l CL,α −CL,β˙ 0  2S  c CM,α lc CM,α c CM,β˙ 0 CA = q∞  0 0 0 0 c 0 0 0 0 



0 cθθ 0 0

0 0 cββ 0

 0  0   0 2ξact ω0act

KSE

 khh  0 =  0 0

 −l CL,α˙ 0 0 lc CM,α˙ 0 0  0 0 0 0 0 0  0 −CL,α −CL,β 0 c CM,α c CM,β KA = q∞ S  0 0 0 0 0 0

−CL,α˙ c CM,α˙   0 0

0 0 kθθ 0 0 kββ 0 0 (8) (9)

 0 0  0 0

(10)

The related data are available in [18]. The actuator state is here represented by the variable z, q∞ = 1/2ρ∞ V∞2 is the dynamic pressure, S is the wing surface, l is the relative distance between the elastic axis and the aerodynamic center of the wing and βC is the control input. The actuator dynamics is included in the model considering also its compliance, according to the frequency-domain formulation: β(s) =

kact ω02act Mβ β (s) − C 2 s2 + 2ξact ω0act s + ω0act kββ 7

(11)

 0 0   −kββ  ω02act

Mβ being the hinge moment acting on the aileron. The compliance 1/kββ has been chosen to be small enough to maintain a good level of actuation precision. Further analyses emphasizing possible effects of such a parameter on the performance of the control system are currently under development. Two capacitive accelerometers are collocated on the wing, one on the leading edge the other slightly behind the elastic axis, located at the mid-chord. Their dynamic model is assumed to be represented by a 2nd order model, characterized by an adequate bandwidth. First of all, a linear flutter analysis is performed varying the flight speed and tracking the pitch and plunge modes. All the simulations are performed at a Mach number M∞ = 0.77, in the heavy gas R12 [18]. The results are shown in figure 5. 8

Plunge Pitch

0.2

Plunge Pitch

7

0.15

6 0.1

5

f [Hz]

g [−]

0.05

0

4

3

−0.05

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2

−0.15

1

−0.2

0 0

20

40

60

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V∞ [m/s]

100

120

140

0

20

40

(a)

60

80

V∞ [m/s]

100

120

140

(b)

Figure 5: Linear flutter analysis The flutter speed is predicted at V∞ = 108.58 [m/s] with an error of 1.3 % with respect to the experimental value. Following the interpretation of [1] we define some standard testing cases for our control law: response to initial conditions, response to input pulses, response to random input with time varying flow velocity and any response evaluated for a slightly different model (variations in mass, inertia, stiffness, etc.). Of particular interest is also the effectiveness of the controller when switched on only after an input perturbation is introduced. We choose to employ the first two cases for tuning the control law, while the others are considered in the verification phase ones. Extensive numerical simulations have been carried out to understand the behaviour of the ID-RNN and CO-RNN and to determine the correct networks topology in terms of number of neurons and value of the various parameters involved. The results of this numerical campaign are summarized in table 1. ID-RNN CO-RNN 6 5 6 · 10−2 8 · 10−2 4 · 10−2 200 [Hz]

nx η ρ Sampling frequency

Table 1: Networks parameters As also reported in [14], the number of neurons affect the networks memory. Increasing nx , the robustness and accuracy of the identified output is increased, as well as the 8

related computational cost is, while the learning rate should be decreased to avoid possible unstable effects. The learning rates have been chosen as a trade-off between fast learning (high η) and its stability (low η). The results obtained in the first two test cases mentioned are displayed in figures 6 and 7, where it is shown the response of the accelerometer located at the leading edge and the control effort involved. The design speed is V∞ = 120 [m/s], well beyond the open loop flutter limit. The initial condition assumed in the first test case is q0 = (0.01, 0, 0, 0)T , while in the second case the initial condition is homogenuous and an impulse of amplitude 1 [deg] is given to the actuator, at time t = 0.1 [s].

Control effort 0.6

BACT model − open loop BACT model − closed loop RNN adaptive controller

0.4

0.3

0.4

0.2

0.2

βc [deg]

y1 [g]

0.1

0

0

−0.2

−0.1

−0.4

−0.2

−0.3 −0.6

−0.4 0

1

2

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t [s]

(a)

(b)

Figure 6: Response to initial condition

Control effort 0.12

BACT model − open loop BACT model − closed loop RNN adaptive controller

0.03

0.1

0.08

0.02

0.06

0.01

β [deg]

0

0.02

c

y1 [g]

0.04

0

−0.01 −0.02

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−0.04

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0

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t [s]

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t [s]

(a)

(b)

Figure 7: Response to input impulse Real and identified output are both shown to prove that a fairly good level of convergence has been obtained by the ID-RNN. The performance indexes considered in the design have been the settling time, which should be as small as possible, and a limited control effort. For the linear model these requirements have been satisfied. Since the closed-loop system is formulated in a discrete time framework, a classical Lyapunov analysis for assess the 9

stability of the whole system is rather difficult. However, as suggested in [2], the run1 time monitoring of the Lyapunov function V = qT q has provided valuable stability 2 information while the system is in operation.

5.2

High fidelity simulation

The high fidelity model is composed by the previous servoelastic model coupled to AeroFoam. After a convergence analysis performed on steady state cases, a computational mesh composed by 103040 cells is employed. The effect of a moving aileron is included by a smooth mesh deformation algorithm based on radial basis functions [17]. At the same design test point of the previous analysis, a sample of the aerodynamic and leading edge acceleration is given in figure 8. As can be seen from figure 8, the effect of a nonlinear 0.04

0.03

0.02

y [g]

0.01

0

−0.01

−0.02

−0.03

−0.04

0

2

4

6

8

10

12

14

16

Time [s]

(a) Sample of numerical solution

(b) Limit cycle oscillation at V∞ = 120 (m/s)

Figure 8: High fidelity representation of the BACT wing aerodynamic model is somewhat beneficial to the dynamics of the whole aeroservoelastic system, because beyond the stability limit the wing oscillations are bounded by a limit cycle and an exponential divergence, typical of linear systems, does not occour. The control law designed in the previous case is now applied without changing any parameter. However, networks training is kept active in order to achieve the required level of adaptiveness in this verification simulation also. The results regarding the response to input noise with varying flight speed are shown in figures 9 and 10, while the response to an input impulse at V∞ = 120 [m/s], switching on the controller at t = 0.5 [s], are shown in figure 11. As it can be seen, the controller designed on the low fidelity model is still able to efficiently stabilize the nonlinear model under operational conditions never seen before, a really appreciable feature of neural network-based controllers. To conclude we present a comparison of response and control effort for the linear and nonlinear models, carried out at V∞ = 130 [m/s]. The case considered is the response to input impulse, applied at t = 0.2 [s] and amplitude 1 [deg]. The results are shown in figure 12. The two responses are quite different, this can be due to the effects of a nonlinear, unsteady aerodynamic versus a linear, quasi-steady approximation, combined with a possible different training convergence history. Such an influence should be investigated by further studies. As a results, the control effort is greater during the first peaks in the nonlinear case, but with a decay that is much more faster than its linear counterpart. However, the stabilization task is carried out very efficiently in both cases, with a maximum control effort well below 10

126

0.04

0.03

124

0.02

122

∆ β [deg]

C

120

∞

V [m/s]

0.01

0

−0.01

118 −0.02

116 −0.03

114

0

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Time [s]

(a) Flight speed ramp

(b) Input noise

Figure 9: Flight conditions 0.04

0.15

Open loop Closed loop

0.03

0.1

0.02 0.05

β [deg]

0

0

C

y [g]

0.01

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Time [s]

(a) Response at leading edge

(b) Control effort

Figure 10: Response to noise with varying flight speed 0.04

1.2

Open loop Control off Control on

1

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βC [deg]

y [g]

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Time [s]

(a) Response at leading edge

(b) Actuator output

Figure 11: Control switch on/off the quite conservative limit of 1 [deg]. Further numerical simulations have been carried out, resulting in a new flutter speed of 134 [m/s] (25 % greater than the nominal case), 11

Control effort 0.4

Nonlinear Linear

0.04

Nonlinear Linear

0.3

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0.2 0.02

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βc [deg]

y1 [g]

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t [s]

t [s]

(a) Response at leading edge

(b) Control effort

Figure 12: Comparison between linear and nonlinear model achieved by assuring the previously cited limit on the control effort. Other analyses are now under consideration to assess the effect of possible actuator saturations.

6

CONCLUDING REMARKS

In the present work a recurrent neural network-based controller is presented. A model hierarchy is considered for the design and verification of the control law, proving that the adaptive controller presented is not affected by any spill-over effects, perfoming fairly efficiently in all the test cases considered. Some of the suggestions proposed in the literature regarding the Validation and Verification (V&V) of adaptive control law are followed, trying to define some peculiar test cases which can be used to asses the performance of the controller since the first steps of the design. Furthermore, a redundant controller design is considered, providing two different learning algorithms for design and verification phases. In this simple case analysed, very interesting results are achieved, in particular the ability of the controller to rapidly adapt to unforseen configuration is proved. However, as stated in different sources, no complete methodology exist to provide a V&V practitioner with the ability to gain an assurance for neural networks in safety and mission-critical systems, and the ultimate goal to provide guarantees for all unexpected cases will probably never be reached.

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