**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 2^{nd} 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,

, andThe 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 a_{1} 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 **a _{1} is 10.78 **and

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 2^{nd} 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!!!

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