The passivity of adaptive output regulation of nonlinear exosystem with application of aircraft motions

R. Ravi Kumar, Jinde Caob,c,1,2, Ahmad Alsaedi Department of Agroforestry, FCRI, MTP, Tamilnadu Agricultural University, Coimbatore, India ravisugankr@gmail.com School of Mathematics, Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing 210096, China jdcao@seu.edu.cn Department of Mathematics, Nonlinear Analysis and Applied Mathematics Research Group, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia aalsaedi@hotmail.com


Introduction
It is well known that the passivity theory of deterministic nonlinear systems was first founded by [21], which is a powerful technique in handling stability issue and has important application in many engineering problems.The nonlinear dissipative control and passivity have showed that some inequality involves supply rates and storage functions [4,6,17].Especially, Hill and Moylan [6] proposed a nonlinear version of the Kalman-Yacubovitch-Popov (KYP) property and a sufficient and necessary condition for an affine nonlinear system to be passive.The principal imposed that the state equations should involve the control vector, which is only linear.In [16], multivalued controls are derived from a special maximally monotone operator, and they suggested that strongly passive linear system (with possible parametric uncertainty and external disturbances) with multivalued control laws are ensuring regulation of the output to be desired value.It can be shown that, for mechanical systems, the energy balancing approach and energy shaping techniques are used port-Hamiltonian passivity with output regulation system.Our methodology is different from port-Hamiltonian passivity with output regulation setting.If an internal model with transfer function is in the feedback loop and the closedloop system is stable, then we obtain tracking and/or disturbance rejection for sinusoidal reference, and disturbance signals of frequency gain exists in exosystem.If the reference and disturbance signals are periodic, then the internal model principle leads to repetitive control.The objective of our study is exosystem with internal model minimum phase based on n regulated outputs that are converted as single input nonlinear adaptive formation, with support of KYP property, this exosystem is derived for passivity.
In [2], passivity combined with geometric nonlinear control theory has been proved.The results developed in [2], the global stabilization of nonlinear systems, robust and adaptive control of minimum phase nonlinear systems, found out parametric uncertainty.The aim of passivity-based synthesis approach controller is to render a nonlinear system to be passive.But in this, any information for nonlinear systems with structural uncertainties or uncertain perturbations were not given.One of the main motivations for studying passivity in control theory context is to prove the stability and structural uncertainties.An important result in this area is KYP lemma, which is used in solving the well-known stability problems.However, these known results are related only to the case of state feedback passivity.In [4,10,17,18], the view of input-output nature of the passivity concept is described, and it seems useful to establish relations between output feedback counterparts of stability and passivity.All the above literature are studied continuous time control schemes with properties of feedback equivalent to passive, it is particular class of interconnected internal model with zero-dynamics, i.e., weak minimum phase condition, but not sufficient for feedback equivalence to nonlinear passive systems.However, as far as we know, a general nonlinear system with output regulation to be passive is still remains open and challenging.The nonlinear system with output regulation of passivity is useful in aircraft handling qualities.
The aircraft motion is regulated by line-of-sight (LOS) angle rates, which will be regulated in some range, and those rates are commanded in aircraft maneuvering [9].The LOS angles are related to output regulation of nonlinear systems [9,11].It is showed; aircraft with follower aircraft communication error signals are interrelated to output regulation minimum phase nonlinear system.These concepts are more useful to design the autopilot, state feedback controller based eigen structure assignment methods, and those are commonly used to aircraft handling qualities etc.The problem of nonlinear systems with output regulation is solved by variable gain feedback law that is more similar to Nonlinear Anal.Model.Control, 22(3):366-385 adaptive learning regulation of uncertain minimum phase systems [1,3,5,7,[12][13][14]19].In this paper, the variable w is an exogenous variable, that is modeling reference has to be tracked and or disturbance has to be rejected.The exogenous w = col(ω, w) will be taken as n harmonic oscillators with low-high frequency gain, where ω = (ω 1 , ω 2 , . . ., ω n ) are output feedback error disturbances and w = (w 1 , w 2 , . . ., w m ) are taken as estimated disturbances.The variable ẇ = s(w) = ( w 0 0 −w ), where w are over-estimated disturbances, −w are under-estimated disturbances.And in this case, under-estimated disturbances will be dominated.The use of conditional servo compensator enables to achieve zero steady-state tracking error without degrading the transient response of the system; it is given in [15].In [20], the problem of robust adaptive output regulation of hybrid adaptive external systems is described by nonlinear differential equations.In those article, information for under-dimensional internal model is not given.It was shown that set of all feed-forward inputs are essential to keep the tracking error identically at zero, which is a subset of solutions of a linear differential equation.
In recent work [8], studied Rayleish oscillator as a back born of aircraft maneuverability.The exogenous signals are transformed to interconnected subsystem with appropriate structures, i.e., one subsystem contains no control law, and whereas the other one is regulated error output is a control input, which satisfies a Lyapunov inequality on behalf of weak minimum phase condition.In particular, in [8], imposed adaptive output regulation for nonlinear systems in the case of exosystem signals with harmonic oscillators was emphasized, it does not give any information about control output equivalence to feedback control.Compared with deterministic case, up to date, there still requires much work of investigating the nonlinear system with output regulation of minimum phase exosystem is global asymptotic stabilization.Based on the above, we investigate global asymptotic stabilization the relationship between output regulation of passive system and corresponding minimum phase of adaptive nonlinear systems.Especially, different from the deterministic exosystem, our results first show that the local asymptotic stability of non zero-dynamics systems under the low-gain frequency gain.Then, sufficient conditions derived for exosystem with n regulated outputs are converted as single input adaptive nonlinear form, and global asymptotic stability is provided via passivity theory.Finally, the physical example aircraft motion shows the applicability of the obtained results.
The paper is organized as follows.Section 2 contains some basic assumptions and problem formulations.The main results and discussion are given in Section 3. In Section 4, the examples are considered, and in Section 5 some conclusions are drawn.
Notation.In this paper, R denotes the real numbers, C 1 is continuous differentiable on some closed time interval.For any x ∈ R n , |x| denotes the Euclidean norm of R n .G is the vehicle's center of mass, m is its mass, J denotes inertia matrix, and I denotes inertial frame with respect to vehicles absolute position is measured.
Furthermore, system (1)-( 3) exists in pair sets the latter is a compact set for any integer p), then the error feedback controllers ξ = α(ξ, e), u = β(ξ, e) and the initial conditions are in a compact set W×Z×Θ×E ⊂ R d×d ×R n ×R p ×R r such that trajectory of resulting closed-loop system (1)-( 4) originating from W × Z × Θ × E are bounded and lim t→∞ e(t) = 0.
The problem of semi global output regulation in which r = 1 and assume the change of variable e i → ϕ = k −(i−1) e i , i = 1, . . ., r − 1, such that system (1)-( 3) can be written as in which k > 1 is a design parameter and the a i , i = 0, . . ., r − 2, are roots of the polynomial The change of variables can be applied for ( 1)-( 3), we have where ϕ = (ϕ 1 , ϕ 2 , . . ., ϕ r−1 ), A H is Hurwitz matrix, and , which is exists in a compact set E, system (6) regarded as a system with input u and output θ have relative degree one and zero dynamics, then Under the condition grap , the classical result [2] showed that system (7) is globally asymptotically stable.
In this setting, we assume that the controller solves the problem of output regulation (7).This controllers are driven by the "dummy" regulated output θ, and it is not actual regulated output ϕ 1 .However, in this case, θ can be constructed as a linear components of ϕ 1 , ϕ 2 , . . ., ϕ r , which is related to the partial state e = (e 1 , e 2 , . . ., e r ).If ϕ coincides with (r − 1)th derivative with respect to time, there is an actual regulated output e.
The following theorem shows that how to solve the problem of output regulation ( 1)-( 4) in an appropriate domain.(10) and τ and ξ are locally asymptotically stablilizable in Ω, then there exists a continuous function k : R → R such that the controller solves the problem of output regulation (1)-( 4) More conventital design and methodologies has been obtained from Marconi et al. [8,9].In that design, the triplet (F (•), G(•), γ(•)) have been fulfilling internal model property, and it can be effectively carried out in the case of φ : R p×1 → R such that for all w ∈ W. In this setting, and let φ c be any locally Lipschitz and bounded function such that φ c in τ (w).It carry out that where G be any vector, ξ = (ξ 0 , . .., ξ p−1 ) and makes (12) fulfilled with γ(ξ) , where (λ 0 , λ 1 , . . ., λ p−1 ) are coefficients of a Hurwitz polynomial, then g > 0 be a high-gain observer such that ( 12) is locally exponentially stabilizable in Ω with domain of attraction W × A.
We need the following assumptions to prove that system (1)-( 3) is passive.
Remark 1. Suppose that Assumption 1 is fulfilled, and system (1)-( 3) is input-to-state stable with respect to state z, the input u and equilibrium at z = 0 of the system ẇ = s(w), ż = f (w, z, 0) are locally exponentially stable for any w ∈ W if ρ 1 and ρ 2 are locally quadratic.
Theorem 2. Consider a system described by equations of the form in which F (ξ) and G(u) are smooth vector fields and F (0) = 0.If there exist the smooth feedback law u = γ(ξ)+v, v = −k(e) and satisfies Assumptions 1-4, then system (1)-( 3) is locally asymptotically stable in W × Z × Θ × E.
We assume that the function φ is a solution of u * , span{(u * (w), then it can be solvable by theory of high-gain observers via Lyapunov function technique.Now consider the positive definite and proper functions and observe that , where the function φ is locally Lipschitz and bounded.
Remark 2. The proof of Theorem 1 is infinitesimal version of dissipative passivity inequality for system (1)-(3).To prove global asymptotic stability in W × Z × Θ × E we need the exact passivity for (1)-( 3), this will be proved via KYP lemma that is the following result.

Distribution with passive system
In this section, we are showing how the distribution w can be passive.In this work, the general theory summarized in Theorem 1 will be applied to the relevant case of passivity concept.The exosystem w = col(ω, w) can be specified as where S(w) = blk diag(S 1 (w 1 ), . . ., S r (w r )), S i = ( 0 w −w 0 ), here r is a set of harmonic oscillators with constant frequencies ω = (ω 1 , ω 2 , . . ., ω n ) with the values ω and w that are unknown but bounded in a compact set w As based on the above situation, the function u * (ω, w) introduced as u * (ω, w) = Γ (w)ω, Γ (w) = (Γ 1 (w),. . .,Γ r (w)) with Γ i (w) ∈ R 1×2 , and the pair (S(w), Γ (w)) is observable for all ω ∈ Ω 1 and w ∈ W .This is the case for steady state control input required to enforce zero regulation error, which is a linear combination of r harmonics with uncertain frequencies, amplitudes and phases.Definition 1. (See [2].)System ( 1)-( 3) with distribution w is said to passive if there exists a C 1 nonnegative definite storage function V mapping from V : R n → R with V (0) = 0 such that, for all u ∈ R, e ∈ R r and w ∈ W, where τ (w) = (u * (w), L 1 S (u * (w)), . . ., L d−1 S u * (w)) is a solution of ( 1)-(3).Assumption 5. Let the matrix S have above expression and is characterized by the polynomial of the block-diagonal if it satisfies the Lie derivative and L 2 -norm such that where S = S(w)w.The next result is KYP property, which described passivity implies global asymptotic stability.
Then integral Hölder's inequality states that 3) is passive with C r , r 1, the storage function V , taking the derivative with respect to w, t (like the above inequality), clearly implies the theorem statement.Therefore, we can guarantee that system (1)-( 3) is globally asymptotically stable in R d×d × R n × R r .

Examples
The equations of motion of aircraft are described by where x i ∈ R 3 and v i ∈ R 3 are the position and linear velocity of ith aircraft with respect to the inertial frame F 0 = ( e 1 , e 2 , e 3 ) and body fixed frame F i = ( e 1i , e 2i , e 3 ).Its angular velocity, expressed in F i = ( e 1i , e 2i , e 3 ) relative to the fixed fame F 0 , is denoted as y = (y 1 , y 2 , y 3 ) T .The orientation (attitude) of the ith aircraft is represented using the four-element unit quaternion Q i = (q T i , η T i ) T composed of a vector component q i ∈ R 3 and scalar component η i ∈ R, which are subject to the unity constraint q T i q i + η 2 i = 1.
The rotation matrix R(Q i ) related to the unit quaternion Q i that brings the initial frame into body frame.I 3 is 3 × 3 identity matrix, and S(q i ) is the skew-symmetric matrix.The m and g are mass of ith aircraft and gravitation, respectively.The matrix I fi ∈ R 3×3 is the symmetric positive definite constant inertia matrix of the ith aircraft.The scalar T and the vector Γ i represent the magnitude of the thrust applied to the ith vehicle in the direction of e 3i and the external torque applied to the system, which is expressed in F i .

Attitude error dynamics
Let the unit quaternion Q di = (q di , η di ) T represent a desired attitude for the ith aircraft, to be determined later through the control design.The attitude tracking error is described by the discrepancy between the vehicle's attitude and its desired attitude, namely , and is governed by the unit-quaternion dynamics.The angular velocity error vector y = y − R(Q i )y d , where y d is the desired angular velocity of the aircraft, which is related to desired attitude Q di = (q T di , η T di ).Then we define where With the above assumptions, our objective in this work is to design the UAV aircraft control schemes, in terms of such that the vehicles convergent to a prescribed stationary formulation in the presence of communication signals.Moreover, our objective is guaranteed by v → 0, |x i − x j | < for each > 0, i, j ∈ N .
To design a thrust and torque input for the class of under actuated UAVs, equation (32) can be rewritten as The main difficulty in using this extraction algorithm, in this paper, the design of residues can be controlled by )) e 3 that achieves the formation along with communication signals.Note that the term )) e 3 can be regulated as a perturbation term that to be translational dynamics in (33).

Control design reduction (position control design)
To simplify the design of the intermediary translation control and the input torque for each aircraft, which are proposed in this section in two preliminary inputs, that satisfies some of the requirements.Let the auxiliary variable θ associates with state x i , then where φ i ∈ R 3 be the regulated input error signal design, γ be the regulated output design and u i ∈ R 3 be additional input vectors to be designed later.We see that γ is not direct solvable, in this regard, we should introduce some saturation function of priori bounds.Therefore, to satisfy the above requirement, we consider the standard formation stabilization control law The variable k ij is the (i, j)th entry of the weighted adjacency matrix k of the communication signals, which are characterized by the information flow between aircraft.We propose that an intermediary control input for each aircraft is γi = −k θ1 φ − k θ2 φ, where k θ1 , k θ2 are strictly positive scalar gains.By using Therorem 2, the extracted value of the thrust will be used as the real input of the translational dynamics.
We assume that the linear velocity vector is not available for feedback.In other words, we would like to design a linear velocity free global control law that guarantees the boundedness and the asymptotic convergence to zero of the following position and linear velocity tracking errors: The desired trajectory along the UAV aircraft (32) to be tracked to the allowing thrust and torque.
To design a torque input of (32) without linear velocity measurements, we introduce ξ, where ξ ∈ R 3 is design variable, which will be determine later.To achieve our objective and solve the above problems, we introduce the following change of variables.
Let χ = e − ξ be an change variable, and also it will be new error signals, which may depends on explicitly of the linear velocity of the aircraft, that is z = χ = ė − ξ = v i − ξ.Then to design the attitude tracking torque, we introduce the following variable: where y = y − R( Q)y d is angular velocity error vector, y d is the desired angular velocity of the aircraft, and β is a design parameter.Exploiting the rotational dynamics in (32) and expression (34), we can easily show that Ω * = ˙ y − β.It is clear that y = y − y d , which is interlink with new error signals z = z − z, this is depends on the explicit linear velocity of the aircraft.
We define the following input torque for each aircraft: Γ i = I fi ẏ + I fi S(y d ) ẏd ẏ − k q1 β − k q2 q.Therefore,

Simulation results
Simulations results are presented to illustrate the effects of proposed control scheme.In this scheme, we consider some basic parameters: v(0) = (0, 0, 0), g = 9.8, k t1 = 0.1, k t2 = 0.2, k θ1 = k θ2 = 0.5, m = 3kg, and I fi = col(0.1,0.1, 0.1).Among three control models, the position stabilization is most advanced one and the simulations are only presented for this model.Theorem 2 is applied for equation (32), the desired yaw angular velocity y d = 0 is set to zero.In [11], the helicopters model with four control parameters like as track vertical position, lateral, longitudinal, and yaw attitude time reference z r (t), y r (t), x r (t), and ψ r (t) were used, attitude with engine dynamics are independent.But the engine dynamics are not only related to attitude, it is also related to control force and communication signals.In our paper, vehicles attitude and angular velocity are related to quaternion term Q di and control force F i .Comparing [11], our result Fig. 2 shows clear performance of take up and landing.In [11], the mass M and inertia matrix J was defined as 8 and diag(0.18,0.34, 0.28) kg m 2 , and PID is maintained as constant value like as K P = 30, K I = 0.007, K D = 0.185, respectively.In our paper, mass m and inertia matrix I is fixed as 3 and diag(0.1,0.1, 0.1, 0.1) kg m 2 , respectively, and no fixed PID is used.The disturbance is a fast ramp to simulate a gust of wind with MATLAB Simulink, which is applied to the position in 3-D. Figure 2(a) demonstrates the UAV ability to

Figure 2 .
Figure 2. (a) 3D plot of the UAV vehicle trajectory with desired trajectory response; (b) The behavior of state x (solid line) with desired velocity x d (dot line) response with disturbance w(t).

Figure 3 .
Figure 3. (a) The behavior of velocity response; (b) The behavior of attitude q (solid line) with desired attitude q d (dot line) response; (c) The behavior of angular velocity y (solid line) and desired angular y d velocity (dot line) response.