Nonlinear adaptive flight control system:Performance enhancement and validation

Yue FENG, Zonghu SUN, Lioni WU,*, Yongshun WANG, Bin XI,Weng Khuen HO, Yncheng YOU

aSchool of Aerospace Engineering, Xiamen University, Xiamen 361102, China

bDepartment of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore

KEYWORDSAdaptive control systems;Control engineering;Flight control systems;Nonlinear control systems;Robustness stability

AbstractThe problem of decreasing stability margins in L1adaptive control systems is discussed and an out-of-loop L1adaptive control scheme based on Lyapunov’s stability theorem is proposed.This scheme enhances the effectiveness of the adaptation, which ensures that the system has sufficient stability margins to achieve the desired performance under parametric uncertainty,additional delays,and actuator faults.The stability of the developed control system is demonstrated through a series of simulations.Compared with an existing control scheme,the constant adjustment of the stability margins by the proposed adaptive scheme allows their range to be extended by a factor of 4–5,bringing the stability margin close to that of variable gain PD control with adaptively scheduled gains.The engineered practicability of adaptive technology is verified.A series of flight tests verify the practicability of the designed adaptive technology.The results of these tests demonstrate the enhanced performance of the proposed control scheme with nonlinear parameter estimations under insufficient stability margins and validate its robustness in the event of actuator failures.

1.Introduction

Adaptive control1has been a well-researched topic in control theory over several decades.It is a promising method in the field of automatic flight control, because it can improve the performance of control systems in the presence of uncertain modeling parameters, various external disturbances, and strong coupling and nonlinearity over a wide flight envelope.Such issues are often encountered in the field of Unmanned Aerial Vehicles (UAVs).2With the rapid development of UAVs in both military and civilian fields, several advances in adaptive flight control theory have helped to improve performance and robustness so that engineering implementations are now possible.In addition to several well-known modifications,3–5many new adaptive control methods have been developed in the past decade, such as neural network adaptive control,6L1adaptive control,7optimal control modification,8adaptive loop transfer recovery,9retrospective cost optimization,10and combined composite model reference adaptive control.11In spite of extensive research on adaptive control theory, the fact remains that few schemes have been subjected to flight validation experiments12,13and many technical problems related to engineering applications remain unresolved.

L1adaptive control,which is an improved Model Reference Adaptive Control (MRAC) scheme,14is one method that has been employed in successful flight experiments.The L1filter is designed with trade-off between performance and robustness, and the adaptive law follows Lyapunov’s stability theorem15or the piecewise constant method.16Many simulations using L1adaptive control have been performed,17–19but very few have been considered in realistic scenarios, such as under computational latency,transport delays,and actuator dynamics.In terms of flight validation, a flight experiment using an L1adaptive control system was conducted on the NASA Air-STAR flight test vehicle in 2009,20,21and subsequent experiments have also been conducted.Flight tests based on the piecewise constant method have been implemented on Calspan’s Learjet22and a commercial autopilot,23and those based on Lyapunov’s stability theorem have been implemented on the Rascal UAV,24a small indoor UAV25(where the nonlinear adaptive parameters were omitted), and the SmallWhite UAV,26which was the initial application of nonlinear L1adaptive control.The oscillations caused by an insufficient timedelay margin in the AirSTAR flight test were first reported in Ref.27and have been further discussed in detail, and it is thought that a problem with the L1control system reduces the stability margins.

An adaptive control system with strong nonlinearity is proposed in this paper.The construction of the L1control system has a design weakness in that the bound of the nonlinearity is limited.Systems based on the piecewise constant method can be approximated as Linear Time-Invariant (LTI) systems in most flight conditions, whereas those based on Lyapunov’s stability theorem have been proved to approximate an implementable LTI controller28despite the nonlinearity introduced through the nonlinear parameters.From a theoretical perspective, nonlinear adaptive control systems can adaptively adjust the control gain to improve the dynamic performance,thereby enhancing the control performance and robustness.However,current L1adaptive control systems can be approximately regarded as linear control systems and cannot ensure consistent performance when the system degenerates.This is the reason for the oscillations reported in Ref.27In the case of UAV systems, the margin requirements may be insufficient because of inaccurate models and changes in the installation center of gravity and load, which could result in system oscillations or even divergence.Therefore, there is an urgent need for a flight control system with strong and effective nonlinearity to satisfy the flight control requirements and ensure flight safety,which is the motivation for this study.

Although the nonlinear parameters were considered in this previous study, the L1adaptive control system still performed as a linear control system,and so the stability margin could be no longer improved when it became insufficient.Nonlinear L1adaptive control systems make small-scale adjustments to the stability margin on the basis of the baseline controller with relatively weak adaptation, and the stability margin inevitably decreases compared with that of the baseline controller.The main factor that limits the ability of nonlinear L1adaptive control systems is the L1filter.The characteristics of this filter imply that its output has a time delay with respect to its input,which causes a phase difference.The feedback signals in nonlinear adaptation are filtered by the L1filter and sent back to the closed-loop system.The phase lag of the feedback signals significantly impacts the phase and time-delay margins.In addition, the phase differences between the feedback signals of the adaptive control law and those of the baseline control law prevent the nonlinear parameters from directly affecting the baseline controller.Based on these considerations, a new L1adaptive control scheme is proposed,in which the phase difference is eliminated and the reduced stability margin is overcome by redesigning the position of the filter in the feedback loop.The problem of decreased stability margins in L1control systems is solved,and the proposed approach enables the control performance to remain consistent with the expected performance in situations where the system has insufficient stability margins or reduced control performance.

This paper is a follow-up study on the application of L1adaptive control theory in the field of flight control system.The main contribution is the proposal of a new L1adaptive control system with enhanced performance.First, compared with the original L1control system,the nonlinearity and adaptation are enhanced and the range in which the system stability margin can be adjusted is greatly expanded.Additionally,based on the fact that traditional Proportional–Derivative(PD)controllers are widely used in engineering,the adaptation for PD feedback gains implemented in this study has significant practicability.From the perspective of engineering validation, the proposed adaptive control system has been used in flight test verifications with nonlinear parameters, which increases the confidence in adaptive control as a possible flight control method.

The remainder of this paper is organized as follows.Taking the SD-25 as the research object,Section 2 presents the design of the nonlinear out-of-loop L1adaptive control system based on a mathematical model for the pitch dynamics.Section 3 demonstrates the stability of the proposed control system.Simulation results and comparative stability analysis are given in Section 4,and the flight test validation is described in Section 5.Finally, Section 6 summarizes the main contributions of this study and states the conclusions.

2.Design of out-of-loop L1adaptive control system

2.1.Problem formulation

UAV motion can be abstracted as six-degree-of-freedom rigid body motion, which can be described in the form of 12-state differential equations.29As the dynamics of UAVs can be decoupled,a simplified mathematical model for the UAV pitch dynamics is used for the design of control system, where the variables of interest are the pitch angle θ and the pitch rate q.Considering straight and level flight,the simplified equations of motion are as follows:

where ρ is the air density,Vais the speed of the UAV through the surrounding air mass, c is the mean chord of the UAV wing, S is the planform area of the UAV wing, Jyis the moment of inertia of the y axis, δeis the elevator deflection,and the coefficients C0m, Cαm, Cqm, and Cδem are stability derivatives and are functions of state variables.

The system described in Eq.(1)is time-variant and difficult to obtain an analytical solution for sophisticated control system design.The small-perturbation equation30around the singular points can be obtained based on certain assumptions,26where the time-varying coefficients are approximated as constants, and finally a second-order linear time-invariant is obtained.

Considering x=[θ q]Tas the system state vector, it follows that the second-order state-space equation with zero initial conditions can be expressed as

where θ is the pitch angle,q is the pitch rate,u(t)is the control signal,y(t)is the regulated output,A is the known state matrix,B is the known control input matrix, with (A, B) controllable,Λ is the known output matrix, Δ(x) is the uncertain part, and a1,a2,b1are the constants coefficient obtained by the linearization of Eq.(1),which are determined by the variables and stability derivatives at singular points.

This paper considers systems in the presence of unknown constant parameters.The uncertain part Δ(x), which is a type of structured uncertainty matched by the control input, is defined as a linear combination of N known basis functions related to state variables with N unknown constants and can be described as

where τ is a constant vector,and Φ(x)is the known regression vector.

Considering the uncertainty caused by the state variables θ and q and the constant deviation of the control surface δΔ,we choose Φ（x）=[θ（t）q（t）δΔ]Tand τ=[τ1τ2τ3]T.

2.2.Out-of-loop L1adaptive control architecture

Fig.1 shows the structure of the out-of-loop L1adaptive control system.The L1adaptive control law15introduces a lowpass L1filter to guarantee robustness in the presence of fast adaptation while reducing the stability margin of the system.The concept of the Out-of-Loop L1Adaptive Control(OLLAC) system is to move the L1filter out of the feedback loop so as to overcome the problem of a decreased system stability margin and guarantee greater robustness.

Remark 1.The filter in the OLLAC system,hereinafter referred to as the Out-of-Loop(OL)filter,is only effective for parameter estimation and can be separated from the closed-loop control module when analyzing the system stability.Therefore, the robustness of the OLLAC system is ensured at a high identification rate and the L1filter has no impact on the stability margins, especially for a low cut-off frequency.

2.2.1.Baseline controller

A feedforward and feedback controller,which is widely used in flight control, is selected as the baseline controller.Neglecting the uncertain component, the control law defined by Eq.(4)yields the closed-loop system with the zero initial conditions in Eq.(5):

where Am?A-BKm, Bm?kcB, kcis the feedforward gain,Km=[kpkd] is the feedback gain, and r(t) is the reference input signal.According to the desired performances for Amand Bm,the baseline controller gains kp,kd,and kcare readily obtained.

2.2.2.State predictor

The closed-loop system defined by Eq.(2) with the feedback control law um（t）=-Kmx（t）under zero initial conditions is

Fig.1 Structure of OLLAC system.

where ua(t) is the adaptive term of the control signal.

We consider the following state predictor under zero initial conditions:

where τ^（t）=[^τ1（t）^τ2（t）^τ3（t）]Tis the estimate of the ideal unknown parameterτ.

2.2.3.Adaptation law

Based on the projection method described in the appendix and introducing the autocorrelation modification26to minimize the impact of noise, the adaptive process is governed by the following adaptation laws:

2.2.4.Control law

The control law for the OLLAC system is composed of the baseline feedback component um(t) and the adaptive component ua(t), which have the form of

where the Laplace transform of η(t)is defined as η（s）?C（s）^τT.Here, C(s) is the OL filter, which is a Bounded-Input/Bounded-Output (BIBO)-stable and strictly proper transfer function with a Direct Current (DC) gain of C（0）=1.

Remark 2.Nonlinearity is mainly reflected in η（t）Φ（x）,which is designed to counteract the uncertain component Δ(x).^τ is the vector of adaptive gains, the first two of which are nonlinear terms and compensate for the feedback gain of the baseline controller so as to adjust the dynamic performance.The third gain is a linear term that compensates for the constant deviation of the control surfaces to realize accurate tracking.

The original adaptive control law in Ref.15 is ua（s）=-C（s）（（η′（s）-kcr（s））,where η′（s）is the Laplace transform of η′（t）?^τT（t）Φ（x）.In contrast, the OL filter does not affect the feedback signals in the regression vector Φ (x).

Finally,the out-of-loop L1adaptive controller is defined by the relationships in Eqs.(7)–(10).

3.Analysis of OLLAC system

3.1.Stability of closed-loop reference system

A closed-loop reference system can be constructed for the OLLAC system described in Section 2.2 as

Lemma 1.A necessary and sufficient condition for the system in Eq.(11) to be Bounded-Input/Bounded-State (BIBS)-stable with respect to r(t) is that H（s）?（sI-Am）-1B is a proper BIBO-stable transfer function.

Proof.The Laplace transforms of the state equation and control law in Eq.(11) can be written as

where xin（s）?（sI-Am）-1x0, x0is the initial value of x(t), and the subscript ‘‘r”represents the variables in the reference system.

As Amis a Hurwitz matrix and both xin(t)and r(t)are uniformly bounded, xr(t) is also uniformly bounded from Eq.(13).Thus, the BIBS stability of the closed-loop system has been proved.Note that this design condition is not required to prove the L1stability31.

Remark 3.The influence of the L1filter on the closed-loop stability is considered in the proof of Lemma 1 in Ref.7.However, the object on which C(s) acts in Eq.(11) is the uncertain parameter estimation, which is independent of the dynamics of the closed-loop system.According to the time-scale separation principle,32C（0）=1 in the above proof,and thus the stability of the closed-loop system is not affected by C(s).

3.2.Steady-state performance of error dynamics

The following error dynamics can be computed by subtracting Eq.(7) from Eq.(6):

Lemma 2.The tracking error xein Eq.(14) is uniformly bounded and asymptotically stable with the adaptation laws described in Eq.(8).

Proof.Consider the global radially unbounded quadratic Lyapunov function candidate of the form

Consequently, Lemma 2 has been proved.

4.Simulations and analysis

4.1.Implementation and time domain simulations

4.1.1.Configuration for simulation

A closed-loop OLLAC system was constructed in MATLAB to verify the proposed system performance with the SD-25 Vertical Take-Off and Landing (VTOL) UAV as the design object.The system and controller are discretized with a sampling time of Ts= 0.01 s.Controller parameters are designed based on the linear model and the OLLAC system will be eventually verified using a nonlinear dynamic model.

We considered the steady-state flight condition with a pitch angle of 3.05°,attack angle of 3.05°,pitch angle rate of 0 rad/s,speed of 30 m/s, and altitude of 100 m as the singular point.The linear small-perturbation equation described in Eq.(2)can then be written as

According to Section 2.2,the baseline controller is designed as BKm=A-Amand kcB=Bm, and so we have negative feedback gains Km=[0.7 0.15] and feedforward gain kc=1.03.

Remark 4.The exact values of the feedback gains are Km=[0.68 0.17].We replace these with the approximate values following the conventions of engineering applications.The cut-off frequency of the OL filter, the design of which is related to the trade-off between control performance and robustness,affects the speed at which the adaptation acts on the system and is limited by the frequency of system oscillations.Theoretically,the adaption rate should be as large as possible to ensure better learning efficiency.In practice,there is an upper limit that can be determined through simulations.The identification process will be unable to converge when the adaption rate is above this upper limit.The range of nonlinear parameters is normalized to make it symmetrical about zero, and the projection method is realized on the normalized range.

4.1.2.Comparative experiment with L1adaptive control system

The time domain responses of the OLLAC system and the original L1adaptive control structure are compared under the same design parameters.The simulation results are compared separately in systems with sufficient and insufficient stability margins.The step responses of nonlinear dynamic model are shown in Fig.2 and Fig.3.The simulation duration is 30 s,and the power of white noise for θ and q is chosen with 0.0001 and 0.2, respectively.To simulate an insufficient stability state in the aircraft,a pure time-delay module has been added to the actuator with dδ=15Ts.

According to Fig.2,when the system margins are adequate,both adaptive control systems exhibit very similar performance.However, in terms of the 0.15 s delay of the actuator,the system with the L1control method diverges, whereas the pitch angle of the OLLAC system tracks the instructions and the adaptive parameter estimate converges, as shown in Fig.3.A comparison of the curves in Fig.3 shows that small oscillations occur in the first 5 s,and the trend of the nonlinear parameters is nearly the same.Over the next 5 s, the system with the L1controller diverges, while that controlled by OLLAC remains stable.The approximate time-delay margins of the systems under OLLAC and L1adaptive control can be obtained from the simulations in time domain with different values of dδ.The time-delay margin of the OLLAC system is about twice the value for L1adaptive control, which are 0.21 s and 0.08 s, respectively.

Fig.2 Comparison of step responses between OLLAC and L1adaptive control with adequate stability margins.

Fig.3 Comparison of step responses between OLLAC and L1adaptive control with insufficient stability margins.

In conclusion,when a system is divergent due to an insufficient stability margin, the adaptation of the L1controller cannot make it stable.In contrast,steady tracking and convergent identification can be achieved by the OLLAC system.As a result, the adaptive performance of OLLAC is significantly better than that of the conventional system, and the stability margins and control performance are further optimized.

Remark 5.Fig.3 provides a more intuitive understanding of the adaptation process.The first 10 s is the period in which the nonlinear parameters are identified; hence, the dynamic performance of the closed-loop system is reconstructed from an unstable state to a stable state.Correspondingly,the pitch angle gradually converges from the oscillating state during this period.The identification gradually converges in the next 10 s, and the system reaches a steady state in which the pitch oscillations disappear and the stability margin is improved.

4.1.3.Comparative experiment with traditional control system

In this section,we describe experiments conducted to illustrate the advantages of the OLLAC system versus traditional PD controllers.The baseline controller of the OLLAC system,which is a traditional PD controller, is noted as Controller 1.Controller 2 is also a traditional PD controller with different control gains to Controller 1, and the gains of Controller 2 are obtained by superimposing the converged estimates of nonlinear parameters in Fig.3 onto the gains of Controller 1.The simulations are carried out under the same conditions as described in Section 4.1.2, and a pure time-delay module is added to the actuator with dδ=15Ts.The open-loop Bode diagram and the pole-zero map used to compare Controllers 1 and 2 are presented in Fig.4.The step responses of the systems controlled by Controllers 1 and 2 and the OLLAC system are shown in Fig.5.

As we can see from Fig.4(a), the 0.15 s delay of the actuator exceeds the margin of Controller 1,but remains within that of Controller 2.This is also illustrated in Fig.5 where the step response curve of Controller 1 diverges while that of Controller 2 indicates smooth tracking.Of more interest is that the pitch angle of the OLLAC system in Fig.5 is consistent with that of Controller i in the first few seconds,and converges in essentially the same way as the curves of Controller 2.The estimations of the OLLAC system enable adjustments toward a more stable state.

In abstract terms, the closed-loop systems of Controllers 1 and 2 can be regarded as two points in different states,having different but unchanged performance.The OLLAC system can be considered equivalent to its baseline controller (Controller 1)at the initial time,because the nonlinear parameters are initially set to 0.Likewise, at the end of the simulation, the OLLAC system can be regarded as essentially equivalent to Controller 2.Therefore, the OLLAC system can be regarded as operating along a line that has one more time axis than the points of Controllers 1 and 2, and it continuously moves from one state (i.e., from Controller 1 or 2) to the other (i.e.,to Controller 2 or 1).

In summary, the OLLAC system can be regarded as an adaptive PD control system with the baseline controller as its foundation.The nonlinear parameters compensate for the feedback gains of the baseline controller to create greater stability margins and better control performance in the presence of system degradation or uncertainties.Compared with the traditional control method,the robustness of the OLLAC system is enhanced.

4.2.Performance analysis and verification

As previously stated, the OLLAC system can move from one point to another.This section discusses the limiting distances between the initial state and the state that can be achieved by the OLLAC system, L1adaptive control, and the PD controller with variable gains.

The closed-loop systems with the OLLAC,L1adaptive control and gain-scheduling PD controller are noted as Systems A,B and C, respectively.The same parameters are selected for adaptation, and the control gains for the baseline controller of Systems A and B are the same as the initial PD gains in System C.

4.2.1.Stability margin and robustness

The stability margin analysis method34for linear systems is not generally suitable for nonlinear systems.To solve this problem,the nonlinear parameters^τ1and^τ2,which introduce nonlinearity into the system, were placed in a mesh.If the mesh is sufficiently dense, the nonlinear parameters in each cell can be processed as constants.As a result, the nonlinear part of the control system is converted to a constant term,and the nonlinear system can be analyzed based on the stability margin analysis of linear systems.

Two-dimensional grid coordinates were constructed based on the ranges of ^τ1and ^τ2to form a grid with a resolution of 0.02.The stability of Systems A and B was then analyzed through Bode diagrams.Likewise, the coordinates of System C were constructed and indicated the increment based on the initial PD control gains.The comparison results for the effects of adaptive parameters on the stability margins are shown in Fig.6 and Table 1.

Fig.4 Frequency domain analysis of Controllers 1 and 2.

Fig.5 Comparison of time-domain response among Controller 1, Controller 2, and OLLAC.

Fig.6 Effects of adaptive parameters on stability margins for Systems A, B, and C.

Table 1 Stability margins analysis of Systems A, B, and C.

Fig.6 shows how the stability margins of the closed-loop Systems A,B,and C change as the gains of the advanced controller vary in response to the nonlinear parameters or gain scheduling.The ranges on the z-axis of Systems A, B, and C indicate the extent to which they can ‘‘move”the system.The traditional controller will be represented as a plane parallel to the x-y plane and the values of z-axis are its stability margins.Apparently, traditional controllers are unable to reconfigure the system dynamic performance.As we can see in Fig.6,Systems A and C display nearly the same trend,while the curvature of the stability margin for System B is significantly smaller than that of the other two systems.This means that,for the same change rate of^τ1and^τ2,the closed-loop performance of System B changes at the smallest rate.From Table 1, when the adaptive parameters change over the same ranges, the phase, gain, and time-delay margins of System A have ranges that are 4, 4.2, and 5 times those of System B,respectively.By extension, the parameters for System B range over a small interval around the figures for the baseline controller under the adaptation.In particular, the upper bounds of the phase and time-delay margins of System B are less than the figure for the baseline controller,which means that the stability margin is reduced when the adaptation of System B is introduced.This explains why System B diverges in Fig.3 under the same parameter identification as System A.

In summary,the adaptation in the original L1adaptive control system only works around a fixed state,or‘‘datum point,”and the system is only adjusted slightly with a small amplitude in the vicinity of this datum point,which leads to limited adaptation performance.In contrast,the OLLAC system moves the datum point to adjust the system quickly over a larger range and produces more efficient adaptation.

4.2.2.Control performance

The control performance of Systems A,B,and C was analyzed under the nominal model.The grid coordinates were consistent with those constructed in Section 4.2.1,and the step responses of Systems A,B and C were compared in the time domain.We define eAas the difference between the step responses of Systems A and C.Similarly, eBis the difference between the step responses of Systems B and C.The mean and variance of eAand eBin each cell are compared in Fig.7.

As can be seen from Fig.7,the control performance of System A is consistent with that of System C,whereas that of System B is significantly affected by nonlinear parameters.As the nonlinear parameters deviate significantly from the initial values,the performance of System B shows obvious differences to that of System C.Therefore, with respect to robustness and control performance, System A is highly consistent with an automatic variable gain controller.

4.3.Effects of matrix Q on control objectives

We now explore the design of the matrix Q in the algebraic Lyapunov equation.According to the analysis in Section 3.2,any positive-definite symmetric matrix Q can be solved to obtain a matrix P that conforms to the stability condition,but the matrix P calculated from different matrices Q will be different.The influence of matrix P on the control system is analyzed in the following.We define the adaptation error ε（t）?xTe（t）PB, and rewrite Eq.(8) as

Fig.7 Comparison of mean and variance results.

The value of λ is determined completely by the matrix P.As P is a symmetric positive-definite matrix,λ>0.Thus,Eq.(25)becomes

where χ=b1/P12is a constant.

According to Eq.(28), the value of λ determines how the different tracking tasks should be weighted.For λ > 1, the adaptation is driven primarily by eq（t）, and the main object of the controller is the pitch rate.For λ < 1, the adaptation is driven primarily by eθ（t）, and the main object of the controller is the pitch.Note that the smooth signal ^q（t）cannot be completely tracked by q（t）due to sensor noise,which makes it difficult to achieve eq（t）≠0.Considering the steady-state system, we have ˙^τ（t）=0, and thus eθ（t）=-λeq（t）.As a result,there is always a pitch tracking error in the case of measurement noise.

Taking the SD-25 VTOL UAV as the object, the matrix Q is designed as a diagonal matrix Q=diag（Q11;Q22）.If we define ?Q11/Q22, the relationship between ζ and λ is shown in Fig.8.The simulation results in the time domain for λ = 0.5, 1.0, and 2.0 are shown in Fig.9 and Table 2.

Fig.8 indicates that ζ is a monotonically decreasing function of λ and the curve is approximately exponential.According to Fig.9 and Table 2,higher values of λ result in a greater pitch tracking error.To achieve better tracking performance,λ < 1 can be ensured through the appropriate design of ζ and the matrix Q.

Fig.8 Curve of ζ - λ.

5.Experimental results

5.1.Flight test overview

An SD-25 VTOL UAV is used to evaluate the OLLAC system,as shown in Fig.10.Past flight tests on the SD-25 have suffered from the problem of an insufficient stability margin,which could potentially lead to oscillations in attitude angles.To solve this problem, the adaptive control system, which is used to track the attitude angles,is designed and implemented in the inner loop and commands are given by the outer loop.

The SD-25 VTOL UAV is equipped with four elevon segments (inner/outer) and four rudder segments for the V-tail.Its wingspan is 3.49 m and its takeoff weight is 25 kg.The adaptive control algorithm was written in the C programming language and was run at a frequency of 100 Hz on a Freescale MPC5674 autopilot.The sensors on the autopilot include an attitude and heading reference system,a GPS receiver(NovAtel),and a digital atmospheric data system.The planned route was a circle with a radius of 400 m.

Under the single-variable method, the flight test was completed in two sorties.Sortie I and Sortie II were designed to test the control performance and stability for the original L1control system in Ref.15 and the OLLAC system,respectively.The robustness of the OLLAC system was verified in Sortie II by reducing the elevator efficiency to 60%to imitate actuator failure (Sortie II - actuator failure).

Fig.9 Time-domain responses for different λ.

Table 2 Analysis of tracking error for different λ.

Fig.10 Photograph of SD-25 VTOL UAV.

5.2.Data analysis

The experimental parameters were selected to be the same as the designed parameters in Section 4.1.1.To compare the proposed control system with the existing control methods,conditions such as the flight status, outer-loop control strategy,control parameters, and commands were the same for Sorties I and II.

Table 3 Analysis of pitch tracking error.

5.2.1.Data analysis for comparative experiment

Fig.11 and Table 3 compare the flight results over 100 s between Sortie I and Sortie II.The altitude commands corresponding to the two groups of data are consistent with descending 50 m and climbing 50 m.

For similar pitch commands, Sortie I exhibits smallamplitude high-frequency pitch oscillations, but these are not observed in Sortie II.The data in Table 3 indicate that the indices of the tracking error in Sortie II are smaller than those in Sortie I, and in particular, the mean pitch tracking error of Sortie I is 14.8 times that of Sortie II.

The flight data show that the problem of the insufficient stability margin in the SD-25 flight still exists under the original L1control system.However, this problem is solved by the OLLAC system, which achieves better control performance and greater stability margin.

Remark 6.It can be seen from Figs.11(c) and (f) that, for similar pitch commands, the adaptive control component is different.This problem was further analyzed and found to be caused by differences in the linear parameter estimation, which can be understood as a difference in the trim elevator deflection of the UAV due to changes in the center of gravity or weight between the two flights.

Fig.11 Aircraft response for original L1adaptive control (Sortie I) and OLLAC (Sortie II) systems.

Fig.12 Aircraft response under OLLAC system with actuator failure.

5.2.2.Data analysis for robustness verification

Fig.12 shows flight data for a period of 80 s from Sortie II in which actuator failure was simulated.The dotted rectangles denote the period (12.0 s ≤t ≤49.8 s) in which the elevator effect was reduced to 60 %.

At the moments of elevator failure and recovery,a tracking error of approximately 1.0°can be observed in Figs.12(a)and(b).It follows that the parameter identification works and converges within the next 7 s (Figs.12(c)-12(f)), and the tracking error gradually decreases to zero within 10 s due to the enhanced adaptation.Therefore, the OLLAC system is robust and can identify and compensate for the uncertainties to achieve zero-error tracking when actuator failure occurs.

6.Conclusions

The design, analysis, simulation, and flight test validation of an OLLAC system have been described in this paper.The problem of the insufficient stability margin15,26has been discussed, and a new structure has been proposed to enhance the nonlinearity and effectiveness of the adaptation process.As the theoretical basis for application, the BIBS stability and uniform boundedness of the tracking error have been proved.A method for stability margin analysis of nonlinear control systems has been established, in which the nonlinear system can be regarded as a linear system within each mesh gridded by the nonlinear parameters.The stability margins of the original L1control system, when adjusted by the adaptive scheme, have obvious boundaries and are smaller than those of the baseline controller.The proposed OLLAC system significantly extends these boundaries and achieves unrestricted stability margins.Comparative experiments focusing on the control performance and robustness have been realized in the frequency and time domains, respectively.Compared with the original control system, the OLLAC system reconstructs the closed-loop system in a wider range and at a faster rate,which leads to a high degree of nonlinearity and enhanced adaptation to compensate for uncertainties.

A practical flight test validation of the proposed control system was realized on a demonstrator with an insufficient stability margin, and the robustness was further verified by imitating an actuator failure.The pitch oscillations observed under a baseline controller and L1adaptive control system were significantly reduced by the proposed system,which exhibits sufficient robustness to ensure accurate tracking and stability.In addition, the influence of the matrix in the Lyapunov equation on the control performance was analyzed in terms of its physical meaning,and a design method was further discussed.The proposed OLLAC scheme adds new capabilities in terms of enhanced performance and robustness for adaptive flight control systems and promotes the inclusion of nonlinear terms in engineering applications, which further increases the viability of adaptive control for UAVs.

The engineering practicability of the OLLAC system comes from the fact that the baseline PD controller is already widely used in engineering.If MIMO systems are involved,there may be implementation difficulties because of the number of parameters,complex control systems,and other issues.Future research will investigate how to implement and optimize the OLLAC system under MIMO systems, such as by designing an appropriate parameter setting method or handling the MIMO system.If possible,we will implement the OLLAC system for the three-axis attitude control of a vehicle to solve the problem of modal coupling under large maneuvers.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China(No.U21B6003)and the China Scholarship Council (CSC, No.202006310096).

Appendix A.A projection method35is introduced in the adaptive law to ensure that the adaptive parameters have a boundary.Considering a parameter vector θ with boundary constraints of the form ‖θ‖≤θmax, we define the continuous projection operator as