# Attitude Optimal Backstepping Controller Based Quaternion for a UAV

Authors: Kaddouri Djamel; Mokhtari Abdellah; Abdelaziz Benallegue
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

A hierarchical controller design based on nonlinearH∞ theory and backstepping technique is developed for a nonlinear and coupled dynamic attitude system using conventional quaternion based method. The derived controller combines the attractive features of H∞ optimal controller and the advantages of the backstepping technique leading to a control law which avoids winding phenomena. Performance issues of the controller are illustrated in a simulation study made for a four-rotor vertical take-off and landing (VTOL) aerial robot prototype known as the quadrotor aircraft.


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## Body

## 1. Introduction

Control of aerial robots is highly complex and has been the subject of many research papers. The research on Mobile Robots with high degree of autonomy becomes possible, due to the development and reduction of costs on computer, electronic, and mechanic systems [1]. Euler-angle based and conventional quaternion based methods have been extensively employed for spacecraft attitude control. However, the first method suffers from singularity that prohibits large orientation maneuvers, while the second exhibits ambiguity and unwinding phenomena; hence the advantage of a four-parameter attitude representation such as quaternions is the avoidance of singular points in the representation, together with better numerical properties. The attitude control problem of a rigid body has been investigated by several researchers and a wide class of controllers has been developed. In this context Fortuna et al. [2] propose an approach based on two parallel controllers derived in quaternion algebra; a PD (proportional and derivative) feedback controller and a feedforward controller implemented by means of a hypercomplex multilayer perceptron neural network. The proposed controller in [3] is based upon the compensation of the Coriolis and gyroscopic torques and the use of a PD2 feedback structure, where the proportional action is in terms of the vector quaternion and the two derivative actions are in terms of the airframe angular velocity and the vector quaternion velocity. Wang et al. in [4] treat how to improve attitude control performances of roll and pitch channels under both small and large amplitude manoeuvre flight conditions. Zhao et al. in [5] discuss trajectory tracking control for vertical take-off and landing (VTOL) Unmanned Aerial Vehicles (UAVs) using the command filtered backstepping technique and a second-order quaternion filter are developed to filter the quaternion and automatically compute its derivative, which determines the commanded angular rate vector. In Fresk and Nikolakopoulos approach in [6] both the quadrotor’s attitude model and the proposed nonlinear proportional squared (P2) control algorithm have been implemented in the quaternion space, without any transformations and calculations in Euler’s angle space or Direct Cosine Matrix. Moreover Honglei et al. [7] derive a backstepping-based inverse optimal attitude controller which has the property of a maximum convergence rate in the sense of a control Lyapunov function under input torque limitation and the inverse optimal approach is employed to circumvent the difficulty of solving the Hamilton-Jacobi-Bellman (HJB) equation. Likun and Qingchao in [8] construct a Lyapunov function based on attitude tracking errors and angular rates. The stabilizing feedback control law is deduced via Lie derivation of the Lyapunov function. A virtual controller and some command references are introduced by Sun et al. in [9] to asymptotically stabilize the system of the tracking error dynamics. Then, the actual controller and command references are derived by solving a system of linear algebraic equations, such as attitude stabilization, and [10] develops a control law that uses both optimal control and finite-time control techniques which can globally stabilize the attitude of spacecraft system to a set of equilibria. A solution for robust spacecraft attitude control using a nonlinear H∞ control law to stabilize the maneuver in the presence of external disturbance is investigated by [11]. Moreover, a nonlinear H∞ output feedback controller is proposed and coupled to a high-order sliding mode estimator to regulate a UAV (Unmanned Aerial Vehicle) in the presence of the unmatched perturbations [12]. Many searchers like [13] used the globally non-singular unit quaternion representation in a Lyapunov function candidate. Guilherme et al. perform a control structure through a nonlinear H∞ controller to stabilize the rotational movements and a control law based on backstepping approach to track the reference trajectory [14].Combining the backstepping technique withH∞ technique and using the conventional quaternion based methods to stabilize the attitude of spacecraft system to a set of equilibria is the goal of this work. The control strategy must overcome some difficulties such as the highly nonlinear and coupled dynamics more over the dynamic, complex, and unstructured environments which may cause unpredictable disturbances to the control system. So dealing with some states in each step will reduce difficulties and make them restrained. Using the optimal condition of Pontryagin and the optimal control approach to design a robust control law for attitude motion control it has been shown that the resulting control law has excellent performance, as demonstrated by simulations. The paper is outlined as follows. The formulation of the problem is developed in Section 2. Optimal attitude control H∞/backstepping with some mathematical preliminaries are developed in Section 3. Simulation results applied for the quadrotor are discussed in Section 4. Finally, Section 5 presents some conclusions.


## 2. Problem Formulation

LetQ be the quaternion and the dynamic attitude system represented as(1)Q˙t=12Qt⊗Ω¯t,JΩ˙t=-Ωt×JΩt+ut+dt,where J∈R3×3 denotes the inertia matrix of the body and satisfies J=JT>0, Ω¯=0,Ω, and Ω∈R3 is the angular velocity vector of the body in the body-fixed frame, u(t)∈R3 is the control torque vector, and d(t)∈R3 is the external disturbance vector. The attitude quaternion Q(t)∈R4 is defined by Q(t)=(q0(t),q1(t),q2(t),q3(t))T=(q0(t),qv(t))T and the Euclidean norm Q(t)2=1, ∀t≥0. If Qd is the desired quaternion written in dynamic form as (2)Q˙dt=12Qdt⊗Ω¯dtwith Ω¯d=0,Ωd, Ωd∈R3 is the desired angular velocity. The quaternion error in multiplicative form is(3)Qet=Qd-1t⊗Qtor(4)Qdt⊗Qet=Qt.Then the derivative of the above equation gives(5)Q˙dt⊗Qet+Qdt⊗Q˙et=Q˙t,which leads to(6)Q˙dt⊗Qet+Qdt⊗Q˙et=12Qt⊗Ω¯t,Q˙et=12Qd-1t⊗Qt⊗Ω¯t-Qd-1t⊗Q˙dt⊗Qet,Q˙et=12Qet⊗Ω¯t-Ω¯dt⊗Qet;then, (7)Q˙et=12Qet⊗Ω¯t-Qe-1t⊗Ω¯dt⊗Qet.Let(8)Qe-1t⊗Ω¯dt⊗Qet=Ω¯d∗twith(9)Ωd∗t=RTQetΩdt.Using Rodriguez formula one can define the rotation matrix in quaternion representation [15, 16]:(10)RTQet=I+2SQet+2S2Qet.Let an auxiliary angular velocity be defined as (11)Ωauxt=Ωt-Ωd∗t,Ω˙auxt=Ω˙t-Ω˙d∗t,so the system in quaternion error can be represented as(12)Q˙et=12Qet⊗Ω¯auxt;from (9) one can write (13)Ω˙d∗t=R˙TQetΩdt+RTQetΩ˙dt,Ω˙d∗=STΩtRTQetΩdt+RTQetΩ˙dt.S is skew-symmetric which satisfies the condition -S=ST. From (11) we get (14)JΩ˙auxt=JΩ˙t-JΩ˙d∗t,JΩ˙auxt=-Ωt×JΩt+ut+dt-JΩ˙d∗t,JΩ˙auxt=-Ωauxt+Ωd∗t×JΩauxt+Ωd∗t+ut+dt-JΩ˙d∗t;× is the cross product. Finally(15)Ω˙auxt=-J-1Ωauxt×JΩauxt-J-1Ωauxt×JΩd∗t-J-1Ωd∗t×JΩauxt-J-1Ωd∗t×JΩd∗t-STΩtRTQetΩdt-RTQetΩ˙dt+J-1ut+J-1dt;using the skew matrix we obtain(16)Ω˙auxt=-J-1SΩauxtJΩauxt-J-1SΩauxtJΩd∗t-J-1SΩd∗tJΩauxt-J-1SΩd∗tJΩd∗t-STΩtRTQetΩdt-RTQetΩ˙dt+J-1ut+J-1dt.Let(17)vt=ut-SΩauxtJΩd∗t-SΩd∗tJΩd∗t-JSTΩtRTQetΩdt-JRTQetΩ˙dt;then (16) can be written as(18)Ω˙auxt=-J-1SΩauxtJΩauxt-J-1SΩd∗tJΩauxt+J-1vt+J-1dt,Ω˙auxt=-J-1SΩauxtJ-J-1SΩd∗tJΩauxt+Jvt+J-1dt.In that case (18) can be written as(19)Ω˙auxt=A2tΩauxt+B2vt+G2twtwith(20)A2t=-J-1SΩauxtJ-J-1SΩd∗tJ,B2t=J-1;G2t=J-1;wt=dt;taking Qe(t)=(qe0(t),qev(t))T, the development of (12) will lead to the kinematic equation of the rigid body motion described in terms of the attitude quaternion: (21)Q˙et=q˙e0tq˙evt=12EQetΩauxt.E(Qe(t)) is the kinematics Jacobian matrix (see [17]) defined as(22)EQe=-qevTt-Sqevt+qe0tI3×3and the matrix S(qev(t))∈R3×3 denotes a skew-symmetric matrix given by(23)Sqev=0qev3-qev2-qev30qev1qev2-qev10.Let(24)x1t=qevt,x2t=Ωauxt,with(25)Qet=Qd-1t⊗Qt=qe0t,qevtT,B1t=12E1x1t.So the system in hierarchical form based on (22) and (20) will be presented as(26)x˙1t=12-Sqevt+qe0tI3×3x2t=12E1x1tx2t=B1tx2t,x˙2t=A2tx2t+B2tvt+G2tw2t.The performance specification in steady state must lead to the equilibrium point(27)xdt=±1,0,0,0,0,0,0T.This section is a necessary introduction for the following section which deals with the optimal backstepping technique based on the model presented in (26), leading to an optimization problem with constraints and a backstepping controller. The Hamiltonian equation and Riccati formula solution is developed meanwhile.


## 3. Optimal Attitude Control

In this section we are interested in optimal control of systems modeled by differential equations, in finite dimension, defined as follows:(28)x˙=fxt,ut,where x∈Rn1 is the state vector, t∈R is the time variable, and u∈Rn2 is the control input. For such systems, the goal is to determine a controller to bring the system from an initial set to a final set, by minimizing criterion called the cost. In this case, we define the optimal control problem, as(29)min⁡Jx,ux˙=fx,uxt0=x0.Let the cost function with constraint be presented as(30)J¯x,u,λ=Jx,u+∫t1t2λTfx,u,t-x˙tdt.With(31)X×U∋x,u⟶Jx,u=Φxt2,t2+∫t1t2Lxt,ut,tdt,where X×U∋(x,u)↦J(x,u)∈R, X×U is a normed space. Here, X is the space of continuous functions with continuous derivatives R⊃t0,tf→Rn1 and U is the space of continuous functions defined on t0,tf→Rn2. We consider a new function J¯, X×U×M∋(x,u,λ)↦J¯(x,u,λ), where M is the space of differentiable functions R→Rn1 and the vector λ is known as the costate vector. This new function allows us to move from solving an optimization problem with constraints to an optimization problem without constraints. Consider the cost function of the form(32)J¯x,u,λ=Jx,u+∫t0tfλtTfx,u-x˙tdt.We note the Hamiltonian function as follows:(33)Hxt,ut,λt=Lx,u+λtTfx,u.The key point of this theory is the Pontryagin Maximum principle formulated in 1956, according to which the optimal control minimizes the functionJ with the following optimality condition:(34)λ˙tT=-∂H∂xx,u,λ,λTtf=∂Φxtf,tf∂xtf,∂H∂u=0,xt0=x0.The principle goal in designing an optimal control law is to make the steady state converge to the closer equilibrium. The controller is established into two steps (as classical backstepping) passing through a virtual controller and a matrix Pi positive definite computed through Riccati equation. The local stabilizability and detectability are then ensured by the existence of a proper solution of the algebraic Riccati equations. Assuming that system (26) is stabilizable, then the following theorem is established.Theorem 1.
Consider the hierarchical system (26) with the assumption that pair Ai,Bi has to be stabilizable; then there exists a virtual control vi and a positive semidefinite matrix Pi such that the subsystem can be represented in the form(35)z˙it=Aitzit-Aitvi-1t+v˙i-1t-Bitvit-Biitwit,i=1 to n. n is the degree of differentiability of the system (i=1,2; n=2 in our case), with the new variable z defined as(36)zit=vi-1t-ϕi-1tand the virtual backstepping controller v(37)Bitvit=BitRi-1BiTtPizit+v˙i-1t-Aitvi-1t+Pi-1BiTtPi-1zi-1twhich asymptotically stabilizes the disturbance free systemProof.
In backstepping controller each step will be treated as a submodel. The link between different submodels is made using backstepping technique. So let model (26) be represented in error form as(38)zi=vi-1-ϕi-1;differentiating (38) gives(39)z˙i=v˙i-1-ϕ˙i-1and ϕi-1=xi for every Ai=0, and then(40)z˙i=v˙i-1-Aixi-Bivi-Giwi,z˙i=v˙i-1-Aivi-1-zi-Bivi-Giwi,z˙i=Aizi-Aivi-1+v˙i-1-Bivi-Giwi.Since the variable of differential zi depends essentially on the variable vi (actual virtual control) and vi-1 (past virtual control), then one can introduce by analogy the variable zi-1 in model (40). Let the model with the new variable control(41)z˙i=Aizi+Biξi-αi-1zi-1with ξi the new variable control defined as(42)Biξi=-Aivi-1-Bivi+v˙i-1+αi-1zi-1-Giwiξi=-vi+Bi-1v˙i-1-Aivi-1+αi-1zi-1-Giwi.Chosing Lk as (43)Lk=∑i=1k12ziTQizi+12ξkTRkξk+∑i=1k-112ziTPiBiRi-1BiTPizi-γk2wkTwk,Φk=∑i=1k12ziTPiziwith k represents the step number. The z˙i for i=1⋯k-1 will be computed by using recurrence formula; hence the Hamiltonian function will be defined as (44)Hk=∑i=1k12ziTQizi+12ξkTRkξk+∑i=1k-112ziTPiBiRi-1BiTPizi-γk2wkTwk+∑i=1k-1λiTz˙i+λkTz˙k;replacing z˙k from (41) gives(45)Hk=∑i=1k12ziTQizi+12ξkTRkξk+∑i=1k-112ziTPiBiRi-1BiTPizi-γk2wkTwk+∑i=1k-1λiTz˙i+λkTAkzk+Bkξk-αi-1zk-1.Using the optimality conditions for control law leads to(46)∂Hk∂ξk=0⟹Rkξk+BkTλk=0⟹ξk=-Rk-1BkTλk;since (47)λk=∂Φk∂zk=Pkzkthen(48)ξk=-Rk-1BkTPkzk;equaling (42) and (48) will give (49)Bkξk=-BkRk-1BkTPkzk=-Bkvk+v˙k-1-Akvk-1+αk-1zk-1-Gkwk.So the virtual control vi can now be computed:(50)Bkvk=BkRk-1BkTPkzk+v˙k-1-Akvk-1+αk-1zk-1-Gkwk.Equation (38) will help to extract the past z˙i from recurrence technique(51)zi=vi-1-ϕi-1;replacing vi-1 from (50) gives(52)Bi-1zi=Bi-1Ri-1-1Bi-1TPi-1zi-1+v˙i-2-Ai-1vi-2+αi-2zi-2-Gi-1wi-1-Bi-1ϕi-1,Bi-1zi=Bi-1Ri-1-1Bi-1TPi-1zi-1-Ai-1xi-1+zi-1+v˙i-2+αi-2zi-2-Bi-1,i-1wi-1-Bi-1ϕi-1,Bi-1zi=Bi-1Ri-1-1Bi-1TPi-1zi-1-Ai-1zi-1+αi-2zi-2+v˙i-2-Ai-1xi-1-Bi-1ϕi-1-Bi-1,i-1wi-1;using (40) leads to (53)Bi-1zi=Bi-1Ri-1-1Bi-1TPi-1zi-1-Ai-1zi-1+αi-2zi-2+z˙i-1;then z˙i-1 is evaluated:(54)z˙i-1=Bi-1zi+Ai-1-Bi-1Ri-1-1Bi-1TPi-1zi-1-αi-2zi-2;by recurrence z˙i is(55)z˙i=Bizi+1+Ai-BiRi-1BiTPizi-αi-1zi-1.Introducing the new computed variables in the Hamiltonian (45) will give(56)Hk=∑i=1k12ziTQizi+12ξkTRkξk+∑i=1k-112ziTPiBiRi-1BiTPizi+∑i=1k-1λiTBizi+1+Ai-BiRi-1BiTPizi-αi-1zi-1+λkTAkzk+Bkξk-αk-1zk-1-γk2wkTwk.Taking the worst case for wk in linear form [18]:(57)wk=12γk2GkTPkzk.So (56) becomes(58)Hk=∑i=1k12ziTQizi+12ξkTRkξk+∑i=1k-112ziTPiBiRi-1BiTPizi+∑i=1k-1λiTBizi+1+Ai-BiRi-1BiTPizi-αi-1zi-1+λkTAkzk+Bkξk-αk-1zk-1-14γk2zkTPkGkGkTPkzk.Applying the optimality conditions from (58) leads to(59)λ˙i=-∂Hk∂ziTi=1⋯k-1=-Qizi-PiBiRi-1BiTPizi-Bi-1Tλi-1-Ai-BiRi-1BiTPiTλi+Pi+1-1BiTPiTλi+1λ˙i=-Qizi-PiBiRi-1BiTPizi-Bi-1TPi-1zi-1-Ai-BiRi-1BiTPiTPizi+Pi+1-1BiTPiTPi+1zi+1,(60)λ˙i=-Qizi-PiBiRi-1BiTPizi-Bi-1TPi-1zi-1-Ai-BiRi-1BiTPiTPizi+PiBizi+1.For the final step λ˙k is computed:(61)λ˙k=-∂Hk∂zkT,λ˙k=-Qkzk-AkTλk-Bk-1Tλk-1-12γk2PkBkkBkkTPkzk,(62)λ˙k=-Qkzk-AkTPkzk-Bk-1TPk-1zk-1-12γk2PkGkGkTPkzk.Taking (47) and differentiating for i and k: (63)λ˙i=P˙izi+Piz˙ii=1⋯k-1,λ˙k=P˙kzk+Pkz˙k;comparing with (60) and (62) leads to(64)P˙izi+Piz˙i=-Qizi-PiBiRi-1BiTPizi-Bi-1TPi-1zi-1-Ai-BiRi-1BiTPiTPizi+Pi+1-1BiTPiTPi+1zi+1,P˙izi=-Qizi-PiBiRi-1BiTPizi-Bi-1TPi-1zi-1-Ai-BiRi-1BiTPiTPizi+Pi+1-1BiTPiTPi+1zi+1-PiBizi+1+Ai-BiRi-1BiTPizi-αi-1zi-1,P˙izi=-Qi-PiBiRi-1BiTPi+AiTPi+PiAizi-Bi-1TPi-1zi-1+Piαi-1zi-1+PiBizi+1-PiBizi+1;taking Piαi-1=Bi-1TPi-1 then(65)αi-1=Pi-1Bi-1TPi-1;finally the Riccati equation before the final step will be(66)-P˙i=Qi-PiBiRi-1BiTPi+AiTPi+PiAii=1⋯k-1.Let us compute the Riccati equation for the final step using the computed value αi-1:(67)P˙kzk+Pkz˙k=-Qkzk-AkTPkzk-12γk2PkBkkBkkTPkzk-Bk-1TPk-1zk-1,P˙kzk=-Qkzk-AkTPkzk-12γk2PkGkGkTPkzk-Bk-1TPk-1zk-1-PkAkzk+Bkξk-Pk-1Bk-1TPk-1zk-1;replacing the value of ξk from (48): (68)P˙kzk=-Qkzk-AkTPkzk-12γk2PkGkGkTPkzk-Bk-1TPk-1zk-1-PkAkzk-BkRk-1BkTPkzk-Pk-1Bk-1TPk-1zk-1then (69)-P˙kzk=Qk+AkTPk+PkAk+12γk2PkGkGkTPkzk-PkBkRk-1BkTPkzk+Bk-1TPk-1zk-1-Bk-1TPk-1zk-1.Finally the Riccati equation is formulated as(70)-P˙k=APk+PkAk+Pk12γk2GkGkT-BkRk-1BkTPk+Qkand the equation of vk becomes(71)Bkvk=BkRk-1BkTPkzk+v˙k-1-Akvk-1+Pk-1Bk-1TPk-1zk-1-12γk2GkGkTPkzk,Bkvk=BkRk-1BkT-12γk2GkGkTPkzk+v˙k-1-Akvk-1+Pk-1Bk-1TPk-1zk-1,Bkvk=ΓkPkzk+v˙k-1-Akvk-1+Pk-1Bk-1TPk-1zk-1;this will conclude the proof.Step 1.
The algorithm (hierarchical) for computing control law needs initialization. Letv0(t)=x1d(t); ϕ0(t)=x1(t); ϕ1(t)=x2(t); P0=0; B0=0; G1=0; and taking the virtual control v1sign⁡(q0(0)) then the submodel will be (72)x˙1=B1ϕ1x2=B1v1sign⁡q00;z1=v0sign⁡q00-ϕ0t=x1dsign⁡q00-x1t,z˙1=x˙1dsign⁡q00-x˙1t=x˙1dsign⁡q00-B1v1sign⁡q00.Let the model with the new variable control(73)z˙1=A1z1+B1ξ1with (74)A1=0⟹B1ξ1=x˙1dsign⁡q00-B1v1sign⁡q00.Taking (75)L1=12z1TQ1z1+12ξ1TR1ξ1+12z1TP1B1R1-1B1TP1z1,Φ1=12z1TP1z1the Hamiltonian will be for this case(76)H1=12z1TQ1z1+12ξ1TR1ξ1+λ1TB1ξ1,Φ1z1,t=12z1TP1z1;applying the optimality conditions:(77)λ˙1=-∂H1∂qevT=-Q1z1,(78)λ1=∂Φ1z1,t∂z1=P1z1control law is deduced from Hamiltonian equation which reflects the H∞ control law (optimality condition): (79)∂H1∂ξ1=0⟹R1ξ1+B1Tλ1=0⟹ξ1=-R1-1B1Tλ1=-R1-1B1TP1z1for x1d=[0,0,0]T;(80)ξ1=R1-1E1TP1qev;differentiating (78) and comparing with (77) lead to(81)P˙1z1=-P1B1-R1-1B1TP1z1-Q1z1;the Riccati equation became(82)-P˙1=-P1E1R1-1E1TP1+Q1with Q1 being a symmetric matrix and R1 being a diagonal matrix; then the control law ξ1 is calculated as(83)B1ξ1=B1R1-1E1TP1qev=x˙1dsign⁡q00-B1v1sign⁡q00;finally the control law v1 is computed:(84)v1=-R1-1E1-1R1-1P1qvsign⁡q00.Step 2.
Taking the second submodel(85)Ω˙aux=A2Ωaux+B2v+B22w2,x˙2=A2x2+B2ϕ2+B22w2;the model presented in z variable form is(86)z˙2=v˙1sign⁡q00-ϕ˙1=v˙1sign⁡q00-x˙2,z˙2=v˙1sign⁡q00-A2x2-B2v2-B22w2,z˙2=v˙1sign⁡q00+A2z2-v1sign⁡q00-B2v2-B22w2;this equation can be written:(87)z˙2=A2z2+B2ξ2-P2-1B1TP1z1taking the worst case for w2(88)w2=12γ22B22TP2z2with(89)B2ξ2=-B2v2+v˙1sign⁡q00-A2v1sign⁡q00+P2-1B1TP1z1-12γ22B22B22TP2z2.Let (90)Φ2=12z1TP1z1+12z2TP2z2;by recurrence the past z˙1 is calculated:(91)z˙1=B1z2-B1R1-1B1TP1z1;the Hamiltonian will be(92)H2=∑i=1212ziTQizi+12ξ2TR2ξ2+12z1TP1B1R1-1B1TP1z1-14γ22z2TP2B22B22TP2z2+λ1Tz˙1+λ2Tz˙2,H2=12z1TQ1z1+12z2TQ2z2+12z1TB1P1R1-1B1TP1z1-14γ22z2TP2B22B22TP2z2+12ξ2TR2ξ2+λ1TB1z2-B1R1-1B1TP1z1+λ2TA2z2+B2ξ2-P2-1B1TP1z1.Applying the optimality conditions for λ˙1 and λ˙2 will give(93)λ˙1=-Q1z1-P1B1R1-1B1TP1z1+B1R1-1B1TP1Tλ1+P2-1B1TP1Tλ2,λ˙1=-Q1z1+P1B1z2,λ˙2=-Q2z2-A2TP2z2-P1z1-12γ22P2B22B22TP2z2knowing that(94)λ1=∂Φ2∂z1=P1z1,λ2=∂Φ2∂z2=P2z2;then the control law is deduced from Hamiltonian equation (optimality condition):(95)∂H2∂ξ2=0;then(96)∂H2∂ξ2=0⟹R2ξ2+B2Tλ2=0⟹ξ2=-R2-1B2TP2z2;hence (97)λ˙1=P1z˙1+P˙1z1=-Q1z1+P1B1z2,P˙1z1=-Q1z1+P1B1z2-P1B1z2-B1R1-1B1TP1z1-P˙1=-P1B1R1-1B1TP1+Q1.Similarly, (98)λ˙2=P2z˙2+P˙2z2=-Q2z2-A2TP2z2-B1TP1z1-12γ22P2G2G2TP2z2,P˙2z2=-Q2z2-A2TP2z2-B1TP1z1-12γ22P2G2G2TP2z2-P2A2z2+B2ξ2-P2-1B1TP1z1,P˙2z2=-Q2z2-A2TP2z2-B1TP1z1-12γ22P2G2G2TP2z2-P2A2z2-P2B2-R2-1B2TP2z2+P2P2-1B1TP1z1,P˙2z2=-Q2z2-A2TP2z2-P2A2z2+P2B2R2-1B2TP2z2-12γ22P2G2G2TP2z2,-P˙2=A2TP2+P2A2+P212γ22G2G2T-B2R2-1B2TP2+Q2-P˙2=A2TP2+P2A2+P2Γ2P2+Q2.Hence the Riccati equation is determined. Q2 is chosen to be a symmetric matrix. So the control law is (99)B2v2=B2R2-1B2TP2z2+v˙1sign⁡q00-A2v1sign⁡q00+P2-1B1TP1z1-12γ22B22B22TP2z2;in other way (100)z2=v1sign⁡q00-x2,z2=v1sign⁡q00-Ωaux.So (101)v2=R2-1J-1P2v1sign⁡q00-Ωaux+Jv˙1sign⁡q00-A2v1sign⁡q00J-P2-1J-1P1qvsign⁡q00-J12γ22B22B22TP2v1sign⁡q00+Ωaux;replacing A2 by its value the real control u is finally computed:(102)A2=-J-1SΩauxJ-J-1SΩd∗J,u=v2+SΩauxJΩd∗+SΩd∗JΩd∗+JSTΩRTQeΩd+JRTQeΩ˙d.


## 4. Simulation Results

The parameters of the model areJ=diag⁡0.0098,0.0098,0.0176, the parameters of the controller are γ2=150, R1=0.1I3×3, R2=0.03I3×3, and Q1=Q2=I3×3. Note the quaternion can be represented as q0(t)=cos⁡(ϕ), q→v(t)=e→sin⁡(ϕ), and hence the initial location can be easily seen through ϕ(0).Case 1.
Let the initial condition of the quaternion error beQe0=0.8224,0.2226,0.4397,0.3604T, which corresponds to an angle ϕ(0)=34.67°; the matrices P1 and P2 are computed through Riccati equation and γ2 is chosen to achieve good performances. For that case q0(0)>0 and will be stabilized to 1. Figures 1, 2, and 3 show the attitude responses and the control inputs, which gives a convergence in short time. These results are nearby the normal reflect when analyzing the initial location of ϕ which lies in the first quadrant.Figure 1
Quaternion error (Case1).Figure 2
Angular velocity (Case1).Figure 3
Control input (Case1).Case 2.
Taking an initial condition of the quaternion errorQe0=-0.02226,-0.8224,-0.3604,-0.4397T, and ϕ=91.34°, this case will give q0(0)<0 and simulation under closed loop control shows stabilization of q0(0) to -1. This is confirmed by Figures 4, 5, and 6; this will emphasize the fact of ϕ(0) which lies in the second quadrant.Figure 4
Quaternion error (Case2).Figure 5
Angular velocity (Case2).Figure 6
Control input (Case2).


## 5. Conclusion

A hierarchicalH∞/backstepping controller is proposed and theory to derive control law is developed. This combination will emphasize not only the robustness performance but also the choice of the adaptation matrix. The local stabilizability/detectability conditions are thus ensured by the existence of the proper solutions of the unperturbed Riccati equations. Theoretical results are supported by numerical simulations that demonstrate efficiency of the proposed controller design. Further investigation is focused on hierarchical observer/controller.



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*Source: 8573235-2016-03-03.xml*