A Feedback Linearization Control of Container Cranes ... - IJCAS

A Feedback Linearization Control of Container Cranes: Varying Rope Length 381where⎡m m m ⎤M m m 0 ,11 12 13=⎢⎥⎢ 21 22 ⎥⎢m31 0 m33⎥⎣G = ⎡0 −mpgcosθmpglsin θ⎤,⎣⎦T⎦V⎡0V V ⎤0 0 ,m12 m13m =⎢V⎥⎢ m23⎥⎢0Vm32 Vm33⎥u = ⎡F x Fl0 ⎤ ,⎣ ⎦m = m + m m12 = m sin θ , m13 = m lcos θ ,11 p t ,m21 = mpsin θ,m22 = mp+ ml,p⎣233 = p + ,m31 = mplcos θ,m m l IVm12 = m pθ cos θ,Vm13 =− mplsin θ θ + mpcos θl,VVm23 mpl θ,=− V 32 = m lθ , andm33 mpll.= mpIt is remarked that the inertia matrix M ( q)and thecentripetal term Vm( q, q ) satisfy the skew-symmetricrelationship. Therefore the following equation holds.T 1ξ ( M ( q) − Vm( q, q)) ξ = 0,2 (9)where M ( q)is the time derivative of M ( q ) , andthen the following inequalities hold.2 T21 ξ ≤ξ ξ ≤ 2 ξk M( q) k ,Tpξ ∈ R 3 , (10)where k1,k 2 are positive constants.In order to separate the unactuated dynamics (θ -dynamics) from the actuated dynamics (( x, l)-dynamics), (7) is rewritten as 1θ = ( −ml p cosθ x−2mllp θ −mglp sin θ).(11)ml + I2pIt can be seen from (11) that a change of the ropelength leads to a change of the sway dynamics. Thiscoupling effect has to be effectively controlled. Using(11), (5) and (6) can be rewritten as2 2 2 2( p + t − p θ p + ) + p2 2= 2mpcos θ l θ( ml p /( ml p + I) −1)2 2θ ( θ θ )m m ml cos /( ml I) x m sinθl+ ml p sin + mgl p cos /( ml p + I) + Fx,2m sin θ x+ ( m + m ) l = m l θ + m gcos θ + F.(13)p p l p p l⎦(12)Now, we introduce a new vector r = [ x l] T for thepurpose of partial feedback linearization and thetracking of both trolley and hoist dynamics at thesame time (see Section 3). Then, the coordinate vectorT T Tcan be written as q = [ xlθ] = [ r θ ] . And, byrearranging the terms in (12)-(13), the r -dynamicsbecome r = P( F + W) = PF + W,(14)where⎡p11 p12 ⎤ ∆'−1P = ⎢ Mp21 p⎥ = ⎡ ⎤⎣ 22 ⎦⎣ ⎦⎡ mp + ml −mpsinθ⎤1 ⎢⎥2 2 2= ⎢ ml p cos θ ⎥ ,det( M ′) ⎢− mpsinθmp + mt−2⎥⎢⎣ml p + I ⎥⎦M ′ is the inertia matrix of ( x, l)-dynamics,2 2 2 2( p t p θ p )det ( M ′) = m + m − m l cos /( m l + I)p l2p2= ⎡ ⎣T⎤x , ⎦= ⎡T⎤1 2 = , andF F F× ( m + m ) −msin θ,W w w PW⎣ ⎦⎡2 22mpcos θ l θ( ml p /( ml p + I) −1⎢)W =⎢2 2+ ml p sin θ ⎢θ + mgl p cos θ /( ml p + I)⎢2⎢ml⎣ p θ + mg p cosθ( )3. CONTROL LAW DESIGNIn this section, a nonlinear control law forsuppressing the sway angle of the suspended load andcontrolling the trolley position and the hoisting ropelength is derived. The novelty of this law lies inimproving the transient performance and alsoguaranteeing the asymptotic stability of the trolleyposition and hoisting rope length tracking errors,irrespective of the variations of the rope length. Theinformation on the sway angle, sway angular velocity,trolley displacement and velocity, hoisting rope lengthand its time rate of change is assumed to be known.Let the tracking errors of the trolley position and thehoisting rope length be defined byTd e e⎤⎥⎥⎥.⎥⎥⎦e= r− r = [ x l ] ,(15)Twhere rd = [ xd ld] , xe= x− xd,le= l − ld,andx d and l d are the desired trolley position andhoisting rope length, respectively.To improve the transient performance in positioningand load hoisting control, and sway suppression, thefollowing new control law is proposed: