Drexel Autonomous Systems Lab

 

                   

 

SLIDING MODE CONTROL OF A SUSPENDED PENDULUM

Theory    Simulations    Real Time Control    References

        Sliding mode control is an efficient tool to control complex high-order dynamic plants operating under uncertainty conditions due to its order reduction property and its low sensitivity to disturbances and plant parameter variations. Its robustness property comes with a price, which is high control activity. The principle of sliding mode control is that; states of the system to be controlled are first taken to a surface (sliding surface) in state space and then kept there with a shifting law based on the system states. Once sliding surface is reached the closed loop system has low sensitivity to matched and bounded disturbances, plant parameter variations [1]. For the theory of sliding mode control click here. Sliding mode control can be conveniently used for both non-linear systems and systems with parameter uncertainties due to its discontinuous controller term. That discontinuous control term is used to negate the effects of non-linearities and/or parameter uncertainties.

        Suspended pendulum is a 2nd order non-linear system due to the sine term:

 The fact that this change is sinusoidal makes sliding mode control attractive to use as a controller for suspended pendulum systems, for sinusoidal functions are bounded. This can be extended to robotics and systems with moving linkages for that kind of systems have inertias that show sinusoidal characteristic.

The parameters of the pendulum system that is controlled with sliding mode control is given below. The period of the suspended pendulum is about 4 seconds.

The simplified form of the equation of motion is, .

where

The sliding mode control  input is,  

, and 

The control signal has two parts; first part is state feedback control law and following that is the discontinuous term that overcomes the sine term. s represents the sliding surface. Due to the chattering phenomenon of the sliding mode control it is convenient to replace the sign function with a continuous approximation. Following is the pseudo sliding mode control law.

The procedure to design the controller parameters is as follows:

1-    Find the bound of the non-linearity or uncertainty and if possible decrease the magnitude of the bound by defining some part of the non-linearity or uncertainty with a linear combination of the states.

2-    Either select a sliding surface and find state feedback parameters or design a state feedback controller that would impose a sliding surface. Note that these can be totally different approaches.

Following are some options to deal with the non-linear term, sine.

i- ρ can be selected as , or rejecting the non-linear term totally.

ii- ρ can be selected as  . In this case the equation of the system would be . Note that this equation is the same as the equation of motion that is linearized around 0 degrees. However sliding mode control is not restricted to work around the equilibrium point because the difference between sin(θ) and θ is bounded and can be rejected with the discontinuous term of the sliding mode control law.

iii- ρ can also be selected by treating the non-linear term as an uncertainty in the system parameter a1 as  in the below equation.

.

Note that sin(θ)/θ is a bounded function. Nominal value can be used as system parameter and max and min values would determine the bound of the uncertainty. For further discussion see reference [1].

        In this study ρ is taken as , where a1 is 10.78 and η is chosen as 1.22. η is a small parameter that would make the resulting system equation insensitive to the non-linear term.

        The state feedback parameters are found by using pole placement and then corresponding sliding surface is obtained. The magnitude of the poles are Φ and m. They are selected as  2 rad/s equal to each other; the linear part of the system would resemble a critically damped second order system. Making the system critically damped has two significances: First is not to cross the sliding surface and second is to have real valued sliding surface. Choosing 2 rad/s as the poles of the two first order decaying systems mean that the pendulum would respond to a step input in 2*(4*(1/2s)) = 4 seconds, (time constants of the two first order systems are 1/(2s)). It should be kept in mind that this is an approximation. The system is a 2nd order system with a discontinuous term.

        Finally δ, which approximates the discontinuous term of the sliding mode control law is taken as 0.1. At around 0.1 the chattering is not observed in the simulations performed.

 

        Integral action is going to be added to the sliding mode control soon!!!

 

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For questions about this tutorial please feel free to contact vefa@drexel.edu.