Nonlinear Dyn DOI 10.1007/s11071-015-2209-8

ORIGINAL PAPER

Robust backstepping decentralized tracking control for a 3-DOF helicopter Yao Yu · Geng Lu · Changyin Sun · Hao Liu

Received: 7 May 2014 / Accepted: 11 June 2015 © Springer Science+Business Media Dordrecht 2015

Abstract A three-degree-of-freedom helicopter attitude tracking control system can be described as a multi-input multi-output strict-feedback form system with unknown parameters, bounded disturbances, and nonlinear uncertain cross-couplings. Both normbounded and nonlinear uncertainties are discussed. The whole nonlinear system is divided into a nominal disturbance-free system and an equivalent disturbance part. A robust control method based on signal compensation technique and backstepping decentralized control is proposed. Robust practical tracking property of closed-loop system is proven, and the tracking error can be made as small as desired with expected convergence rate. Experimental results demonstrate the improved tracking performance of the attitude tracking control Y. Yu · C. Sun (B) School of Automation and Electrical Engineering, University of Science and Technology Beijing, Beijing 100083, People’s Republic of China e-mail: [email protected] Y. Yu e-mail: [email protected] Y. Yu · G. Lu Department of Automation, TNList, Tsinghua University, Beijing 100084, People’s Republic of China G. Lu e-mail: [email protected] H. Liu School of Astronautics, Beihang University, Beijing 100191, People’s Republic of China e-mail: [email protected]

system with different types of reference signals, such as step signals, ramp signals, and nonstationary sinusoidal signals. Keywords 3-DOF helicopter · Decentralized control · Backstepping · Robust control · Uncertain system

1 Introduction Unmanned helicopters play very important roles in military and civil applications, especially in restricted areas and dangerous environments. The main difficulties in designing tracking controllers for helicopters arise from the high nonlinearities, strong crosscouplings, and large uncertainty of dynamics [1,2]. To date, researchers have paid significant attention to the control problem of autonomous unmanned helicopters. Various control algorithms are investigated, from classical control to advanced control, for the motion control of helicopters. Robust controllers are used widely to take into account of uncertainties and couplings [3– 7]. Dynamical sliding-mode control approach has been often adopted for plant subjects to uncertainties [8,9]. Adaptive theory is also applied to overcome an uncertainty problem [10–12]. In [9–12], experimental results of laboratory helicopters have been presented. Furthermore, formation control problem of multiple unmanned aerial vehicle (UAV) systems is considered [13,14].

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In this paper, a robust decentralized control method is applied to an actual helicopter test bed produced by Quanser Consulting Inc., for laboratory use. Since it possesses many real-life peculiarities such as unmodeled dynamics, nonlinearity, uncertainty, couplings, disturbances, and measurement noise, the device of a 3DOF helicopter is a typical and challenging experimental platform and is widely used for evaluating the effectiveness of the proposed control strategies. For example, an adaptive output feedback control method was applied to a single-input single-output (SISO) model which involves only the pitch angle of the laboratory helicopter [10]. A synchronized trajectory-tracking control strategy was proposed for only elevation axis of multiple 3-DOF laboratory helicopters [15]. Later, MIMO models of the 3-DOF laboratory helicopters were studied. In [9], robust regulation for helicopters with external perturbations was addressed. In [11], an adaptive controller was implemented to make the travel and elevation angles track the outputs of a reference model. A passification-based adaptive control law was designed for travel and elevation angles [16]. In [17], a neural network-based approximate predictive control was experimentally tested by motion control of elevation and pitch angles. In [18], a robust output feedback attitude controller of a 3-DOF helicopter with multioperation points was proposed. In [19], two robust control strategies for a 3-DOF helicopter via sliding-mode techniques were presented: both quasi-continuous controllers along with a sliding-mode differentiator and classical proportional–integral–derivative (PID) controllers in combination with a second-order slidingmode observer were presented. In [20], input and state constraints were systematically accounted for within the control design procedure. In [21], a robust control method for attitude regulation of the laboratory helicopter was considered. In [22], robust linear quadratic regulator (LQR) attitude control for the laboratory helicopter was investigated. In [23], a robust H-infinity attitude controller was proposed for the laboratory helicopter. In [24,25], the robust position control problem for the laboratory helicopter was discussed. However, due to the complexity of the MIMO systems, it is challenging to realize highly dynamic flight maneuvers with a variety of types of reference signals. Moreover, few researches focused on decentralized tracking control laws for MIMO models of helicopters. In the last 20 years, backstepping control has become one of the most popular control methods for some

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special classes of nonlinear systems, since it provides a systematic procedure for designing a controller by a step-by-step recursive algorithm. For better control performance, different control strategies are combined together by taking advantage of their strengths, respectively [26–28]. Combined with backstepping control strategy, many effective methods have been proposed for stability analysis and controller design. However, backstepping control has the drawback of the phenomenon of “explosion of complexity” in the control law due to repeated differentiations of the virtual control functions. Another important issue associated with the control of nonlinear systems concerns convergence rate and steady-state tracking error bounds. A new robust adaptive controller for MIMO feedback linearizable nonlinear systems, capable of guaranteeing a prescribed performance, was developed in [29]. By prescribed performance, the tracking error should be made as small as desired, with a maximum overshoot less than a sufficiently small prespecified constant, exhibiting convergence rate no less than a prespecified value. Visualizing the prescribed performance characteristics as tracking error constraints, the key idea of the technique in [29] was to provide an error transformation function that transforms the original “constrained” nonlinear system into an equivalent “unconstrained” one. Stabilizing the equivalent “unconstrained” system was sufficient to achieve prescribed performance guarantees. However, a tangent hyperbolic function which was generally used as the transformation function, combined with prescribed smooth function to transform the tracking error, made the controller design very complex. Furthermore, this technique had a singularity problem for a certain prescribed performance condition [29,30]. In this paper, the results have the following features: (1) We are devoted to linear decentralized backstepping tracking control strategies for MIMO models with unknown parameters, bounded disturbances, nonlinear uncertain cross-couplings, and measurement noise. (2) By means of Lagrange formalism, the 3-DOF laboratory helicopter’s uncertainties, which are mainly caused by parameter perturbation and external disturbance, are norm-bounded [9,11,16,21–24]. In this paper, aerodynamic influence with nonlinear bounds is further considered. (3) Furthermore, compared to previous studies on the laboratory helicopters [16,18,19,21–24], better

Robust backstepping decentralized tracking control

performance for the closed-loop system, i.e., transient and steady-state properties, is explored. (4) The same linear decentralized controller is applied to the 3-DOF helicopter with different types of reference signals, such as step signals, ramp signals, and nonstationary sinusoidal signals. The experimental results are presented. (5) The tracking error can be made as small as desired, with convergence rate no less than a given value. For the 3-DOF helicopter system, a design method in combination with the signal compensation technique and the backstepping method is applied. Fig. 1 3-DOF helicopter system

The basic idea of signal compensation method was first introduced in [31] for linear time-invariant systems with parameter perturbations. Then, we utilized it to deal with robust output tracking problem for SISO nonlinear systems [32]. Now, it is further developed to treat the control problem for MIMO systems with norm-bounded uncertainties [33,34]. By this method, the controller design is performed in a step-by-step procedure. For each subsystem, a robust controller is composed of a nominal controller for the nominal subsystem and a robust compensator to restrain the uncertainties involved in the subsystem. The trait of the proposed method is that the designed controllers are linear time-invariant, and for each local subsystem, only local information is used in the design procedure. So the controller can be easily realized, and the explosion of terms, which generally occurs in the backstepping control law, is avoided. The organization of this paper is as follows. In Sect. 2, a helicopter laboratory experimental setup and its detailed mathematic model, as well as a formulation of the control task, are introduced. In Sect. 3, a robust backstepping decentralized tracking controller is proposed. The closed-loop robust control properties are proven in Sect. 4. Experimental results are presented in Sect. 5. Finally, the brief closes with a short conclusion in Sect. 6.

2 Problem description 2.1 Helicopter benchmark The helicopter setup is manufactured by Quanser Consulting Inc., [35]. Figure 1 shows a photograph of the helicopter equipment with 3-DOF: the elevation, pitch,

and travel. Figure 2 gives a detailed schematic diagram of the acting forces on the system. The 3-DOF helicopter consists of a base upon which a long arm is mounted. The arm carries a counterweight on one end and the helicopter body on the other. The arm can tilt on an elevation axis as well as swivel on a travel axis. Two DC motors with propellers are installed on the helicopter body, which is free to roll on a pitch axis. Three encoders fixed on these axes measure the elevation, pitch, and travel angles of the helicopter body. The voltages to the front and back motors are control inputs to generate thrusts acting on the helicopter body. A positive voltage to either motor causes an elevation of the body. When the force generated by the front motor is higher than the force generated by the back motor, the helicopter body will also pitch up. If the body pitches, the thrust vectors result in a travel of the body as well [35].

2.2 Nomenclature Following notation is used through the paper(see Fig. 2). (t), p(t), γ (t)—Elevation angle, pitch angle, travel angle, deg; 0 —Initial angle between helicopter arm and its base, deg; u f (t), u b (t)—Control voltages for front and back motors, V; J1 , J2 , J3 —Moments of inertia about elevation, pitch, and travel axis, kg · m2 ; K f —Propeller force-thrust coefficient, N/V; m—Effective mass of helicopter, kg;

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will be denoted as ϕi (t). ϕ1 (t), ϕ2 (t), ϕ3 (t) are nonlinear uncertainties, which involve aerodynamic torque and couplings between control channels. di (t)(i = 1, 2, 3) are the disturbances describing the influence of external environment. Since only two control inputs are available for controlling 3-DOF, the helicopter represents an underactuated mechanical system. This gives us the freedom to select two outputs only. Here, the elevation and pitch motions of the helicopter are considered. The control system can be described as a two-input two-output nonlinear uncertain system, whose state-space model can be rewritten as: x˙11 = x12 x˙12 = b1 (t)u 1 + φ1 (t) Fig. 2 Schematic of the 3-DOF helicopter [21,22]

x˙21 = x22

g—Gravity constant, 9.8m/s2 ; L a —Distance from the travel axis to the helicopter body, m; L h —Distance from the pitch axis to each motor, m; Remark 1 With three incremental encoders, only relative angles (t), p(t), γ (t) can be obtained. So the initial values of (t), p(t), γ (t) are always 0◦ . From Fig.1, one has that, without control inputs, the helicopter body with two propellers in a symmetrical structure is parallel to the horizontal plane, and there exists initial angle 0 between helicopter arm and its base as Fig. 2 shows.

2.3 Mathematic model of the 3-DOF helicopter The mathematic model of the helicopter laboratory experimental setup can be derived by means of Lagrange formalism. Based on the diagram Fig. 2, the dynamic equations for 3-DOF such as elevation, pitch, and travel of a helicopter can be obtained as follows (see also in [15,16])   J1 ¨ = K f L a cos p u f + u b − mgL a sin( + 0 ) + ϕ1 (, ˙ , p, p, ˙ γ , γ˙ ) + d1 (t)   J2 p¨ = K f L h u f − u b +ϕ2 (, , ˙ p, p, ˙ γ , γ˙ ) + d2 (t)   J3 γ¨ = K f L a sin p u f + u b + ϕ3 (, ˙ , p, p, ˙ γ , γ˙ ) + d3 (t)

(1)

where 0 is the initial angle between helicopter arm and its base. For simplicity of statement, ϕi (, , ˙ p, p, ˙ γ , γ˙ )

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x˙22 = b2 (t)u 2 + φ2 (t) y1 = x11 y2 = x21

(2)

where x T = [x11 x12 x21 x22 ] = [ ˙ p p] ˙ is the state T vector, u = [u 1 u 2 ] = [u f + u b u f − u b ] the control input vector, and yT = [y1 y2 ] = [ p] the output vector. The values of bi (i = 1, 2) are given by K f La K f Lh cos x21 , and b2 = . The uncertainb1 = J1 J2 ties φ1 (t), φ2 (t) of the system, which include the aerodynamic forces, the pitch and elevation channel couplings, the parameter perturbations, and time-varying disturbances, are defined as follows:   1 mgb1 ϕ1 (t) − sin(x11 + 0 ) + d1 (t) φ1 (t) = J1 Kf 1 (3) φ2 (t) = [ϕ2 (t) + d2 (t)] J2 Remark2 The pitch angle is mechanically limited π π during experiment. Without loss of within − , 2 2  π π generality, assume that x21 ∈ − + θ0 , − θ0 , 2 2 with θ0 a positive constant. Therefore, sin θ0 ≤ cos x21 ≤ 1. In this paper, the following assumptions are introduced. They are reasonable assumptions in this kind of systems to consider bounded uncertain parameters, bounded external disturbances, and bounded reference inputs, as well as linear norm-bounded uncertainties.

Robust backstepping decentralized tracking control

Assumption A (A1) According to Remark 2 and definition of bi (t)(i = 1, 2), the lower bounds of uncertain time-varying parameters bi (t) are existent, so there are known positive constants bi0 (i = 1, 2), such that bi (t) > bi0 , i = 1, 2. (A2) The external disturbance d(t) is bounded, and there exists a known and positive constant η1 such that d(t)∞ ≤ η1 . (A3) The reference input r(t) belongs to C 1 , and there exists known and positive constants η2 , η3 , such that r(t)∞ ≤ η2 and ˙r (t)∞ ≤ η3 . (A4) There are known positive constants ξi j ( j = 1, 2, 3, 4) and nonnegative-valued functions ςϕi (t), such that |φi (t)| ≤ ξi1 |x11 (t)| + ξi2 |x12 (t)| + ξi3 |x21 (t)| + ξi4 |x22 (t)| + ςϕi (t),

i = 1, 2 (4)

Remark 3 For this kind of simplified experimental setup, according to Lagrange formalism, the nonlinear uncertainty, which is mainly caused by parameter perturbation, are norm-bounded [9,11,16,21–24]. 2.4 Control task Currently, most previous works considered full state feedback strategies for the motion control problem of helicopters. In this paper, we aim at designing a linear decentralized controller to have the attitude x11 , x21 of the helicopter track the reference signals, denoted by yd1 , yd2 , which are given by the following reference models: y˙di (t) = −αi ydi (t) + βi ri (t), i = 1, 2

(5)

where αi and βi (i = 1, 2) are given positive constants, and ri (t)(i = 1, 2) are given reference inputs. In the experiment, a wide range of types of timevarying reference inputs, such as step signals, ramp signals, and nonstationary sinusoidal signals, are considered.

3 Control design procedure In this section, a robust design method of robust decentralized controllers based on signal compensation is proposed. For each subsystem, the tracking control input u i (t) consists of a nominal controller and a robust

compensator. A nominal controller is first designed for the disturbance-free helicopter model, and then, a robust compensator is introduced to attenuate the effects of uncertainties and disturbance involved in the helicopter. In the following, we will give the backstepping design procedure for the ith subsystem. Step 1: Define the tracking error of the ith subsystem z i1 = xi1 − ydi

(6)

(2) and (5) yield the error equation z˙ i1 = xi2 − y˙di = xi2 + αi ydi − βi ri

(7)

To stabilize the subsystem (7), a virtual controller xˆi2 (t) is constructed as xˆi2 = −ai1 z i1 + y˙di = −ai1 z i1 − αi ydi + βi ri

(8)

Define z i2 = xi2 − xˆi2 , it follows that z˙ i1 = −ai1 z i1 + z i2

(9)

Step 2: By the definition of z i2 and from Eqs. (2), (7), and (9), one has z˙ i2 = bi u i + φ˜ i (10) where φ˜ i = φi − y¨di + ai1 (−ai1 z i1 + z i2 )

(11)

The robust control input u i consists of two parts: a nominal control input u i0 designed to stabilize the nominal subsystem z˙ i2 = bi0 u i0 , and a robust compensating input wi to restrain the effects of system uncertainties. The control input u i (t) is constructed as fi wi (12) u i = u i0 + bi0 with the nominal control input ai2 (13) u i0 = − z i2 bi0 where ai2 and f i are positive constants to be determined. Substituting the control input (12) and (13) into the subsystem (10), it follows that (14) z˙ i2 = −ai2 z i2 + f i wi + φˆ i where φˆ i (t) = φ˜ i (t)+(bi −bi0 )u i (t). The robust compensating input wi is designed to restrain the uncertainties φˆ i (t). If f i wi = −φˆ i (t), the desired tracking property of the closed-loop system can be ensured. However, φˆ i (t) includes the nonlinear uncertainties, parameter perturbations, and external disturbances. So the robust compensating input wi is constructed as wi (s) = −Fi (s)φˆ i (s) 1 Fi (s) = (15) s + fi

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If the initial values z(t0 ) and w(t0 ) are zero, then z(t) ≤ ε, w(t) ≤ ε, t ≥ t0

(19)

where t0 is the initial time. In order to prove Theorem 1, the following lemma is needed. Let   bi ˆ − 1 f i wi , i = 1, 2 (20) ψi = φi − bi0

Fig. 3 Block diagram of the robust nonlinear control system

f i wi is a low-pass filter. If f i is positive and sufficiently large, one can expect that f i wi would approximate −φˆ i and restrain the effect of φˆ i to obtain robust property. wi is realizable and can be expressed as  ai2  z i2 (s) wi (s) = − 1 + s 

bi0 bi0 u i0 (s) = + (16) ai2 s Remark 4 In the procedure of controller design, positive constants ai1 , ai2 are chosen so that the tracking errors z i1 (t)(i = 1, 2) have desired convergence speed to zero, if φˆ i (t) = 0(i = 1, 2). Positive constants f i (i = 1, 2) are needed to be determined to compensate the uncertainties φˆ i (t)(i = 1, 2). The whole configuration of this robust nonlinear control scheme is depicted in Fig. 3. 4 Robust property Let z(t)T = [z 11 z 12 z 21 z 22 ] w(t)T = [w1 w2 ]

(17)

Theorem 1 Under Assumption A, the closed-loop system, composed of the 3-DOF helicopter plant (2) and robust decentralized controllers (12), (6), (8), (13), and (16), has global robust control property, that is, for any given bounded initial states and any given positive constant ε, one can find sufficiently large constants f i (i = 1, 2) and constant T ≥ t0 , such that all the states are bounded, and moreover z(t) ≤ ε, w(t) ≤ ε, t ≥ T

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(18)

Lemma 1 For any given positive constant εψ , one can find sufficiently large positive constants f i (i = 1, 2), such that, for ∀t > t0 , one has 2  εψ ψ12 ψ22 z2 + 1 + ≤ (21) f1 f2 2 Proof From the definition of ψi , one has |ψi | = φi − y¨di + ai1 (−ai1 z i1 + z i2 )

 bi − 1 −ai2 z i2 bi0 2 ≤ Σk=1 Σ 2j=1 μik j |z k j | + ςψi (t), i = 1, 2 (22)

where, under the assumption A, the following definitions are set ξ11 + a11 ξ12 2 μ111 = + a11 J1

 b1 ξ12 μ112 = + a11 + a12 −1 J1 b10 ξ13 + a21 ξ14 ξ14 , μ122 = μ121 = J1 J1 ξ21 + a21 ξ22 ξ22 μ211 = , μ212 = J2 J2 ξ23 + a21 ξ24 2 μ221 = + a21 J2

 b2 ξ24 μ222 = + a21 + a22 −1 J2 b20 mgb1 + η1 ςϕ1 (t) + Kf ςψ1 (t) = + 2α1 β1 η2 + β1 η3 J1 

ξ11 β1 ξ13 β2 + + 2ξ12 β1 + 2ξ14 β2 η2 α1 α2 + J1 ςϕ2 (t) + η1 ςψ2 (t) = + 2α2 β2 η2 + β2 η3 J 

1 ξ23 β2 ξ21 β1 + + 2ξ22 β1 + 2ξ24 β2 η2 α1 α2 + J2 (23)

Robust backstepping decentralized tracking control 2   ai1 2 2 z + ai2 z i2 aVi + 2 i1 i=1

 2 +2 + 1 wi ψi a2  



2 bi 2 +2 + 1 f − a i i2 wi a2 bi0 2    a 2 2 z i1 + z i2 aVi + ≤− 2 i=1



 2 ψi 2 2 2 + + 1 ( f i − 2ai2 ) wi − +1 2 2 a a fi

  2 a f − 2a w2 ≤ −aV − z2 − +1 2 a2

 2  εψ 2 2 z + + 1 + 1 (30) a2 2

μik j are positive constants, ςψi (t) positive valued and bounded functions. Therefore, for any given εφ , one can find sufficiently large positive constant f i , such that

=−

μik j ςψi (t) ≤ εφ , ≤ εφ , i = 1, 2; k = 1, 2; j = 1, 2 fi fi (24) then, one has |ψi | √ ≤ εφ [|z 11 | + |z 12 | + |z 21 | + |z 22 |+1], i = 1, 2 fi (25) If we choose √ 20εφ < εψ ,

(26)

then ψ12 ψ2 + 2 ≤ 2εφ2 [|z 11 | + |z 12 | + |z 21 | + |z 22 | + 1]2 f1 f2  2 2 2 2 ≤ 10εφ2 z 11 + z 12 + z 21 + z 22 +1 ≤

2  εψ z2 + 1 2

(27) 

Proof of Theorem 1 Consider the following positive function V = V1 + V2 where

(28)

   

2 z i2 (t) 2 + 1 z i2 (t) wi (t) P Vi = z i1 (t) + wi (t) a2   a = min a11 a12 a21 a22   11 (29) P= 12

The time derivative of V along the trajectories of the closed-loop system satisfies that

 2 2    2 2 ˙ ˙ ai1 z i1 − z i1 z i2 + V = +1 Vi = −2 a2 i=1 i=1   2 + ai2 wi z i2 + wi φˆ i + f i wi2 ai2 z i2

 2   2 ai1 2 2 aVi + z i1 +ai2 z i2 +2 + 1 wi φˆ i ≤− 2 a2 i=1 

  2 2 +2 + 1 − a w f ) ( i i2 i a2

where f = min{ f 1 , f 2 }, a = max{a11 , a12 , a21 , a22 }.  a3 , then from (21) and If f > 2a and εψ ≤ a2 + 2 (30), it follows that

 2 εψ 2 ˙ (31) V ≤ −aV + + 1 a2 2 Therefore, one has V (t) ≤ e If εψ ≤



−a(t−t0 )

V (t0 ) +

 2 εψ 2 +1 2 a 2a

(32)

λa 3 ε +2

(33)

a2

where λ = λmin ( P), then  e−a(t−t0 ) V (t ) ε2 0 2 2 2 + z i1 Σi=1 + z i2 + wi2 ≤ λ 2

(34)

Thus, the inequality above implies that for any bounded initial conditions, one can always find sufficiently large constants f i and a finite time T ≥ t0 such that inequality (18) holds. If the initial values z(t0 ) and w(t0 ) are zero, i.e., V (t0 ) = 0, then z(t) ≤ ε, w(t) ≤ 

ε, t ≥ t0 . Remark 5 The steady-state tracing error ε and convergence rate a can be chosen arbitrarily. If the uncertainties are norm-bounded (Assumption A4), then global tracking transient performance can be achieved.

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Consider the following assumption, which is milder than Assumption A4. Assumption A5 There are nonnegative-valued functions ρi (x11 , x12 , x21 , x22 , t), such that |φi (t)| ≤ ρi (x11 , x12 , x21 , x22 , t)

i = 1, 2

(35)

Remark 6 In our work, aerodynamic torque is also considered for the 3-DOF helicopter model. So the uncertainties may have nonlinear bounds. Because the same robust linear controller is applied, only semiglobal robust practical tracking property can be ensured if the system involves nonlinear uncertainties. Lemma 2 Under Assumption A5, for any given positive constant εψ , one can find sufficiently large positive constants f i (i = 1, 2), such that 2  εψ ψ12 ψ2 + 2 ≤ ψ(z) z2 + 1 f1 f2 2

(36)

where ψ(z) is a nonnegative function. Proof Following similar steps in proving Lemma 1, under Assumption A5, the conclusions of Lemma 2 hold. 

Theorem 2 Suppose Assumption A5 holds, the closedloop system, composed of the controlled plant (2) and controller (12), (6), (8), (13), and (16), has semiglobal robust tracking property, that is, for any given constants ε > 0, r z ≥ 0 and rw ≥ 0, if z(t0 ) ≤ r z and w(t0 ) ≤ rw , one can find sufficiently large constants f i (i = 1, 2) and constant T ≥ t0 , such that the states x(t), z(t), and w(t) are bounded, and moreover, z(t) ≤ ε, w(t) ≤ ε, t ≥ T

(37)

If the initial value z(t0 ) and w(t0 ) is zero, then z(t) ≤ ε, w(t) ≤ ε, t ≥ t0

(38)

Proof This theorem can be shown by using the same Lyapunov function V in the proof of Theorem 1. According to Theorem 1, the time derivative of V along the trajectories of the closed-loop system satisfies that

  2 a 2 ˙ f + 1 − 2a w2 V ≤ −aV − z − 2 a2

 2  εψ 2 + + 1 ψ(z) z2 + 1 (39) 2 a 2

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Consider a set Ω(ra , rb ) in R 6 defined as Ω(ra , rb )  

 z(t) 4 2 r ≤ V ≤ r , z(t) ∈ R , w(t) ∈ R b a w(t) (40)   where rb = λε2 , ra ≥ max rb , λ p2 (r z2 + rw2 ) , and λ p2 = λmax (P). Let χ = max(z,w)∈Ω(ra ,rb ) ψ(z). If we choose f and εψ satisfying f > 2a + and

a 4 ⎧ ⎪ ⎪ ⎨

(41) ⎫ ⎪ ⎪ ⎬

arb a

,

 (42) ⎪ 2 2 ⎪ ⎭ + 1 2χ + 1 2 2 a a 

z(t) ∈ Ω(ra , rb ), one respectively, then for any w(t) has

2 εψ ≤ min

⎪ ⎪ ⎩ 2χ λ p2

  a 2 f + 1 − 2a w2 z2 + 2 a2

 2  εψ 2 2 − + 1 ψ(z) z + 1 a2 2

 2  εψ 2 a a 2 2 2 χ z + 1 + 1 ≥ z + w − 2 4 a2 2

 2 ε 2 a ψ χ V− +1 ≥ 4λ p2 a2 2   2χ

2ε a 2 ψ + z2 1 − +1 4 a a2 ≥0

(43)

Therefore, for any given constants ε > 0, r z ≥ 0, and rw ≥ 0, if we choose rb = λε2

  ra ≥ max rb , λ p2 (r z2 + rw2 ),

(44)

then V (t0 ) ≤ ra , and one can find sufficiently large positive constant f satisfying inequality (41) and sufficiently small positive constant εψ satisfying inequality (42) such that 

z(t) ∈ Ω (ra , rb ) V˙ (t) ≤ −aV (t), ∀ (45) w(t)

Robust backstepping decentralized tracking control

which implies that z(t) and w(t) are bounded, converge exponentially to the following domain, and stay in it   

z(t) z(t) ≤ ε, w(t) ≤ ε (46) w(t) 

Remark 7 The tracking error can be made as small as a desired number ε, with convergence rate no less than a given value a. So with different types of time-varying reference inputs, both the steady-state and transient properties of the closed-loop system can be guaranteed.

5 Experimental results In this section, our proposed control strategy is applied to the 3-DOF Quanser helicopter. The laboratory equipment is comprised of a 3-DOF helicopter plant, two universal power modules(UPMs), a dSPACE system and a personal computer(PC) equipped with WinConSimulink-RTX configuration. The UPM has a maximum continuous voltage of ±24 V and a maximum continuous current of 5A. The 3-DOF helicopter experiment has three encoders. The attitude angles are measured by the encoders with a resolution of 4096 counts per revolution. Thus, the effective position resolution is 0.0879◦ . The dSPACE system, which has a sample rate of 10kHZ, is used to access control voltages and angles information. From (12), (6), (8), (13), and (16), the backstepping decentralized controller is obtained as  ai2  + ai2 fi 1 + s u i (t) = − bi0 (47) [ai1 (xi1 − ydi ) + xi2 + αi ydi − βi ri ] As can be seen from the above equation, the local controller uses only local information, even if the system contains uncertainties, couplings, and disturbances. Also, the controller is a linear time-invariant one. The system parameters are given by the Quanser Reference Manual [35]. In the reference manual [35], the nominal values of control gain are provided as b10 = 0.0858, b20 = 0.581. αi , βi (i = 1, 2) are parameters of the reference models, and a = min{a11 a12 a21 a22 } is the convergence rate. So αi , βi , and ai j (i = 1, 2; j = 1, 2) can be chosen according to the time constant of the controlled plant and expected dynamic characteristics of the closedloop system. According to the existing experiment

results [16,18,19,21–24], it would be great performance if the settling times of elevation and pitch angles could be within 3s and 2s (for the 5 % criterion) in the step response. So we choose the control parameters ai j = 1.1(i = 1, 2; j = 1, 2) and the reference models of Eq. (5), with α1 = β1 = 1.1, α2 = β2 = 1.5 based on the expected settling times in the step response. The mathematic model of the helicopter laboratory experimental setup can be derived by means of Lagrange formalism. Based on Lagrange formalism, the 3-DOF laboratory helicopter’s uncertainties, which are mainly caused by parameter perturbation and external disturbance, are norm-bounded [9,11,16,21– 24,36]. The adaptive identification of pitch and elevation model parameters is studied in [36], which provides the bounds of uncertainties with ξ11 = 0, ξ12 = 0.1, ξ13 = 0, ξ14 = 0, ςϕ1 (t) = 3, ξ21 = 0, ξ22 = 0, ξ23 = 0, ξ24 = 0.09, ςϕ2 (t) = 0.23. In the reference manual [35], main parameters associated with the 3DOF helicopter are listed as J1 = 0.9138kg · m2 , J2 = 0.0364kg · m2 , K f = 0.1188N/V, m = 1kg, L a = 0.66m, L h = 0.178m. Based on Eq. (23), μ111 = 1.3304, μ112 = 6.1094, μ121 = 0, μ122 = 0, μ211 = 0, μ212 = 0, μ221 = 2.7198, μ222 = 4.6725, ςψ1 (t) = 72.4384, ςψ2 (t) = 266.1208. From inequalities (24), (26), and (33), the control parameters f i (i = 1, 2) can be chosen with respect to the control performance index ε. Remark 8 The proof of Theorem points out that when the control performance index ε is decreased, f i (i = 1, 2) are generally increased. This will lead to large control effort u(t) and even saturation phenomenon during the transient process, which may damage the stability of the system. So we have to make trade-off between the control performance and the control effort by appropriately choosing the value f i (i = 1, 2). The controller parameters f i (i = 1, 2) should be selected with consideration of actuator saturation constraints. With incremental encoders, only relative angles can be obtained. So the initial values of states can be set to zero. If αi , βi , and ai j (i = 1, 2; j = 1, 2) are chosen appropriately with respect to the time constant of the controlled plant, actuator saturation phenomenon could be also reduced. Remark 9 f i (i = 1, 2) determined by the above theoretical analysis with sufficient condition may be conservative. In practical applications, if f i (i = 1, 2) are far

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less than the theoretical values, robust control property may be still achieved [34,37]. Remark 10 In this paper, aerodynamic influence with nonlinear bounds is further considered. It is difficult to acquire the accurate knowledge of bounds of the aerodynamic influence for aggressive maneuvers, since research on flight dynamics modeling is still in its initial stage [38]. Actually, for the ith control channel, we can tune f i online mono-directionally, that is, set it to some initial value and run the closed-loop control system; if the ith subsystem does not perform satisfactory performance and saturation phenomenon does not occur, set f i to some larger values until satisfactory performance is obtained. In our experiment, the control parameters are tuned to f 1 = 6, f 2 = 14 with good performance. The flight data indicate that the saturation phenomenon is avoided.

Fig. 5 Steady-state errors of elevation angle and pitch angle with step responses

To verify the validity of the robust controller, the same controller is applied to the 3-DOF Quanser helicopter with different types of reference signals. In our experiment, four types of reference inputs are considered as follows: A. Step responses In this case, the reference input commands are step signals with amplitudes of 20◦ . The transient responses and steady-state errors of step signals are shown in Figs. 4 and 5, respectively. In our results, the settling times of elevation and pitch angles are approximate to 2.9 and 2 s (for the 5 % criterion). The steady-state errors of elevation and pitch Fig. 6 Control voltages for front and back motors with step responses

Fig. 4 Transient responses of elevation angle and pitch angle with step responses

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angles are <0.05◦ and 0.1◦ . Besides, the step response overshoots of both angles are <1 %. The control voltages for front and back motors with step responses are presented in Fig. 6. Compared to previous experimental results of 3DOF helicopters, in [16], an adaptive controller was implemented, and the settling time of elevation and pitch angles was more than 6 s. In [18], an output feedback controller got the result of 4.9 and 3.8 s settling time. In [19], PID controllers with SMO and compensation were proposed, and more than 40 s settling time was achieved. In [21], a robust attitude controller achieved step response with 4.6 and 3.8 s settling time and with 0.2◦ and 0.4◦ steady-state errors. In [22], the

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settling times are 2.83s and 2.36s for elevation and pitch angles with 5 % criterion, respectively, whereas the steady-state errors of elevation and pitch channels are <1◦ and 0.2◦ , respectively. In [23], for the elevation channel, the settling time was about 6.5s with 5 % criterion, and the steady-state error is about 0.3◦ ; for the pitch channel, the settling time was about 2.9s, and the steady-state error was about 0.3◦ . In [24], for the elevation channel, the steady-state error was about 0.1◦ , and the pitch motion was not discussed. According to the above results, the transient property and steady-state property of pitch and elevation angles to step waves are evidently improved by our method.

Fig. 7 Tracking curves with square-wave reference inputs

Fig. 8 Control voltages for front and back motors with squarewave reference inputs

B. Responses to square-wave reference inputs In this experiment, the reference inputs are squarewave signals with period of 60 s and magnitude of 20◦ . The tracking curves with square-wave reference inputs are shown in Fig. 7. It can be seen that the 3-DOF helicopter has robust tracking property under aggressive maneuvers in serious coupling condition. The control voltages for front and back motors with square-wave reference inputs are depicted in Fig. 8. C. Responses to trapezoidal reference inputs The desired reference inputs are trapezoidal signals with unknown slopes and amplitudes. We set the reference inputs as ⎧ ⎪ t, for t ≤ 10 ⎪ ⎪ ⎪ ⎪ 10, for 10 < t ≤ 30 ⎪ ⎪ ⎪ t ⎪ ⎨ − + 17.5, for 30 < t ≤ 50 4 (48) r1 (t) = 5, for 50 < t ≤ 60 ⎪ ⎪ ⎪ ⎪ t ⎪ ⎪ − 25, for 60 < t ≤ 90 ⎪ ⎪ 2 ⎪ ⎩ 20, for 90 < t ≤ 100 ⎧ t, for t ≤ 20 ⎪ ⎪ ⎪ ⎪ 20, for 20 < t ≤ 30 ⎪ ⎪ ⎪ ⎪ 3t ⎪ ⎨ − + 42.5, for 30 < t ≤ 50 4 r2 (t) = (49) 5, for 50 < t ≤ 70 ⎪ ⎪ ⎪ ⎪ 3t ⎪ ⎪ − 47.5, for 70 < t ≤ 90 ⎪ ⎪ ⎪ 4 ⎩ 20, for 90 < t ≤ 100 Figures. 9 and 10 show the experiment results with trapezoidal reference inputs. One sees that the tracking errors of elevation and pitch angles are both <0.3◦ .

Fig. 9 Tracking curves with trapezoidal reference inputs

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Fig. 10 Tracking errors of elevation angle and pitch angle with trapezoidal reference inputs

Fig. 11 Control voltages for front and back motors with trapezoidal reference inputs

In contrast, the tracking errors were guaranteed to be <0.5◦ [21]. The control voltages for front and back motors with trapezoidal reference inputs are presented in Fig. 11. D. Responses to nonstationary sinusoidal reference inputs In this case, we set the reference inputs as   r1 (t) = 10 sin(0.1t) + 2 sin 0.01t + 0.0002t 2   +2 sin 0.02t + 0.0002t 2 + 10

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Fig. 12 Tracking curves with nonstationary sinusoidal reference inputs

Fig. 13 Tracking errors of elevation angle and pitch angle with nonstationary sinusoidal reference inputs

  r2 (t) = 10 sin(0.1t + π ) + 2 sin 0.01t + 0.0002t 2   +2 sin 0.02t + 0.0002t 2 + 10 (50) Figures. 12 and 13 show the tracking curves of pitch and elevation angles with nonstationary sinusoidal signals. From Figs. 12 and 13, one sees the tracking errors are <0.4◦ . The control voltages for front and back motors with nonstationary sinusoidal reference inputs are presented in Fig. 14. In the experiments A, B, C, D, the same controller is applied to the 3-DOF helicopter with different types

Robust backstepping decentralized tracking control

References

Fig. 14 Control voltages for front and back motors with nonstationary sinusoidal reference inputs

of reference inputs, namely step signals, square-wave signals, trapezoidal signals and sinusoidal signals with unknown slopes, unknown amplitudes, and unknown nonstationary period. It is obvious that the transient and steady-state properties are evidently improved compared to previous studies, and the decentralized controller designed by the proposed method is robust and effective.

6 Conclusions In this paper, a robust design method of robust decentralized controllers was proposed for the attitude control of a 3-DOF helicopter. For each subsystem, the controller of this method consists of a nominal controller and a robust compensator. The trait of our method is that for each subsystem, only local information is used in the design procedure; moreover, the controller is a linear time-invariant one. Robust practical tracking property of closed-loop system is proven. Furthermore, experimental results demonstrated the effectiveness of the proposed controller. Acknowledgments This work is supported by National Natural Science Foundation (61473324), National Outstanding Youth Science Foundation (61125306), Specialized Research Fund for the Doctoral Program of Higher Education (20130006 120027), Beijing Higher Education Young Elite Teacher Project ( YETP0378), Beijing Natural Science Foundation(4154079).

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