Transcript of "Backstepping control of cart pole system"
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Backstepping Control of Cart Pole System Presented by Shubhobrata RudraMaster in Control System Engineering Roll No: M4CTL 10-03 Under the Supervision of Dr. Ranjit Kumar Barai
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Content Objectives of the Research Modeling of the Physical Systems Difficulties of the Controller Design Backstepping Control Stabilization of Inverted Pendulum Anti Swing Operation of Overhead Crane Adaptive Backstepping Control & its application on Inverted Pendulum Conclusion & Scope of Future Research References
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Objective of the Research Maintain the stability of an inverted pendulum mounted on a moving cart which is travelling through a rail of finite length. Enhance tracking control of an overhead crane (cart pole system in its stable equilibrium) with guaranteed anti-swing operation
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Modeling of Cart Pole System F T f x ,x V d2 M m 2 x l sin F T dt
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Contd. State Model of Inverted Pendulum: If the angle of on Hence Based the Angular position pendulum is ofMost of thein Pendulum quite small we Nonlinearities can space it those replace is nonlinear the except divide possible toterms. friction T are the Hence total the we can functions region realize a of the operatingLinear pendulum small Model for angle in two different x2 zone angle deviation!!!
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Difficulties of the Controller Design The system Model is quite complicated and nonlinear. It is almost impossible to obtain a true model of the real system and if it is achieved by means of some tedious modeling, the model will be too complex to design a control algorithm for it. The system has got two output, namely the motion of the cart and the angle of the pendulum. It is a quite complicated design challenge to reshape the control input in such a manner that can control both output of the cart pole system simultaneously.
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CONTENT What is Backstepping? Why Backstepping? Different Cases of Stabilization Achieved by Backstepping Backstepping: A Recursive Control Design Algorithm New Research Ideas
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What is Backstepping? Stabilization Problem of Dynamical System Design objective is to construct a control input u which ensures the regulation of the state variables x(t) and z(t), for all x(0) and z(0). Equilibrium point: x=0, z=0 Design objective can be achieved by making the above mentioned equilibrium a GAS.
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Contd. First step of the design is to construct a control input for the scalar subsystem z can be considered as a control input to the scalar subsystem Construction of CLF for the scalar subsystem Control Law: But z is only a state variable, it is not the control input.
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Contd. Only one can conclude the desired value of z as Definition of Error variable e: z is termed as the Virtual Control Desired Value of z, αs(x) is termed as stabilizing function. System Dynamics in ( x, e) Coordinate:
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Contd. Modified Block Diagram Feedback Control Law Backstepping αs Signal -αs
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Contd. So the signal αs(x) serve the purpose of feedback control law inside the block and “backstep” -αs(x) through an integrator. Feedback loopBackstepping of Signal -αs(x) with + αs(x) Through integrator
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Contd. Construction of CLF for the overall 2nd order system: Derivative of Va A simple choice of Control Input u is: With this control input derivative of CLF becomes:
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Why Backstepping? Consider the scalar nonlinear system Not at all!!!! This is an Useful Control Law( using Feedback Linearization): Nonlinearity, it is it essential to has an Stabilizing cancel out the effect on the term ? system. Resultant System: Edurado D. Sontag Proposed a formula to avoid the Cancellation of these useful nonlinearities.
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Contd. Sontags Formula: 2 4 V V V f f g x x x V for g 0 u V x g x V 0 for g 0 x So this control But this Control Law (Sontag’s Formula): For large values formula leads the law avoids a of x, the complicated of cancellation control law useful control input becomes nonlinearities! for u≈sinx For higher intermediate Control Law (using Backstepping): values ofof x values x
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Contd. Simulation Results: Stabilization of the Nonlinear Scalar plant Control Effort of x with time time Variation variation with Feedback Feedback Linearization Linearization ***Sontag’s Formula +++Backstepping Control law Sontag’s Formula Backstepping Control Law
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Contd. IEEE Explore 1990-2003 Backstepping in title Conference Paper Journal PaperOla Harkegard Internal seminar on Backstepping January 27, 2005
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Different Cases of Stabilization Achieved by Backstepping Integrator Backstepping Nonlinear Systems Augmented by a Chain of Integrator Stable Nonlinear System Cascaded with a Dynamic System Input Subsystem is a Linear System Input Subsystem is a Nonlinear System Nonlinear System connected with a Dynamic Block Dynamic block connected with the system is a linear one Dynamic block connected with the system is a Nonlinear one
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Integrator Backstepping Theorem of Integrator Backstepping: Nonlinear System Integrator If the nonlinear system satisfies certain assumption with z Є R as its control then The CLF depicts the control input u renders the equilibrium point x=0, z=0 is GAS.
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Chain of Integrator Chain of integrator: ∫ ∫ ∫ Nonlinear System K th integrator CLF
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Integrator Backstepping Example Stabilization ofResults Simulation an unstable system x x2 xz z u Stabilizing Function: Choice of Control law: The equilibrium point x=0, z=0 is a GAS.
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Stabilization of Cascaded System Stable nonlinear system cascaded with a Linear system u z Az Bu u y Cz ∫ x f x gxy CLF A, B, C are The Control Law: FPR Ensures the Equilibrium (x=0, z=0) is a GAS.
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Contd. Stable nonlinear system cascaded with a Nonlinear system u zz= η(Az+ βBuu z) (z) x = ff (x , z )g x (y, z ) y x x +g x y y C(z) = Cz Feedback Passive u=K(z)+r(z)v CLF SystemFeedback is a with U(z) as a +ve Definite Transformation Storage Function Such that the Control Law resulting system is Passive with Storage Function U(z) Ensures the Equilibrium (x=0,z=0) is GAS.
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Stabilization with Passivity an Example System Dynamics: x x 1 ez x2 z 4 u z z 3u z 3u z x x 1 ez x2 z 4 Feedback Law: u z2 v t t y v d U zt U z0 z6 d Storage function: 0 0 Derivative of Storage Function: U z z5 z 3v z6 z 4v
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Contd. Control law u z2 x3 Simulation Results: Phase-Plane Portrait 10 5 x2(t) 0 -5 -10 -10 -5 0 5 10 x1(t)The equilibrium point x=0, z=0 is a GAS.
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Block Backstepping Nonlinear system cascaded with a Linear Dynamic Block u z Az Bu x f x gxy y Cz Using the feedback transformation Stable/Unstable The State equation of the system becomes Nonlinear system Control Law Minimum Phase Linear Systemof the A0 are the Eigen values with relative degree one function zeros of the transfer Zero Dynamics Ensures the equilibrium point x=0, z=0 is GAS.
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Contd. Nonlinear system cascaded with a Nonlinear Dynamic Block u z x, z x, z u x f x gxy y C z Control Law: Nonlinear System with relative degree one And the zero dynamics subsystems is globally defined and it is Input to state stable Ensures the equilibrium x=0, z=0 is GAS.
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Backstepping: A Recursive Control Design Algorithm Backstepping Control law is a Constructive Nonlinear Design Algorithm It is a Recursive control design algorithm. It is applicable for the class of Systems which can be represented by means of a lower triangular form. In order of increasing complexity these type of nonlinear system can be classified as Strict Feedback System Semi –Strict Feedback System Block Strict Feedback Systems
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Contd. Block Strict Feedback forms: x f x g x y1 X 1 F1 x, X 1 G1 x, X 1 y2 y1 C1 X 1 X2 F2 x, X 1 , X 2 G2 x, X 1 , X 2 y3 y2 C2 X 2 Xk Fk x, X 1 , X 2 ,, X k G2 x, X 1 , X 2 ,, X k yk 1 yk Ck X k X m 1 Fm 1 x, X 1 , X 2 ,, X m 1 Gm 1 x, X 1 , X 2 ,, X m 1 ym ym 1 Cm 1 X m 1 Xm Fm x, X 1 , X 2 ,, X m Gm x, X 1 , X 2 ,, X m u ym Cm X m
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Contd. Assumptions: n Each K subsystem with state X k and yk ,and input yk 1 satisfies: BSF-1: Its relative degree is one uniformly in x, X1 ,, X k 1 BSF-2: Its zero dynamics subsystem is ISS w.r.to x, X1 ,, X k 1 , yk Sub-System Dynamics in transformed Co ordinate: Ck yk X k Fk x, X 1 ,, X k Gk x, X 1 ,, X k yk 1 Xk f k x, y1 , 1 ,, xk , k g k x, y1 , 1 ,, xk , k yk 1 k 1 k x, X 1 ,, X k Fk x, X 1 ,, X k Gk x, X 1 ,, X k yk k 1 i 1 Xi k x, X 1 ,, X k Fk x, X 1 ,, X k Xk x, X 1 ,, X k 1 , yk , k x, y1 , 1 ,, yk 1 , k 1 , yk , k
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Contd. The change of Coordinate Results: Strict Feedback x x x x f f g x y1y1 g x Form X 1 F1, x, X,1 λ G1 x, X 1 x2 y y1 F x y 1 1 G y, 1 1 1 , λ1 y2 y1 C1 X 1 y2 F x, y , λ , y2 , λ2 G2 x, y1 , λ1 , y2 , λ2 y3 2 1 1 X 2 F2 x, X 1 , X 2 G2 x, X 1 , X 2 y3 y2 C2 X 2 ym F x, y1 , λ1 , , ym 1 , λm Gk x, y1 , λ1 , , ym 1 , λm ym 1 m 1 1 1 1 Xk Fk x, X 1 , X 2 ,, X k G2 x, X 1 , X 2 ,, X k yk Fm x, y1 , λ1 , , ym , λm Gk x, y1 , λ1 , , ym , λm u 1 ym yk Ck X k X m 1 x,Fy 1, x, X 1 , X 2 ,, X m 1 Gm 1 x, X 1 , X 2 ,, X m 1 ym λ1 m λ 1 1 Zero Dynamics ym 1 Cm 1 X m 1 Xm Fm x, X 1 , X 2 ,, X m Gm x, X 1 , X 2 ,, X m u λm ym Cm, X1 , λ1 , , ym x ym , λm 1 , ym , λm 1
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New Research IdeasIn 1993, I. Kanellakopoulos and P. T. Krein introduced the use of Integralaction along with the Backstepping control algorithm, which considerablyimproves the steady-state controller performance [2].It is possible to represent a complex nonlinear system as a combination oftwo nonlinear subsystem, while each subsystem is in Block Strict Feedbackform. And if the zero dynamics of input subsystem is Input to State Stable(ISS). Then it is possible to stabilize the system using Backstepping algorithm.Integral Action along with Block Backstepping algorithm may gives a bettertransient as well as steady state performance of the controller for complexnonlinear plant.
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Content Control Objective Two Zone Control Theory of Inverted Pendulum Design of Control Algorithm for Stabilization zone Design of Control Algorithm for Swinging Zone Schematic Diagram of Controller Results of Real Time Experiment Comparative Study and Conclusion
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Control Objective Design a control system Stability of Maintain the that keeps the Inverted Pendulum the pendulum balanced and tracks the when it is suffering with cart to a commanded external disturbances. position!!!
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Two Zone Control Theory Most of the nonlinearities (present in the state model of Inverted Pendulum) are the function of pendulum angle in space. Stabilization Unstable Zone Equilibrium Point Swinging Zone
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Features of Two Zone Control Theory Two independent controller can be used for two different zones. One can use a linearize model of Inverted Pendulum in Stabilization zone Linear model of the pendulum facilitates the design of more complex control algorithm, which enhance the steady state performance of the inverted pendulum. While a less complicated algorithm can be used for the swinging zone operation. Designer can modify the algorithm independently for each zone and get a optimal combination of controller for swinging and stabilization zone.
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Design of Control Algorithm for Stabilization Zone Linearize model of Inverted Pendulum It is possible to represent the system as a The state model combination of of the system two dynamic not allows the block each of designer to them in block implement strict feedback backstepping system algorithm on it Choice of Control Variable::
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Contd. Choice of Stabilizing Function: Choice of second error variable: Derivative of z1 and z2 Integral action is introduced to enhance the controller performance in steady state operation
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Contd. Choice of CLF: Control Input:Where Integral action reduces the steady state error of the system. Derivative of CLF:
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Contd. List of the controller parametersWhere d1=c1+c2 & d2=c1c2. k1=1, c1=c2=50, c=0.001
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Design of Control Algorithm for Swinging Zone State model of the Inverted Pendulum: Choice of Control variable:
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Contd. Choice of Stabilizing function: Choice of second error variable: Derivative of z3 and z4
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Contd. Choice of CLF: Control Input: Derivative of CLF:
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Contd. List of Controller’s Parametersk2=0.1, d3=c3+c4 and d4=c3c4+1, where c3=c4=20
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Schematic Diagram of ControllerController for Stabilization Zone Control Input Linear BacksteppingReference Controller Input Switch Switching ing Inverted Law Mecha Pendulum nism Nonlinear Backstepping Controller Controller for Swinging Zone
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Results of Real Time Experiment Angle of the Inverted Pendulum Pendulum reach its stable position within 4 seconds
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Contd. Angular Velocity of the Inverted Pendulum
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Contd. The cart is able to Cart Movement with time track the reference trajectory within 15 seconds
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Comparative Study and Conclusions Comparative study on the Pendulum angular position in space
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Contd. Comparison of Cart tracking Performance
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Conclusion Backstepping controller along with Integral action enhance the performance of the steady state operation of the controller. Nonlinear Backstepping controller ensure the enhance swing operation of the Inverted Pendulum. The Backstepping control algorithm has an ability of quickly achieving the control objectives and an excellent stabilizing ability for inverted pendulum system suffering with an external impact. The use of integral-action in backstepping allows us to deal with an approximate (less informative and less complex) model of the original system; as a result it reduces the computation task of the designer, but offering a controller which is able to provide successful control operation in spite of the presence of modeling error
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Content Control Objective Two Zone Control Theory of Over Head Crane Design of Control Algorithm for Stabile Tracking zone Design of Control Algorithm for Anti-Swinging Zone Schematic Diagram of Controller Results of Real Time Experiment Comparative Study and Conclusion
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Control Objective Proper tracking of The Cart Motion along a reference/desired Proper Antiswing trajectory. operation of pay load during travel
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Two Zone Control Theory Most of the nonlinearities (present in the state model of Overhead Crane) are the function of payload angle in space. Anti Swing Zone Stable Tracking Zone
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Design of Control Algorithm for Stable Tracking Zone Linearize model of Overhead Crane The Primary objective of design is to control the motion of the cart along with a reference trajectory Choice of Control Variable:
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Contd. Choice of Stabilizing Function: Choice of second error variable: Derivative of z1 and z2 Integral action is introduced to enhance the controller performance in steady state operation
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Contd. Choice of CLF: Control Input:Where Integral action reduces the steady state error of the system. Derivative of CLF:
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Contd. List of Controller Parameters Where d1=c1+c2 & d2=c1c2. k1=1, c1=c2=50, c=0.001
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Design of Control Algorithm for Anti-Swinging Zone In case of Anti swing operation the primary concern of the controller is to reduce the oscillation of the pay load, & brings it back inside the stable region. In case of Inverted Pendulum the controller tries to launch the pendulum inside its stabilization zone. So in case of Anti-swing operation the same controller which has been used for Swinging operation can be utilized!!!!!!!
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Contd. Same Control Algorithm is being used to serve the Anti SwingSwinging opposite purpose!!! Zone ZoneInverted Pendulum Overhead Crane
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Schematic Diagram of ControllerController for Stable Tracking Zone Control Input Linear Backstepping Reference Controller Input Switch Overhead Switching ing Inverted Crane Law Mecha Pendulum nism Nonlinear Backstepping Controller Controller for Anti Swing Zone
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Results of Real Time Experiment Motion of the Cart Steady state Tracking error reduces with time
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Contd. Cart Motion of the pendulum when suffering with an external impact The cart is able to track the reference trajectory within 15 seconds
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Contd. Cart Velocity when suffering with an external impact
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Contd. Angle of the Payload when suffering with an external impact The angle of the payload reduces within 15 seconds
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Contd. Angular Velocity of the Payload when suffering with an external impact
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Conclusion Backstepping controller along with Integral action enhance the performance of the steady state operation of the controller. Nonlinear Backstepping controller ensures the proper anti-swing operation of overhead crane. Here one can reuse the nonlinear controller which has been used for swinging purpose of inverted pendulum. Though the total control scheme is little bit complex than that of classical PID controller. But in case of classical PID control it is not able to address the problem of anti-swing operation properly.
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Adaptive Backstepping Controland its Application on Inverted Pendulum
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Content Adaptation as Dynamic Feedback Adaptive Integrator Backstepping Stabilization of an Inverted Pendulum Robust Adaptive Backstepping Simulation Results Conclusion
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Adaptation as Dynamic Feedback Stabilization problem of a nonlinear system: x x u Static Control Law: Dynamic Control Law u x c1 x Θ is an unknown γ is adaptation constant parameter gain Θ ~ an unknown is ˆ Can use a Is the Oneparameter so it is Augmented Lyapunov function: parameter error impossible to use certainty equivalence form of this expression control where θ is replaced law, containing by an estimate of θ, ˆ unknown parameter
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Contd. Derivative of Augmented Lyapunov function: 1 ~~ Va xx 2 ~ 1~ c1 x x x Update law: ˆ ~ x x Which ensures the negative definiteness of Va. System dynamics: ~ x c1 x x ~ x x
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Contd. Block diagram of the Closed loop Adaptive system
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Adaptive Backstepping Stabilization of 2nd order nonlinear system: x1 x2 1 x1 x2 2 x u θ is an Stabilizing Function: unknown x1 c1 x1 x1 parameter. So s 1 θ should be replaced by its CLF: estimated 1 2 11 2 1 2 2 value. Vc Vc x x x1 z x2 sz2 x 2 2 21 2 Control law: u c u x2 c2 x2 s x1 x 1 s s x 2 x2 1 2 x ˆ x 2 s x1 1 2 x1
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Contd. Error Dynamics: d z1 c1 1 z1 0 ~ dt z2 1 c2 z2 2 x Construction of Augmented Lyapunov Function: ~ 1 2 1 2 1 ~2 Va z , z1 z2 2 2 2 Derivative of Augmented Lyapunov function: Update Law : a z1 , z2 , ~ ˆ c1 z1 2 zc2 z2 2 2 ~ 1ˆ V 2 z2 2
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Contd. Block diagram of the closed loop Adaptive System:
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Adaptive Backstepping Control of Inverted Pendulum (6.3.5.a) Dynamics of the Cart Pole system: M m cx x ml cos ml sin 2 u (t ) (I ml 2 )θ mgl sin θ mlcos θ x Dynamics of the Pendulum Angle: cos 2 sin 1 sec 2 tan 3 ut Model is beingWhere obtained State Space Representation: I ml 2 Lagrangian M1 m Dynamics` ml z1 z2 (M m) g 2 g z1 z2=u - k z 3 mlg z1 1 sec z1 3 cos z1 & k z 2 tan z1 - 2 3 z 2 sin z1
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Contd. Reformed Equation of Control Input : u g ( z1 ) z2 h Definition of 1st error variable: k (z) e1 - h= ref g(z) Stabilizing Function: zref c1e1 ref Choice of 2nd error variable: e2 zref - z2 Control Lyapunov Function: 1 2 1 2 V2 e1 e2 2 2
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Contd. Derivative of Lyapunov Function: ref u V2 e1e1 e2e2 e1 c1e1 e2 e2 c1 c1e1 e2 h g Control Input: ˆ u g z1 1 - c12 e1 c1 c2 e2 ref h Augmented Lyapunov Function: 1 2 1 2 1 1 2 Va e1 e2 g2 h 2 2 2 1g 2 2
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Contd. Derivative of the Lyapunov function: 2 2 g 2 ˆ ˆ 1 dg } h (e - 1 dh ) Va -c1e1 - c2e2 {e2 ((1 - c1 )e1 (c1 c2 )e2 ref h) - 2 g 1 dt 2 dt Parameter Update Law: ˆ dg 2 ˆ 1e2 ((1 - c1 )e1 (c1 c2 )e2 ref h) dt ˆ dh e 2 2 dt
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Robust Adaptive Backstepping Difficulties for the designer of Adaptive Control Mathematical Models are not free from Unmodeled Dynamics Parameter Drift may occur in the time of real world implementation Noises are unavoidable in real time application. Bounded disturbances may cause a high rate of adaptation which leads to an unstable/undesirable system performance.
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Contd. A continuous Switching function is use to implement the Robustification Different type of switching measures : techniques can be used to prevent the abnormal ˆ g 2 c2Robust Adaptive e2 ˆ ˆ 1e2 1 c1 e1 c1 h variation of the rate of 1 ref gs g Control!!!!! adaptation ˆ h ˆ 2 e2 2 shhwhere 0 ˆ if h h0 0 ˆ if g g 0 ˆ h h0 ˆ g g0 if h 0 ˆ h 2h0gs g0 if g 0 ˆ g 2 g0 hs h0 ˆ h ˆ g ˆ if h 2h0 g0 ˆ if g 2 g 0 h0
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Simulation Results Angular variation of Pendulum
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Conclusion & Scope of Future Research This research presents an idea of using integral action along with the backstepping control algorithms and achieves quite satisfactory results in real time experiment. One can employ Adaptive Block Backstepping algorithm to obtain a more generalize controller for the cart pole system A Robust Adaptive Block Backstepping control algorithm can be designed to address the problem of motion control of a cart pole system on inclined rail.
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QuestionsPolygonia interrogationis known as Question Mark
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References M. Krstic, I. Kanellakopoulos, and P. V. Kokotovic, Nonlinear and Adaptive Control Design, New York; Wiley Interscience, 1995. I. Kanellakopoulos and P. T. Krein, “Integral-action nonlinear control of induction motors,” Proceedings of the 12th IFAC World Congress, pp. 251- 254, Sydney, Australia, July 1993. H. K. Khalil, Nonlinear Systems, Prentice Hall, 1996. J.J.E Slotine and W. LI, Applied Nonlinear Control, Prentice Hall, 1991 Jhou J. and Wen. C, Adaptive Backstepping Control of Uncertain Systems, Springer-Verlag, Berlin Heidelberg, 2008. A Isidori, Nonlinear control Systems, Second Edition, Berlin: Springer Verlag, 1989.
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References K. J. Astrőm and K. Futura, “Swinging up a pendulum by energy control,” Preprints 13th IFAC World Congress, pp: 37-42, 1996. Furuta, K.: “Control of pendulum: From super mechano-system to human adaptive mechatronics,” Proceedings of 42th IEEE Conference on Decision and Control, pp. 1498–1507 (2003) Angeli, D.: “Almost global stabilization of the inverted pendulum via continuous state feedback,” Automatica, vol: 37 issue 7, pp 1103–1108 2001. Aström, K.J., Furuta, K.: “Swing up a pendulum by energy control,” Automatica, Vol: 36, issue 2, pp 287–295, 2000 Chung, C.C., Hauser, J.: “Nonlinear control of a swinging pendulum”.
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References Gordillo, F., Aracil, J.: “A new controller for the inverted pendulum on a cart,”. Int. J. Robust Nonlinear Control Vol: 18, pp 1607–1621, 2008 S. J. Huang and C. L. Huang, “Control of an inverted pendulum using grey prediction model,” IEEE Transaction on Industry Applications, Vol: 36 Issue: 2, pp 452-458, 2000 R. oltafi Saber, “Fixed point controllers and stabilization of the cart pole system and the rotating pendulum,” Proceedings of the 38th IEEE Conference on Decision and Control, Vol: 2, pp 1174-1181, 1999. Q. Wei, et al, “Nonlinear controller for an inverted pendulum having restricted travel,” Automatica, vol. 31, no 6, pp 841-850, 1995 Ebrahim. A and Murphy, G.V, “Adaptive Backstepping Controller Design of an inverted Pendulum,” IEEE Proceedings of the Thirty-Seventh Symposium
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References Lee, H.-H., 1998, “Modeling and Control of a Three-Dimensional Overhead Crane,” ASME J. Dyn. Syst., Meas., Control, 120, pp. 471–476. Kiss, B., Levine, J., and Mullhaupt, P., 2000, “A Simple Output Feedback PD Controller for Nonlinear Cranes,” Proc. of the 39th IEEE Conf. on Decision and Control, Sydney, Australia, pp. 5097–5101 Yang, Y., Zergeroglu, E., Dixon, W., and Dawson, D., 2001, “Nonlinear Coupling Control Laws for an Overhead Crane System,” Proc. of the 2001 IEEE Conf. on Control Applications, Mexico City, Mexico, pp. 639–644. Joaquin Collado, Rogelio Lozano, Isabelle Fantoni, “Control of convey- crane based on passivity,” Proceedings of the American Control Conference Chicago, Illinois, pp 1260-1264 June 2000
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Taken from Feedback Manual of Inverted Pendulum
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Taken from Feedback Manual of Inverted Pendulum
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Feedback Positive Real• The triple (A,B,C) is feedback positive real (FPR) if there exist a linear feedback transformation u = Kz + v such that the following two conditions hold good• A + BK is Hurwitz• And there are matrices P > 0, Q ≥ 0 which satisfy A sufficient condition for FPR is that there exists a gain row vector K such that A + BK is Hurwitz, in other words the transfer function is appositive real one , and the pair (A + BK, C) is observable.
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Passivity The system (i) z z z u, y C z , C 0 0, z Rn , u RIs said to be feedback passive (FP) if there exists a feedback transformation. u K z r zv (ii)such that the resulting system, y = C(z) is passive with a storage function U(z) which is positive definite and radically unbounded: t y v d U zt U z0 0The system of (i) is said to be feedback strictly passive (FSP) if the feedback transformation of equation (ii) renders it strictly passive:
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The system of (3.5.35) is said to be feedback strictly passive (FSP) if the feedback transformation of equation (3.5.36) renders it strictly passive: t t y v d U zt U z0 z d 0 0
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