Robust output tracking control of VTOL aircraft without velocity measurements

Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011

Robust output tracking control of VTOL aircraft without velocity measurements ⋆ Shanwei Su, Yan Lin School of Automation, Beijing University of Aeronautics and Astronautics, Beijing, 100191, China (e-mail: sushanwei@ yahoo.com.cn, [email protected]). Abstract: In this paper, we develop a nonlinear controller to achieve causal output tracking for a non-minimum phase vertical take-off and landing (VTOL) aircraft without velocity measurements. Due to the presence of disturbances, auxiliary control inputs are introduced in the state observer to attenuate their effects. By decomposing the original system into one minimum phase and one non-minimum phase subsystems, the corresponding sub-control laws are respectively designed. After control transformation, the resulting controller not only forces the VTOL aircraft to asymptotically track the desired trajectories, but also drives the unstable internal dynamics to follow the causal ideal internal dynamics solved via stable system center method. Numerical simulation results illustrate the effectiveness of the proposed controller. Keywords: Vertical take-off and landing aircraft, output-feedback control, output tracking, non-minimum phase, internal dynamics. 1. INTRODUCTION Over the past few decades, VTOL aircraft has been a benchmark of nonlinear non-minimum phase control systems. The main difficulty of VTOL aircraft control lies in its non-minimum phase nature which limits the straightforward application of some powerful nonlinear control techniques such as feedback linearization and sliding mode control (SMC)(Shkolnikov,2002). The bulk of the existing work with respect to VTOL aircraft control covers two main branches: the stabilization control (Olfati-Saber, 2002; Chemori & Marchand, 2008;) and the trajectory tracking control (Huang & Yuan, 2002;Al-Hiddabi & McClamroch,2002;), while the latter is more challenging since it not only needs to stabilize the controlled system but also pursues to achieve the output tracking of the reference signals. Besides, how to tackle the internal dynamics which is asymptotically unstable, is another notable problem. Hauser, Sastry and Meyer (1992) applied an approximate input-output linearization procedure developed for slightly non-minimum phase nonlinear systems to VTOL aircraft by assuming that no zero dynamics exists. Martin, Devasia and Paden (1996) proposed a scheme for output tracking of non-minimum phase flat systems, which can be applied well to either the slightly or the strongly non-minimum phase systems. However, it is not a feasible job to determine flat outputs from the system model. Huang and Yuan (2002) treated output tracking problem of VTOL aircraft based on a Lyapunov-based technique and a minimum-norm strategy. Unfortunately, the internal dynamics in Huang and Yuan (2002) is stabilized to zero instead of tracking its bounded ideal internal ⋆ Work supported by NSF of China under Grant 60874044 and Research Foundation for Key Disciplines of Beijing Municipal Commission of Education under Grant XK100060422.

978-3-902661-93-7/11/$20.00 © 2011 IFAC

dynamics (IID)(Gopalswamy,1993), which stands for the desired roll angle and the corresponding angle velocity of VTOL aircraft. From the viewpoint of physics, a fact that an aircraft flies around a curve while its roll angle keeps unchanged does not conform flight laws. To find a bounded solution for the unstable internal dynamics, by applying the noncausal stable inversion approach in Devasia, Chen and Paden (1996), Al-Hiddabi and McClamroch (2002) studied the problem of tracking control for a class of MIMO nonlinear non-minimum phase control systems without uncertainties, and then the controller was testified via VTOL flight control. In fact, an aircraft often unavoidably suffers from uncertainties and disturbances, such as modeling error and wind disturbances which may affect the aircraft performance greatly and cannot be neglected in controller design. Lu and Spurgeon (1997) and Zhu, Wang and Cai (2010) employed sliding mode control to deal with output tracking of VTOL aircraft in the presence of unknown uncertainties and input disturbances. However, the SMC technique needs the bounds on disturbances and results in a discontinuous control. It should be pointed out that in the aforementioned papers, the VTOL aircraft states are all required to be available for feedback. On the other hand, Do, Jiang and Pan (2003) and Wang, Liu and Cai (2009) presented nonlinear output-feedback controllers to force the velocity-sensorless VTOL aircraft to track a desired vertical trajectory without considering the lateral movement. As a result, the roll attitude is kept horizontal, i.e., the asymptotically unstable internal dynamics is stabilized to be zero. Under this circumstance, the tracking problem is greatly simplified, as it is only during lateral movements that the non-minimum phase property reflected by the coupling between lateral and vertical thrusts is problematic (Conlini, et, al., 2010).

13846

10.3182/20110828-6-IT-1002.00912

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

In this paper, a robust output tracking control of a VTOL aircraft without velocity measurements is proposed with the assumption that some unknown but bounded input disturbances exist. We first construct a modified state observer including auxiliary control inputs to attenuate the effect of the disturbances. Based on which, an outputfeedback control scheme is proposed. By taking the lateral movement into account, the corresponding internal dynamics of the aircraft is forced to follow its causal and bounded IID solved via the stable system center (SSC) method proposed by Shkolnikov and Shtessel (2002), rather than noncausal IID solved via the stable inversion approach in Devasia, Chen and Paden (1996). It should be pointed out that, to the best of our knowledge, this is the first time to apply the SSC method to the VTOL aircraft control. Consequently, the proposed scheme guarantees the causal output tracking of the VTOL aircraft even in the presence of unexpected changes of desired trajectories. The paper is organized as follows. In section 2, the control problem of the VTOL aircraft is formulated. In section 3, the design procedure of controller is presented. In section 4, the stability analysis is provided. In section 5, the numerical simulation is given to show the effectiveness of the proposed design method. Section 6 draws the conclusions. 2. PROBLEM FORMULATION The mathematical model of the VTOL aircraft moving in the vertical-lateral plane is described as (Hauser, Sastry & Meyer, 1992) x˙ 1 = x2 , x˙ 2 = −(u1 + ξ1 ) sin x5 + (u2 + ξ2 )ε cos x5 ,

end, to avoid noise-sensitivity caused by reduced-order observer, the following full-order observer is designed for the purpose of the tracking control: x ˆ˙ 1 = x ˆ2 + k11 (x1 − x ˆ1 ), ˙x ˆ2 = −u1 sin x5 + u2 ε cos x5 − ud1 + k12 (x1 − x ˆ1 ), x ˆ˙ 3 = x ˆ4 + k21 (x3 − x ˆ3 ), ˙x ˆ4 = u1 cos x5 + u2 ε sin x5 − ud2 − 1 + k22 (x3 − x ˆ3 ), x ˆ˙ 5 = x ˆ6 + k31 (x5 − x ˆ5 ), ˙x ˆ6 = u2 − ud3 + k32 (x5 − x ˆ5 ),

(2) where kij , 1 ≤ i ≤ 3, 1 ≤ j ≤ 2, are positive constant observer gains, and the auxiliary inputs udi , i = 1, 2, 3, will be designed to attenuate the effect of the disturbances. ˜ i = (˜ Let the observer errors be defined as X x2i−1 , x ˜2i )T = (x2i−1 − x ˆ2i−1 , x2i − x ˆ2i )T , i = 1, 2, 3. And subtracting (2) from (1), we have µ ¶ µ ¶ ¶µ ¶ µ x ˜˙ 2i−1 −ki1 1 0 x ˜2i−1 = , + −ki2 0 udi + di x ˜2i x ˜˙ 2i | {z } Ai0

(3) i = 1, 2, 3, where the matrix Ai0 is Hurwitz for any positive constants ki1 , ki2 , and the lumped disturbances d1 = −ξ1 sin x5 + εξ2 cos x5 , d2 = ξ1 cos x5 + εξ2 sin x5 , d3 = ξ2 . (4) Further, define the following error coordinate transformation

x˙ 3 = x4 ,

e1 = x ˆ1 − y1d ,

x˙ 4 = (u1 + ξ1 ) cos x5 + (u2 + ξ2 )ε sin x5 − 1,

e2 = x ˆ2 − y˙ 1d ,

x˙ 5 = x6 ,

e3 = x ˆ3 − y2d ,

x˙ 6 = u2 + ξ2 (1) with the outputs y1 = x1 , y2 = x3 , where x1 and x3 denote, respectively, the horizontal position and vertical position of the aircraft center of mass in the body-fixed reference frame shown in Fig.1; x5 denotes the roll angle of the aircraft, x2 , x4 and x6 are the corresponding velocities, respectively; the control inputs are the thrust (directed out the bottom of the aircraft), u1 , and the rolling moment about the aircraft center of mass, u2 ; ξ1 and ξ2 are thrust and rolling moment disturbances which are unknown smooth, bounded functions of time, respectively; ε is a small coefficient that characterizes the coupling between the rolling moment and the lateral force; ‘−1’ denotes the acceleration of gravity. By setting x1 = x2 = x3 = x4 = 0 in (1), it can be seen that the resulting zero dynamics x ¨5 = 1ε sin x5 is asymptotically unstable for ε > 0. Therefore, the VTOL system is nonminimum phase. Under the assumption that without velocity measurements, say, x2 , x4 and x6 are not available for measurement, the control objective is to design a feedback control law so that the outputs y1 and y2 can track the smooth reference trajectories y1d and y2d , respectively. To this

e4 = x ˆ4 − y˙ 2d , η1 = x ˆ5 , η2 = εˆ x6 − e2 cos x ˆ5 − e4 sin x ˆ5 , whose time derivative yields

(5)

e˙ 1 = e2 + k11 x ˜1 , e˙ 2 = −u1 sin x5 + u2 ε cos x5 − ud1 + k12 x ˜1 − y¨1d , e˙ 3 = e4 + k21 x ˜3 , e˙ 4 = u1 cos x5 + u2 ε sin x5 − ud2 − 1 + k22 x ˜3 − y¨2d , 1 η˙ 1 = (η2 + e2 cos η1 + e4 sin η1 ) + k31 x ˜5 , ε 1 η˙ 2 = (η2 + e2 cos η1 + e4 sin η1 )(e2 sin η1 − e4 cos η1 ) ε +¨ y1d cos η1 + (1 + y¨2d ) sin η1 − k12 x ˜1 cos η1 −k22 x ˜3 sin η1 + (e2 sin η1 − e4 cos η1 )k31 x ˜5 + εk32 x ˜5 (6) +ud1 cos η1 + ud2 sin η1 − εud3 . The internal dynamics expressed by the last two equations of (6) can be described as ˜ Yd ), η˙ = Φ(e, η, ud , X, (7)

13847

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

where η = (η1 , η2 )T , e = (e1 , e2 , e3 , e4 )T , ud = (ud1 , ud2 , ud3 )T , T ˜ = (˜ X x1 , x ˜3 , x ˜5 ) , Yd = (y1d , y˙ 1d , y¨1d , y2d , y˙ 2d , y¨2d )T . (8) Rewriting (7) by separating its linear part from its nonlinear part yields ˜ Yd ), η˙ = Aη η + Ae e + q(e, η, ud , X, (9) where ˜ Yd ) ∂Φ(e, η, ud , X, (0, 0, 0, 0, 0), ∂η ˜ Yd ) ∂Φ(e, η, ud , X, (0, 0, 0, 0, 0), Ae = ∂e

Aη =

(10)

and ˜ Yd ) = Φ(e, η, ud , X, ˜ Yd ) − Ae e − Aη η. (11) q(e, η, ud , X, Hence, the specific form of (9) shows that µ ¶ Ã 1 !µ ¶ Ã 1 !µ ¶ µ ¶ q e1 η˙ 1 η1 + 1 , (12) + 0 ε = 0 ε q2 e η˙ 2 η 2 2 1 0 0 0 | {z } | {z } Aη

method is conducted backward in time, the resulting IID is noncausal. On the contrary, the SSC method does not necessarily require the future information of the desired trajectories, the online solving procedure for bounded IID is performed forward in time. Therefore, we turn to the SSC method to get the causal IID of (14) that will be given in Section 5 in detail. Let η˜1 = η1 − η1d , η˜2 = η2 − η2d . (15) And subtracting (14) from (12) results in 1 1 η˜˙ 1 = e2 + η˜2 + q˜1, η˜˙ 2 = η˜1 + q˜2 , (16) ε ε where q˜1 = q1 , q˜2 = q2 − ψ. Define the new inputs v1 = −u1 sin x5 + u2 ε cos x5 − ud1 − y¨1d , (17) v2 = u1 cos x5 + u2 ε sin x5 − ud2 − y¨2d − 1. Then the system (6) with its last two equations replaced by (16) can be rewritten as e˙ 1 = e2 + k11 x ˜1 , e˙ 2 = v1 + k12 x ˜1 ,

Ae

where

1 ˜5 , q1 = (e2 cos η1 + e4 sin η1 − e2 ) + k31 x ε 1 q2 = (η2 + e2 cos η1 + e4 sin η1 )(e2 sin η1 − e4 cos η1 ) ε +¨ y1d cos η1 + (1 + y¨2d ) sin η1 − η1 − k12 x ˜1 cos η1 −k22 x ˜3 sin η1 + (e2 sin η1 − e4 cos η1 )k31 x ˜5 + εk32 x ˜5 (13) +ud1 cos η1 + ud2 sin η1 − εud3 . If ε > 0, the matrix Aη is non-Hurwitz, the internal dynamics is asymptotically unstable. To curb this phenomenon, the way out is to force the unstable internal dynamics η to track its bounded IID ηd , which can be ˜ ud to their desired states in (12). obtained by setting e, X, Under ideal conditions, udi = −di in (3), which means that the effects of the disturbances are completely eliminated, ˜ → 0, and e = 0 is our control aim. By assuming thus X η1 = x ˆ5 = x5 , and noticing (4), the following term in q2 of (13) ud1 cos η1 + ud2 sin η1 − εud3 = 0, which illustrates that the IID is not influenced by the disturbances. Consequently, the differential equation about ηd appears ¶ µ ¶ ¶ Ã 1 !µ µ 0 η1d η˙ 1d 0 + ψ, (14) = ε 1 η η˙ 2d 2d 1 0 where ψ = y¨1d cos η1d + (1 + y¨2d ) sin η1d − η1d . Since Aη is a non-Hurwitz matrix for ε > 0, (14) has no stable numerical solutions. However, this does not mean that a bounded solution cannot be found for such an unstable system (Al-Hiddabi & McClamroch, 2002). In fact, under suitable assumptions, via the noncausal stable inversion approach (Devasia, Chen & Paden, 1996) or SSC method (Shkolnikov & Shtessel, 2002), a bounded solution can be computed. It is worth noting that stable inversion approach is of limited practical use, due to the fact that the desired trajectories and their any changes must be exactly known in advance, and the bounded solution of unstable internal dynamics needs to be pre-calculated offline. What is more, the computing procedure of stable inversion

e˙ 3 = e4 + k21 x ˜3 , e˙ 4 = v2 + k22 x ˜3 , 1 1 η˜˙ 1 = e2 + η˜2 + q˜1 , ε ε η˜˙ 2 = η˜1 + q˜2 . (18) From (12), we see that the internal dynamics is linear relative with e1 , e2 , and linear irrelative with e3 , e4 . Thus the system (18) can be decomposed into the following two parts (Al-Hiddabi,2002): minimum phase part e˙ 3 = e4 + k21 x ˜3 , e˙ 4 = v2 + k22 x ˜3 , and non-minimum phase part

(19)

e˙ 1 = e2 + k11 x ˜1 , e˙ 2 = v1 + k12 x ˜1 , 1 1 η˜˙ 1 = e2 + η˜2 + q˜1 , ε ε η˜˙ 2 = η˜1 + q˜2 .

(20)

Now, the tracking problem for the original system (1) has been converted into a stabilization problem for the new subsystems (19) and (20), which is beneficial to the control law design that will be discussed in our next section. 3. CONTROL LAW DESIGN 3.1 Control law of the auxiliary inputs Inspired by the uncertainty and disturbance estimation (UDE) method in Talole and Phadke (2009), we obtain the estimation of di in (3) as ¡ ¢ dˆi = Gi (s)di = Gi (s) x ˜˙ 2i − udi + ki2 x ˜2i−1 , (21)

1 where Gi (s) = 1+τ is a first-order low pass filter with is time constant τi . Let ¡ ¢ udi = −dˆi = −Gi (s) x ˜˙ 2i − udi + ki2 x ˜2i−1 . (22)

13848

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

Solving udi , i = 1, 2, 3, from (22) yields Z t 1 x ˜2i−1 (τ )dτ ). (23) x2i + ki2 udi = − (˜ τi 0 Since x ˜2i is unaccessible, by noticing (3), x ˜2i can be obtained as x ˜2i = x ˜˙ 2i−1 + ki1 x ˜2i−1 (24) Subsituting (24) into (23) results in Z t 1 ˙ udi = − (x x ˜2i−1 (τ )dτ ). (25) ˜2i−1 + ki1 x ˜2i−1 + ki2 τi 0 From the designed auxiliary control law udi , we can see that it performs a PID control action. 3.2 Control law of the minimum phase part The control law of (19) can be designed as v2 = −l1 e3 − l2 e4 , (26) where l1 , l2 are constants to be chosen such that the polynomial s2 + l2 s + l1 = 0 is Hurwitz.

4. STABILITY ANALYSIS In this section, we present the stability analysis for the proposed control law (35). Firstly, define the disturbance estimation error as d˜i = di − dˆi , i = 1, 2, 3. By notic1 ing (21), dˆi = Gi (s)di = 1+τ di . The time derivative is ˙ ˙ of dˆi is dˆi = τ1i (di − dˆi ) = τ1i d˜i . Hence d˜i = d˙i − ˙ dˆi = − τ1i d˜i + d˙i . Taking (4) into account, we can ob¯ ¯ ¯ ¯ tain that ¯d˙i ¯ ≤ Bi (ξ1 , ξ2 , ξ˙1 , ξ˙2 , x5 , x˙ 5 ), where Bi , for i = 1, 2, 3, are some continuous positive functions. Fur¯ther, ¯ by noticing (5),(15),(23),(26),(34) and (35), hence, ¯˙¯ ˜ ξ, ηd ), where ξ = (ξ1 , ξ˙1 , ξ2 , ξ˙2 )T . Note ¯di ¯ ≤ Bi (e, η˜, X, ¯ ¯ ¯ ¯ B2 ˙ that d˜i d˜i ≤ − τ1i d˜2i + Bi ¯d˜i ¯ ≤ ( 2αii − τ1i )d˜2i + 21 αi , where αi is any positive constant. Define the compact set ˜ ξ1 , ξ2 , ξ˙1 , ξ˙2 , ηd ) : eT e + η˜T η˜ Ω := {(e, η˜, X, ˜T X ˜ + (ξ 2 + ξ 2 ) + (ξ˙2 + ξ˙2 ) + η T ηd ≤ R0 },(36) +X

3.3 Control law of the non-minimum phase part

1

T

Letting the variable z = (e1 , e2 , η˜1 , η˜2 ) , the system (20) becomes z˙ = Az + bv1 + q˜, (27) where       0 1 0 0 0 k11 x ˜1 0 0 0 0  ˜1   1 k x 1 A= 1 (28)  , b =  0  , q˜ =  12 . q ˜ 0 0 1   ε ε 0 q˜2 0 0 1 0 Replacing v1 = vnl − Kz in (27) yields z˙ = A0 z + bvnl + q˜, (29) where K is selected so that A0 = A − bK is a Hurwitz matrix; hence, for any given symmetric positive definite matrix Q, there exists a unique symmetric positive definite matrix P satisfying the Lyapunov equation AT0 P + P A0 = −Q. (30) Pre-multiplying the vector z T P on both sides of (29) results in z T P z˙ = z T P A0 z + z T P bvnl + z T P q˜. (31) T Because z P bvnl and z T P q˜ are all scalars, we select z T P bvnl = −z T P q˜. (32) The minimum-norm solution can be obtained as δ (z T P b)T z T P q˜ =− , (33) vnl = − T (z P b)(z T P b)T σ where for convenience, we use δ and σ to denote the numerator and denominator of the above equation, respectively. To avoid the problem of singulairty, we redesign the control law vnl as    −δ, σ > ǫ, σ (34) vnl = λσ δ   − tanh2 ( ), σ ≤ ǫ, σ ǫ where ǫ is a small positive constant. By noticing (17), the practical inputs are obtained via the input transformation ¶−1 µ ¶ µ ¶ µ v1 + ud1 + y¨1d − sin x5 ε cos x5 u1 = . v2 + ud2 + y¨2d + 1 cos x5 ε sin x5 u2 (35)

2

1

2

d

where R0 is a given positive constant. To proceed, by noticing (3) and (22), ˜˙ i = Ai0 X ˜ i + bd˜i , i = 1, 2, 3, X (37) µ ¶ −ki1 1 ˜ i = (˜ where X x2i−1 , x ˜2i )T , Ai0 = , b = (0, 1)T . −ki2 0 Let the Lyapunov function for the observer error system (37) ˜ iT Pi X ˜ i + γi d˜2i . (38) Vi := X 2 We are now in a position to present our main theorem. Theorem 1: Let the closed-loop system consists of the controlled plant (1) and the full order observer (2). Let the control law be given by (35), and the Lyapunov functions be defined by (38). Then for the given compact set Ω, if Vi (0) < R0 , i = 1, 2, 3, there exist γi and τi , i = 1, 2, 3, such that all signals of the closed-loop system are uniformly semiglobally bounded and the tracking error converges to a residual set that can be made arbitrarily small by properly choosing some design parameters. Proof. We first prove the stability of the observer error system. Since Ai0 is a Hurwitz matrix, then for any given symmetric positive definite matrix Qi , there exists a unique symmetric positive definite matrix Pi satisfying the Lyapunov equation ATi0 Pi + Pi Ai0 = −Qi . Since Bi is continuous, it has maximum on Ω, say, Mi . Therefore, the time derivative of Vi is ˜ T Qi X ˜ i + 2X ˜ T Pi bd˜i + γi d˜i d˜˙i V˙ i = −X i i 1 T ˜ T Pi bbT PiT X ˜i ˜ i Qi X ˜i + X ≤ −X γi i B2 1 γi +( i + 1 − )γi d˜2i + αi 2αi τi 2 ¶ µ λmin (Qi ) λmax (Pi bbT PiT ) ˜ T ˜ Xi Pi Xi − ≤− λmax (Pi ) γi λmin (Pi ) M2 γi γi 2 (39) −( − i − 2) d˜2i + αi . τi αi 2 2 By choosing ri , γi and τi such that

13849

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

½

We then prove the stability of the minimum phase system. Substituting the control law (26) into the minimum phase system (19) yieldsµe˙ m = A¶ ˜3 ,where em = m0 em + bm x 0 1 , bm = (k21, k22 )T . Since (e3, e4 )T , Am0 = −l1 −l2 Am0 is a Hurwitz matrix and we have proved in the above stability analysis that x ˜3 is bounded on the compact set Ω, it is easy to obtain the boundedness of em . Besides, the deeper the eigenvalues of Am0 lie in the left half-plane, the smaller the tracking error em . Next, we prove the stability of the non-minimum phase system. Two cases are considered. Case 1: σ > ǫ. By (29)-(32), z T P z˙ = z T P A0 z. Select the Lyapunov function Vn = z T P z, whose time derivative of min (Q) T Vn yields V˙ n = −z T Qz ≤ − λλmax (P ) z P z, from which it is clear that the tracking error z exponentially converges to a small residual set that depends on the designer, thus the control law vnl is bounded. Case 2: σ ≤ ǫ. Since σ = (z T P b)(z T P b)T is bounded by construction, it can be concluded that the tracking error z and the control law vnl are bounded too. From the above stability analysis and by noticing the boundness of ηd , Yd , it follows that both the states x and x ˆ are bounded. By (35), it can be deduced that the practical control laws u1 , u2 are bounded. Hence, all the signals of the closed-loop system are bounded. This completes the proof. 5. SIMULATION RESULTS 5.1 VTOL IID (ε = 0.5) By (14), the IID of the VTOL can be rewritten in the following¶form η˙ d = Aη ηd + bψ, where ηd = (η1d , η2d )T , Aη = µ 0 2 , b = (0, 1)T , ψ = y¨1d cos η1d +(1+¨ y2d ) sin η1d −η1d . 1 0 In the simulation, the desired output trajectories are y1d = R cos(ωt), y2d = R sin(ωt). The sine signals can be gener-

η 1d

0.02 0.01 0 −0.01

0

20

40

60 80 time(s)

100

120

140

40

60 80 time(s)

100

120

140

(a) −3

2 η 2d

¾ M2 λmin (Qi ) λmax (Pi bbT PiT ) 2 − , − i −2 , λmax (Pi ) γi λmin (Pi ) τi αi (40) where ri is a positive design constant, then the inequality (39) satisfies γi V˙ i ≤ −ri Vi + αi . (41) 2 Hence, V˙ i ≤ −ri R0 + γ2i αi when Vi = R0 . That is, if ri is i αi , we have V˙ i < 0 on Vi = R0 , chosen such that ri > γ2R 0 which implies that if Vi (0) ≤ R0 , then Vi (t) ≤ R0 , ∀t ≥ 0, ˜ i and d˜i are i.e., Vi ≤ R0 is an invariant set. Hence, X bounded, for i = 1, 2, 3 and the auxiliary control law udi are bounded. As a result, all signals of the observer errors are bounded. Moreover, solving (41) yields γi αi −ri t γi αi + (Vi (0) − )e . (42) 0 ≤ Vi ≤ 2ri 2ri i αi Hence, limt→0 Vi (t) ≤ γ2r . That is, for fixed γi and ri , i by choosing αi sufficiently small, which can be done by letting time constants τi sufficiently small, Vi (t) as well as ˜ i can converge to a small residual set. X ri = min

x 10

1 0 −1

0 (b)

20

Fig. 1. (a) IID η1d (Desired roll angle in physical meaning). (b) IID η2d . µ ¶ 0 ω ated by the exosystem w˙ = Sw, where S = .The −ω 0 characteristic polynomial P (λ) = |λI − S| = λ2 + ω 2 , from which we can determine the polynomial order k = 2, the coefficients p1 = 0, p0 = ω 2 . By setting the desired eigenvalue s1,2 = −1, the corresponding characteristic 2 equation (s + 1) = s2 + 2s + 1 = 0; hence, the parameters c1 = 2, c0 = 1. According to the formula in Shkolnikov and Shtessel (2002), we get −2 −1 −2 −1 P1 = (I + c1 Q−1 −I 1 + c0 Q1 )(I + p1 Q1 + p0 Q1 ) µ ¶ 2 1 0.5(1 − ω ) 2 = , (43) 1 0.5(1 − ω 2 ) 1 + 0.5ω 2 −1 P0 = c0 Q−1 1 − (P1 + I)p0 Q1 µ ¶ 2 1 −ω 1 − ω2 (44) . = 1 + 0.5ω 2 0.5(1 − ω 2 ) −ω 2 The IID ηd can be solved from the following matrix differential equation η¨d + c1 η˙ d + c0 ηd = −(P1 θ˙d + P0 θd ), T where θd = ( 0 1 ) ψ. Fig.1 give the IID of the VTOL (η1d and η2d ) solved via the SSC method with R = 1 and ω = 0.1.

5.2 Simulation results In the simulation, to show the advantage of the SSC method, the amplitude R and the frequency ω of the desired output trajectories y1d and y2d switch from 1 to 1.2 and 0.1 to 0.2, respectively, at the time T = 62.8 + 5 · rand(1). Such switches may occur in the case of obstacle avoidance. Note that y1d and y2d correspond to an aggressive manoeuver in the sense that linear controllers will fail to work. The input disturbances are chosen as ξ1 = 0.1 cos 2t, ξ2 = 0.2 sin t. The coupling coefficient ε is selected to be 0.5, which means that the VTOL aircraft is a strongly non-minimum phase system. The initial conditions are chosen as x(0) = (1.5, 0, −0.5, 0.2, 0.28, 0)T , x ˆ(0) = (1.3, 0.1, −0.3, 0.1, 0.1, 0.1)T . The observer gain ki1 = 2, ki2 = 1, i = 1, 2, 3. The controller gains of the minimum phase subsystem are l1 = 1, l2 = 2. For the non-minimum phase subsystem, we choose K = (−0.5, 4, 4.25, 6), such that the eigenvalues of A0 = A−bK are all placed at −1. The corresponding matrix P can be computed from (30) with Q = I4×4 and ǫ = 0.1, λ = 6. To

13850

18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011

2

loop system is semiglobally stable. Simulation results have verified the validity of the proposed controller. Finally, we point out that in this paper, the disturbances matched with the control inputs do not appear in the IID of the VTOL aircraft, otherwise, it will pose a challenge that the IID is affected by the disturbances, which falls into our current study.

desired path actual path (τ=0.1)

1.5

actual path (τ=0.01)

y2

1 0.5 0 −0.5

REFERENCES

−1 −1.5

−1

0

(a)

1

2

y1

0.3 η1d

0.25

η1 (τ=0.1)

0.2

η1 (τ=0.01)

Roll angle

0.15 0.1 0.05 0 −0.05 −0.1 −0.15 (b)

0

20

40

60

80

100

time(s)

Fig. 2. (a) The desired path and the actual path. (b) The desired roll angle and the actual roll angle. illustrate the effectiveness of the proposed control scheme, the simulation is done by considering two cases: the filter time constant τ = 0.1 and τ = 0.01, respectively. From the simulation results (see Fig.2.(a)), we can conclude that with smaller filter time constant τ, the tracking performance can be improved significantly. From Fig.2.(b), when τ = 0.01, it can be seen that the actual roll angle can perfectly track the desired roll angle, the IID, solved via the SSC method, which, in turn, verifies the correctness of Fig.1. The simulation also shows the advantage of the utilization of the SSC method over the noncausal stable inversion adopted by current scheme dealing with VTOL in Al-Hiddabi and McClamroch (2002) when some unexpected changes of desired trajectories occur. Indeed, Fig.2.(b) demonstrates that the roll angle of our scheme is able to adapt the pre-unknown change of the desired roll angle, while the noncausal stable inversion method fails under this circumstance. 6. CONCLUSION A nonlinear controller to achieve the causal output tracking for a non-minimum phase VTOL aircraft without velocity measurements has been proposed. By introducing auxiliary control inputs in the modified state observer, the effects of the disturbances have been attenuated. Based on the observer and the state transformation, the control laws have been respectively designed for both the minimum phase and the non-minimum phase subsystems, which has the ability to force the VTOL aircraft not only to asymptotically track the desired trajectories, but also to drive the unstable internal dynamics to follow the causal IID of the VTOL aircraft. It has been proved that the overall closed-

Al-Hiddabi, S. A., & McClamroch, N. H. (2002). Tracking and maneuver regulation control for nonlinear nonminimum phase systems: application to flight control. IEEE Transactions on Control Systems Technology, 10(6), 780-792. Chemori, A., & Marchand, N. (2008). A prediction-based nonlinear controller for stabilization of a non-minimum phase PVTOL aircraft. International Journal of Robust and Nonlinear Control, 18(8), 876-889. Consolini, L., Maggiore, M., Nielsen, C., & Tosques M. (2010). Path following for the VTOL aircraft. Automatica, 46(8), 1284-1296. Devasia, S., Chen, D. G., & Paden, B. (1996). Nonlinear inversion-based output tracking. IEEE Transactions on Automatic Control, 41(7), 930-942. Do, K. D., Jiang, Z. P., & Pan, J. (2003). On global tracking control of a VTOL aircraft without velocity measurements. IEEE Transactions on Automatic Control, 48(12), 2212-2217. Gopalswamy S., & Hedrick J. K. Tracking nonlinear nonminimum phase systems using sliding control. International Journal of Control, 57(5), 1141-1158. Hauser, J., Sastry, S., & Meyer, G. (1992). Nonlinear control design for slightly non-minimum phase systems: application to V/STOL aircraft.Automatica, 28(4), 665679. Huang, C. S., & Yuan, K. (2002).Output tracking of a nonlinear non-minimum phase PVTOL aircraft based on non-linear state feedback control. International Journal of Control, 75(6), 466-473. Lu, X. Y., & Spurgeon, S. K. (1997). Robust variable structure control of a PVTOL aircraft. International Journal of Systems Science. 28(6), 547-558. Martin, P., Devasia, S., & Paden, B. (1996). A different look at output tracking: control of a VTOL aircraft. Automatica, 32(1), 101-107. Olfati-Saber, R. (2002). Global configuration stabilization for the VTOL aircraft with strong input coupling.IEEE Transactions on Automatic control , 47(11), 1949-1952. Shkolnikov, I. A., & Shtessel, Y. B. (2002). Tracking in a class of non-minimum phase systems with nonlinear internal dynamics via sliding mode control using method of system center. Automatica, 38(5), 837-842. Talole, S. E., & Phadke. S. B. (2009). Robust inputoutput linearisation using uncertainty and disturbance estimation. International Journal of Control, 82(10), 1794-1803. Wang, X. H., Liu, J. K., & Cai, K. Y. (2009). Tracking control for a velocity-sensorless VTOL aircraft with delayed outputs. Automatica, 45(12), 2876-2882. Zhu,B., Wang, X. H., & Cai, K. Y. (2010). Approximate trajectory tracking of input-disturbed PVTOL aircraft with delayed attitude measurements.International Journal of Robust and Nonlinear Control, 20(14), 1610-1621.

13851