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Construction of terminal control for one nonlinear system⁎
17th IFAC Workshop on Control Applications of Optimization 17th IFAC Workshop on Control Applications of Optimization Yekaterinburg, Russia, 2018 of online 17th IFAC Workshop onOctober Control 15-19, Applications Optimization Available at www.sciencedirect.com Yekaterinburg, Russia, October 15-19, 2018 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 of Optimization Yekaterinburg, Russia, October 15-19, 2018
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IFAC PapersOnLine 51-32 (2018) 166–168
Construction of terminal control Construction of terminal control Construction of terminal control nonlinear system Construction of terminal control nonlinear system nonlinear system nonlinear system L. N. Lukianova. ∗∗
for for for for
one one one one
L. N. Lukianova. ∗ L. N. Lukianova. ∗ L. N. Lukianova. ∗ ∗ Lomonosov Moscow State University, Moscow, 620990 Russia, State University, Moscow, 620990 Russia, ∗ Lomonosov Moscow (e-mail: [email protected]). State University, Moscow, 620990 Russia, ∗ Lomonosov Moscow (e-mail: [email protected]). Lomonosov Moscow State University, Moscow, 620990 Russia, (e-mail: [email protected]). (e-mail: [email protected]). Abstract: A A mathematical mathematical model model of of motion motion of of aa nonlinear nonlinear controlled controlled system system describing describing the the Abstract: Abstract: A mathematical model of motion of a nonlinear controlled system describing the dynamics of a four-screw helicopter with a rotary device for engines is considered. The equations dynamics of A a four-screw helicopter a rotary device for engines is considered. The equations Abstract: mathematical model with of motion a nonlinear controlled system describing of motion motion of ofathe the model in in question differ fromofsimilar similar models in the the order of the the ones in the the dynamics four-screw helicopter with a rotary device for engines is considered. The equations of of model question differ from models in order of ones in the dynamics of a four-screw helicopter with a rotary device for engines is considered. The equations equations for the dynamics of a number of phase variables of the model. The problem of terminal of motion of model in question differ from similar models in the order of the ones in the equations for dynamics of a number of phase variables of the model. The problem of terminal of motion of the the model of in phase question differ from similar models in the of the ones in the control in for the presence constraints is considered. considered. To solve it, order the method method of dynamic equations the dynamics of a number of phase variables ofTo thesolve model. The problem of dynamic terminal control in the presence of phase constraints is it, the of equations for the dynamics ofofa the number of phase variables ofTo theThe model. The problem of dynamic terminal control in the presence of phase constraints is considered. solve it, the method of linearization of the location control function is applied. values of the parameters are linearization the location of theconstraints control function is applied. values the parameters are control in theof presence of phase is considered. To The solve it,not theof method of dynamic found under which the solution of the problem by this method does contain singularities linearization of the location of the control function is applied. The values of the parameters are found under of which the solution the problem by isthis method notofcontain singularities linearization location of theof applied. Thedoes values thephase parameters are and satisfies satisfies thetheimposed imposed condition of problem thefunction problem with constraints on the the variables. found under which the solution ofcontrol the by this method does not contain singularities and the condition of the problem with constraints on phase variables. found under which the solution of the problem by this method does not contain singularities For parameters parameters satisfying these conditions conditions numerically Terminal controls andphase corresponding and satisfies thesatisfying imposed condition of the problem withTerminal constraints on the variables. For these numerically controls and corresponding and satisfiesare theconstructed. imposed condition of the problem withTerminal constraints on the variables. trajectories For parameters satisfying these conditions numerically controls andphase corresponding trajectories are constructed. For parameters satisfying these conditions numerically Terminal controls and corresponding trajectories are constructed. © 2018, IFACare (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. trajectories constructed. Keywords: terminal terminal control, control, nonlinear nonlinear control control problem, problem, boundary boundary value value problem. problem. Keywords: Keywords: terminal control, nonlinear control problem, boundary value problem. Keywords: terminal control, nonlinear control problem, boundary value problem. 1. INTRODUCTION INTRODUCTION The 1. The components components of of the the vector vector (u (u11 ,, u u22 ,, u u33 )) are are controllable controllable 1 1. INTRODUCTION parameters; g = 9.8 is a constant, T > 0, T The components of the vector (u , u , u ) are controllable 1 2 3 parameters; g = 9.8 is a constant, T > 0, T ∈ ∈ R R11 .. 1. INTRODUCTION The components of the vector (u , u , u ) are controllable 1 2 3 A mathematical model of the motion of a nonlinear conparameters; g = 9.8 is a constant, T > 0, T ∈ R . The controls consists of piecewise A mathematical model of the motion of a nonlinear con- parameters; The class class of ofg admissible admissible consists = 9.8 is acontrols constant, T initial > of 0,position Tpiecewise ∈ R1is. A mathematical model of the motion of a nonlinear controllable system describing the dynamics of a four-screw differentiable functions t, t ∈ [0, T ]. The The class of admissible controls consists of piecewise trollable system describing themotion dynamics a four-screw differentiable t,π tcontrols ∈ [0, T ]. consists The position is A mathematical model the of a of π πinitial class of functions admissible trollable system the dynamics ofnonlinear a four-screw helicopter with aadescribing rotaryofdevice device for motors motors [1,2,3] is concon- The given: x (− ), x ˙˙ 0of,, yyposition ˙˙ 0piecewise ,, zz˙˙0 ,, and π differentiable functions t,π2 t,, ππ∈ [0, T ].(− The is 0 ,, y 0 ,, z 0 ,, θ 0 ∈ 0 ∈ helicopter with rotary for [1,2,3] is con2ϕ 2 ,, π 2initial given: x y z θ ∈ (− ϕ ∈ (− ), x and 0 0 0 0 0 0 0 0 trollable system describing the dynamics of a four-screw π π differentiable functions t, ∈ T ].(− The 2,ty˙π 2 , [0, 2 π 2initial position is helicopter with a rotary motors [1,2,3] is con- end sidered. The The equations of device motionfor of the the model in question question position: x , y , z , x ˙ z ˙ . given: x , y , z , θ ∈ (− , ϕ ∈ , ), x ˙ , y ˙ , z ˙ , and 1 1 1 1 1 1 0 0 0 0 0 0 0 0 sidered. equations of motion of model in 2 2 2 2 end position: , y , z , x˙ 1π,,y˙π1 ,ϕz˙01 .∈ (− π , π ), x˙ 0 , y˙ 0 , z˙0 , and helicopter a rotary device motors [1,2,3] is congiven: x0 , y0 , zx sidered. equations motion of the model in differ in in The thewith order of the theofones ones in for the equations forquestion the dy- end position: x011,,θy011 ,∈z11(− , x˙ 12, y˙21 , z˙1 . 2 2 differ the order of in the equations for the dysidered. The equations of motion of the model in question Problem of terminal of in end position: x ˙ 1 , y˙ 1 consists , z˙1 . 1 , y1 , zcontrol 1, x differ in the order of the ones in the equations for the dynamics of a number of phase variables of the model. When Problem of terminal control consists of in the the determinadeterminanamics ofthe a number ofthe phase variables of the model. When differ in order of ones in the equations for the dy(t), u (t), u (t)) of the of tion of the control (u Problem of terminal control consists of in thetrajectory determina1 2 3 namics of a number of phase variables of the model. When solving the problem of terminal control by the dynamic (t), u (t), u (t)) of the trajectory of tion of the control (u 1 2 3 Problem of control terminal consists of inthe the determinasolving thea number problemofofphase terminal control by model. the dynamic (t),initial u2 (t), uposition trajectory of tion system of the (ucontrol namics of variables ofathe When the (1) from the the finite one 1 3 (t)) of to solving the problem of terminal control by the dynamic linearization method [4,5,6] for such system, reverse the system (1) from the initial position to the finite one π π (t), u (t), u (t)) of the trajectory of tion of the control (u linearization method [4,5,6] for such a system, reverse 1 2 3 θ(t) ∈ to solving the problem of terminal control by the dynamic in time T , when the constraints (− , ), ϕ(t) π π the system (1) from the initial position the finite one linearization method [4,5,6] for such a system, reverse in 2 , 2 ), ϕ(t) ∈ selection occurs, occurs, which depends on the the phase variables variables time thethe constraints θ(t) ∈ to (−the ∈ π T , when π π π the system (1) from initial position finite one 2 2 selection which depends on phase linearization method [4,5,6] forforsuch a system, reverse (− [0, T ]. π in time Ttt ,∈ when the constraints θ(t) ∈ (− π2 , π2 ), ϕ(t) ∈ selection occurs, which depends on the phase variables 2 ), 2 ,, π and does not have the inverse all values of the vari(− ), ∈ [0, T ]. π time 2 π and does occurs, not havewhich the inverse foronallthe values of variables the vari- in selection depends , 2 ),Tt ,∈when [0, T ].the constraints θ(t) ∈ (− 2 , 2 ), ϕ(t) ∈ and not have the inverse for all of the vari- (− ablesdoes [5]. The The boundary conditions forvalues thephase initial system (− π22 , π22 ), t ∈ [0, T ]. ables [5]. boundary conditions for the initial system and does not have the inverse for all values of the variables [5].partially The boundary conditions for conditions the initial for system and the the partially selectable boundary conditions for the 2.1 Translation of the system (1) to a position with zero and selectable boundary the 2.1 Translation of the system (1) to a position with zero ables [5].partially The boundary conditions for the initial system and the selectable boundary conditions for the auxiliary linearized system, under the control construction values variables ϕ.the system (1) to a position with zero 2.1 Translation auxiliary linearizedselectable system, under the control construction values variables θ, θ,of and the partially boundary conditions for the 2.1 Translation ofϕ. the system (1) to a position with zero auxiliary linearized system, under the control construction method under consideration, can generate controls and values variables θ, method under consideration, canthe generate and values variables θ, ϕ. auxiliary linearized system, under control controls construction ϕ. method under consideration, can generate controls and trajectories that do not satisfy the imposed constraints. trajectories thatconsideration, do not satisfycan thegenerate imposedcontrols constraints. controls u uii (t), (t), ii = = 1, 1, 2, 2, 3 3 chosen chosen Lemma 1. 1. The The tt ∈ ∈ [0, [0, T T11 ]] controls method under and Lemma trajectories that do not satisfy thecondition imposedfor constraints. These circumstances, circumstances, as well well as the the condition for the phase phase for the time interval, in the shape of ] controls u (t), i = 1, 2, 3 chosen Lemma 1. The t ∈ [0, T These as as the 1 i for the time interval, in the shape of trajectories that do not satisfy the imposed constraints. ] controls ui (t), i = 1, 2, 3 chosen Lemma 1. The t ∈ [0,inT1the These circumstances, as require well as the condition for the phase variables of the the model, model, require finding the parameters parameters of for the time interval, shape of variables of finding the of These circumstances, as require well as conditions the condition for theIn phase for the time interval, in the shape of variables of the model, finding the parameters of the system under which these are met. the the system whichrequire these conditions are met. In the g variables of under thewemodel, finding parameters of the system under which these conditions are met.and In the present paper give solution These the questions and g present paper we give aa solution These questions the u1 (t) the system under which these conditions are met. In the gθθ00 present we give a solution for These questions and the results of ofpaper numerical calculations for the model model parameters u1 (t) − gθ0 results numerical calculations the parameters = max{|θ0 |, |ϕ0 |}. (2) (t) u − present paper we give a solution These questions and the 2 1 = 1 ,, T T u (t) results of numerical calculations for the model parameters (2) = 1 of the nonlinear system under consideration. 2 1 = max{|θ0 |, |ϕ0 |}. − T (t) = u θ01 ,T ϕ of the nonlinear system under consideration. 1 (t) T = max{|θ |, |ϕ |}. (2) u results of numerical calculations for the model parameters 3 2 1 0 0 ϕ T 0 − u32 (t) (t) = − of the nonlinear system under consideration. 1 , , T = max{|θ |, |ϕ |}. u (2) − , 1 0 0 ϕ T 0 u (t) T of nonlinear systemOF under consideration. 3 2. the THE PROBLEM TERMINAL CONTROL FOR FOR T1101 , −ϕ 2. THE PROBLEM OF TERMINAL CONTROL u3 (t) − T1 , of the system (1) into a state 2. THE PROBLEMMODEL OF TERMINAL CONTROL FOR NONLINEAR MODEL WITH CONTROLS CONTROLS translate NONLINEAR WITH translate the the phase phase vector vector T1 of the system (1) into a state 2. THE PROBLEMMODEL OF TERMINAL CONTROL FOR NONLINEAR WITH CONTROLS translate (T the1 ),phase vector of the system (1) into a state y(T z(T θ(T = 0, = 0, 1 ), 1 ), 1) 1) NONLINEAR MODEL WITH CONTROLS translate the phase vector the system into ), y(T ), z(T θ(T 0, ϕ(T ϕ(T(1) 0, a state (T 1 1 1 ),of 1) = 1) = Consider the motion of the vector (x, y, z, θ, ϕ) as a funcConsider the motion of the vector (x, y, z, θ, ϕ) as a funcz(T),1 ), )= (T1 ), y(T1x), (T 0, ϕ(T1 ) = 0, yy θ(T (T11), (3) (T Consider thett motion the vector y, z, θ, ϕ) as a function of of time time ∈ [0, [0, T T ]] of satisfying the(x, equations: 1 ), z(T θ(T 0,1 ). ϕ(T ) = 0, (T1 ), y(T1x), (T ),)zz= (3) tion ∈ satisfying the equations: 1 1 ), (T1 1 (T1 ). 1 Consider the motion of the vector (x, y, z, θ, ϕ) as a func(T ), y (T ), z (T ). (3) x tion of time t ∈ [0, T ] satisfying the equations: 1 1 1 Here we have the relation Here we have the relation (3) x (T1 ), y (T1 ), z (T1 ). tion of time t ∈ [0, T ] satisfying the equations: Here we have the relation x x = − − sin sin θu θu1 ,, (4) (T1 ) = y (T1 ) = z (T1 ) = 0, x00 ,, x x˙˙ 00 ,, x x11 ,, x x˙˙ 11 ,, x ¨¨ = Here we have x relation (T ) = y (T ) = z (T ) = 0, (4) xthe 1 1 1 1 , 1 ,, x = cos − sin θu1ϕu yy¨¨ = θθ sin yyx00,, yyx˙˙00,,, yyx11,, yyx˙˙11,,, (4) x (T1 ) = y (T1 ) = z (T1 ) = 0, cos sin ϕu 1 00 , x 00 , x11 , x 11 , x ¨ = − sin θu , x ˙ ˙ (T ) = y (T ) = z (T = 0, (4) x 1 1 1 1 )integrating Boundary conditions (3) are found by (1) ununz cos ϕu − g, z , z ˙ , z , z ˙ , y ¨ = cos θ sin ϕu , y y y y (1) P : 1 − g, z0 , z˙0 , z1 , z˙1 , 1 Boundary conditions (3) are found by integrating (1) z ¨ = cos θ cos ϕu (1) P : y¨˙ = cos θ sin ϕu11, yz000,, zy˙˙000,, yz111,, zy˙˙111,, der controls (2). The relations (4) are found by substituBoundary conditions (3) are found by integrating (1) unθ cos ϕu − g, P : zθ¨˙ = cos 1 (1) , θ u controls (2). The (3) relations (4) are found by substituu22 , θ cos ϕu1 − g, θz000 , z˙0 , z1 , z˙1 , Boundary are found by integrating (1) unzθ¨˙ = cos (1) der P : ϕ tion controls (3) in in conditions (1). The relations der (2). (4) are found by substituu θϕ 2 00,.. 3 ,, tion (3) (1). ϕθθ˙˙ = = u ϕ der controls (2). The relations (4) are found by substitu3, 0, θ = u tion (3) in (1). 2 0 = u3 ,out at financial support ϕ0 . of the RNF, project Due to the stationarity of the system (1), we next consider The work wasϕ˙ carried (3)the in (1). The work wasϕ˙ carried = u3 ,out at financial support ϕ0 . of the RNF, project tion Due to stationarity of the system (1), we next consider the system (1) under the initial condition (3)next andconsider T1 = 14–11–00539 Due to the stationarity of initial the system (1), we The work was carried out at financial support of the RNF, project = 0. 0. the system (1) under the condition (3) and T 14–11–00539 Due to the stationarity of initial the system (1), we The work was carried out at financial support of the RNF, project the system (1) under the condition (3)next andconsider T11 = 0. 14–11–00539 = 0. the system (1) under the initial condition (3) and T 14–11–00539 1 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 166 Copyright 2018 IFAC 166 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 166 10.1016/j.ifacol.2018.11.374 Copyright © 2018 IFAC 166
IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018 L.N. Lukianova / IFAC PapersOnLine 51-32 (2018) 166–168
2.2 The linearization of order one for the first three equations of the system (1). Consider the first three equations of the system (1) and will, at this stage, consider (u1 , θ, ϕ) as controlled parameters: 0 x ¨ − sin(θ) 0 0 u1 y¨ = cos(θ) sin(ϕ) 0 0 u1 + 0 . −g z¨ cos(θ) cos(ϕ) 0 0 u1
(5)
We differentiate both sides of the equation (5) in time (2; 3): ... x − cos(θ)θ u1 , ... y ... = − sin(θ)θ sin(ϕ)u1 + z − sin(θ)θ cos(ϕ)u1 − sin(θ)u 1 + + cos(θ) cos(ϕ)ϕ u1 + u 1 cos(θ) sin(ϕ) , − cos(θ) sin(ϕ)ϕ u1 + u 1 cos(θ) cos(ϕ)
and write the right-hand side in the matrix form ... x u1 ... y = A θ , (6) ... z ϕ 0 − sin(θ) − cos(θ)θ A = cos(θ) sin(ϕ) − sin(θ) sin(ϕ)u1 cos(θ) cos(ϕ)u1 , cos(θ) cos(ϕ) − sin(θ) cos(ϕ)u1 − cos(θ) sin(ϕ)u1
where A is the 3×3 matrix, det A = − cos(θ)u21 . Assuming cos(θ)u1 = 0, calculating A−1 and introducing the vector (v1 , v2 , v3 ): ... x v1 ... y = v2 , ... z v3
(7)
from (6) we have − sin(θ) cos(θ) sin(ϕ) cos(θ) cos(ϕ) u˙ 1 cos(θ) sin(θ) sin(ϕ) sin(θ) cos(ϕ) v1 − − − θ˙ = v2 , u1 u1 u1 v sin(ϕ) cos(ϕ) ϕ˙ 3 − 0 u1 cos(θ) u1 cos(θ) (8) with initial condition u1 (0) θ(0) . ϕ(0) The vector (v1 , v2 , v3 ) is chosen from (7) for the ”reference” trajectory w = (x, y, z) ∈ R3 , satisfying the boundary conditions (3): t2 w(t) = w0 + w˙ 0 t + w ¨ 0 + d 0 t 3 + d 1 t4 , (9) 2 4A − BT BT − 3A , d1 = . d0 = T3 T4 A = w1 − w0 + w˙ 0 T − w ¨0
(10)
T2 , 2
¨0 T. B = w˙ 1 − w˙ 0 − w ... w(t) = v(t) = 6d0 (T ) + 24td1 (T ). (11) ¨0 = 0. On the Here u1 (0) = g, T > 0 is a free parameter, w time interval t ∈ [0, T ], the controls ui (t), i = 1, 2, 3 will 167
167
be chosen as the solutions of the system (8) for a specially chosen initial condition u1 (0) = g. Lemma 2. For the function v(t) (11), the relation |vi (t)| ≤ c1 T 2 , t ∈ [0, T ], where the constant c1 depends on the boundary conditions (3). Proof. From the relations (9) - (11), for w ¨0 = 0, T ≥ 1 we have 6 ||v|| = 6||d0 || + 24t||d1 || ≤ 3 (4||A|| + ||B||T )+ T 24 72t 6 24t 24t + 4 (3||A||+||B||T ) = A||( 3 + 4 )+||B||( 2 + 3 ) ≤ T T T T T T 2 96 ≤ ||w1 − w0 + w˙ 0 T − w ¨0 || 3 + 2 T 30 c1 ¨0 T 2 ≤ 2 . + w˙ 1 − w˙ 0 − w T T The following assertions show that for u1 (0) = g there exists T > 0 for which the solution of the system (8) exists and is nonlocally extendible. Consider the first equation of the system (8). Lemma 3. When choosing the initial value u1 (0) = g and 3c1 T > g− for which u1 (0) ≥ 3cT1 + , where the constant c1 > 0 depends on the boundary conditions (3), > 0, the solution of the first equation of the system (8) satisfies the condition u1 (0) + 3cT1 ≥ u1 (t) ≥ u1 (0) − 3cT1 ≥ . Proof. Consider the first equation of the system (8). By 1 Lemma 1, |u 1 (t)| ≤ 3c T 2 . Consequently u1 (t) ≥ u1 (0) − 3c1 3c1 ≥ if t ∈ [0, T ], u1 (0) ≥ 3cT1 + T 2 t ≥ u1 (0) − T psilon, where > 0. On the other hand u1 (t) ≤ u1 (0) + 3c1 3c1 T 2 t ≤ u1 (0) + T if t ∈ [0, T ]. Thus, the first equation (8) has a solution, under the indicated constraints, for any continuous functions θ(t), ϕ(t), v(t) and u1 (0) ≥ 3cT1 + . Solution the first equation (8), estimates 3c1 3c1 ≥ u1 (t) ≥ u1 (0) − ≥ , t ∈ [0, T ]. 2g − ≥ u1 (0) + T T Lemma 4. When the conditions of Lemma 2 are satisfied, the solution of the second equation of the system (8) satisfies the condition θ0 = 0, θ(t) ∈ (− π2 , π2 ), t ∈ [0, T ]. Proof. Consider the second equation of the system (8). As 1 u1 (t) ≥ , t ∈ [0, T ], then |θ (t)| ≤ 1 3c T 2 . Consequently 3c1 3c1 1 − T ≤ θ(t) ≤ + T , t ∈ [0, T ] and with c1 > 0, T > 6c π we have θ(t) ∈ (− π2 , π2 ). Lemma 5. Under the conditions of Lemmas 1-4, the solution of the third equations of the system (8) for ϕ0 = 0 satisfies the condition ϕ(t) ∈ (− π2 , π2 ), t ∈ [0, T ]. Proof. Consider the third equation of the system (8). As u1 (t) ≥ , cos(θ(t)) ≥ cos(1) > 0, t ∈ [0, T ], then 2c1 1 |ϕ (t)| ≤ cos(1) T 2 . Consequently 2c1 2c1 ≤ ϕ(t) ≤ cos(1)T , t ∈ [0, T ]. When c1 > 0, T > − cos(1)T 4c1 π π π cos(1) we have ϕ(t) ∈ (− 2 , 2 ).
The values of the system parameters (8): u1 (0) = g, 6c1 3c1 4c1 , , }. (12) T ≥ max{ π cos(1) π g −
IFAC CAO 2018 168 Yekaterinburg, Russia, October 15-19, 2018 L.N. Lukianova / IFAC PapersOnLine 51-32 (2018) 166–168
Fig. 1. x(t)
Fig. 2. y(t)
Fig. 5. ϕ(t)
Fig. 6. u2 (t)
Fig. 3. z(t)
Fig. 4. u1 (t)
Fig. 7. u3 (t)
Fig. 8. θ(t)
Theorem. The control u1 (t), u2 (t), u3 (t), which has the form (2) for t ∈ [0, T1 ] and is obtained from the solutions of system (8) for t ∈ [T1 , T1 + T ], with u1 (0) = g and T satisfying (1.12) solves the problem of terminal control. 2.3 Solution Algorithm 1. Construction of the reference curve (9) according to the boundary conditions (3). 2. Finding the parameter v(t) ∈ R3 (11).
3. Solution of the system (8) for u1 (0) = g and T . 4. The solution of the system (1) for u1 (t), θ(t), ϕ(t) of item 3. ˙ u3 (t) = ϕ(t) ˙ from the 5. Finding the controls u2 (t) = θ(t), second and third equations of the system (8). We present the results of calculations of the reference trajectory, the linearized system and the initial system using the linearization method for the following boundary conditions: x0 = −6; x˙ 0 = −1; x1 = 6; x˙ 1 = 1; y0 = −5; y˙ 0 = 0.5; y1 = 5; y˙ 1 = −0.4; z0 = 4; z˙0 = 1; z1 = 4; z˙1 = −0.1; ϕ0 = 0; θ0 = −0.1; T = 10; u0 = 4; REFERENCES 1.Gur’yanov A. Ye. Simulation of quadrocopter control, Engineering Bulletin, 08, August 2014, c 522-534, http://engbul.bmstu.ru/doc/723331.html 2.S.Salazar-Cruz, F.Kendoul, R.Lozano, I.Fantoni RealTime Stabilization of a Small Three-Rotor Aircraft, IEEE Transactions on aerospace and electronic systems, Vol. 44, No. 2 April 2008 p.783-794. 168
Fig. 9. (xt , yt , zt ) 3.P.Castillo, R.Lozano, A.Dzul Modeling and control of mini-flying machines, Springer, 2005, - 259 p. 4.Isidori A. Nonlinear control Systems, Springer, 1994. 5.Khalil H.K. Nonlinear Systems, Prentice Hall, 2002. 6.Kim D.P. Theory of automatic control. T. 2. Multidimensional, nonlinear, optimal and adaptive systems: Proc. allowance. - Moscow: Fizmatlit, 2004. - 464 p.
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IFAC PapersOnLine 51-32 (2018) 166–168
Construction of terminal control Construction of terminal control Construction of terminal control nonlinear system Construction of terminal control nonlinear system nonlinear system nonlinear system L. N. Lukianova. ∗∗
for for for for
one one one one
L. N. Lukianova. ∗ L. N. Lukianova. ∗ L. N. Lukianova. ∗ ∗ Lomonosov Moscow State University, Moscow, 620990 Russia, State University, Moscow, 620990 Russia, ∗ Lomonosov Moscow (e-mail: [email protected]). State University, Moscow, 620990 Russia, ∗ Lomonosov Moscow (e-mail: [email protected]). Lomonosov Moscow State University, Moscow, 620990 Russia, (e-mail: [email protected]). (e-mail: [email protected]). Abstract: A A mathematical mathematical model model of of motion motion of of aa nonlinear nonlinear controlled controlled system system describing describing the the Abstract: Abstract: A mathematical model of motion of a nonlinear controlled system describing the dynamics of a four-screw helicopter with a rotary device for engines is considered. The equations dynamics of A a four-screw helicopter a rotary device for engines is considered. The equations Abstract: mathematical model with of motion a nonlinear controlled system describing of motion motion of ofathe the model in in question differ fromofsimilar similar models in the the order of the the ones in the the dynamics four-screw helicopter with a rotary device for engines is considered. The equations of of model question differ from models in order of ones in the dynamics of a four-screw helicopter with a rotary device for engines is considered. The equations equations for the dynamics of a number of phase variables of the model. The problem of terminal of motion of model in question differ from similar models in the order of the ones in the equations for dynamics of a number of phase variables of the model. The problem of terminal of motion of the the model of in phase question differ from similar models in the of the ones in the control in for the presence constraints is considered. considered. To solve it, order the method method of dynamic equations the dynamics of a number of phase variables ofTo thesolve model. The problem of dynamic terminal control in the presence of phase constraints is it, the of equations for the dynamics ofofa the number of phase variables ofTo theThe model. The problem of dynamic terminal control in the presence of phase constraints is considered. solve it, the method of linearization of the location control function is applied. values of the parameters are linearization the location of theconstraints control function is applied. values the parameters are control in theof presence of phase is considered. To The solve it,not theof method of dynamic found under which the solution of the problem by this method does contain singularities linearization of the location of the control function is applied. The values of the parameters are found under of which the solution the problem by isthis method notofcontain singularities linearization location of theof applied. Thedoes values thephase parameters are and satisfies satisfies thetheimposed imposed condition of problem thefunction problem with constraints on the the variables. found under which the solution ofcontrol the by this method does not contain singularities and the condition of the problem with constraints on phase variables. found under which the solution of the problem by this method does not contain singularities For parameters parameters satisfying these conditions conditions numerically Terminal controls andphase corresponding and satisfies thesatisfying imposed condition of the problem withTerminal constraints on the variables. For these numerically controls and corresponding and satisfiesare theconstructed. imposed condition of the problem withTerminal constraints on the variables. trajectories For parameters satisfying these conditions numerically controls andphase corresponding trajectories are constructed. For parameters satisfying these conditions numerically Terminal controls and corresponding trajectories are constructed. © 2018, IFACare (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. trajectories constructed. Keywords: terminal terminal control, control, nonlinear nonlinear control control problem, problem, boundary boundary value value problem. problem. Keywords: Keywords: terminal control, nonlinear control problem, boundary value problem. Keywords: terminal control, nonlinear control problem, boundary value problem. 1. INTRODUCTION INTRODUCTION The 1. The components components of of the the vector vector (u (u11 ,, u u22 ,, u u33 )) are are controllable controllable 1 1. INTRODUCTION parameters; g = 9.8 is a constant, T > 0, T The components of the vector (u , u , u ) are controllable 1 2 3 parameters; g = 9.8 is a constant, T > 0, T ∈ ∈ R R11 .. 1. INTRODUCTION The components of the vector (u , u , u ) are controllable 1 2 3 A mathematical model of the motion of a nonlinear conparameters; g = 9.8 is a constant, T > 0, T ∈ R . The controls consists of piecewise A mathematical model of the motion of a nonlinear con- parameters; The class class of ofg admissible admissible consists = 9.8 is acontrols constant, T initial > of 0,position Tpiecewise ∈ R1is. A mathematical model of the motion of a nonlinear controllable system describing the dynamics of a four-screw differentiable functions t, t ∈ [0, T ]. The The class of admissible controls consists of piecewise trollable system describing themotion dynamics a four-screw differentiable t,π tcontrols ∈ [0, T ]. consists The position is A mathematical model the of a of π πinitial class of functions admissible trollable system the dynamics ofnonlinear a four-screw helicopter with aadescribing rotaryofdevice device for motors motors [1,2,3] is concon- The given: x (− ), x ˙˙ 0of,, yyposition ˙˙ 0piecewise ,, zz˙˙0 ,, and π differentiable functions t,π2 t,, ππ∈ [0, T ].(− The is 0 ,, y 0 ,, z 0 ,, θ 0 ∈ 0 ∈ helicopter with rotary for [1,2,3] is con2ϕ 2 ,, π 2initial given: x y z θ ∈ (− ϕ ∈ (− ), x and 0 0 0 0 0 0 0 0 trollable system describing the dynamics of a four-screw π π differentiable functions t, ∈ T ].(− The 2,ty˙π 2 , [0, 2 π 2initial position is helicopter with a rotary motors [1,2,3] is con- end sidered. The The equations of device motionfor of the the model in question question position: x , y , z , x ˙ z ˙ . given: x , y , z , θ ∈ (− , ϕ ∈ , ), x ˙ , y ˙ , z ˙ , and 1 1 1 1 1 1 0 0 0 0 0 0 0 0 sidered. equations of motion of model in 2 2 2 2 end position: , y , z , x˙ 1π,,y˙π1 ,ϕz˙01 .∈ (− π , π ), x˙ 0 , y˙ 0 , z˙0 , and helicopter a rotary device motors [1,2,3] is congiven: x0 , y0 , zx sidered. equations motion of the model in differ in in The thewith order of the theofones ones in for the equations forquestion the dy- end position: x011,,θy011 ,∈z11(− , x˙ 12, y˙21 , z˙1 . 2 2 differ the order of in the equations for the dysidered. The equations of motion of the model in question Problem of terminal of in end position: x ˙ 1 , y˙ 1 consists , z˙1 . 1 , y1 , zcontrol 1, x differ in the order of the ones in the equations for the dynamics of a number of phase variables of the model. When Problem of terminal control consists of in the the determinadeterminanamics ofthe a number ofthe phase variables of the model. When differ in order of ones in the equations for the dy(t), u (t), u (t)) of the of tion of the control (u Problem of terminal control consists of in thetrajectory determina1 2 3 namics of a number of phase variables of the model. When solving the problem of terminal control by the dynamic (t), u (t), u (t)) of the trajectory of tion of the control (u 1 2 3 Problem of control terminal consists of inthe the determinasolving thea number problemofofphase terminal control by model. the dynamic (t),initial u2 (t), uposition trajectory of tion system of the (ucontrol namics of variables ofathe When the (1) from the the finite one 1 3 (t)) of to solving the problem of terminal control by the dynamic linearization method [4,5,6] for such system, reverse the system (1) from the initial position to the finite one π π (t), u (t), u (t)) of the trajectory of tion of the control (u linearization method [4,5,6] for such a system, reverse 1 2 3 θ(t) ∈ to solving the problem of terminal control by the dynamic in time T , when the constraints (− , ), ϕ(t) π π the system (1) from the initial position the finite one linearization method [4,5,6] for such a system, reverse in 2 , 2 ), ϕ(t) ∈ selection occurs, occurs, which depends on the the phase variables variables time thethe constraints θ(t) ∈ to (−the ∈ π T , when π π π the system (1) from initial position finite one 2 2 selection which depends on phase linearization method [4,5,6] forforsuch a system, reverse (− [0, T ]. π in time Ttt ,∈ when the constraints θ(t) ∈ (− π2 , π2 ), ϕ(t) ∈ selection occurs, which depends on the phase variables 2 ), 2 ,, π and does not have the inverse all values of the vari(− ), ∈ [0, T ]. π time 2 π and does occurs, not havewhich the inverse foronallthe values of variables the vari- in selection depends , 2 ),Tt ,∈when [0, T ].the constraints θ(t) ∈ (− 2 , 2 ), ϕ(t) ∈ and not have the inverse for all of the vari- (− ablesdoes [5]. The The boundary conditions forvalues thephase initial system (− π22 , π22 ), t ∈ [0, T ]. ables [5]. boundary conditions for the initial system and does not have the inverse for all values of the variables [5].partially The boundary conditions for conditions the initial for system and the the partially selectable boundary conditions for the 2.1 Translation of the system (1) to a position with zero and selectable boundary the 2.1 Translation of the system (1) to a position with zero ables [5].partially The boundary conditions for the initial system and the selectable boundary conditions for the auxiliary linearized system, under the control construction values variables ϕ.the system (1) to a position with zero 2.1 Translation auxiliary linearizedselectable system, under the control construction values variables θ, θ,of and the partially boundary conditions for the 2.1 Translation ofϕ. the system (1) to a position with zero auxiliary linearized system, under the control construction method under consideration, can generate controls and values variables θ, method under consideration, canthe generate and values variables θ, ϕ. auxiliary linearized system, under control controls construction ϕ. method under consideration, can generate controls and trajectories that do not satisfy the imposed constraints. trajectories thatconsideration, do not satisfycan thegenerate imposedcontrols constraints. controls u uii (t), (t), ii = = 1, 1, 2, 2, 3 3 chosen chosen Lemma 1. 1. The The tt ∈ ∈ [0, [0, T T11 ]] controls method under and Lemma trajectories that do not satisfy thecondition imposedfor constraints. These circumstances, circumstances, as well well as the the condition for the phase phase for the time interval, in the shape of ] controls u (t), i = 1, 2, 3 chosen Lemma 1. The t ∈ [0, T These as as the 1 i for the time interval, in the shape of trajectories that do not satisfy the imposed constraints. ] controls ui (t), i = 1, 2, 3 chosen Lemma 1. The t ∈ [0,inT1the These circumstances, as require well as the condition for the phase variables of the the model, model, require finding the parameters parameters of for the time interval, shape of variables of finding the of These circumstances, as require well as conditions the condition for theIn phase for the time interval, in the shape of variables of the model, finding the parameters of the system under which these are met. the the system whichrequire these conditions are met. In the g variables of under thewemodel, finding parameters of the system under which these conditions are met.and In the present paper give solution These the questions and g present paper we give aa solution These questions the u1 (t) the system under which these conditions are met. In the gθθ00 present we give a solution for These questions and the results of ofpaper numerical calculations for the model model parameters u1 (t) − gθ0 results numerical calculations the parameters = max{|θ0 |, |ϕ0 |}. (2) (t) u − present paper we give a solution These questions and the 2 1 = 1 ,, T T u (t) results of numerical calculations for the model parameters (2) = 1 of the nonlinear system under consideration. 2 1 = max{|θ0 |, |ϕ0 |}. − T (t) = u θ01 ,T ϕ of the nonlinear system under consideration. 1 (t) T = max{|θ |, |ϕ |}. (2) u results of numerical calculations for the model parameters 3 2 1 0 0 ϕ T 0 − u32 (t) (t) = − of the nonlinear system under consideration. 1 , , T = max{|θ |, |ϕ |}. u (2) − , 1 0 0 ϕ T 0 u (t) T of nonlinear systemOF under consideration. 3 2. the THE PROBLEM TERMINAL CONTROL FOR FOR T1101 , −ϕ 2. THE PROBLEM OF TERMINAL CONTROL u3 (t) − T1 , of the system (1) into a state 2. THE PROBLEMMODEL OF TERMINAL CONTROL FOR NONLINEAR MODEL WITH CONTROLS CONTROLS translate NONLINEAR WITH translate the the phase phase vector vector T1 of the system (1) into a state 2. THE PROBLEMMODEL OF TERMINAL CONTROL FOR NONLINEAR WITH CONTROLS translate (T the1 ),phase vector of the system (1) into a state y(T z(T θ(T = 0, = 0, 1 ), 1 ), 1) 1) NONLINEAR MODEL WITH CONTROLS translate the phase vector the system into ), y(T ), z(T θ(T 0, ϕ(T ϕ(T(1) 0, a state (T 1 1 1 ),of 1) = 1) = Consider the motion of the vector (x, y, z, θ, ϕ) as a funcConsider the motion of the vector (x, y, z, θ, ϕ) as a funcz(T),1 ), )= (T1 ), y(T1x), (T 0, ϕ(T1 ) = 0, yy θ(T (T11), (3) (T Consider thett motion the vector y, z, θ, ϕ) as a function of of time time ∈ [0, [0, T T ]] of satisfying the(x, equations: 1 ), z(T θ(T 0,1 ). ϕ(T ) = 0, (T1 ), y(T1x), (T ),)zz= (3) tion ∈ satisfying the equations: 1 1 ), (T1 1 (T1 ). 1 Consider the motion of the vector (x, y, z, θ, ϕ) as a func(T ), y (T ), z (T ). (3) x tion of time t ∈ [0, T ] satisfying the equations: 1 1 1 Here we have the relation Here we have the relation (3) x (T1 ), y (T1 ), z (T1 ). tion of time t ∈ [0, T ] satisfying the equations: Here we have the relation x x = − − sin sin θu θu1 ,, (4) (T1 ) = y (T1 ) = z (T1 ) = 0, x00 ,, x x˙˙ 00 ,, x x11 ,, x x˙˙ 11 ,, x ¨¨ = Here we have x relation (T ) = y (T ) = z (T ) = 0, (4) xthe 1 1 1 1 , 1 ,, x = cos − sin θu1ϕu yy¨¨ = θθ sin yyx00,, yyx˙˙00,,, yyx11,, yyx˙˙11,,, (4) x (T1 ) = y (T1 ) = z (T1 ) = 0, cos sin ϕu 1 00 , x 00 , x11 , x 11 , x ¨ = − sin θu , x ˙ ˙ (T ) = y (T ) = z (T = 0, (4) x 1 1 1 1 )integrating Boundary conditions (3) are found by (1) ununz cos ϕu − g, z , z ˙ , z , z ˙ , y ¨ = cos θ sin ϕu , y y y y (1) P : 1 − g, z0 , z˙0 , z1 , z˙1 , 1 Boundary conditions (3) are found by integrating (1) z ¨ = cos θ cos ϕu (1) P : y¨˙ = cos θ sin ϕu11, yz000,, zy˙˙000,, yz111,, zy˙˙111,, der controls (2). The relations (4) are found by substituBoundary conditions (3) are found by integrating (1) unθ cos ϕu − g, P : zθ¨˙ = cos 1 (1) , θ u controls (2). The (3) relations (4) are found by substituu22 , θ cos ϕu1 − g, θz000 , z˙0 , z1 , z˙1 , Boundary are found by integrating (1) unzθ¨˙ = cos (1) der P : ϕ tion controls (3) in in conditions (1). The relations der (2). (4) are found by substituu θϕ 2 00,.. 3 ,, tion (3) (1). ϕθθ˙˙ = = u ϕ der controls (2). The relations (4) are found by substitu3, 0, θ = u tion (3) in (1). 2 0 = u3 ,out at financial support ϕ0 . of the RNF, project Due to the stationarity of the system (1), we next consider The work wasϕ˙ carried (3)the in (1). The work wasϕ˙ carried = u3 ,out at financial support ϕ0 . of the RNF, project tion Due to stationarity of the system (1), we next consider the system (1) under the initial condition (3)next andconsider T1 = 14–11–00539 Due to the stationarity of initial the system (1), we The work was carried out at financial support of the RNF, project = 0. 0. the system (1) under the condition (3) and T 14–11–00539 Due to the stationarity of initial the system (1), we The work was carried out at financial support of the RNF, project the system (1) under the condition (3)next andconsider T11 = 0. 14–11–00539 = 0. the system (1) under the initial condition (3) and T 14–11–00539 1 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 166 Copyright 2018 IFAC 166 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2018 IFAC 166 10.1016/j.ifacol.2018.11.374 Copyright © 2018 IFAC 166
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2.2 The linearization of order one for the first three equations of the system (1). Consider the first three equations of the system (1) and will, at this stage, consider (u1 , θ, ϕ) as controlled parameters: 0 x ¨ − sin(θ) 0 0 u1 y¨ = cos(θ) sin(ϕ) 0 0 u1 + 0 . −g z¨ cos(θ) cos(ϕ) 0 0 u1
(5)
We differentiate both sides of the equation (5) in time (2; 3): ... x − cos(θ)θ u1 , ... y ... = − sin(θ)θ sin(ϕ)u1 + z − sin(θ)θ cos(ϕ)u1 − sin(θ)u 1 + + cos(θ) cos(ϕ)ϕ u1 + u 1 cos(θ) sin(ϕ) , − cos(θ) sin(ϕ)ϕ u1 + u 1 cos(θ) cos(ϕ)
and write the right-hand side in the matrix form ... x u1 ... y = A θ , (6) ... z ϕ 0 − sin(θ) − cos(θ)θ A = cos(θ) sin(ϕ) − sin(θ) sin(ϕ)u1 cos(θ) cos(ϕ)u1 , cos(θ) cos(ϕ) − sin(θ) cos(ϕ)u1 − cos(θ) sin(ϕ)u1
where A is the 3×3 matrix, det A = − cos(θ)u21 . Assuming cos(θ)u1 = 0, calculating A−1 and introducing the vector (v1 , v2 , v3 ): ... x v1 ... y = v2 , ... z v3
(7)
from (6) we have − sin(θ) cos(θ) sin(ϕ) cos(θ) cos(ϕ) u˙ 1 cos(θ) sin(θ) sin(ϕ) sin(θ) cos(ϕ) v1 − − − θ˙ = v2 , u1 u1 u1 v sin(ϕ) cos(ϕ) ϕ˙ 3 − 0 u1 cos(θ) u1 cos(θ) (8) with initial condition u1 (0) θ(0) . ϕ(0) The vector (v1 , v2 , v3 ) is chosen from (7) for the ”reference” trajectory w = (x, y, z) ∈ R3 , satisfying the boundary conditions (3): t2 w(t) = w0 + w˙ 0 t + w ¨ 0 + d 0 t 3 + d 1 t4 , (9) 2 4A − BT BT − 3A , d1 = . d0 = T3 T4 A = w1 − w0 + w˙ 0 T − w ¨0
(10)
T2 , 2
¨0 T. B = w˙ 1 − w˙ 0 − w ... w(t) = v(t) = 6d0 (T ) + 24td1 (T ). (11) ¨0 = 0. On the Here u1 (0) = g, T > 0 is a free parameter, w time interval t ∈ [0, T ], the controls ui (t), i = 1, 2, 3 will 167
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be chosen as the solutions of the system (8) for a specially chosen initial condition u1 (0) = g. Lemma 2. For the function v(t) (11), the relation |vi (t)| ≤ c1 T 2 , t ∈ [0, T ], where the constant c1 depends on the boundary conditions (3). Proof. From the relations (9) - (11), for w ¨0 = 0, T ≥ 1 we have 6 ||v|| = 6||d0 || + 24t||d1 || ≤ 3 (4||A|| + ||B||T )+ T 24 72t 6 24t 24t + 4 (3||A||+||B||T ) = A||( 3 + 4 )+||B||( 2 + 3 ) ≤ T T T T T T 2 96 ≤ ||w1 − w0 + w˙ 0 T − w ¨0 || 3 + 2 T 30 c1 ¨0 T 2 ≤ 2 . + w˙ 1 − w˙ 0 − w T T The following assertions show that for u1 (0) = g there exists T > 0 for which the solution of the system (8) exists and is nonlocally extendible. Consider the first equation of the system (8). Lemma 3. When choosing the initial value u1 (0) = g and 3c1 T > g− for which u1 (0) ≥ 3cT1 + , where the constant c1 > 0 depends on the boundary conditions (3), > 0, the solution of the first equation of the system (8) satisfies the condition u1 (0) + 3cT1 ≥ u1 (t) ≥ u1 (0) − 3cT1 ≥ . Proof. Consider the first equation of the system (8). By 1 Lemma 1, |u 1 (t)| ≤ 3c T 2 . Consequently u1 (t) ≥ u1 (0) − 3c1 3c1 ≥ if t ∈ [0, T ], u1 (0) ≥ 3cT1 + T 2 t ≥ u1 (0) − T psilon, where > 0. On the other hand u1 (t) ≤ u1 (0) + 3c1 3c1 T 2 t ≤ u1 (0) + T if t ∈ [0, T ]. Thus, the first equation (8) has a solution, under the indicated constraints, for any continuous functions θ(t), ϕ(t), v(t) and u1 (0) ≥ 3cT1 + . Solution the first equation (8), estimates 3c1 3c1 ≥ u1 (t) ≥ u1 (0) − ≥ , t ∈ [0, T ]. 2g − ≥ u1 (0) + T T Lemma 4. When the conditions of Lemma 2 are satisfied, the solution of the second equation of the system (8) satisfies the condition θ0 = 0, θ(t) ∈ (− π2 , π2 ), t ∈ [0, T ]. Proof. Consider the second equation of the system (8). As 1 u1 (t) ≥ , t ∈ [0, T ], then |θ (t)| ≤ 1 3c T 2 . Consequently 3c1 3c1 1 − T ≤ θ(t) ≤ + T , t ∈ [0, T ] and with c1 > 0, T > 6c π we have θ(t) ∈ (− π2 , π2 ). Lemma 5. Under the conditions of Lemmas 1-4, the solution of the third equations of the system (8) for ϕ0 = 0 satisfies the condition ϕ(t) ∈ (− π2 , π2 ), t ∈ [0, T ]. Proof. Consider the third equation of the system (8). As u1 (t) ≥ , cos(θ(t)) ≥ cos(1) > 0, t ∈ [0, T ], then 2c1 1 |ϕ (t)| ≤ cos(1) T 2 . Consequently 2c1 2c1 ≤ ϕ(t) ≤ cos(1)T , t ∈ [0, T ]. When c1 > 0, T > − cos(1)T 4c1 π π π cos(1) we have ϕ(t) ∈ (− 2 , 2 ).
The values of the system parameters (8): u1 (0) = g, 6c1 3c1 4c1 , , }. (12) T ≥ max{ π cos(1) π g −
IFAC CAO 2018 168 Yekaterinburg, Russia, October 15-19, 2018 L.N. Lukianova / IFAC PapersOnLine 51-32 (2018) 166–168
Fig. 1. x(t)
Fig. 2. y(t)
Fig. 5. ϕ(t)
Fig. 6. u2 (t)
Fig. 3. z(t)
Fig. 4. u1 (t)
Fig. 7. u3 (t)
Fig. 8. θ(t)
Theorem. The control u1 (t), u2 (t), u3 (t), which has the form (2) for t ∈ [0, T1 ] and is obtained from the solutions of system (8) for t ∈ [T1 , T1 + T ], with u1 (0) = g and T satisfying (1.12) solves the problem of terminal control. 2.3 Solution Algorithm 1. Construction of the reference curve (9) according to the boundary conditions (3). 2. Finding the parameter v(t) ∈ R3 (11).
3. Solution of the system (8) for u1 (0) = g and T . 4. The solution of the system (1) for u1 (t), θ(t), ϕ(t) of item 3. ˙ u3 (t) = ϕ(t) ˙ from the 5. Finding the controls u2 (t) = θ(t), second and third equations of the system (8). We present the results of calculations of the reference trajectory, the linearized system and the initial system using the linearization method for the following boundary conditions: x0 = −6; x˙ 0 = −1; x1 = 6; x˙ 1 = 1; y0 = −5; y˙ 0 = 0.5; y1 = 5; y˙ 1 = −0.4; z0 = 4; z˙0 = 1; z1 = 4; z˙1 = −0.1; ϕ0 = 0; θ0 = −0.1; T = 10; u0 = 4; REFERENCES 1.Gur’yanov A. Ye. Simulation of quadrocopter control, Engineering Bulletin, 08, August 2014, c 522-534, http://engbul.bmstu.ru/doc/723331.html 2.S.Salazar-Cruz, F.Kendoul, R.Lozano, I.Fantoni RealTime Stabilization of a Small Three-Rotor Aircraft, IEEE Transactions on aerospace and electronic systems, Vol. 44, No. 2 April 2008 p.783-794. 168
Fig. 9. (xt , yt , zt ) 3.P.Castillo, R.Lozano, A.Dzul Modeling and control of mini-flying machines, Springer, 2005, - 259 p. 4.Isidori A. Nonlinear control Systems, Springer, 1994. 5.Khalil H.K. Nonlinear Systems, Prentice Hall, 2002. 6.Kim D.P. Theory of automatic control. T. 2. Multidimensional, nonlinear, optimal and adaptive systems: Proc. allowance. - Moscow: Fizmatlit, 2004. - 464 p.