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# Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control Technique

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## Stabilization of Inertia Wheel Pendulum using Multiple Sliding Surface Control TechniquePresentation Transcript

• Stabilization of Inertia WheelPendulum using Multiple Sliding Surface Control Technique A paper from Multtopic Conference, 2006. INMIC’ 06. IEEE By Nadeem Qaiser, Naeem Iqbal, and Naeem Qaiser Dept. of Electrical Engineering, Dept. of Computer Science and Information technology, PIEAS Islamabad, PakistanCONTROL OF ROBOT ANDMoonmangmee Speaker: Ittidej VIBRATION LABORATORY No.6 Student ID: 5317500117 January 17, 2012
• 2/18Key references for this presentationProceedings:[1] M.W. Spong, P. Corke, and R. Lozano, “Nonlinear Control of the Inertia Wheel Pendulum”, Automatica, 2000.PhD Thesis:[2] Reza Olfati-Saber, Nonlinear Control of Undeactuated Mechanical Systems with Application to Robotics and Aerospace Vehicles, MIT, PhD Thesis 2001.Textbook:[3] Bongsob Song and J. Karl Hedrick, Dynamic Surface Control of Uncertain Nonlinear Systems: An LMI Approach, Springer, New York, 2011.
• 3/18Classifications & Styles of Control Paper Control Application (or Control Engineering)  Dynamic model  Controller and/or observer design Engineers  Experimental setup  Simulations vs. experimental resultsControl Theory (or Mathematical Type 2:Control Theory or Control Sciences)  Problem formulation & Assumptions Type 1:  Mathematical proofs (rigorously):  Dynamic model (a benchmark) definition, lemma, proposition,  Controller and/or observer theorem, corollary, etc. design  No experiments  Computer simulation via  Illustrated examples compare with other methods  Sometimes has no simulations Engineers & Mathematicians Mathematicians are in a majority
• 4/18Control Sciences Stabilization of Inertia WheelPendulum using Multiple Sliding Surface Control Technique Control Theory (or Mathematical Control Theory or Control Sciences)
• 5/18Outline Underactuated Mechanical Systems Overview of Control System Design Dynamic Model Controller Design Stability Analysis Simulation Results Concluding Remarks
• 6/18Underactuated Mechanical Systems q2 Fully actuated: #Control I/P = #DOF. q2 Underactuated: #Control I/P < #DOF. q1S q1E Pendubot Inverted PendulumLP q2 q2M q1 q1 Rotational Inverted q3A Acrobot Pendulum q1 q2X q2 q2E q1 q1 q4 Rotary Prismatic Perpendicular Rotational “Fish Robot” System Inverted Pendulum [Mason and Burdick, 2000]
• 7/18Underactuated Mechanical Systems
• 8/18The Inertia-Wheel Pendulum I2 q2S q1E I1 , L 1 gLPM  Single-Input-Single-Output (SISO)  Nonlinear time-invariantA  Underactuated mechanical systemX  Simple mechanical system [Spong et al, 2000] Euler-Lagrange (EL) equationsE of motion
• 9/18Control System Architecture Step 2: Step 1: Step 3:
• 10/18 Dynamic Model é m 11 m 12 ùé& ù q& é ( m l + m L )g sin (q ) ù - é0 ùStep 1: ê úê 1 ú+ ê 1 1 2 1 1 ú ê út I2 q2 ê úê& ú q& ê ú= ê1 ú ê 21 m 22 úê 2 ú m ë 42 44443 û ûë ê ë 0 ú û ê ú ë û 1444 144444444442 44444444443 { M (q ) g (q ) Q (q ) q1 2 2 I1 , L 1 w h e r e m 11 = m 1l 1 + m 2 L 1 + I 1 + I 2 g a n d m 12 = m 21 = I 2 a r e con st a n t s í m q& + m q& - ( m l + m L )g sin (q ) = 0 ï & & ï 11 1 12 2 1 1 2 1 1 W: ì ï m 21q& + m 22q& = t & & ï î 1 2  & & L e t x 1 = q1, x 2 = q1, x 3 = q 2, a n d x 4 = q 2 í ï x& = x 2 ï 1 ï ï x& = ..... + t ï ï 2 S t a t e e qu a t io n : ì ï x& = x 4 ï 3 ï ï x& = ..... + t ï ï î 4
• 11/18 Collocated Partial Feedback Linearization General Form of the EL Equations of Motion for an Underactuated Mechanical System: [Spong et al, 2000] Proposition There exists a global é (q ) invertible change of control in the form m m 1 2 ( q ) ù é& ù q& é (q , q ) ù h & é0 ù ê 11 W: ê ú ê 1 ú+ ê1 ú= ê ú m 2 2 (q ) ú ê& ú ê (q , q ) ú êt ú & t = a (q )u + b (q , q ) ê 2 1 (q ) m q& úê 2 ú h ê2 & ú ê ú ë ûë û ë û ë û whereC on figu r a t ion v ect or : - 1 C on t r ol v ect or : a ( q ) = m 2 2 ( q ) - m 2 1 ( q )m 1 1 ( q )m 1 2 ( q ) T (n - m ) mq = [q 1 , q 2 ] Î ¡ ´ ¡ ,t Î ¡ m ( m ) con t r ols & & - 1 & b ( q , q ) = h 2 ( q , q ) - m 2 1 ( q )m 1 1 ( q )h 1 ( q , q )( n - m ) u n d er a ct u a t ed coor d in a t es such that the dynamics of transformed ( m ) a ct u a t ed coor d in a t es to the partially linearized system.Remarks: fully linearized system (using a change of control)  Impossible partially linearized system (q2 transform into a double integrator)  Possible after that, the new control u appears in the both (q1, p1) & (q2, p2) subsystems this procedure is called collocated partial linearization
• 12/18 Collocated Partial Feedback Linearization í ï & q1 = p1 ü ï ï ï ï ý ( q 1 , p 1 ) n o n lin e a r s u bs y s t e m ï ï ï & p 1 = f 0 ( q , p ) + g 0 ( q )u ï Wn ew : ï ì þ ï & q2 = p2 ü ï ï ï ï ý ( q 2 , p 2 ) lin e a r s u bs y s t e m ï ï ï & p2 = u ï ï î þ - 1 where  is an m m positive definite symmetric matrix and g 0 (q ) = - m 11 (1)m 12 (q ) (q)Step 2:Transform to the Partial Feedback Linearization form í a (q , q ) = ( m m - m 2 ) / m ï & ï 11 22 21 11 t = a (q , q )u + b (q ) w h er e ì & ï b (q ) = ( m 21 / m 11 ) ( m 1l1 + m 2 L 1 )g sin ( q 1 ) ï ï îDefine new state variables New state equation in the Strict Feedback Form í & ï z = (m l + m L ) g sin ( z )íz = m q + m qï & & ï 1 ï 1 1 2 1 2 } N o n lin a e r (C o r e o r R e d u c e d )ï 1 ïï 11 1 12 2 ï m 12 ü ïï z = q ( p en d u lu m a n gle) ï 1 ïì 2 ì z& = z1 - z3 ï ï L in e a r (o r O u t e r )ï 1 ï 2 m 11 m 11 ýï z = q ( w h eel v elocit y ) ïï 3 & ï ïïî 2 ï z& = u ï ï 3 ï ï ï î þ
• 13/18 Controller DesignStep 3: Outer Subsystem Controller DesignGoal: Stabilizes z 2 ® 0, z 3 ® 0 Second define the sliding surface ì S 1 = z 2 - z 2d ï ï ï z& = 1 (m 1l 1 + m 2L 1 ) g sin ( z 2 ) } C ore 1 m 12 ï ï ü ï S& = z& - z&d = z1 - z 3 - z&d ï 1 m ï 1 2 2 m 11 m 11 2 í z& = z1 - 12 z3 ï ï O uter ï 2 m m ý ï ï 11 11 ï To achieve this condition, we choose ï z& = u ï ï ï î 3 ï ï þ m 11 æ çK S + 1 z - z& ÷ ö z 3d = ç 1 1 ÷ ç 2d ÷ Core Subsystem Controller Design m 12 ç è m 11 1 ÷ øFirst Design the synthetic inputs z2d for thecore subsystem achieves the Lyapunov stability Third Design again the synthetic I/P, z3d 2 V ( z i ) = 1 z i > 0 ( p osit iv e d efin it e) S 2 = z 3 - z 3d 2 Þ V& z i ) = z i z& < 0 ( n ega t iv e d efin it e) ( S& = z& - z&d = u - z&d 2 3 3 3 iTo achieve this condition, we choose Finally, the control law chosen to drive - 1 S20z 2d = - a t a n (cz 1 ) , 0 < a £ p 2 ,c > 0 u = z&d - K 2S 2 3
• 14/18Stability ResultsTheorem 1: Theorem 3: í S& = - K S - ( m / m )S ï 1 SN : { z& = sin ( z 2 d + S 1 ) 1 SL ï :ì & 1 1 12 11 2 í S& = - K S - ( m / m )S ï 1 ï S 2 = - K 2S 2 ï 1 1 12 11 2 ï î SL : ì ï S& = - K 2S 2  is global asymptotic stability ï 2 î L ( ,  ) is global asymptotic stability N L  is global exponential stability L SN S1= 0 : { z& = 1 sin (a t a n - 1 (cz 1 ) ) Proposition 1:  |S1 = 0 is globally Lipschitz. N Theorem 2:  |S1 = 0 if 0 < a ≤  and c > 0 N /2 then z1 = 0 is global asymptotic stability.Remark: we left out all of the proofs from the presentation
• 15/18Simulation Results I2 Initial state: q2(T) = 0(q1(0), q2(0)) = ( 0) , Plant parameters: m11 = 4.83 10-3 m12 = m21 = m22 I1 , L 1 g = 32 10-6 g q1(T) = 0 w = 379.26 10-3 q1(0) =  Controller parameters: I1 , L 1 a =  c = 9, /2, Final state: K1 = 4, K2 = 6, and  = 0.001 T (q1(T), q2(T)) = (0, 0) where the plant parameters are setted I2 as same as in Olfati-Saber (2001) and Spong (2000).
• 16/18Simulation Results Pendulum angle, velocity Pendulum angle, velocityMSS Controller [Olfati-Saber, 2001] 2.2 sec 3.6 sec time (second) time (second) Wheel velocity VS Wheel velocity 3 sec 3.7 sec time (second) time (second)
• 17/18Simulation Results Control effort (Nm) Control effort (Nm) 0.43 Nm 0.33 Nm VS time (second) time (second) MSS Controller [Olfati-Saber, 2001]
• 18/18Concluding Remarks The collocated partial feedback linearization was presented for transform a nonlinear underactuated mechanical system into the strict feedback form A Multiple Sliding Surface controller is designed to achieves global asymptotically stable of the pendulum angle and the wheel velocity (neglect the wheel angle) The MSS has advantages that the two controllers, i.e.  no supervisory switching required as in Spong’s design (2000) (more simple structure)  the response is faster than the designs by Olfati-Saber (2001) (more better performance) However, more control effort required for MSS
• Thank you Please comments and suggests!CONTROL OF ROBOT AND VIBRATION LABORATORY