Feedback Linearisation for Nonlinear Vibration Problems

Abstract

Feedback linearisation is a well-known technique in the controls community but has not been widely taken up in the vibrations community. It has the advantage of linearising nonlinear system models, thereby enabling the avoidance of the complicated mathematics associated with nonlinear problems. A particular and common class of problems is considered, where the nonlinearity is present in a system parameter and a formulation in terms of the usual second-order matrix differential equation is presented. The classical texts all cast the feedback linearisation problem in first-order form, requiring repeated differentiation of the output, usually presented in the Lie algebra notation. This becomes unnecessary when using second-order matrix equations of the problem class considered herein. Analysis is presented for the general multidegree of freedom system for those cases when a full set of sensors and actuators is available at every degree of freedom and when the number of sensors and actuators is fewer than the number of degrees of freedom. Adaptive feedback linearisation is used to address the problem of nonlinearity that is not known precisely. The theory is illustrated by means of a three-degree-of-freedom nonlinear aeroelastic model, with results demonstrating the effectiveness of the method in suppressing flutter.

1. Introduction

The effects of nonlinearity are everywhere to be seen in nature and it is true to say that many problems in structural dynamics are nonlinear, but in the past it has been convenient and easier to assume linearity. This linear analysis approach has necessarily led to conservative design of engineering structures that operate dynamically. In the last few decades the need to assume linearity has been increasingly challenged, mainly because of the high cost of fuel and materials and the need to preserve the earth’s valuable natural resources. This is particularly true of the aerospace industries where the need to produce lightweight, fuel efficient aircraft is an unremitting pressure on design engineers. One problem is, of course, that nonlinear dynamic analysis is complicated and the useful linear-analysis methods, such as modal decomposition, are not applicable.

Inevitably active control will increasingly be seen as a solution for problems of nonlinearity in elastomechanics and aeroelasticity. One option is to design lightweight, fuel efficient aircraft and use active control to nullify the effects of the nonlinearity. By this approach, well-understood linear analysis methods can still be used. Feedback linearisation, described, for example, by Isidori [1], is a technique now well established in the active control community. Depending upon the implementation it is able to completely or partially linearise the nonlinear system. In the latter case there remains a (generally nonlinear) subsystem with dynamics that must be investigated to ensure stability. Of course, the linear part can be used for the attainment of a chosen control objective using conventional linear time invariant (LTI) techniques. An important aspect of feedback linearisation is that for the desired control objectives to be met precisely, the nonlinearity must be known in terms of its physical location, its type (e.g. cubic, quartic, free-play, etc.), and the numerical values of its parameters. This seems initially to pose a serious restriction on its application, but fortunately there exist techniques, such as adaptive feedback linearisation, that are able to account for incorrectly estimated nonlinear terms.

It is apparent from the literature that the feedback linearisation method, which holds much promise, has found only very limited application in active vibration suppression. Examples include Fossen and Paulsen [2] who applied adaptive feedback linearisation to the automatic steering of ships. In aeroelasticity, Ko and his colleagues [3–5] used feedback linearisation methods (including adaptation) and carried out a series of wind-tunnel tests on a two degree of freedom aerofoil with either one or two control surfaces. They found that with an erroneous nonlinear parameter, but without adaptation, their system reached a nonzero equilibrium, rather than the zero equilibrium that is usually sought. Monahemi and Krstic [6] employed adaptive feedback linearisation to update the aerodynamic parameters in their nonlinear model and thereby suppress wing-rock motion, a phenomenon triggered primarily by aerodynamic nonlinearities. Poursamad [7] implemented a hybrid neural-network controller for antilock braking with adaptive feedback linearisation to handle nonlinear and time-varying brake parameters. Bechlioulis and Rovithakis [8] developed a multiple-input, multiple-output tracking controller with adaptive feedback linearisation and Shojaei et al. [9] demonstrated the ability of adaptive feedback linearisation in aiding effective trajectory tracking in the presence of both parametric and nonparametric uncertainties in wheeled robots. Tuan et al. [10] designed a controller based on partial feedback linearisation of the nonlinear dynamics of a 3D overhead crane.

In this paper the problem of active control of nonlinear systems of the form 𝐀₁𝐱̈+𝐀₂𝐱̇+𝐀₃𝐱+𝐟_nl(𝐱,𝐱̇)=𝐁𝐮, (1) representative of nonlinear vibration problems in elastomechanics and aeroelasticity, is considered. The vectors 𝐱, 𝐱̇, and 𝐮 typically contain the state variables and inputs, respectively. The nonlinearity 𝐟_nl is given as a function of 𝐱 and 𝐱̇ and the matrices 𝐀₁, 𝐀₂, 𝐀₃, 𝐁 represent the usual system parameters. This class of problem, characterised by the second-order matrix differential equation with additional nonlinearity confined to the left-hand side of (1), prevails to a very large extent in engineering mechanics and is therefore worthy of the special attention devoted to it in this paper. The classical output feedback linearisation [1] may be greatly simplified in the case of elastomechanical or aeroelastic systems described by (1). In particular, (i)the essential theory is carried out entirely using the second-order matrix differential equation familiar to structural and aero-structural dynamicists, with the result that the need for repeated differentiation, usually described using the Lie-algebra notation, is rendered unnecessary,(ii)a linear transformation applies between the state variable in (1) and the coordinates of the linearised system,(iii)cancellation of the whole of the open-loop system dynamics (not just the nonlinear terms) results in a set of independent linear single degree of freedom systems for the application of conventional LTI control methods,(iv)complete linearisation may be achieved with an equal number of actuators and sensors. If the number of actuators and sensors is less than the dimension of the system, then there will remain a nonlinear subsystem of dimension equal to the difference between the dimension of the full system and the number of sensors (and actuators). This subsystem can be made independent of the linearised part and methods are described to check its stability.(v)Adaptive feedback linearisation is described for the treatment of an incorrect estimate of the nonlinearity. This makes use of Lyapunov stability criteria and results in a parameter update rule that evolves with time to ensure stability of the system.

The method is illustrated by means of a three degree of freedom aeroelastic system consisting of a flexible wing described in terms of two assumed modes in bending and torsion and a third degree of freedom that describes the angular motion of an underslung pylon-engine assembly. The parameters and dimensions of the system are carefully chosen to have realistic values.

2. Active Feedback Linearisation

Feedback linearisation [1] is a process whereby a nonlinear system is rendered linear by virtue of active control. Unlike Jacobian linearisation, it is exact, and does not entail an approximation at any stage. The method is implemented by transforming a nonlinear system into a linear one. For the class of second-order systems given by (1) considered in this paper, 𝐱̈=𝐟(𝐱,𝐱̇)+𝐆𝐮,𝐟(𝐱,𝐱̇)=Ψ𝐱+Φ𝐱̇+Ω𝐟_nl,Ψ=−𝐀₁⁻¹𝐀₃,Φ=−𝐀₁⁻¹𝐀₂,Ω=−𝐀₁⁻¹,𝐆=𝐀₁⁻¹𝐁, (2) becomes 𝐳̈=Ψ𝐳+Φ𝐳̇+𝐆𝐮. (3) In these equations 𝐮 and 𝐮 are, respectively, the actual (or physical) input applied to the nonlinear system and the so-called “artificial” input to the linear system. The matrices Ψ, Φ, 𝐆 are dependent on the chosen inputs 𝐮. The mapping from the nonlinear domain to the linearised domain is achieved through a nonsingular, linear, coordinate transformation. The actual input is designed to neutralise the effect of the nonlinearity, a procedure which can sometimes be achieved in full, and sometimes partially, as will be explained theoretically and by means of illustrative examples.

The process is quite straightforward for elastomechanical and aeroelastic systems described by second-order matrix differential equations with nonlinearity in the system parameters. The method described by Isidori [1] allows for greater generality, including nonlinearity in the input and output as well as in the system parameters, which is not required here and its omission leads to simplifications which aid the understanding of an otherwise fairly complicated procedure.

The feedback linearisation procedure classically, using the Lie algebra, entails repeated differentiation of each of the outputs with respect to time, until the input terms appear. The classical procedure is greatly simplified in the case of second-order matrix systems such as those in elastomechanics or aeroelasticity, as explained in the sequel. The present work addresses two cases. The first case is that of complete input-output linearisation, meaning a full complement of outputs and inputs at every degree of freedom of the system. In this case, since the number of outputs is equal to the dimension of the system, it is possible to linearise the entire nonlinear system. The complete dynamics of the original system are preserved during the transformation. In the second case, an incomplete system of equal numbers of inputs and outputs is assumed. In this case, as the number of inputs and outputs is less than the dimension of the overall model only a partial linearisation of the system can be achieved. The portion that remains untransformed will contribute to what is known as the internal dynamics, whose stability must be ensured for stability of the overall closed-loop system. This is achieved by examining the stability of the zero dynamics [1], which is obtained by setting all coordinates of the linearised subsystem to zero in the expressions for the internal dynamics. Expressions for the latter are obtained such that their time-derivatives are orthogonal to the inputs, rendering the zero dynamics uncontrollable. The zero dynamics may be either linear or nonlinear.

2.1. Complete Input-Output Linearisation (𝑛-Inputs, 𝑛-Outputs)

In the present case the number of inputs and outputs is equal to the dimension of the system. The matrices and vectors of the original nonlinear system, given in (2), have the dimensions Ψ, Φ, Ω, 𝐆∈ℝ^𝑛×𝑛; 𝐱, 𝐮, 𝐟∈ℝ^𝑛×1 and those given by the desired linearised equation (3) by Ψ, Φ, 𝐆∈ℝ^𝑛×𝑛; 𝐳, 𝐮∈ℝ^𝑛×1. The first step is to choose the vector of actual inputs that cancels the nonlinearity, 𝐮=𝐆⁻¹(𝐮−𝐟(𝐱,𝐱̇)). (4) It can be seen that the nonlinearity is indeed cancelled by substituting (4) into (2). In fact, this is a special case where not only the nonlinearity, but also the complete open loop dynamics is cancelled by the choice of actual input. The result is the linearised system of independent, second-order, single degree of freedom equations 𝑥̈₁𝑥̈₂⋮𝑥̈_𝑛=𝑢₁𝑢₂⋮𝑢_𝑛. (5) In fact, these equations are a special case of single degree of freedom equations, where each equation is simply a double integrator. Equation (5) is a particular form of (3), where also it is seen particularly that 𝐱=𝐳. The simplicity of (5) is an advantage of the complete cancellation of the open-loop dynamics in (4). The choice of the artificial input 𝐮 necessarily depends upon the control objective, for example, the assignment of a pair of complex conjugate poles in each of the systems in (5) to avoid resonances. Whatever the control objective is, it will result in the determination of gains defined here in terms of negative feedback as 𝑢₁=−𝜍₁𝜈₁𝑥₁𝑥̇₁𝑢₂=−𝜍₂𝜈₂𝑥₂𝑥̇₂⋮𝑢_𝑛=−𝜍_𝑛𝜈_𝑛𝑥_𝑛𝑥̇_𝑛. (6) In this special case where the entire open loop dynamics are cancelled, the control results in a closed-loop system that is comprised of 𝑛 decoupled, single degree of freedom subsystems. Then, having defined the artificial inputs, the actual inputs that provide the desired linearisation are determined from (4). It is seen that the nonlinearity in (2) is neutralised and the closed-loop system is indeed linear with the required dynamics.

If the actual input were chosen to cancel the nonlinearity alone (not the complete open-loop dynamics), then (4) would be replaced by 𝐮=𝐆⁻¹𝐮−Ω𝐟_nl, (7) and the linearised equations would remain coupled (unlike (5)) and consequently the gains in (6) would take a different form, as 𝑢₁𝑢₂⋮𝑢_𝑛𝜍=−¹¹𝜈₁₁𝜍₁₂𝜈₁₂⋯𝜍_1𝑛𝜈_1𝑛𝜍₂₁𝜈₂₁𝜍₂₂𝜈₂₂⋯𝜍_2𝑛𝜈_2𝑛𝜍⋮⋮⋮⋮⋱⋮⋮_𝑛1𝜈_𝑛1𝜍_𝑛2𝜈_𝑛2⋯𝜍_𝑛𝑛𝜈_𝑛𝑛×𝑥₁𝑥̇₁𝑥₂𝑥̇₂⋮𝑥_𝑛𝑥̇_𝑛. (8) Clearly, there are a greater number of control gains in (8) than in (6), which means that there is more control flexibility, which might be used, for example, to assign the eigenvectors as well as the eigenvalues. This may be readily achieved using methods such as that presented in [11] and is illustrated through a numerical example later on.

2.2. Partial Input-Output Linearisation (𝑚-Inputs, 𝑚-Outputs, 𝑚<𝑛)

The inputs and outputs (actuators and sensors), 𝐮, 𝐱_(1∶𝑚)∈ℝ^𝑚×1, in equal numbers are now fewer than the dimension of the system. Linearisation results in similar expressions to those obtained for the complete input-output case presented above. Equation (2) is now rewritten as 𝑥̈₁⋮𝑥̈_𝑚𝑥̈_𝑚+1⋮𝑥̈_𝑛=𝑓₁(𝐱̇)⋮𝑓𝐱,_𝑚(𝐱,𝐱̇)𝑓_𝑚+1(𝐱̇)⋮𝑓𝐱,_𝑛(𝐱̇)+𝑔𝐱,_1,1⋯𝑔_1,𝑚𝑔⋮⋱⋮_𝑚,1⋯𝑔_𝑚,𝑚𝑔_𝑚+1,1⋯𝑔_𝑚+1,𝑚𝑔⋮⋱⋮_𝑛,1⋯𝑔_𝑛,𝑚𝑢₁⋮𝑢_𝑚, (9) and the coordinate system which maps the original nonlinear system to the partially linearised system may be expressed as 𝑧₁𝑧₂⋯𝑧_𝑚^𝑇=𝑥₁𝑥₂⋯𝑥_𝑚^𝑇, (10) which is identical to the full output feedback case, except of course that there are now only 𝑚 outputs. Further (𝑛-𝑚) coordinates are needed and are chosen as coefficients of the orthonormal basis of the null space of 𝐆^𝑇_{(1∶𝑛,1∶𝑚)} so that 𝑥₁⋮𝑥_𝑛𝑧=𝐕⋮^𝑚+1𝑧_𝑛,𝐕^𝑇𝐕=𝐈_{(𝑛−𝑚)×(𝑛−𝑚)},𝐕^𝑇𝐆_{(1∶𝑛,1∶𝑚)}=𝟎,𝐕∈ℝ^{𝑛×(𝑛−𝑚)}. (11) As with the full output feedback case, it is now necessary to choose actual inputs so that the nonlinearity is eliminated. This is achieved by 𝐮=𝐆_{1∶𝑚,1∶𝑚}⁻¹𝐮−𝐟(𝐱,𝐱̇)_(1∶𝑚,1),𝐟(𝐱,𝐱̇)_(1∶𝑚,1)=Ψ_{(1∶𝑚,∶)}𝐱+Φ_{(1∶𝑚,∶)}𝐱̇+Ω_{(1∶𝑚,∶)}𝐟_nl, (12) and substitution of (12) into the upper partition of (9) leads to 𝑚 independent linear second-order systems with artificial inputs 𝐮={𝑢₁⋯𝑢_𝑚}^𝑇, expressed as 𝑧̈₁𝑧̈₂⋮𝑧̈_𝑚=𝑥̈₁𝑥̈₂⋮𝑥̈_𝑚=𝑢₁𝑢₂⋮𝑢_𝑚. (13) Then by combining (9), (10), and (11) it is found that 𝑧̈_𝑚+1⋮𝑧̈_𝑛=𝐕^𝑇𝑓₁(𝐳̇)⋮𝑓𝐳,_𝑛(𝐳,𝐳̇)+𝐕^𝑇𝐆_{(1∶𝑛,1∶𝑚)}𝑢₁⋮𝑢_𝑚, (14) so that from (11), 𝑧̈_𝑚+1⋮𝑧̈_𝑛=𝐕^𝑇𝑓₁(𝐳̇)⋮𝑓𝐳,_𝑛(𝐳̇),𝐳, (15) which ensures uncontrollability of the nonlinear internal dynamics (15).

The stability of the complete system is then determined by the zero dynamics, which are generally nonlinear, obtained by setting to zero in (15) the external coordinates (𝑧₁,…,𝑧_𝑚) of the partially linearised system in (13). The equations of the zero dynamics and their stability will be addressed for a specific aeroelastic example in the sequel.

As before, the artificial inputs in (13) may be chosen as a linear combination of the instantaneous displacement and velocity to fulfil a control objective. When the zero dynamics are found to be globally stable, then the desired control behaviour is unaffected by the nonlinearity confined to the internal dynamics.

3. Aeroservoelastic Model

The governing equation of aeroservoelastic systems takes the usual form [12] given by 𝐀𝐪̈+𝜌𝑉𝐁+𝐃𝐪̇+𝜌𝑉²𝐂+𝐄𝐪+𝐟_nl(𝐪)=𝐟_ext, (16) where 𝐀, 𝐃, 𝐄 are the inertia, structural damping, and structural stiffness matrices, respectively, 𝐁, 𝐂 are the aerodynamic damping and aerodynamic stiffness matrices, respectively, and 𝜌,𝑉 are air density and velocity, respectively, (the present 𝐁 is different from the input distribution matrix, also denoted by 𝐁, in Section 1). The vector 𝐪 contains generalised coordinates describing the motion of the system, whereas the vector 𝐟_ext contains externally applied generalised forcing terms, including control forces and gusts. The nonlinearity is confined to 𝐟_nl(𝐪). Modified aerodynamic strip theory is used to compute the lift and pitch moment and an additional unsteady aerodynamic derivative term is included to account for significant unsteady effects [12], which appears in the matrix 𝐁. In practice, the aerodynamic matrices are frequency dependent [12] and the time domain model would need to include aerodynamic states to account for this. The approach used is perfectly adequate for low speed, high aspect ratio wings. The system consists of a wing with an underslung engine attached by a pylon. Aerodynamic forces/moments acting on the pylon-engine arrangement are assumed to be negligible compared to those acting on the wing.

In the analysis presented here the wing deflection is described in terms of two assumed modes and a further generalised coordinate representing the angular displacement of the engine about the axis of the wing. The assumed modes are depicted in Figure 1, which also shows the coordinate system with its origin at the root, leading edge of the wing.

Figure 1 

The two deflection patterns assumed for the flexible wing.

The vertical deflection of the wing 𝜁(𝑥,𝑦) is then given by 𝜁=𝑦²𝑞₁+𝑦𝑥−𝑥_𝑓𝑞₂, (17) where 𝑞₁, 𝑞₂ are generalised coordinates that quantify the amount of bending and torsion modes present in the overall deflection, and 𝑥_𝑓 is the 𝑥-coordinate of the wing flexural axis. The pylon-engine is modelled as a rigid body, connected to the wing by a nonlinear stiffness. A sketch of the pylon-engine and also the complete wing-pylon-engine system may be found in [13].

The pylon-engine rotational degree of freedom 𝜗_pe is the deflection of the pylon-engine relative to the wing. The absolute rotation 𝜃_pe may be obtained by adding the wing twist angle at the engine attachment location, 𝜒₂, to 𝜗_pe. The absolute pylon-engine rotation becomes a coordinate in the assumed-modes domain, 𝑞_pe=𝜃_pe. The coordinate transformation matrix T is defined by 𝜁₁𝜒₂𝜗_pe=𝑦²₁000𝑦₂00−𝑦₂1𝑞₁𝑞₂𝑞_pe, (18) where 𝜁₁ and 𝜒₂ denote the vertical displacement at point “1” (the intersection of the wing flexural axis with the local 𝑥-axis at the point of attachment of the pylon to the wing) and the angular wing twist at point “2” (the intersection of the quarter-semi-span with the quarter-chord). The 𝑦-coordinates of points 1 and 2 are identical (𝑦₁=𝑦₂). The system matrices, 𝐀, 𝐁, 𝐂, and 𝐄 are given in terms of the parameters of the wing and pylon-engine system in [13], whereas vectors 𝐟_𝑐 and 𝐟_nl may be found in Appendix A in this paper. Note that in the present text, 𝑘_𝑇, 𝐾_𝑇 are used to describe the coefficients of the linear and cubic components of coupling stiffness, respectively, which is different to the notation used in [13].

4. Numerical Simulation

The dimensions and parameters chosen for the model are given in [13], with 𝐾_𝑇=300×𝑘_𝑇. For the flexible wing, the values chosen are based on those used in a numerical example given in [12]. The dimensions and mass of the pylon-engine arrangement are chosen to be representative of a real aircraft (e.g. [14]).

The three degree of freedom aeroservoelastic model is used initially to determine the flutter speed of the linear system. Then, the cubic hardening term, 𝐾_𝑇, is included in the torsional spring connecting the wing to the pylon-engine, and the nonlinear time-domain response to initial conditions is produced at an air speed just above the flutter speed.

The aeroservoelastic matrices are given in general terms in [13] and expressed here in terms of specific parameter values in consistent units (to three significant figures) as 𝐀_𝑤=10⁵,𝐄9.490.063300.06330.09420000_𝑤=10⁸,𝐀1.100000.1420000_pe=10³,𝐄4.7601.510001.5100.647_pe=10⁶,00001.90−0.9580−0.9580.511𝐁=10⁴,2.9800−0.2290.01690000𝐂=10³.04.9700−0.4060000 (19) A simplification in the matrix subscripts previously defined in [13] has been made here. The control force distribution matrix is given (as a function of air speed) by 𝐟_𝑐=10²−3.19𝑉²−2.89𝑉²0−0.302𝑉²−0.120𝑉²×𝛽−0.0188000.01₁𝛽₂𝑇_pe, (20) where 𝛽₁ and 𝛽₂ are control surface (flap) angles and 𝑇_pe is a control torque applied directly to the pylon-engine assembly. The nonlinear internal force is given by 𝐟_nl=10⁸0−2.87𝜗1.53³,𝑦₂=𝑦₁=1.875. (21) Further information on the general form of the control and nonlinear spring forces can be found in Appendix A.

4.1. Uncontrolled Linear System: Natural Frequencies, Damping and Flutter Speed

The structural modes in the linear model (nonlinearity neglected) occur at 1.71 Hz (bending), 4.06 Hz (pylon-engine mode), and 6.83 Hz (torsional). It is evident from Figure 2 that flutter occurs at an airspeed of 77.6 ms⁻¹, involving coupling of the pylon-engine mode and wing bending modes.

(a)

(b)

Figure 2 

𝑉-𝜔 and 𝑉-𝜁 plots for the wing-pylon-engine model.

4.2. Nonlinear Time-Domain Response

The nonlinear system is simulated at an airspeed of 80 ms⁻¹, just above the flutter point, under the application of the initial conditions 𝜁₁ = 0.333 mm, 𝜒₂ = 0.00333 rad, and 𝜗_pe = 0.05 rad. These values have been chosen as they are representative of typical physical displacements one might expect in practice, for the wing-pylon-engine system specified in [13]. The resulting response of the system clearly exhibits limit cycle oscillation (LCO). A sample of the response for the 𝜗_pe coordinate is shown in Figure 3.

Figure 3 

Steady-state LCO in 𝜗pe response.

For comparison, the simulated response for the 𝜗_pe coordinate just below the linear flutter speed, at an airspeed of 75 ms⁻¹, is shown in Figure 4. As expected the response continues to decay and converges to the origin.

Figure 4 

Decaying response, just below linear flutter speed.

In the following section, linearising feedback control is applied to the nonlinear model in the assumed-modes domain. In the examples considered, the control objective is the assignment of the poles of the system. Judicious placement of the poles can be used to increase the flutter speed of the system, thereby increasing the flight envelope. Of course, other control objectives could be used instead. The resulting linearised system consists of the uncoupled single degree of freedom subsystems referred to previously in (5). Two cases are considered: those of 3 inputs and 3 outputs (denoted 3I3O) and of 2 inputs and 2 outputs (denoted 2I2O) which are representative of the analysis in Section 2.

4.3. Feedback Linearisation: 3I3O Case

The poles of the system are assigned for the uncoupled linearised system in (5) (by cancellation of the entire open-loop dynamics) and with feedback law (6), so that the natural frequencies and damping values are given by 𝜔₁=0.93Hz,𝜉₁𝜔=0.01,₂=2.9Hz,𝜉₂𝜔=0.01,₃=4.95Hz,𝜉₃=0.01, (22) at an air speed of 80 ms⁻¹. The first mode is predominantly wing bending, the second is pylon-engine deformation and the third is mainly the twisting motion of the wing. With initial conditions the same as the uncontrolled case (Section 4.2), namely 𝜁₁ = 0.333 mm, 𝜒₂ = 0.00333 rad, and 𝜗_pe = 0.05 rad and in the absence of any nonlinear parameter error the poles of the linearised system are assigned exactly. The system response now decays to zero as shown in Figure 5, whereas the uncontrolled nonlinear system exhibited a limit cycle. The required control surface and actuator inputs are shown in Figure 6, where it can be seen that the input magnitudes are feasible in practice.

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Figure 5 

Feedback linearisation: response at 80 ms−1 (assumed-modes coordinates).

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Figure 6 

Control surface deflection angles and actuator torques for exact feedback linearization.

Now, suppose the linearising control is chosen to cancel the nonlinearity alone, in which case the linearised equations generally remain coupled, resulting in a greater number of control gains in (8) and greater control flexibility as explained in Section 2.1. One is then able to assign eigenvectors corresponding to the assigned poles. The same pole-placement above is now implemented, with the following respective eigenvectors assigned: 𝐯₁=110^𝑇,𝐯₂=1−10^𝑇,𝐯₃=001^𝑇. (23) Thus, it is desired to deliberately couple the bending and torsion in the first two modes, whilst leaving the pylon-engine mode decoupled. The resulting closed-loop responses and control inputs are shown in Figures 7 and 8, respectively.

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Figure 7 

Feedback linearisation with cancellation of nonlinearity only and with eigenvector assignment: response at 80 ms−1 (assumed-modes coordinates).

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Figure 8 

Control surface deflection angles and actuator torques for exact feedback linearisation: cancellation of nonlinearity only, with assignment of eigenvectors.

It is evident from the first two plots in Figure 7 that a mixing of modes has occurred, as these two responses now contain multiple harmonics. The third coordinate, however, remains unchanged. This is expected, as the eigenvector assignment commanded that the pylon-engine mode should remain uncoupled. It can also be seen from the first two plots in Figure 8 that the required control surface deflections have increased; it is easier to control the uncoupled modes than the deliberately coupled ones. Thus, in the case of the three degree of freedom wing-pylon-engine system there is some merit in using feedback linearisation to cancel the entire open loop dynamics despite the loss of control flexibility. For the remainder of this simulation and those appearing later on, only the case where the entire open loop dynamics are cancelled will be considered.

When a +40% error in 𝐾_nl is incorporated, and the closed-loop response is simulated based on the above feedback parameters and with the same initial conditions, an unstable response sets in from the very beginning, as seen in Figure 9. The nonlinear spring, connected between wing twist and rotation of the pylon-engine assembly, is not capable of suppressing the response (in the assumed-modes coordinates) into a limit cycle. The problem of error in the nonlinear parameter is of practical importance, since it is seldom possible to be precise in estimating the magnitude of nonlinear terms. This problem is addressed in the sequel.

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Figure 9 

Feedback-linearisation with nonlinear parameter error: response at 80 ms−1.

4.4. Feedback Linearisation: 2I2O Case

The three degree of freedom system is now considered to be instrumented with only two inputs, trailing edge control-surface (flap) angles 𝛽₁ and 𝛽₂, and two outputs, discrete wing displacement 𝜁₁ and twist 𝜒₂. In this case there is no control torque 𝑇_pe. The measurement of 𝜗_pe is necessary, but only for determination of 𝐟(𝐳,𝐳̇). The zero dynamics of the 2I2O aeroservoelastic problem are given in Appendix B and stabilised by the introduction of structural damping 𝐃_mod=10⁵,2.050.01030.002200.01030.0767−0.002820.00220−0.002820.00356 (24) which corresponds to 1% of modal damping. The introduction of structural damping increases the flutter speed of the system from 77.6 ms⁻¹ to 95.05 ms⁻¹, and thus the simulation is carried out above the new speed, at 97.5 ms⁻¹. Pole-placement is specified such that 𝜔₁=0.93Hz,𝜉₁𝜔=0.01,₂=4.95Hz,𝜉₂=0.01. (25) As before, feedback linearisation successfully places the poles to the desired values and the transient response shown in Figure 10 is obtained.

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Figure 10 

Feedback-linearisation: response at 97.5 ms−1 (assumed-modes coordinates).

It is evident from the third subplot in Figure 10 that the coordinate 𝑞_pe eventually stabilises because the zero dynamics of the system are stable. When a +40% error in 𝐾_nl is incorporated and the closed-loop response is simulated based on the same feedback parameters instability does not occur (as in the 3I3O case) but a degradation in performance is observed in Figure 11 where it is seen that the control fails to eliminate an LCO.

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Figure 11 

Feedback-linearisation with nonlinear parameter error: response at 97.5 ms−1.

5. Treatment of Nonlinear Parameter Error: Adaptive Feedback Linearisation

The treatment of error in the numerical value of a nonlinear parameter is carried out using adaptive feedback linearisation, which makes use of Lyapunov stability theory. It results in an updating rule for the erroneous parameter, which evolves with time. This approach ensures that the system remains stable; the original control objectives might be compromised because of evolution of the nonlinear parameter estimate. This might be acceptable in many cases depending upon engineering judgment.

An erroneous estimate, 𝐟^′_nl, is now assumed in place of the true nonlinear force vector, 𝐟_nl. The corresponding erroneous input is then determined, by comparison with (2) and (4), as 𝐮^′=𝐆⁻¹𝐮−𝐟^′(𝐱),𝐟^′(𝐱)=Ψ𝐱+Φ𝐱̇+Ω𝐟^′_nl, (26) and substituting (26) into (2) leads to 𝐱̈=𝐮+Ω𝜺,𝜺=𝐟_nl−𝐟^′_nl. (27) Recall (6) from Section 2.1 earlier. This may be combined into a single equation 𝐱𝐱̇𝐮=Γ,Γ∈ℝ^𝑛×2𝑛. (28) The artificial inputs may be expressed in terms of the state variables, 𝐱, by using the above equation together with the coordinate transformation 𝐳=𝐓_𝑧𝑥𝐱 (29) defined in (10) and (11).

Thus, 𝐮=Γ^′𝐱𝐱̇=Γ^′₁Γ^′₂𝐱𝐱̇,Γ^′=Γ_𝑧𝐓_𝑧𝑥𝟎𝟎𝐓_𝑧𝑥, (30) and the second-order equation of the closed-loop system is then cast as 𝐱̈=Γ^′₁𝐱+Γ^′₂𝐱̇+Ω𝜺. (31) Equation (31) shows that the nonlinear parameter error results in an input to the closed-loop system. It may be readily shown, for the 3I3O system, that Γ^′_1,𝑧𝜍=−¹𝜍₂𝜍₂,Γ^′_2,𝑧𝜈=−¹𝜈₂𝜈₂. (32) The system is then represented in state-space as 𝐳̇‌=Α𝐳+Δ𝐾_𝑇𝜗³_pe𝐁Ω𝐓_(3,∶)^𝑇,Γ𝚨=𝟎𝐈_′1,𝑧Γ^′_2,𝑧,𝟎𝐈𝐁=,𝐳𝐳̇𝐳=, (33) where 𝐓 is defined in (18) and where the erroneous nonlinear parameter is given by 𝐾^′_𝑇=𝐾_𝑇+Δ𝐾_𝑇,𝜺=Δ𝐾_𝑇𝜗³_peΩ𝐓_()3,∶^𝑇. (34)(35) It is conceivable that this additional (unknown) input may potentially destabilise the system, or at least degrade the control performance. This possibility may be eliminated by accounting for the nonlinearity errors using an adaptive algorithm. Such an algorithm seeks to guarantee asymptotic stability of the closed-loop response.

A scalar Lyapunov function 𝑉 in 𝐳, Δ𝐾_𝑇 may be defined, such that asymptotic stability of the closed-loop system is guaranteed by ensuring that 𝑉>0 and its time-derivative 𝑉̇<0 [15]. Such a function may be used as a basis for computing a parameter update law. The Lyapunov function, 𝑉𝐳,Δ𝐾_𝑇=𝐳^𝑇𝐏𝐳+Δ𝐾²_𝑇>0 (36) is considered, where 𝐏=𝐏^𝑇≻0. Differentiating the above equation with respect to time, one obtains 𝑉̇𝐳,Δ𝐾_𝑇=𝐳̇^𝑇𝐏+𝐏^𝑇𝐳+2Δ𝐾_𝑇dΔ𝐾_𝑇.d𝑡 (37) Now, by combining (33) and (37) and expanding and rearranging 𝑉̇𝐳,Δ𝐾_𝑇=𝐳^𝑇𝐀^𝑇𝐏+𝐏𝐀𝐳+2Δ𝐾_𝑇dΔ𝐾_𝑇d𝑡+𝜗³_pe𝐓_(3,∶)Ω^𝑇𝐁^𝑇𝐏𝐳, (38) it becomes evident that dΔ𝐾_𝑇d𝑡=−𝜗³_pe𝐓_(3,∶)Ω^𝑇𝐁^𝑇𝐏𝐳 (39) eliminates the second term on the right-hand-side of (38), resulting in 𝑉̇𝐳,Δ𝐾_𝑇=𝐳^𝑇𝐀^𝑇𝐏+𝐏𝐀𝐳. (40) Now, from the definition of Δ𝐾_𝑇 in (34) and knowing that the actual nonlinear parameter 𝐾_𝑇 is constant, it is seen that dΔ𝐾_𝑇=d𝐾d𝑡^𝑇−d𝐾d𝑡^′𝑇d𝐾d𝑡=−^′𝑇.d𝑡 (41) Combining (41) and (39), an update law for 𝐾^′_𝑇 corresponding to the latter equation is obtained as d𝐾^′_𝑇d𝑡=𝜗³_pe𝐓_(3,∶)Ω^𝑇𝐁^𝑇𝐏𝐳. (42) Thus, the initially assumed value 𝐾^′_𝑇 of the nonlinear parameter is continually updated at each time step. Effectively, this increases the dimension of the state vector by 1, as 𝐾^′_𝑇 becomes part of the system state. Now, choosing 𝐏 such that for arbitrary 𝐐≻0, 𝐐=−𝐀^𝑇𝐏+𝐏𝐀, (43) it becomes clear that by substituting the above equation into (40), 𝑉̇ is rendered negative-definite, as required.

For the 2I2O case, an identical approach may be followed to obtain similar expressions. The definition of 𝐳 changes to 𝐳={𝐳_(1∶2)𝐳̇_(1∶2)}^𝑇, and the parameter update rate becomes d𝐾^′_𝑇d𝑡=𝜗³_pe𝐓_(3,∶)Ω^𝑇_()1∶2,∶𝐁^𝑇𝐏𝐳. (44) The matrices 𝐀 and 𝐁 become 𝚨=𝟎𝐈_2×2Γ^′_{1,𝑧(1∶2,1∶2)}Γ^′_{2,𝑧(1∶2,1∶2)},𝟎𝐈𝐁=_2×2. (45) An important consideration in the 2I2O case is the asymptotic stability of the zero dynamics, considered in Appendix B, which is required for the adaptive scheme to work. For the particular 2I2O configuration considered, the presence of structural damping ensures stability of the zero dynamics, thus enabling application of the adaptive method.

Example: Stabilisation of the Aeroelastic System Using Adaptive Feedback Linearisation. The instability shown in Figure 9 may be avoided altogether by implementing the adaptive controller described in Section 5 above. In the 3I3O case, the resulting controlled response, for the same initial conditions, is shown in Figure 12.

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Figure 12 

3I3O system with adaptive feedback linearisation: response at 80 ms−1.

A comparison of Figure 12 with Figure 5 shows that the controlled responses are similar, but not identical. Although adaptive feedback linearisation destroys the original pole-placement, it can be seen that the controlled response is stable. In fact, the adaptive controller successfully drives the responses to zero for values of |𝜗_pe|_max (set as an initial condition) up to around 0.0735 rad. The required control in this case is accomplished through achievable control surface deflection angles (≈11°) and actuator torque magnitude (55 kNm).

In the case of the 2I2O system, the system response with adaptation shown in Figure 13 should be compared to that in Figure 11, for feedback linearisation without adaptation and the same error in the cubic stiffness parameter. The degradation of the system to LCO is completely removed in Figure 13.

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Figure 13 

2I2O system with feedback linearisation: response at 97.5 ms−1.

It can also be seen that the controlled responses, as well as the response of the uncontrolled coordinate converge to zero when the adaptive controller is implemented. In fact, this convergence takes place rapidly. When converted to the physical domain, the resulting magnitudes of the coordinates occur within acceptable limits (maximum values of 0.33 mm, 0.24°, 2.92° for 𝜁₁,𝜒₂,𝜗_pe, resp.). As before, it is found that the control surface deflections required to achieve the responses shown in Figure 13 are feasible (𝛽₁, 𝛽₂ ≈ 30°).

6. Conclusions

Nonlinear systems are ubiquitous in vibrations engineering and aeroelasticity, but the analysis is mathematically intricate and complicated. The paper presents the feedback linearisation methodology, whereby a nonlinear system is rendered linear by means of active control. Having neutralised the nonlinearity, the system may be treated using well-understood linear analysis methods such as modal decomposition, which generally cannot be applied to nonlinear systems directly. The technique is formulated using the second-order representation of elastomechanical and aeroelastic systems with stiffness nonlinearity, familiar to the vibrations community. This has certain advantages over the conventional state-space formulation in that repeated output differentiation, usually described using the Lie algebra notation, is unnecessary. The purpose of the linearising controller is to cancel the nonlinearity completely and therefore it results in a truly linear system, rather than linearisation about an operating point limited to small perturbations, or quasi-linearisation as with describing functions. The controller may be designed to cancel the nonlinearity only or to cancel the complete open-loop dynamics. In the former case there is shown to be greater control flexibility, but the latter case is found to have some merit in the particular example considered of a flexible wing-pylon-engine system with decoupled bending, torsional, and pylon-engine modes.

Exact feedback linearisation requires knowledge of the nonlinearity and when every degree of freedom of the system is available for measurement (and actuation) and then linearisation may be achieved completely. When fewer than the full set of degrees of freedom is available for measurement and actuation, then the system can generally be partitioned into independent linear and nonlinear subsystems, with the dimension of the latter being the difference between the number of degrees of freedom and the number of sensors and actuators. If the nonlinear subsystem is stable, then the dynamics of the linear subsystem may be controlled as required. The problem of an imprecisely known nonlinear term was addressed using adaptive feedback linearisation, resulting in a parameter update rule that evolves in time to ensure stability. It requires an additional state variable to do this and is likely to be more expensive than using feedback linearisation without adaptivity.

Feedback linearisation techniques were illustrated using a three degree of freedom aeroelastic model consisting of a flexible wing and a rigid pylon-engine system attached to the wing via a nonlinear torsional spring with pole placement of the fully linearised and partially linearised system corresponding to the measurement (and actuation) at three degrees of freedom (3I3O) and two degrees of freedom (2I2O) respectively. The parameters of the system were chosen carefully to represent a real aircraft. Adaptive linearisation was applied successfully to compensate for a nonlinear parameter error in the case of 3I3O, and in the 2I2O case the nonlinear subsystem was found to be stable. The magnitudes of the aileron deflection angles were physically realisable.

Appendices

A. Control Forces and Nonlinear Spring Forces

Control forces are applied to the wing-pylon-engine system by means of control surfaces. It is assumed in this work that two control surfaces (different from [14]) are available, the first (closest to the wing root) spanning 85% of the length of the wing and the second spanning the remaining length (the contribution of the control surfaces to the dynamics of the overall system is neglected). The widths of the first and second control surfaces are assumed to be 20% and 33.33% of the chord length, respectively. These particular dimensions have been chosen so as to optimise the distribution of work performed by each control surface. In addition to the ailerons, it is assumed that a separate actuator is available to apply a torque 𝑇_pe directly on the engine rotational degree of freedom. The forcing vector is found to be 𝐟_𝑐=−16𝑟𝛼̃³𝑎_𝐶,1𝑠_𝑤−16𝑟̃1−𝛼³𝑎_𝐶,2𝑠_𝑤014𝑟𝛼̃²𝑏_𝐶,1𝑐_𝑤14𝑟̃1−𝛼²𝑏_𝐶,2𝑐_𝑤−𝑦₂×𝛽001₁𝛽₂𝑇_pe=𝐁𝐮, (A.1) where 𝑟̃=𝜌𝑉²𝑐_𝑤𝑠²_𝑤, 𝛼=0.85 and each surface will have its own deflection angle 𝛽₁, 𝛽₂ and set of aerodynamic parameters 𝑎_𝐶, 𝑏_𝐶, which are the rates of change of lift coefficient and moment coefficient, respectively, with respect to control surface deflection angle.

A cubic hardening nonlinearity is assumed in the torsional spring connecting the pylon-engine to the wing. The nonlinear force developed in the spring may be expressed as 𝑓_nl=𝐾_𝑇𝜗³_pe, (A.2) where 𝐾_𝑇 is the stiffness coefficient of the cubic nonlinearity. The overall nonlinear force vector then takes the form 𝐟_nl=𝐓^𝑇00𝑓_nl=0−𝑦₂𝑓_nl𝑓_nl, (A.3) with 𝐓 defined in (18).

B. Zero Dynamics Expressions (2I2O Case)

In the 2I2O configuration and using (11), 𝑧₁𝑧₂𝑧₃=𝜎100010₁𝜎₂𝜎₃𝑥₁𝑥₂𝑥₃, (B.1) where 𝜎₁=𝑔₂₁𝑔₃₂−𝑔₃₁𝑔₂₂,𝜎₂=𝑔₃₁𝑔₁₂−𝑔₁₁𝑔₃₂,𝜎₃=𝑔₁₁𝑔₂₂−𝑔₂₁𝑔₁₂, (B.2) and 𝑔_𝑖𝑗 denotes the 𝑖𝑗th term of the matrix 𝐆. Then, by inverting the transformation matrix 𝐓_𝑧𝑥, the following expressions are obtained: 𝑥₁=𝑧₁,𝑥₂=𝑧₂,𝑥₃𝜎=−¹𝜎₃𝑧₁−𝜎₂𝜎₃𝑧₂+1𝜎₃𝑧₃. (B.3) Then, from (15) and differentiating the bottom row of (B.1) it is found that 𝑧̈₃=𝝈^𝑇𝐟(𝐳,𝐳̇),𝝈^𝑇=𝜎₁𝜎₂𝜎₃, (B.4) where 𝐟(𝐳,𝐳̇)=Φ𝑧̇₁𝑧̇₂−𝜎₁𝜎₃𝑧̇₁−𝜎₂𝜎₃𝑧̇₂+1𝜎₃𝑧̇₃^𝑇+Ψ𝑧₁𝑧₂−𝜎₁𝜎₃𝑧₁−𝜎₂𝜎₃𝑧₂+1𝜎₃𝑧₃^𝑇+Ω𝐟_nl(𝐳),𝐟_nl(𝐳)=𝐾_𝑇−𝑦₂𝑧₂+−𝜎₁𝜎₃𝑧₁−𝜎₂𝜎₃𝑧₂+1𝜎₃𝑧₃³×𝐓_(3,∶)^𝑇. (B.5) Evidently, the internal dynamics expressions in (B.4) are nonlinear. Now, the zero dynamics may be obtained by setting to zero the coordinates corresponding to the external dynamics (i.e. the partially linearised system). In this case, the external coordinates are 𝑧₁, 𝑧₂, 𝑧̇₁, 𝑧̇₂ so that the zero dynamics are given by 𝑧̈_3,𝑧𝑑=1𝜎₃𝝈^𝑇Ψ_(∶,3)𝑧_3,𝑧𝑑+1𝜎₃𝝈^𝑇Φ_(∶,3)𝑧̇_3,𝑧𝑑+1𝜎³₃𝐾_𝑇𝝈^𝑇Ω𝐓_(3,∶)^𝑇𝑧³_3,𝑧𝑑, (B.6) where the subscript “𝑧𝑑” signifies that the coordinates are specified under zero dynamics conditions.

Zero Dynamics Equilibrium Point Analysis. To obtain the equilibrium points of the zero dynamics, it is necessary to set 𝑧̈_3,𝑧𝑑=𝑧̇_3,𝑧𝑑=0. Equation (B.6) then produces 𝝈^𝑇Ψ_(∶,3)+𝐾_𝑇𝝈^𝑇Ω𝐓_(3,∶)^𝑇𝑧²_3,𝑧𝑑𝑧_3,𝑧𝑑=0, (B.7) which provides the two solutions 𝑧_3,𝑧𝑑=0,𝑧_3,𝑧𝑑=±−𝝈^𝑇Ψ_(∶,3)𝐾_𝑇𝝈^𝑇Ω𝐓_(3,∶)^𝑇. (B.8) It is found that the term inside the square root is always negative, and therefore the second solution for 𝑧_3,𝑧𝑑 is inadmissible. Thus, the only possible equilibrium point of the zero dynamics is the trivial solution 𝑧₃𝑧̇_3𝑧𝑑=00. (B.9) The nature of the above equilibrium point may be determined by examining the eigenvalues of (B.6), evaluated at the equilibrium point, 𝑠²−1𝜎₃𝝈^𝑇Φ_(∶,3)1𝑠−𝜎₃𝝈^𝑇Ψ_(∶,3)−1𝜎³₃𝐾_𝑇𝝈^𝑇Ω𝐓_(3,∶)^𝑇𝑧²_3,𝑧𝑑𝑧_3,𝑧𝑑𝑧̇=0_3,𝑧𝑑=0=0, (B.10) or 𝑠²−1𝜎₃𝝈^𝑇Φ_(∶,3)1𝑠−𝜎₃𝝈^𝑇Ψ_(∶,3)=0. (B.11) The roots are found to be complex with negative real parts, thus revealing the type of equilibrium point to be a stable focus.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors wish to acknowledge the support of EPSRC Grant no. EP/J004987/1 on Nonlinear Active Vibration Suppression in Aeroelasticity.