Experimental Validation of a Metrics Driven L1 ...

Experimental Validation of a Metrics Driven L1 Adaptive Control in the Presence of General Unmodeled Dynamics Ioannis Kitsios∗, Vladimir Dobrokhodov†, Isaac Kaminer‡, Kevin D. Jones§ Naval Postgraduate School, Monterey, CA 93943

Enric Xargay¶, Naira Hovakimyan University of Illinois at Urbana-Champaign, Urbana, IL 61801

Chengyu Cao

∗∗

Mariano I. Lizarraga

University of Connecticut, Stoors, CT 06269

††

UC Santa Cruz, Santa Cruz, CA 95064

Irene M. Gregory

‡‡

NASA Langley Research Center, Hampton, VA 23681

Nhan T. Nguyen §§, Kalmanje S. Krishnakumar¶¶ NASA Ames Research Center, Moffett Field, CA 94035

The paper summarizes the results of an ongoing effort in the development and flight validation and verification of the metrics driven L1 adaptive flight control system. In particular, the paper develops a unified framework for design, implementation, validation and verification of flight critical control systems including: (i) definition of experimental control validation technique that accounts for control surface failures and generalized plant uncertainties or unmodeled dynamics; (ii) tuning the developed L1 adaptive controller to explicitly address performance metrics in the presence of control actuator/surface failures and modeling uncertainties under adverse flight conditions; (iii) development of a flight control system testing environment that includes both hardware and software setup for implementation of the resulting algorithms onboard of a small unmanned aerial vehicle; and (iv) designing and conducting of a comprehensive flight test validation and verification program that demonstrates performance of the proposed adaptive control algorithms in the presence of failures.

I.

Introduction

This paper addresses the objectives posed by NASA IRAC-1.5 (Integrated Resilient Aircraft Control) Program that focus primarily on the design, analysis and flight validation of novel adaptive flight control algorithms capable of fast and guaranteed safe recovery of a suddenly impaired aircraft. To meet these ∗ Postdoctoral

Research Fellow, Dept. of Mech. & Astronautical Eng., Cpt (HAF); [email protected]. Assistant Professor, Dept. of Mech. & Astronautical Eng., AIAA Senior Member; [email protected]. ‡ Professor, Dept. of Mech. & Astronautical Eng., AIAA Senior Member; [email protected]. § Research Associate Professor, Dept. of Mech. & Astronautical Eng., AIAA Associate Fellow; [email protected]. ¶ Graduate Student, Dept. of Aerospace Engineering, AIAA Student Member; [email protected]. Professor, Dept. of Mechanical Science and Engineering, AIAA Associate Fellow; [email protected]. ∗∗ Assistant Professor, Dept. of Mechanical Engineering, AIAA Member; [email protected]. †† Graduate Student, Baskin School of Engineering, AIAA Student Member; [email protected]. ‡‡ Senior Research Engineer, Dynamic Systems and Controls Branch, AIAA Senior Member; [email protected]. §§ Senior Research Scientist, Intelligent Systems Division, AIAA Associate Fellow; [email protected]. ¶¶ Senior Research Scientist, Intelligent Systems Division, AIAA Senior Member; [email protected]. † Research

1 of 36 American Institute of Aeronautics and Astronautics

objectives the paper develops experimental tools and methods for hardware-in-the-loop (HIL) and flight Validation and Verification (V&V) of the recently developed Theory of Fast and Robust Adaptation since it explicitly addresses the goals of the IRAC-1.5. Specifically, it guarantees 1) fast adaptation, which proves to be instrumental for improving both performance and robustness throughout the entire flight envelope, including emergency landing; 2) gain and time-delay margins similar to linear systems; and 3) a priori prediction of the uniform performance bounds. The ultimate goal of implementing L1 adaptive controller onboard is to guarantee that an airplane flying inside the nominal flight envelope and suddenly experiencing an adverse flight regime or control surface failures, such as instantaneous actuator failures, structural damage or unusual attitude, will not “escape” its nominal flight envelope provided that some control redundancy remains. This fact alone will allow for a decrease in fatality rate of commercial aircraft of the future. Furthermore, if active in adverse flight conditions, the L1 adaptive controller allows for maximum possible recovery of any remaining flight control authority by automatically reallocating and combining any available control channels. This is in sharp contrast to Fault Detection and Isolation (FDI) technique. Note that successful recovery from a failure, if possible at all, can be achieved only during the first few seconds after it occurs (there is simply no time for learning). Thus the guaranteed fast rate of adaptation of the L1 adaptive controller makes it ideally suited for such eventualities and may lead to the decrease in fatality rates. This directly contributes to the general mission of the IRAC Program. The literature on adaptive control of both linear and nonlinear systems is vast, see for example Ref.1, 2 . The adaptive control techniques proposed include multiple model switching and tuning, multiple model adaptive control, interactive multiple models, control allocation, sliding mode control, model predictive control, eigenstructure assignment, model referenced adaptive control (MRAC), model reference with dynamic inversion, neural networks, to name a few. In application to the adaptive flight control systems design, one of the historically most influential and effective control techniques is MRAC3–5 . MRAC and its numerous modifications were also successfully used in a number of flight tests for recovering nominal performance in the presence of modeling and environmental uncertainties3, 4 . However, a major challenge in the V&V of these systems was how to determine their stability margins, especially the time-delay margin which is directly dependent upon the adaptation gain. This issue has been usually resolved in Monte-Carlo simulations. It has been observed that increasing the speed of adaptation in MRAC leads to high-frequency oscillations in the control signal and reduces system’s tolerance to the time-delay in control and sensor channels. The lack of robustness exhibited by conventional MRAC was first pointed out in 1979 by Egardt6 and analyzed in greater detail by Rohrs et al, see Ref.7 . In particular, Rohrs and coauthors constructed a counterexample where a first order stable plant with two highly-damped unmodeled poles went unstable when driven by a reference sinusoid at the phase crossover frequency. Although the paper included a rigorous proof of the existence of two infinite-gain operators in the adaptive closed-loop system, the explanation given for the cause of instability was not completely satisfactory. The right explanation was provided in later papers by ˚ Aström8 and Anderson9 . The results and conclusions of this paper led to an ideological controversy, and robustness and convergence of adaptive controllers started to be investigated by other authors. A good survey on the topic can be found in Ref.10 . Special mention deserves the work in this field by Ioannou and Kokotovic11–13 , Peterson and Narendra14, Kresselmeier and Narendra15 , and Narendra and Annaswamy16 . In these papers, the authors not only analyzed the causes of instability but also proposed damping-type modifications of the adaptive laws to prevent them. The basic idea of all the modifications was to limit the gain of the adaptation loop and to eliminate its integral action. Examples of these modifications are the σ-modification13 , the -modification16 , and the parameter projection17 . All these modifications solved the problem of parameter drift ; however, the proposed adaptive control architectures left unsolved the real problem, which was the incapability to achieve uniform transient performance for different reference commands and initial conditions. Also, introducing these changes in search of robustness has it own price the ideal properties of the adaptive law are degraded as does the performance. The setup adopted in Ref.7 was applied to L1 adaptive control in Ref.18, 19 where it was shown that the L1 adaptive controller adapts both in gain and phase. Furthermore, in Ref.20–26 , it was shown that L1 adaptive control theory guarantees robustness (including gain and time-delay margins) in the presence of fast adaptation, achieved via continuous feedback. The separation between adaptation and robustness, inherent to all L1 adaptive control architectures, is made possible by appropriate modification of the control objective with the understanding that uncertainties in any feedback loop can be compensated for only within the available bandwidth in each control channel. This problem reformulation led to a solution that allows


for tracking both in transient and steady-state the input and the output of an auxiliary linear closed-loop reference system, which defines the new and less ambitious, but achievable control objective from the time instant t = 0. In this sense, the L1 adaptive control philosophy seems to follow the insight by Athans, who, in a NASA report in 1985 (Ref.27 ), suggested that the robustness issues of adaptive control should be solved using frequency-domain and non-adaptive control techniques. In brief, the L1 adaptive control theory shifts the tuning issue from the selection of the adaptive gain to determining the structure/bandwidth for a linear filter. In this paper we extend the setup introduced in Ref.7 to flight test environment, i.e we replace the first order nominal plant used there with a small unmanned aerial vehicle (UAV) controlled by a commercial autopilot (AP). The issues raised in Ref.7 coincide with the objectives of IRAC-1.5 and, therefore, form an ideal outline to follow. Thus, initially the conventional MRAC is used to augment the nominal plant and is asked to track a sinusoidal signal of a specific frequency. This is followed by adding modeling uncertainties and repeating the previous scenario. The latter is done to verify correctness of the flight test setup - the results obtained should be similar to the ones obtained in Ref.7 . The same steps are then used to test the L1 adaptive control system. This paper also provides a detailed account of the development of the experimental procedure first in the HIL environment with seamless transition to the flight testing. Each step includes (i) determination of the frequency response of the plant, then (ii) “injection” of failures or unmodeled dynamics with (iii) identification of the transient and performance characteristics of MRAC and L1 control algorithms. This experimental evaluation procedure represents a unified framework for in-flight V&V of the stability, robustness, and performance characteristics of the adaptive control algorithms under consideration. The paper is organized as follows. Section II discusses in-flight frequency response analysis and control V&V. This section also provides some theoretical details necessary for understanding and for an objective evaluation of the adaptive control algorithms. Section III describes the HIL and flight test results of three principal experiments providing the detailed explanation and analysis of the demonstrated performance characteristics. The first experiment demonstrates performance of the adaptive control algorithms in nominal conditions by varying frequency of the reference signal. The second experiment extends the results obtained by Rohrs and shows the performance of the L1 adaptive control algorithm under the same conditions. The third experiment is motivated by the ideas of the previous experiment but extends it to the modeling of the bending mode of an airplane. Finally, the fourth experiment illustrates the capabilities of the adaptive algorithms when recovering from an instantaneous control surface failure while the plant is driven by a sinusoidal reference signal. In conclusion, Section IV provides analysis of the results and defines the directions of future research.

II.

Preliminaries of the Study

The flight test setup used in this paper was first reported in Ref.28 , where the authors provided preliminary results on flight testing of L1 adaptive control system in the presence of UAV control surface failures. The control architecture included a commercial autopilot augmented by the output feedback L1 controller. The results obtained demonstrated that the L1 augmented system provides fast recovery to the sudden locked-inplace failures in either one of the ailerons or in rudder, while the unaugmented system goes unstable. In the present paper, the results for the nominal plant consisting of a UAV and a commercial AP (see Figure 1) are extended to include “unmodeled dynamics” “injected” at the output of the plant motivated by the approach outlined in Ref.7 . Two cases of unmodeled dynamics are considered. In the first case, these unmodeled dynamics are represented by a second-order transfer function that introduces significant uncertainty both at high and low frequencies, see Figure 2a. In the second case, uncertainty is given by a very lightly damped second-order system that represents, for example, a flexible body mode of an airplane, see Figure 2b (the natural frequency of the uncertainty is selected to be outside of the nominal system bandwidth). This setup, in turn, allows us to validate if a small disturbance at the resonant frequency of the transfer function can propagate to the control signal and excite the bending mode. In order to achieve the objectives outlined above, it is first necessary to obtain an adequate representation of the frequency response of the nominal plant. The prototyping system29 developed uses a commercial off the shelf (COTS) autopilot (Piccolo Plus AP30 ) and a remotely controlled (RC) UAV Sig Rascal. The avionics suite also includes an onboard PC104 CPU that communicates with the autopilot over a full duplex serial link at 20Hz. For the experiment at hand the capability to execute turn rate commands and to read


Gp u

y A/P

UAV

Figure 1: Nominal plant consisting of a UAV and a commercial AP.

Gp u

y A/P

UAV 2nd order Uncert.

Adaptive Augmentation (a) Second-order transfer function at phase-crossover frequency.

Disturbance

Gp u

y A/P

UAV Resonant

Adaptive Augmentation (b) High-Frequency Bending Mode

Figure 2: System with two cases of unmodeled dynamics and output disturbance


the turn rate measurements has been utilized. We should point out that the AP also has some “adapting” capability due to the existence of integral path in the lateral control law. More details on the experiment architecture can be found in Ref.28 . A.

Application of Lissajous Graphs to Frequency Response Analysis

To safely obtain the frequency response of the plant in flight and process data in timely manner, we used the Bowditch-Lissajous curves. This technique is named after two scientists, first - the American mathematician Nathaniel Bowditch who, in 1815, described the types of curves that could be drawn by a swinging pendulum31, 32 ; and second - the French mathematician Jules-Antonie Lissajous, who, later in 1857 , extensively studied these curves using a pendulum which drew a line in fine sand as it swung. The Bowditch-Lissajous curve is an XY parametric plot of the input and output of a system with varying values of amplitudes A and B, frequencies ω1 and ω2 , and a phase shift φ: x(t) = A sin(ω1 t) y(t) = B sin(ω2 t + φ)

(1) (2)

The general shape of two dimensional XY plot represented by the above equations is a function of the phase shift φ, the ratios of frequencies ω1 /ω2 and of ratio A/B. The curve is closed if and only if the ratio ω1 /ω2 is rational32 . These properties make interpreting32 Bowditch-Lissajous plot surprisingly quick and simple, and lead to the precise analysis of the results, therefore eliminating any ambiguities in the noisy experimental data. An example of the data interpretation is presented in Figure 3 for the case of linear systems subject to sinusoidal input at frequency ω and a number of phase shifts. Here, the reference input signal is plotted along the horizontal X axis with the corresponding output of the plant plotted along vertical Y axis. For an LTI plant driven by a sinusoidal input the output is also sinusoidal at the same frequency but with different magnitude and phase shift, both functions of frequency. Simple observation suggests that Bowditch-Lissajous plot in general case is represented by an ellipse whose eccentricity is related to the phase shift between nput and output. The ratio of projections of its body onto Y and X axis correspondingly defines the magnitude of the response at a given frequency. B.

Adaptive Control Augmentation Algorithms

In this paper the autopilot mounted onboard the UAV is augmented with an adaptive output feedback controller that modifies the turn rate reference signal based on actual turn rate measurements and sends the augmented command to the autopilot. The objective of wrapping an adaptive augmentation around the autopilot is to study whether turn rate tracking performance and aircraft safety in the event of control surface failures and vehicle damage can be improved. In this section we present two different adaptive algorithms that were implemented onboard and tested both in HIL simulation and in flight test. First, we introduce a conventional output feedback MRAC algorithm with properties similar to the adaptive controller used in Ref.7 . In addition, we also introduce some of the modifications developed to overcome the problem of parameter drift in conventional MRAC. These adaptive algorithms are used to extend Rohrs’ results7 in flight and to verify that the modifications recover stability of the closed-loop adaptive system. Finally, we present an L1 adaptive output feedback architecture for strictly positive real (SPR) reference systems that has been proven to enhance the capabilities of the autopilot33 and to guarantee stability of the UAV in the presence of control surface failures28 . 1.

MRAC Augmentation Algorithm

Several MRAC algorithms are available in the literature with similar robustness properties. References34, 35 provide a good overview of these adaptive controllers. For the purpose of the paper, we use one of the algorithms presented in Ref.34 . The structure of this controller and some of the assumptions that need to be verified are introduced next. The reader is referred to Ref.34 for more details on this algorithm. The single-input single-output plant is assumed to have the following form: Gp (s)

= kp

Zp (s) , Rp (s)


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where Zp (s) and Rp (s) are unknown monic polynomials, and thus kp represents the unknown high-frequency gain. The plant is assumed to be minimum-phase, while an upper bound n on the number of poles, the relative degree of the plant n∗ , and the sign of the high-frequency gain are assumed to be known. The reference model that describes the desired dynamics is given by: Gm (s) =

km

Zm (s) , Rm (s)

where Zm (s) and Rm (s) are monic Hurwitz polynomials, while km is a constant. The reference model Gm (s) is assumed to be SPR and with the same relative degree n∗ as the plant Gp (s). The control law u(t) that solves the model reference problem can be formulated using the following state-space realization: ω˙ 1 (t) =

F ω1 (t) + gu(t) ,

ω1 (0) = 0

ω˙ 2 (t) = u(t) =

F ω2 (t) + gy(t) , θ (t)ω(t) ,

ω2 (0) = 0 (3)

where the Hurwitz matrix F ∈ Rn−1×n−1 and g ∈ Rn−1 are degrees of freedom available to the controller designer, ω1 (t), ω2 (t) ∈ Rn−1 , and θ(t) ∈ R2n is the vector of parameter estimates. Let ω(t) ∈ R2n be given by , ω(t) = ω1 (t) ω2 (t) y(t) r(t) 6 of 36 American Institute of Aeronautics and Astronautics

Figure 4: MRAC augmentation block diagram for unknown high frequency gain Ref34, 35 .

where y(t) and r(t) represent the output of the plant and the reference signal respectively. Then the adaptation law for θ(t) is given by km ˙ (4) , θ(0) = θ0 , θ(t) = −Γe(t)ω(t)sgn kp where Γ > 0 and e(t) = y(t) − ym (t), ym (t) is the output of the reference model Gm (s) to the reference signal r(t). The block diagram for this MRAC algorithm is presented in Figure 4. For details on the choice of the controller parameters F , g, and Γ, the reader is again referred to Ref.34 . This formulation of the MRAC algorithm guarantees that: 1. all signals of the closed-loop adaptive system are bounded and the tracking error e(t) converges to zero asymptotically for any bounded reference signal r(t). 2. if the reference signal r(t) is sufficiently rich, r(t) ˙ is bounded, and the polynomials Zp (s) and Rp (s) are relatively coprime, then the parameter error and the tracking error converge to zero exponentially fast. To improve robustness properties of this algorithm, we consider the following modifications of the MRAC adaptive control law: • Projection operator: ˙ θ(t)

=

km , Γ Proj θ(t), − e(t)ω(t)sgn kp

θ(0) = θ0 ,

where Proj represents the projector operator36 defined over a given compact set Θ. • σ-modification:

˙ θ(t)

= −Γe(t)ω(t)sgn

km kp

− σθ(t) ,

θ(0) = θ0 ,

where σ is a tunable positive parameter. • -modification:

˙ θ(t)

= −Γe(t)ω(t)sgn

km kp

− γθ(t)|e(t)| ,

where γ is a tunable positive parameter.


θ(0) = θ0 ,

2.

L1 Augmentation Algorithm

This section provides an overview of the L1 adaptive output feedback controller for systems of unknown dimension in the presence of unmodeled dynamics and time-varying uncertainties. The reader is referred to Ref.24 for a more detailed explanation of this architecture, as well as for the main results and their proofs. Consider the system y(s) =

Gp (s) (u(s) + z(s)) ,

y(0) = y0 ,

(5)

where u(t) ∈ R is the system’s input, y(t) ∈ R is the system’s output, Gp (s) is assumed to be an unknown strictly proper transfer function, z(s) is the Laplace transform of the time-varying uncertainties and disturbances d(t) = f (t, y(t)), while f is an unknown map, subject to the following assumptions: Assumption 1 There exist constants L > 0 and L0 > 0 such that the following inequalities |f (t, y1 ) − f (t, y2 )| ≤ L |y1 − y2 | |f (t, y)| ≤ L|y| + L0 , hold uniformly in t ≥ 0. Assumption 2 There exist constants L1 > 0, L2 > 0 and L3 > 0 such that for all t ≥ 0: |d(t)|

≤ L1 |y(t)| ˙ + L2 |y(t)| + L3 .

We note that the numbers L, L0 , L1 , L2 , and L3 can be arbitrarily large. The control objective is to design an adaptive output feedback controller u(t) such that the system output y(t) tracks the output of a reference model M (s) to a bounded reference input r(t), i.e. y(s) ≈

M (s)r(s) .

In this paper, we consider a first order system: M (s) =

m , s+m

m > 0.

(6)

As we will see in Section B, this reference model can also be used for the MRAC algorithm as it satisfies all the required assumptions. In order to provide an intuitive explanation behind the main idea of the L1 adaptive augmentation used in this paper, observe that the system in (5) can be rewritten in terms of the desired system M (s) as y(s) =

M (s) (u(s) + σ(s)) ,

y(0) = y0 ,

(7)

where the uncertainties due to Gp (s) and z(s) are lumped into the signal σ(s), which is defined as σ(s)

=

(Gp (s) − M (s)) u(s) + Gp (s)z(s) . M (s)

(8)

The philosophy of the L1 adaptive output feedback controller is to obtain an estimate of the unknown signal σ(t), and define a control signal which compensates for these uncertainties within the bandwidth of a low-pass filter C(s) introduced in the feedback loop. This filter represents the key difference between L1 adaptive control from conventional MRAC, and guarantees that the output of the L1 adaptive controller stays in the low-frequency range even in the presence of high adaptive gains and large reference inputs. The choice of C(s) defines the trade-off between performance and robustness37 . Adaptation is based on the projection operator, ensuring boundedness of the adaptive parameters by definition36 , and uses the output of a state predictor to update the estimate of σ(t), σ ˆ (t). This state predictor is defined to have the structure of the open-loop system (7), using the estimate σ ˆ (t) instead of σ(t) itself, which is unknown. The L1 adaptive control architecture is represented in Figure 5 and its elements are introduced below. State Predictor: We consider the state predictor yˆ˙ (t) = −mˆ y(t) + m (u(t) + σ ˆ (t)) ,

yˆ(0) = y(0) ,


(9)

L1 Augmentation r(t) Control Law

State Predictor

u(t)

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y(t)

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Figure 5: L1 adaptive augmentation loop.

where the adaptive estimate σ ˆ (t) is governed by the following adaptation law. Adaptive Law: The adaptation of σ ˆ (t) is defined as σ ˆ˙ (t) = Γc Proj(ˆ σ (t), −˜ y (t)), σ ˆ (0) = 0,

(10)

where y˜(t) = yˆ(t) − y(t) is the error signal between the state predictor in (9) and the output of the system in (5), Γc ∈ R+ is the adaptation rate subject to a computable lower bound, and Proj denotes the projection operator. Control Law: The control signal is generated by u(s) =

r(s) − C(s)ˆ σ (s) ,

(11)

where C(s) is a strictly proper low-pass filer with C(0) = 1. In this paper, we consider the simplest choice of a first order filter ω , ω > 0. C(s) = s+ω The complete L1 adaptive output feedback controller consists of (9), (10) and (11) subject to the following stability condition: the design of C(s) and M (s) must ensure that H(s) =

Gp (s)M (s) C(s)Gp (s) + (1 − C(s))M (s)

(12)

is stable, and that the following L1 -norm condition holds H(s)(1 − C(s))L1 L
0,

which, at the same time, is an SPR transfer function. One can check that this reference model verifies all the assumptions stated in the description of MRAC. The bandwidth of this reference model was chosen to be 1.5 rad s , which is similar to the bandwidth of the nominal airplane as it can be seen in Table 1.


For the design of the MRAC algorithm, we will assume that the upper bound on the number of poles in the plant is n = 2. With this assumption, both ω1 (t) and ω2 (t) are just scalar signals, while the parameters F and g are scalar positive constants. Therefore, ω(t) belongs to R4 and we will have a total of 4 adaptive parameters:

θ(t)

=

θ1 (t) θ2 (t)

ky (t)

kr (t)

,

where θ1 (t), θ2 (t), ky (t), kr (t) ∈ R. The matrix of adaptive gains will be chosen to be a diagonal 4 × 4 matrix with positive entries. The tunable parameters of the MRAC algorithm have been adjusted in HIL experiments, with the objective of achieving a satisfactory convergence rate for the parameters without compromising the robustness of the closed-loop adaptive system. The values of the parameters are given below: F g

= =

−2 1

Γ

=

diag

5

10 2

2.5

,

while the initial parameter estimate was set to θ0

=

0

0

0 1

.

For this set of parameters, the time-delay margin (defined at the input of the system) obtained at the speed of 22 ms and altitude of 550 m is τ ∗ = 0.5 s. Figure 8 shows the results of one of the experiments. In particular, it shows the response of the system to a set of biased sinusoidal reference signals at different frequencies. As one can see, the MRAC algorithm is able to asymptotically track the output of the reference system to the different reference signals. It is worthwhile to mention that, when tuning any adaptive controller, Lissajous curves may represent a very useful tool to analyze degradation in performance, rather than time history plots; for sinusoidal reference signals, a straight line with slope 1 on a Lissajous plot indicates that desired tracking performance is achieved. In the case of adaptive control, where the objective is to track the output of a reference system, the appropriate Lissajous curve to check performance is the one with the output of the system on the y-axis plotted versus the output of the reference system on the x-axis. Figure 9 shows this Lissajous curve for the previous experiment, together with the Lissajous curve with the output of the system plotted versus the reference signal; one can observe that in Figure 9a, the curves at different frequencies are almost straight lines with slope 1, which implies that the tracking objective of MRAC is asymptotically achieved. On the other hand, Figure 9b shows that there is an evident phase shift (“inflation” of the ideal straight line to an ellipse) between the output of the system and the output of the reference signal, which is consistent with the fact that this is not the goal of adaptive control. Together with the tunable parameters of the MRAC algorithm, the different modifications of the adaptive laws were also adjusted in HIL. The objective was to improve the robustness of the standard MRAC algorithm without sacrificing significantly the performance of the closed-loop adaptive system for reference signals in the low frequency range. The parameters chosen for the different modifications are detailed next: • Projection operator: the bounds for the adaptive parameters were chosen as follows: θ1 (t), θ2 (t), ky (t) ∈ [−0.3, 0.3] ,

and

kr (t) ∈ [0.1, 2] .

• σ-modification: the value of the tunable parameter σ was chosen to be σ = 0.075. • -modification: the value of the tunable parameter γ was chosen to be γ = 0.25. 2.

L1 Adaptive Controller

The L1 augmentation scheme was tuned to achieve a similar level of performance as compared to the MRAC algorithm for reference signals in the low frequency range, and at the same time have similar robustness characteristics. The reference model was chosen to be exactly the same as for the MRAC algorithm, that is, 12 of 36 American Institute of Aeronautics and Astronautics

20

25

θ

2

2

r

ky

cmd

MRAC Adaptive parameters

15 10 Yaw rate, deg/sec

Measured yaw rate r(t), deg/sec

θ1

yaw rate r rm

20

15

5

0

10 5 0 −5

−5

kr

1.5

1

0.5

0

−10 −10 −15 −15 −15

−10

−5 0 5 10 15 Reference system output rm(t), deg/sec

20

−0.5 20

(a) Lissajous r(rm )

40

60

80 100 t, sec

120

140

160

20

(b) Tracking performance

40

60

80 100 t, sec

120

θ

yaw rate r rm

20

1

θ2

2

rcmd

k MRAC Adaptive parameters

15 10 Yaw rate, deg/sec


rad s .

25

15

5

0

10 5 0 −5

−5

160

(c) MRAC parameters

Figure 8: MRAC. Closed-loop response to biased sinusoidal reference signal at ω = 0.1

20

140

y

kr

1.5

1

0.5

0

−10 −10 −15 −15 −15

−10


20

−0.5 20

(d) Lissajous r(rm )

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80 100 t, sec

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(e) Tracking performance

40

60

80 100 t, sec

25

k

cmd

y


Yaw rate, deg/sec


1

15 10

0

10 5 0 −5

−5

rad s .

θ2

2

r

5

160

θ

yaw rate r rm

20

15

140

(f) MRAC parameters


20

120

k

r

1.5

1

0.5

0

−10 −10 −15 −15 −15

−10


(g) Lissajous r(rm )

20

−0.5 20

40

60

80 t, sec

100

120

140

(h) Tracking performance

20

40

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80 t, sec

100

(i) MRAC parameters



120

rad s .

140

20

15

15



20

10

5

0

−5 ω=0.5 rad/s ω=0.3 rad/s ω=0.1 rad/s

−10

−15 −15

−10

−5 0 5 10 Reference input r (t), deg/sec

15

10

5

0

−5 ω=0.5rad/s ω=0.3 rad/s ω=0.1 rad/s

−10

−15 −15

20

c

−10

−5 0 5 10 15 Reference system output r (t), deg/sec

20

m

(a) Lissajous r(rcmd )

(b) Lissajous r(rm )

Figure 9: MRAC. Lissajous curves for the response of the closed-loop system to biased sinusoidal reference signals at different frequencies.

a stable first-order system with bandwidth 1.5 rad s . The adaptive gain was set to 30, 000, while the bandwidth to satisfy the desired trade-off between adaptation and robustness. The of the filter was set to ω = 0.6 rad s . The time-delay observed in HIL for the same condition of 22 ms projection bounds were chosen to be ±1 rad s and 550 m was τ ∗ = 0.5 s. Figure 10 shows the results from one of the experiments. One can observe the degradation of tracking performance as the frequency of the reference signal increases above the bandwidth of the low-pass filter C(s) in the control law. C. 1.

Rohrs’ Counterexample In Flight HIL Simulation Results

Second-Order Unmodeled Dynamics In this section, we introduce artificially at the output of the plant (closed-loop UAV and AP) a second-order system representing unmodeled dynamics, similar to Rohrs’ counterexample. In his counterexample, Rohrs considered the case of very well damped unmodeled dynamics at high frequencies to show that, even in the presence of apparently “harmless” uncertainties, the stability of adaptive controllers was not guaranteed. In this paper, nevertheless, we consider the case of more challenging uncertainties, with the objective of evaluating the performance of the L1 adaptive augmentation in the presence of unmodeled dynamics. In particular, we choose a low-damped second-order system with natural frequency equal to the bandwidth of the plant, i.e. Δ(s)

=

ωn2 , s2 + 2ζωn s + ωn2

with ωn = 1.5 rad s and ζ = 0.45. With the addition of this second-order system, the new plant has the phase crossover frequency at 1.6 rad s . Figure 11 shows the Lissajous curve of the system output versus the reference signal for the cascaded system in response to the biased sinusoidal reference signal at the phase crossover frequency. As one can observe, the Lissajous curve exhibits the typical shape for a phase shift of −180 deg (see Figure 3), which confirms that 1.6 rad s corresponds to the phase crossover frequency of the cascaded system. At this point, having identified the phase crossover of the system, we can extend Rohrs’ results in flight. To this end, we consider the MRAC algorithm presented in the previous section. Figure 12 shows the response of the plant with the second-order system representing unmodeled dynamics to biased sinusoidal reference signals at different frequencies. It can be observed that the closed-loop adaptive system is able to track the rad output of the reference model for reference signals at low frequencies (ω = 0.1 rad s and ω = 0.5 s ). When the system is driven by a reference signal at ω = 1 rad s , which is within the bandwidth of the plant, bursting 14 of 36 American Institute of Aeronautics and Astronautics

20

25

45

15

30

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0

L1 contribution, deg/sec

10 Yaw rate, deg/sec


60

20

15

10 5 0 −5

−5

0 −15 −30

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yaw rate r r

−10

−45

m

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r

cmd

−15 −15

15

−10


20

70


80

90 t, sec

100

−60

110

70

80

90 t, sec

100

(c) L1 contribution


Figure 10: L1 . Closed-loop response to biased sinusoidal reference signal at ω = 0.1 20

25

15

30

0




45

rcmd

5

10 5 0 −5

−5

rad s .

60

yaw rate r rm

20

15

110

15 0 −15 −30

−10 −10

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−15 −15

−60 −10


20

60


70

80 t, sec

90

100

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60

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80 t, sec

90


45

rcmd 15

30

5

0




15

10 5 0 −5

−5

rad s .

60

yaw rate r rm

20

110

(f) L1 contribution


20

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15 0 −15 −30

−10 −10

−45 −15

−15 −15

−60 −10


20

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cmd

30

0




45

r 15

5

10 5 0 −5

−5

rad s .

60

yaw rate r rm

20

15

180

(i) L1 contribution


20

170

15 0 −15 −30

−10 −10

−45 −15

−15 −15

−60 −10


(j) Lissajous r(rm )

20

30

40

50 t, sec

60

70

80

(k) Tracking performance

30

40

50 t, sec

60

(l) L1 contribution

Figure 10: L1 . Closed-loop response to biased sinusoidal reference signal at ω = 5.0 15 of 36 American Institute of Aeronautics and Astronautics

70

rad s .

80

takes place (t = [330 s, 350 s]). During this period, the MRAC algorithm sends commands to the AP up to ±100 deg s , and the output of the plant presents several overshoots (note that, for safety reasons, the AP has internal saturations which limit the commands that the AP sends to the UAV; without these internal saturations, the UAV would probably have gone upside down). After this period, the adaptive parameters of MRAC converge to new values that stabilize the system and achieve tracking of the output of the reference model. Finally, when the system is driven by a reference signal at the phase crossover frequency, ω = 1.6 rad s , parameter drift takes place and the closed-loop system becomes unstable. These results are not new, and just extend the well-known problems of conventional MRAC algorithms in flight. As we previously mentioned, several modifications of the adaptive laws were developed during the last thirty years to solve the problem of parameter drift. Figure 13 shows the results of the same experiment obtained for the MRAC algorithm with the -modification. It can be seen that the parameters stay bounded and the system remains stable for the whole experiment. However, as expected, significant degradation in tracking performance is observed, and the plant barely tracks the output of the reference model even for reference signals at low frequencies. Similar results were observed for the MRAC algorithm with the projection operator and with σ-modification. In the set of experiments run for the conventional MRAC algorithm as well as the MRAC algorithm with its modifications, it has been observed that the convergence of the parameters and thus the transient characteristics are not predictable, and the performance of the closed-loop adaptive system has completely different behavior depending on initial conditions and nature of the reference signals. It is also important to note that further tuning of the MRAC algorithm and its modifications can be done, which might result in improved tracking performance. The purpose of this experiment was to extend Rohrs’ results in-flight and verify (also in-flight) that some of the different modifications introduced during the years were able to solve the problem of parameter drift; but it was not the purpose of the paper to tune MRAC. However, the authors want to emphasize as well that fine-tuning will not change the nature of MRAC algorithms, and therefore predictability of the system response with this kind of algorithms will only be achieved, if possible, through an extensive gain-scheduling, which defeats the purpose of adaptation. At the same time, the trade-off between performance and robustness will always need to be assessed through numerous Monte-Carlo runs. Next we present the results obtained for the same experiment with the L1 adaptive augmentation (Figure 14). As it can be seen, the closed-loop adaptive system is able to track precisely the output of the reference system to the reference signal at ω = 0.1 rad s . As the frequency of the reference signal increases and goes beyond the bandwidth of the low-pass filter C(s), the tracking performance degrades as expected. The L1 adaptive controller keeps the DC gain of the closed-loop adaptive system close to 1 for the referrad rad ence signals at frequencies ω = 0.1 rad s , ω = 0.5 s , and ω = 1 s , while the commands sent to the AP remain always bounded and without high-frequency content. Should one want to reduce the amplitude of the command from the adaptive controller to the AP, it would be enough to shrink the projection bounds for the adaptive law. This would limit as well the capabilities of the L1 adaptive controller. Also, one can observe that the response of the closed-loop adaptive system for every reference signal is very consistent and describes a well-defined and “clear” shape in the Lissajous curve. Bending Mode In this section we extend the philosophy of the previous experiment to the case of highfrequency unmodeled dynamics with very-low damping. This experiment has as a goal to reproduce the effects of bending modes in an airplane, and investigate the possible interaction between adaptive control and these modes. To this end, we choose a very-low damped second-order transfer function with natural frequency equal to approximately 7 times the bandwidth of the plant, i.e. Δ(s)

=

s2

ωn2 , + 2ζωn s + ωn2

with ωn = 10 rad s and ζ = 0.01, and at the same time we introduce a perturbation at the output of the artificially “injected” bending mode at exactly ω = 10 rad s d(t) =

20 sin(10t)

deg . s

Figure 15 shows the response of the system with the conventional MRAC algorithm to a biased sinusoidal reference signal at ω = 0.5 rad s . Initially, we wait for the MRAC parameters to converge, so the MRAC 16 of 36 American Institute of Aeronautics and Astronautics

20

Measured yaw rate r(t), deg/s

15 10 5 0 −5 −10 −15 −15

−10

−5 0 5 10 Reference input r (t), deg/s

15

20

c

Figure 11: Lissajous curve for the nominal system with unmodeled dynamics at the phase crossover frequency.

20

25 m

rcmd AD N

15

O

10

Yaw rate, deg/sec


yaw rate r r

20

15

5

0

10 5 0 −5

−5 ω=1.6 rad/s ω=1 rad/s ω=0.5 rad/s ω=0.1 rad/s

−10

−15 −15

−10


−10 −15 20

100


15 r

cmd ON

y

k ω=0.1rad/s

ω=0.5rad/s

ω=1rad/s

ω=1.6rad/s

20 0 −20 −40

Bursting −80

200

ON

0

−5

−10

Unstable

100

r

AD

5

Parameter Drift

−60

−100

k

10

AD


Yaw rate command to A/P, deg/sec

40

500

θ2

r

60

400

θ1

to A/P

ad

300 t, sec


100 80

200

300 t, sec

400

−15

500

(c) Adaptive command to AP

100

200

300 t, sec

400

500

(d) MRAC parameters

Figure 12: MRAC. Closed-loop response in the presence of second-order unmodeled dynamics to biased sinusoidal reference signals.


20

25

15

m

rcmd ADON

15 10

Yaw rate, deg/sec


yaw rate r r

20

5

0

5 0 −5


−10

−15 −15

10

−10


−10 −15 20

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200

m


300 t, sec

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25 r

ad

1

θ

2

2

ky

ADON 15



θ

to A/P

rcmd

20

600

10 5 0 −5

kr

1.5

ADON

1

0.5

0

−10 ω=0.1 rad/s −15

ω=0.5 rad/s

ω=1 rad/s

ω=1.6 rad/s −0.5

100

200

300 t, sec

400

500

600

100


200

300 t, sec

400

500

600

(d) MRAC parameters

Figure 13: MRAC with -modification. Closed-loop response in the presence of second-order unmodeled dynamics to biased sinusoidal reference signals.

algorithm achieves desired performance, that is, the output of the system tracks the output of the reference model. At t = 50 s, the dynamics representing the bending mode are “injected”, and the system runs in this configuration for 30 s. The “injection” of the bending mode results in small oscillations at the output of the plant, which is probably due to the fact that the noise present in the system is exciting these dynamics. We can see that initially the parameter ky is close to zero, which implies that these oscillations at the output of the plant are well attenuated at the control signal that MRAC sends to the AP. Also, notice that, extrapolating the frequency response analysis in Table 1, the closed-loop system consisting of the UAV with the AP should have around 15 − 20 dB of attenuation at 10 rad s . Therefore, the bending mode is not excited and, during these 30 s, the MRAC algorithm is able to track the output of the reference model without exciting the bending mode. At t = 120 s, we inject the disturbance at the output of the plant. One can see that there is a 10 s transient with high-frequency oscillations at the output of the plant. These initial oscillations damp down and MRAC recovers a similar level of performance as the one before the injection of the disturbance. A similar result is obtained when the disturbance is disabled at t = 180 s. However, when we inject the disturbance for second time at t = 200 s, the parameters start drifting immediately. In particular, we see that the parameter ky grows significantly, which implies that the high-frequency content at the output of the plant is amplified by the MRAC algorithm and sent back to the AP, resulting in an unstable closed-loop adaptive system. This experiment illustrates that conventional MRAC is not able to cope with bending modes, as it generates a counteracting control signal that tries to “fight” the attenuating capabilities of the AP in the high-frequency range. The result is an unstable closed-loop system that increasingly excites


20

25

15

m

rcmd ADON

15 10

Yaw rate, deg/sec


yaw rate r r

20

5

0

5 0 −5


−10

−15 −15

10

−10


−10 −15 20

100

200

m


300 t, sec

400

500


30 r

ad

to A/P

1

rcmd

0.8

ADON

20

0.6

15



25

10 5 0 −5

0.4 0.2 0 −0.2 −0.4 −0.6

−10 −15

ω=0.1 rad/s 100

ω=0.5 rad/s 200

ω=1 rad/s 300 t, sec

−0.8

ω=1.6 rad/s 400

−1

500

100

200

300 t, sec

400

500

(d) L1 contribution


Figure 14: L1 . Closed-loop response in the presence of second-order unmodeled dynamics to biased sinusoidal reference signals.

the bending mode. When the MRAC algorithm is implemented with the -modification, the system remains stable along the whole experiment, Figure 16. Nevertheless, the slow adaptation in the parameters due to the deficient adaptive law results in poor high-frequency disturbance attenuation characteristics. The time segments t = [80, 150] s and t = [150, 220] s explicitly illustrate this phenomenon: as the bending mode and the disturbance are “injected”, the parameter ky jumps but the damping term present in the adaptive law prevents this parameter to converge back to zero, and thus the high-frequency oscillations are not sufficiently attenuated. This results in similar destabilizing dynamics as in conventional MRAC, which eventually lead to significant oscillations at the output of the plant. Next we present the results obtained with the L1 adaptive controller. At t = 50 s the bending mode is “injected”, which -similar to MRAC- results in small amplitude oscillations at the output of the system due to the noise (see Figure 17). It can be seen that the contribution of the L1 adaptive controller stays in the low-frequency range. The disturbance is “injected” at t = 108 s. One can see that the L1 adaptive controller generates small oscillations in the control channel, which causes a 5 s transient at the output of the system with high-frequency oscillations. These oscillations in the contribution of the L1 controller are due to the fact that the projection bounds in the adaptive law are too tight to handle the initial transient produced by the disturbance. Figure 18 shows that the uncertainty estimate σ ˆ (t) hits the bounds of projection, and as a consequence, the error between the output of the state-predictor rˆ(t) is not able to track the output of the plant r(t). These oscillations could be easily avoided by increasing the bounds of the


25 yaw rate r rm

20

r

cmd

15

Yaw rate, deg/sec

10 5 0 −5 −10 −15 50

100

150

200

t, sec

(a) Tracking performance

25

5 r

ad

20 15



to A/P

rcmd

10 5 0 −5

0

−5

θ

−10

1

θ2

−10

ky −15

k

r

50

100

150

−15

200

50

100

t, sec

150

200

t, sec

(b) Adaptive command to AP

(c) MRAC parameters

Figure 15: MRAC. Closed-loop response in the presence of bending modes to biased sinusoidal reference signals.

projection operator and, if necessary, increasing the adaptation gain to reduce the tracking error between the output of the state-predictor and the output of the plant. In the experiment, after these 5 s of transient, the oscillations damp down and the L1 adaptive controller recovers its initial performance. Although the L1 adaptive controller senses this high-frequency disturbance through the feedback signal, fast adaptation allows for rapid estimation of the uncertainties, σ ˆ (t). This estimation goes to the low-pass filter in the control law, which effectively attenuates this high-frequency signal, and therefore the L1 contribution is within the low-frequency range. This results in the fact that the L1 controller is not “fighting” the AP, which is in contrast to the two MRAC algorithms presented above. This experiment shows that, if properly tuned, the L1 adaptive controller guarantees that its contribution is in the low-frequency range and preserves the attenuating properties of the nominal inner-loop, and thus the bending mode is not excited. Control Surface Failure In this section, instead of artificially introducing unmodeled dynamics at the output of the plant, we will consider the case of a sudden control surface failure affecting the nominal plant. In particular, the left aileron will get locked at −6 deg and then, after some time, we will recover full lateral control. This experiment is directly in line with the objectives of the IRAC-1.5. First we show that the nominal plant with the AP is able to recover from the failure and track the reference signal with the remaining control authoritya . Figure 19 shows the response of the plant to a biased a In Ref.,28 the autopilot was intentionally (de)tuned to demonstrate performance recovery capabilities of the L adaptive 1 controller.


25 yaw rate r rm

20

r

cmd

Yaw rate, deg/sec

15 10 5 0 −5 −10 −15 50

100

150 t, sec

200

250


25 r

ad

θ

1.4

1

θ2

1.2

15

ky

1



to A/P

rcmd

20

10 5 0 −5

kr

0.8 0.6 0.4 0.2 0 −0.2

−10

−0.4 −15 −0.6 50

100

150 t, sec

200

250

50


100

150 t, sec

200

250

(c) MRAC parameters

Figure 16: MRAC with -modification. Closed-loop response in the presence of bending modes to biased sinusoidal reference signals.

sinusoidal reference signal at ω = 0.5 rad s . The failure is “injected” at t = 375 s, and the AP takes around 20 s to recover from the failure, which is the time that the integrators of the AP need to readjust their contribution. The Lissajous curve shows that, in the presence of the failure, the AP is able to recover almost full performance with respect to the “healthy” UAV. Next we consider the case of the conventional MRAC algorithm. Figure 20 shows the response of the closed-loop adaptive system. For the sake of clarity, we did not add on the Lissajous curve the third phase in which the UAV recovers full lateral control authority. In the figure, one can see that initially the MRAC algorithm is able to track perfectly the output of the reference system. When the failure is introduced at t = 190 s, the MRAC algorithm is able to recover full performance in around 20 s and keep tracking the output of the reference system perfectly using the remaining control authority. As it can be seen in the figure, the transient is smooth and, as the integrators of the AP readjust, the command from the adaptive controller to the AP recover the nominal adaptive contribution. However, when the UAV recovers full control authority (t = 290 s), the parameters drift and the closed-loop system with the MRAC algorithm becomes unstable. Again, this result is not new. The only objective of this experiment was to verify that the same instability results could be obtained for the nominal plant in the event of a control surface failure. Figure 21 shows that parameter drift can be avoided with the introduction of the -modification at the price of degraded performance. The aileron failure is introduced at t = 175 s, and the initial performance is recovered in about 35 s. Then, when the UAV regains full control authority at t = 250 s, the UAV remains stable and


25 yaw rate r rm

20

r

cmd

15

Yaw rate, deg/sec

10 5 0 −5 −10 −15 40

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120 t, sec

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40 15 10 5 0 −5

20

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ad

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to A/P

r

cmd

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100

120 t, sec

140

160

180

−60

200

40

60

80

100

120 t, sec

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(c) L1 contribution


Figure 17: L1 . Closed-loop response in the presence of bending modes to biased sinusoidal reference signals.

25

60 rhat

15

r

20

Yaw rate, deg/sec

10 5 0 −5 −10

L1 Estimated uncertainties, deg/sec

L1 Predictor tracking error, deg/sec

40 15

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0

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107

108

109

110 111 t, sec

112

113

114

−60 107

(b) Tracking error r˜(t)

108

American Institute of Aeronautics and Astronautics

110 111 t, sec

112

113

(c) uncertainty estimate σ ˆ (t)

Figure 18: L1 . Saturation in the adaptive law.

22 of 36

109

114

20

25 ’Healthy’ UAV Impaired UAV

m

rcmd 15

10

5

Yaw rate, deg/sec


yaw rate r r

20

15

5

0

3 1

10 5 0 −5

−5

2 −10

−10 −15

4 −15 −15

−10

−5 0 5 10 Reference input r (t), deg/sec

15

20

340

360

380 t, sec

c

(a) Lissajous r(rcmd )

400

420


Figure 19: AP. “Open-loop” response to a biased sinusoidal reference signal in the event of a left aileron failure.

20


30

yaw rate r rm r

cmd

20 10

3

Yaw rate, deg/sec


15

5

0

1

−5

10 0 −10

5 −20 2

−10

−15 −15

4

−10

−30


−40

20

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300

350

t, sec



50

15 r

ad

40

θ

to A/P

1

θ2

rcmd 10

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20 10 0 −10 −20 −30

ky kr

5

0

−5

Parameter drift

−10

−40 −50

200

250

300

−15

350

200

250

t, sec

300

350

t, sec


(d) MRAC parameters

Figure 20: MRAC. Closed-loop response to a biased sinusoidal reference signal in the event of a left aileron failure and later recovery of full control authority.


20


20 15

10

Yaw rate, deg/sec


15

5 5 0

1

3

5 0 −5

2

−5

10

4 −10

yaw rate r rm

−10 −15

r

cmd

−15 −15

−10


20

100

150

m


200 t, sec

250

300


25 r

ad

1

θ2

1

ky 15

0.8



θ

to A/P

rcmd

20

10 5 0 −5 −10

kr

0.6 0.4 0.2 0 −0.2

−15

−0.4 100

150

200 t, sec

250

300

100


150

200 t, sec

250

300

(d) MRAC parameters

Figure 21: MRAC with -modification. Closed-loop response to a biased sinusoidal reference signal in the event of a left aileron failure and later recovery of full control authority.

eventually recovers the original performance after a transient of 50 s. The authors want to emphasize again that the performance of the MRAC algorithm with the -modification could be improved with further tuning of the parameters of the controller. However, as pointed in original solicitation of IRAC project, that there are no clear guidelines as how to tune MRAC systematically to enable accurate performance prediction. As we pointed out earlier, the purpose of the paper was not to tune MRAC and its modifications, but just to extend in flight the results obtained by Rohrs et al. in Ref.7 and evaluate the performance of the L1 adaptive augmentation. Figure 22 presents the results obtained for this experiment when the L1 controller is wrapped around the UAV+AP. The failure is introduced at t = 100 s and the L1 adaptive controller takes 15 s to regain a level of performance similar to the initial one using the remaining control authority. Note that even though the UAV has recovered the desired performance in 15 s, the integrators inside the AP keep readjusting during approximately 15 more seconds, and therefore the contribution of the L1 adaptive controller also has to readjust accordingly. The Lissajous curve shows that the L1 adaptive controller is able to maintain the initial phase shift of the output of the plant with respect to the output of the reference system, while the gain slightly increases for the impaired UAV. When the UAV regains full control authority at t = 170 s, the closed-loop adaptive system is able to stabilize the plant and achieve desired performance in again 15 s. After 15 more seconds, when the integrators inside the AP readjust, the command of the L1 controller to the AP is the same as it was for the initial “healthy” UAV. Note that the transient characteristics of the closed-loop adaptive system when (i) the failure is introduced and (ii) the UAV regains full control authority,


are similar. This consistent behavior of the L1 adaptive controller, which can be extended to the steady-state regime as well, has been observed in several other similar simulations, and results in a predictable response of the closed-loop adaptive system with the L1 adaptive augmentation. To conclude this experiment, Figure 23 shows the way how the three different adaptive controllers used in this paper modify the original biased sinusoidal reference signal to achieve the desired control objective. While in Figure 23a, one can see that some oscillations appear in the command the MRAC algorithm sends to the AP for the impaired UAV, and eventually the system becomes unstable when full control authority is recovered. When the MRAC algorithm is implemented with the -modification, the contribution of the adaptive controller does not change significantly during the whole experiment (see Figure 23b), which makes evident that the -modification limits the capabilities of the adaptive algorithm to improve the transient characteristics and recover full performance of the plant. Finally, Figure 23c shows the contribution of the L1 adaptive controller. During the transient phase, the adaptive controller contributes significantly in order to stabilize the plant and track the output of the reference system with the available control authority, while guaranteeing a smooth recovery. Eventually, when the integrators in the AP readjust and are able to compensate for the failure, the command from the adaptive controller converges to the initial contribution. This Figure 23c also shows that the contribution of the L1 controller is slightly greater for the impaired UAV in order to compensate for the reduction of control authority. Furthermore, numerous experiments confirmed that the L1 adaptive controller can be tuned systematically, and therefore its performance is predictable. 2.

Flight Test Results

In this section we present the results of flight tests for conventional MRAC, MRAC implemented with the -modification, and L1 adaptive control. The setup28 used is identical to the HIL setup described in the introduction of Section III with the Rascal UAV replacing the 6DOF simulation30, 40 . Nominal plant As in the previous section, the frequency response analysis for the Rascal UAV was obtained first, verifying that its bandwidth was in fact ωn = 1.5 rad s . The conventional MRAC algorithm and the L1 adaptive augmentation were tuned in flight to satisfy the same trade-off requirements between performance and robustness as in HIL experiments. This tuning phase consisted of a series of flight tests with the nominal plant consisting of the UAV with the AP augmented with the adaptive controllers. The closedloop adaptive systems were driven by a biased sinusoidal reference signal of amplitude 7 deg s at different frequencies. Figures 24 and 25 present the results obtained in two of these experiments. Both algorithms achieve satisfactory performance, with responses comparable to the ones obtained in HIL simulations. The existence of turbulence makes the quality of the data acquired in flight poorer than in HIL. It is important to mention that, even for the nominal plant, the MRAC parameters slowly diverge, illustrating that in a real system one will always encounter unmodeled dynamics. In the flight with the L1 adaptive controller, one can observe that at t = 70 s the L1 adaptive controller generates small oscillations (1 Hz) in the control channel during approximately 3 s. Again, as mentioned in the HIL section when describing the bending mode experiment, these oscillations are due to internal saturation of the L1 controller. The levels of projection were too tight for the level of turbulence during the flight. This suggests that the L1 controller requires further fine-tuning. Nevertheless, the stability of the closed-loop adaptive system is not compromised. Second-order unmodeled dynamics The same second-order transfer function with natural frequency ωn = 1.5 rad s and damping ratio ζ = 0.45 was “injected” at the output of the actual plant. Using the Lissajous curves, the phase crossover frequency of the real plant with the unmodeled dynamics was determined in realtime with data received through telemetry on the ground. Figure 26 shows the Lissajous curve r(rcmd ) for the system driven with a biased sinusoidal reference signal at ω = 1.6 rad s ; one can see that -similar to the experiments in HIL- this Lissajous curve presents the typical shape for a phase shift of −180 deg between reference input and system’s output. Next, the conventional MRAC algorithm is implemented onboard in order to verify that we are able to reproduce in real flight the same instability observed in HIL. Figure 27 shows the response of the system in the presence of the second-order unmodeled dynamics to a biased sinusoidal reference signal at the phase crossover frequency. One can see that the parameters drift slowly, generating command signals larger than ±100 deg s , which are sent to the AP. Again, it is important to note that for safety reasons, the AP limits the commands received from the adaptive controller to avoid undesirable attitudes that might lead to the loss of the UAV. This same experiment was repeated for different values of the adaptive gains. The objective was to analyze


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the sensitivity of the response of the closed-loop adaptive system to these parameters. Figure 28 shows the response of the system to the same reference signal with adaptive gains 150% larger than the original ones. After an initial slow drift, the parameters blow up at t = 110 s. This result is consistent with MRAC theory: as one tries to increase the speed of adaptation the robustness of the adaptive system is significantly reduced. The same experiment (with the original adaptive gains) was conducted for the MRAC algorithm with the -modification (see Figure 29). As expected, the parameters do not drift, and the system retains stability during the whole flight. However, similar to HIL results, the performance of the closed-loop adaptive system is significantly degraded due to the addition of the damping term in the adaptive law. Finally, in Figure 30, we present results obtained for the L1 adaptive augmentation. The system maintains stability during the whole flight and the control signal remains inside reasonable bounds during the experiment. As one would expect, since the frequency of the reference signal is well beyond the bandwidth of the low-pas filter C(s) in the control law, the L1 adaptive controller is not able to recover desired performance of the closed-loop adaptive system. The response with L1 adaptive controller is consistent during the entire flight, and does not exhibit undesirable characteristics like bursting. In order to illustrate the degradation of performance as the frequency of the reference signal increases beyond the bandwidth of the low-pass filter, the closed-loop system with the second-order unmodeled dynamics and the L1 controller implemented onboard was driven with a set of biased sinusoidal reference signals at different frequencies. Figure 31 shows the results of these experiments. It can be seen that the output of the closed-loop adaptive system is able to track the output of the reference system for reference signals at low frequencies (ω = 0.1 rad s ), and, as the frequency of the reference signal increases, the ideal Lissajous curve with slope-1 “inflates” progressively. This graceful degradation in the performance of the system is consistent with the theory of fast and robust adaptation, and results in a predictable response of the closed-loop adaptive system, as opposed to MRAC


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algorithms. To conclude this section, the last set of figures (Figures 32a and 32b) describes a similar experiment where, after “injecting” the second-order dynamics and enabling MRAC, the closed-loop system became unstable, and the control command from the adaptive algorithm eventually hit the saturation limit of the AP. Then, at t = 56 s, the adaptive controller was switched from conventional MRAC to L1 adaptive augmentation, and the UAV recovered stability in “no-time” (around 1.5 s, therefore confirming the claims of guaranteed and robust adaptation.

IV.

Conclusion

This paper presented a flight test verification and validation of two different adaptive control algorithms in flight test, MRAC and L1 Adaptive Control in the presence of unmodeled dynamics. To keep the results in historical context, for testing the performance in the presence of unmodeled dynamics, we referred to Rohrs’ example, proposed in early 80s. A detailed analysis of HIL and flight test results is included in the paper. This analysis showed that conventional MRAC loses stability in the presence of unmodeled dynamics in HIL and during flight, verifying that the setup adopted qualitatively reproduces results obtained by Rohrs. In addition, it was shown that modifications of MRAC proposed in the literature do maintain stability, albeit the transient performance cannot be predicted. Finally, the L1 output feedback adaptive controller was also tested in the same setup. Both HIL and flight test results confirmed that the L1 controller maintains stability and predictable performance in the presence of unmodeled dynamics and control surface failures. Finally, numerous flight test experiments confirmed that the performance of the L1 adaptive controller can be tuned systematically, and therefore its real performance is predictable.


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V.

Acknowledgments

This work was sponsored in part by NASA Grants NNX08AB97A, NNX08AC81A, and NNL08AA12I, and Hellenic Air Force Research Grant (KAE 0482/EF11-410).

References 1 Zhang, Y. and Jiang, J., “Bibliographical review on reconfigurable fault-tolerant control systems,” Annual Reviews in Control , Vol. 32, No. 2, 2008, pp. 229–252. 2 G.Ducard, Fault-tolerant Flight Control and Guidance Systems: Practical Methods for Small Unmanned Aerial Vehicles, Springer, 2009. 3 Wise, K., Lavretsky, E., and et. al., “Adaptive Flight Control of a Sensor Guided Munition.” 2005. 4 Wise, K., Lavretsky, E., and Hovakimyan, N., “Adaptive Control in Flight: Theory, Application, and Open Problems,” Proc. of American Control Conf., Minneapolis, MN, 2006, pp. 5966–5971. 5 Sharma, M., Lavretsky, E., and Wise, K., “Application and Flight Testing of an Adaptive Autopilot on Precision Guided Munitions,” AIAA Guidance, Navigation, and Control Conference, AIAA-2006-6568 , 2006. 6 Fidan, B., Adaptive control tutorial, Society for Industrial Mathematics, 2006. 7 Rohrs, C., Valavani, L., Athans, M., and Stein, G., “Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics,” Automatic Control, IEEE Transactions on, Vol. 30, No. 9, 1985, pp. 881–889. 8˚ Astr¨ om, K. J., “Interactions between Excitation and Unmodeled Dynamics in Adaptive Control,” 23rd Conference on Decision and Control , Las Vegas, NV, December 1984. 9 Anderson, B. D. O., “Failures of Adaptive Control Theory and their Resolution,” Communications in Information and Systems, Vol. 5, No. 1, 2005, pp. 1–20. 10 Ortega, R. and Tang, Y., “Robustness of Adaptive Controllers– A Survey,” Automatica, Vol. 25, No. 5, September 1989, pp. 651–677. 11 Ioannou, P. A. and Kokotovic, P. V., “An Asymptotic Error Analysis of Identifiers and Adaptive Observers in the presence of Parasitics,” IEEE Transactions on Automatic Control , Vol. 27, No. 4, August 1982, pp. 921–927. 12 Ioannou, P. A. and Kokotovic, P. V., Adaptive Systems with Reduced Models, Springer Verlag New York, Inc., 1983. 13 Ioannou, P. A. and Kokotovic, P. V., “Robust Redesign of Adaptive Control,” IEEE Transactions on Automatic Control , Vol. 29, No. 3, March 1984, pp. 202–211. 14 Narendra, K. S. and Perterson, B. B., “Bounded Error Adaptive Control,” IEEE Transactions on Automatic Control , Vol. 27, No. 6, December 1982, pp. 1161–1168. 15 Kresselmeier, G. and Narendra, K. S., “Stable Model Reference Adaptive Control in the presence of Bounded Disturbances,” IEEE Transactions on Automatic Control , Vol. 27, No. 6, December 1982, pp. 1169–1176. 16 Narendra, K. S. and Annaswamy, A., “A New Adaptive Law for Robust Adaptive Control without Persistent Excitation,” IEEE Transactions on Automatic Control , Vol. 32, No. 2, February 1987, pp. 134–145. 17 Naik, S. M., Kumar, P. R., and Ydstie, B. E., “Robust Continuous-Time Adaptive Control by Parameter Projection,” Vol. 37, No. 2, February 1992, pp. 182–197. 18 Hovakimyan, N., Patel, V. V., and Cao, C., “Rohrs Example Revistited with L Adaptive Controller: The Classical 1 Control Perspective,” Proc. of AIAA Guidance, Navigation & Control Conf., Hilton Head Island, SC, 2007, AIAA 2007-6597. 19 Xargay, E., Hovakimyan, N., and Cao, C., “Benchmark problems of adaptive control revisited by L1 adaptive control,” Control and Automation, 2009. MED ’09. 17th Mediterranean Conference on, June 2009, pp. 31–36. 20 C. Cao and N. Hovakimyan, “Design and Analysis of a Novel L Adaptive Control Architecture with Guaranteed Transient 1 Performance,” IEEE Transactions on Automatic Control , Vol. 53, No. 2, 2008, pp. 586–591. 21 Cao, C. and Hovakimyan, N., “Guaranteed Transient Performance with L Adaptive Controller for Systems with Unknown 1 Time-Varying Parameters and Bounded Disturbances: Part I,” New York, NY, July 2007, pp. 3925–3930. 22 Cao, C. and Hovakimyan, N., “Stability Margins of L Adaptive Controller: Part II,” New York, NY, July 2007, pp. 1 3931–3936. 23 Cao, C. and Hovakimyan, N., “L Adaptive Controller for Systems in the Presence of Unmodelled Actuator Dynamics,” 1 46th Conference on Decision and Control , New Orleans, LA, Dec. 2007, pp. 891–896. 24 Cao, C. and Hovakimyan, N., “L Adaptive Output Feedback Controller for Systems of Unknown Dimension,” IEEE 1 Transactions on Automatic Control , Vol. 53, No. 3, April 2008, pp. 815–821. 25 Cao, C. and Hovakimyan, N., “L Adaptive Controller for Nonlinear Systems in the Presence of Unmodelled Dynamics: 1 Part II,” Seattle, WA, June 2008, pp. 4099–4104. 26 Cao, C. and Hovakimyan, N., “L Adaptive Output Feedback Controller for Non-Strictly Positive Real Reference Systems 1 with Applications to Aerospace Examples,” AIAA Guidance, Navigation, and Control Conference and Exhibit, Honolulu, HI, August 2008. 27 Athans, M., “Research on Optimal Control, Stabilization, and Computational Algorithms for Aerospace Application,” NASA contract report 172528, Massachussetts Institute of Technology, 1985. 28 Dobrokhodov, V., Kitsios, I., Kaminer, I., Jones, K., Xargay, E., Hovakimyan, N., Cao, C., Lizarraga, M., and Gregory, I., “Flight Validation of a Metrics Driven L1 Adaptive Control,” Proceedings of AIAA Guidance, Navigation, and Control Conference and Exhibit, 2008. 29 Dobrokhodov, V., Yakimenko, O., Jones, K., Kaminer, I., Bourakov, E., Kitsios, I., and Lizarraga, M., “New Generation of Rapid Flight Test Prototyping System for Small Unmanned Air Vehicles,” AIAA Modeling and Simulation Technologies Conference Proceedings, 2007.


30 Vaglienti, B., Hoag, R., and Niculescu, M., Piccolo System Documentation, Cloud Cap Technologies, April 2005, http://www.cloudcaptech.com. 31 Kent, A., Lancour, H., and Nasri, W., Encyclopedia of library and information science, Dekker, 1968. 32 Weisstein, E. W., CRC concise encyclopedia of mathematics, Chapman & Hall/CRC, 2003. 33 Kaminer, I., Pascoal, A., Xargay, E., Cao, C., Hovakimyan, N., and Dobrokhodov, V., “3D Path Following for Small UAVs using Commercial Autopilots augmented by L1 Adaptive Control,” Submitted to Journal of Guidance, Control and Dynamics. 34 P.Ioannou and Sun, J., Robust Adaptive Control, Prentice Hall, New Jersey, 1996. 35 Narendra, K. and Annaswamy, A., Stable Adaptive Systems, Dover Publications Inc, Mineola, NY, 1989. 36 Pomet, J. B. and Praly, L., “Adaptive Nonlinear Regulation: Estimation from the Lyapunov Equation,” Vol. 37(6), June 1992, pp. 729–740. 37 Cao, C. and Hovakimyan, N., “Stability Margins of L Adaptive Controller: Part II,” New York, NY, July 2007, pp. 1 3931–3936. 38 Microbotics, Inc, Microbotics Inc. 28 Research Drive, Suite G,Hampton, VA 23666-1364, Servo Switch/Controller Users Manual , August 2008, http://www.microboticsinc.com. 39 Persistent Systems, The Wave RelayTM Quad Radio Router , August 2008, http://www.persistentsystems.com. 40 Lizarraga, M. I., Dobrokhodov, V., Elkaim, G. H., Curry, R., and Kaminer, I., “Simulink Based Hardware-in-the-Loop Simulator for Rapid Prototyping of UAV Control Algorithms,” AIAA Infotech@Aerospace conference, 2008.


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Figure 32: MRAC and L1 algorithms in flight reproducing Rohrs’ setup.


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