Backstepping control of cart pole system

Slide 1

Backstepping Control of Cart Pole SystemPresented by Shubhobrata RudraMaster in Control System EngineeringRoll No: M4CTL 10-03

Under the Supervision ofDr. Ranjit Kumar Barai

ContentObjectives of the ResearchModeling of the Physical SystemsDifficulties of the Controller DesignBackstepping Control Stabilization of Inverted PendulumAnti Swing Operation of Overhead CraneAdaptive Backstepping Control & its application on Inverted Pendulum Conclusion & Scope of Future ResearchReferences

Objective of the ResearchMaintain 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

Modeling of Cart Pole System

F

V

Contd. State Model of Inverted Pendulum:

Most of the Nonlinearities except the friction T are the functions of the pendulum angle x2If the angle of the pendulum is quite small we can replace those nonlinear terms. Hence we can realize a Linear Model for small angle deviation!!!Hence Based on Angular position of Pendulum in space it is possible to divide the total operating region in two different zone

Difficulties of the Controller DesignThe 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.

BACKSTEPPING CONTROL

CONTENTWhat is Backstepping?

Why Backstepping?

Different Cases of Stabilization Achieved by Backstepping

Backstepping: A Recursive Control Design Algorithm

New Research Ideas

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.

Contd.Block Diagram of the system:

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.

Contd.Only one can conclude the desired value of z as

Definition of Error variable e:

z is termed as the Virtual ControlDesired Value of z, s(x) is termed as stabilizing function.System Dynamics in ( x, e) Coordinate:

Modified Block Diagram

Contd.

Feedback Control Law sBackstepping Signal -s

So the signal s(x) serve the purpose of feedback control law inside the block and backstep -s(x) through an integrator.

Contd.

Feedback loop with + s(x) Backstepping of Signal -s(x)Through integrator

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:Contd.

Consider the scalar nonlinear system

Control Law( using Feedback Linearization):

Resultant System:

Edurado D. Sontag Proposed a formula to avoid the Cancellation of these useful nonlinearities.

Why Backstepping?

is it essential to cancel out the term

Not at all!!!!This is an Useful Nonlinearity, it has an Stabilizing effect on the system.

Sontag's Formula:

Control Law (Sontags Formula):

Control Law (using Backstepping):

Contd.

For large values of x, the control law becomesusinx

So this control law avoids the cancellation of useful nonlinearities! For higher values of x But this formula leads a complicated control input for intermediate values of x

Simulation Results: Stabilization of the Nonlinear Scalar plant

Contd.

Variation of x with timeFeedback LinearizationSontags FormulaBackstepping Control Law

Feedback Linearization***Sontags Formula +++Backstepping Control law Control Effort variation with time

Contd. IEEE Explore 1990-2003 Backstepping in title

Conference Paper

Journal PaperOla Harkegard Internal seminar on Backstepping January 27, 2005

Different Cases of StabilizationAchieved by BacksteppingIntegrator BacksteppingNonlinear Systems Augmented by a Chain of Integrator

Stable Nonlinear System Cascaded with a Dynamic SystemInput Subsystem is a Linear SystemInput Subsystem is a Nonlinear System

Nonlinear System connected with a Dynamic BlockDynamic block connected with the system is a linear oneDynamic block connected with the system is a Nonlinear one

Integrator BacksteppingTheorem of Integrator Backstepping:

If the nonlinear system satisfies certain assumption with z R as its control then

The CLF

depicts the control input u

renders the equilibrium point x=0, z=0 is GAS.

Nonlinear SystemIntegrator

Chain of IntegratorChain of integrator:

CLF

Nonlinear System

K th integrator

Stabilization of an unstable system

Stabilizing Function:

Choice of Control law:Integrator Backstepping Example

Simulation Results

The equilibrium point x=0, z=0 is a GAS.

uStabilization of Cascaded SystemStable nonlinear system cascaded with a Linear system

CLF

The Control Law:

Ensures the Equilibrium (x=0, z=0) is a GAS.

u

A, B, C are FPR

Stable nonlinear system cascaded with a Nonlinear system

CLF

Control Law

Ensures the Equilibrium (x=0,z=0) is GAS.

u

Feedback Passive System with U(z) as a +ve Definite Storage Function

u=K(z)+r(z)v is a Feedback TransformationSuch that the resulting system isPassive withStorage Function U(z) Contd.

System Dynamics:

Feedback Law:

Storage function:

Derivative of Storage Function: Stabilization with Passivity an Example

u

Control law

Simulation Results:

The equilibrium point x=0, z=0 is a GAS.

Contd.

Block BacksteppingNonlinear system cascaded with a Linear Dynamic Block

Using the feedback transformation The State equation of the system becomesControl Law

Ensures the equilibrium point x=0, z=0 is GAS.

u

Eigen values of the are the zeros of the transfer function

Zero Dynamics

Stable/Unstable Nonlinear system

Minimum Phase Linear System with relative degree one

Nonlinear system cascaded with a Nonlinear Dynamic Block

Control Law:

Ensures the equilibrium x=0, z=0 is GAS.

Contd.

uNonlinear System with relative degree oneAnd the zero dynamics subsystems is globally defined and it is Input to state stable

Backstepping: A Recursive Control Design AlgorithmBackstepping 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 SystemSemi Strict Feedback SystemBlock Strict Feedback Systems

Strict Feedback Systems:

Control Input:CLFContd.

Lower Triangular Form

Semi Strict Feedback Systems:

CLF:Control Input:

Contd.

Lower Triangular Form

Block Strict Feedback forms:

Contd.

Assumptions:Each K subsystem with state and ,and input satisfies: BSF-1: Its relative degree is one uniformly inBSF-2:Its zero dynamics subsystem is ISS w.r.to

Sub-System Dynamics in transformed Co ordinate:

Contd.

The change of Coordinate Results:

Contd.

Strict Feedback FormZero Dynamics

In 1993, I. Kanellakopoulos and P. T. Krein introduced the use of Integral action along with the Backstepping control algorithm, which considerably improves the steady-state controller performance [2].

It is possible to represent a complex nonlinear system as a combination of two nonlinear subsystem, while each subsystem is in Block Strict Feedback form. 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 better transient as well as steady state performance of the controller for complex nonlinear plant.New Research Ideas

STABILIZATION OF INVERTED PENDULUM

ContentControl 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

Control Objective

Design a control systemthat keeps the pendulumbalanced and tracks thecart to a commandedposition!!!

Maintain the Stability of the Inverted Pendulum when it is suffering with external disturbances.

Two Zone Control TheoryMost of the nonlinearities (present in the state model of Inverted Pendulum) are the function of pendulum angle in space.

Stabilization Zone

Swinging Zone

Unstable Equilibrium Point

Features of Two Zone Control TheoryTwo 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.

Linearize model of Inverted Pendulum

Choice of Control Variable::

Design of Control Algorithm for Stabilization Zone

The state model of the system not allows the designer to implement backstepping algorithm on it

It is possible to represent the system as a combination of two dynamic block each of them in block strict feedback system

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

Choice of CLF:

Control Input:

Where

Derivative of CLF:

Contd.

Integral action reduces the steady state error of the system.

List of the controller parameters

Where d1=c1+c2 & d2=c1c2. k1=1, c1=c2=50, c=0.001

Contd.

State model of the Inverted Pendulum:

Choice of Control variable:

Design of Control Algorithm for Swinging Zone

Choice of Stabilizing function:



Contd.

Choice of CLF:

Control Input:

Derivative of CLF:

Contd.

List of Controllers Parameters

Contd.

List of Controllers Parameters

k2=0.1, d3=c3+c4 and d4=c3c4+1, where c3=c4=20

Contd.


ReferenceInputLinear Backstepping Controller Nonlinear Backstepping Controller Controller for Stabilization Zone Controller for Swinging ZoneInverted Pendulum

SwitchingMechanismControl InputSwitchingLaw

Results of Real Time ExperimentAngle of the Inverted Pendulum

Pendulum reach its stable position within 4 seconds

Angular Velocity of the Inverted Pendulum

Contd.

Cart Movement with time

Contd.

The cart is able to track the reference trajectory within 15 seconds

Cart Velocity Contd.

Voltage applied on the actuator motor

Contd.

Moderate Variation of voltage reduces the stress on actuator motor

Angle of the Inverted Pendulum when it is suffering with external impact

Contd.

Pendulum regain its inverted position within 3 seconds

Angular Velocity of the Pendulum

Contd.

Cart Position of the pendulum (suffering with an external impact)

Contd.

Cart Regain its Desired trajectory within 12 seconds

Cart Position of the pendulum (suffering with an external impact)

Contd.

Voltage applied on the actuator motor

Contd.

Comparative Study and Conclusions Comparative study on the Pendulum angular position in space

Comparison of Cart tracking Performance

Contd.

ConclusionBackstepping 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

ANTISWING OPERATION OF OVERHEAD CRANE

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



Comparative Study and Conclusion

Content

Control ObjectiveProper tracking of The Cart Motion along a reference/desired trajectory.Proper Antiswing operation of pay load during travel

Most of the nonlinearities (present in the state model of Overhead Crane) are the function of payload angle in space.

Two Zone Control Theory

Stable Tracking Zone

Anti Swing Zone

Linearize model of Overhead Crane

Choice of Control Variable:

Design of Control Algorithm for Stable Tracking Zone

The Primary objective of design is to control the motion of the cart along with a reference trajectory

Contd.Choice of Stabilizing Function:



Integral action is introduced to enhance the controller performance in steady state operation

Contd.Choice of CLF:

Control Input:

Where

Derivative of CLF:

Integral action reduces the steady state error of the system.

List of Controller Parameters

Where d1=c1+c2 & d2=c1c2. k1=1, c1=c2=50, c=0.001

Contd.

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!!!!!!!

Design of Control Algorithm for Anti-Swinging Zone

Contd.Same Control Algorithm is being used to serve the opposite purpose!!!

Swinging ZoneAnti Swing ZoneInverted PendulumOverhead Crane


ReferenceInputLinear Backstepping Controller Nonlinear Backstepping Controller Controller for Stable Tracking Zone Controller for Anti Swing ZoneInverted Pendulum

SwitchingMechanismControl InputSwitchingLaw

OverheadCrane

Motion of the Cart


Steady state Tracking error reduces with time

Cart Velocity

Contd.

Payload Angular PositionContd.

3.15

Payload Angular Velocity Contd.

Cart Motion of the pendulum when suffering with an external impact

Contd.

The cart is able to track the reference trajectory within 15 seconds

Cart Velocity when suffering with an external impact

Contd.

Angle of the Payload when suffering with an external impact

Contd.

The angle of the payload reduces within 15 seconds

Angular Velocity of the Payload when suffering with an external impact

Contd.

ConclusionBackstepping 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.

Adaptive Backstepping Controland its Application on Inverted Pendulum

ContentAdaptation as Dynamic FeedbackAdaptive Integrator BacksteppingStabilization of an Inverted PendulumRobust Adaptive BacksteppingSimulation ResultsConclusion

Adaptation as Dynamic FeedbackStabilization problem of a nonlinear system:

Static Control Law:

Augmented Lyapunov function:

is an unknown constant parameter

is an unknown parameter so it is impossible to use this expression of control law, containing unknown parameterOne Can use a certainty equivalence form where is replaced by an estimate of ,

Dynamic Control Law

is adaptation gain Is the parameter error

Derivative of Augmented Lyapunov function:

Update law:

Which ensures the negative definiteness of .

System dynamics:

Contd.

Block diagram of the Closed loop Adaptive system

Contd.

Adaptive BacksteppingStabilization of 2nd order nonlinear system:


CLF:

Control law:

is an unknown parameter. So should be replaced by its estimated value.

Error Dynamics:

Construction of Augmented Lyapunov Function:

Derivative of Augmented Lyapunov function:Update Law :

Contd.

Block diagram of the closed loop Adaptive System:

Contd.

Adaptive Backstepping Control of Inverted PendulumDynamics of the Cart Pole system:

Dynamics of the Pendulum Angle:

WhereState Space Representation:

Model is being obtained Lagrangian Dynamics`

(6.3.5.a)

Contd. Reformed Equation of Control Input :

Definition of 1st error variable:


Choice of 2nd error variable:

Control Lyapunov Function:

Derivative of Lyapunov Function:

Control Input:

Augmented Lyapunov Function:

Contd.

Derivative of the Lyapunov function:

Parameter Update Law:

Contd.

Robust Adaptive BacksteppingDifficulties 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.

Contd. Robust Adaptive Control!!!!!Different type of switching techniques can be used to prevent the abnormal variation of the rate of adaptationA continuous Switching function is use to implement the Robustification measures :

where

Simulation ResultsAngular variation of Pendulum

Disturbances Signal:

Contd.

Estimation of the Parameter g

Contd.

Parameter Estimation of h with time

Contd.

Conclusion & Scope of Future ResearchThis 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.

Questions

Polygonia interrogationis known as Question Mark

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.

Fu-Kuei Tsai and Jung-Shan-Lin, Nonlinear Backstepping control Design of Furuta Pendulum, Proceedings of the 2003 IEEE Conference on Control Applications, Toronto, Canada, pp. 100-107, August 24-28, 2003.

Yung-Chih Fu and Jung-Shan Lin, Nonlinear Backstepping control Design of Furuta Pendulum, Proceedings of the 2005 IEEE Conference on Control Applications, Toronto, Canada, pp. 96-101, August 28-31, 2005.

C.I.Byrnes, A Isidori, J C Willems, Passivity, Feedback equivalence, and the global stabilization of minimum phase nonlinear systems, IEEE Transactions on Automatic Control,Vol: 36 issue 11, pp 1228-1240, 1991.

R. Ortega, Passivity properties for the stabilization of cascaded nonlinear systems, Automatica, Vol. 27, pp. 423-424, 1991

A. ohsumi and T. Izumikawa, Nonlinear control swing up and stabilization of an inverted pendulum, Proceedings of the 34th IEEE Conference on Decision and Control, vol: 4, pp: 3873-3880, 1995.

References

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References

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References

Thank you

Taken from Feedback Manual of Inverted Pendulum

Taken from Feedback Manual of Inverted Pendulum

Feedback Positive RealThe 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.

PassivityThe system(i)

Is said to be feedback passive (FP) if there exists a feedback transformation.

(ii)

such that the resulting system, y = C(z) is passive with a storage function U(z) which is positive definite and radically unbounded:

The system of (i) is said to be feedback strictly passive (FSP) if the feedback transformation of equation (ii) renders it strictly passive:

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:

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A0

Technology

Backstepping control of cart pole system