sliding mode control,extraction of equivalent control

i was using the sliding mode control for position control of ultrasonic motor. actually the motor operation is rather simple. It has a piezo
material, that is sinusoidally excited, the vibration generated is transmitted to a moving stage through a spacer. The motion equation is M d^2(x)/dt^2+ Friction=Kf*Vc where M is the mass of the moving stage and x is the position, Frixtion is frictional force , Kf is the force constant, and Vc is the control voltage. The control voltage is a dc voltage given to the power electronics driver which generates a sine voltage at fixed frequency but its magnitude proportional to the Vc. I use the sliding surface as below S=lamda*e+de/dt where e si the tracking error e=xd-x; when i use this way the equivalent control obtained from the condition ds/dt=0 is equivalent=M/kf(lamda*de/dt+d^2(xd)/dt^2) when i use this , the de/dt part creates the problem. if i keep lamda small, this rsults in S to change sign so rapidly as de/dt is doing so, results in control signal being changed sign rapidly..thus system screams to die.. if i keep it high so as to supress the effect of de/dt in S , the effect comes in equivalent control, I am just lost how to solve this problem..and its been a long time i am trying it out...can anyone suggest me how to get rid of this problem... jenish
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junu wrote:

The biggest problem of sliding mode control is, that "real" sliding mode is based on infinite fast switching. It can be imagined like a proportional controller with an infinite gain and constraints on the control variable. Thus sliding mode control is applied when the control variable can be changed very fast without negative effects on the system.
By your choice of the sliding surface, you will force the tracking error to follow the first order dynamics defined by s=0: de/dt=-lambda e. Thus your choice of lambda controls the time constant of the target error dynamics.
The equivalent control is the expected mean control signal on the sliding surface, while in reality the control signal is switching very fast. The equivalent control can be used to predict the mean control signal. In this analysis, measurement noise is neglected. So the term de/dt in the equivalent control is not a problem, as long as xd(t) is two times differentiable.
The problem of this control concept is, that chattering (fast oscillations of the control variable) will occur in time discrete implementations. This is not a flaw of your design, but a property of sliding mode control.
perhaps you have a look at: K. David Young, Vadim I. Utkin and ĻUmit ĻOzgĻuner,"A Control Engineer’s Guide to Sliding Mode Control", IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 3, MAY 1999, p.328-342
I hope this helps a bit.
Jan
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junu wrote:

The biggest problem of sliding mode control is, that "real" sliding mode is based on infinite fast switching. It can be imagined like a proportional controller with an infinite gain and constraints on the control variable. Thus sliding mode control is applied when the control variable can be changed very fast without negative effects on the system.
By your choice of the sliding surface, you will force the tracking error to follow the first order dynamics defined by s=0: de/dt=-lambda e. Thus your choice of lambda controls the time constant of the target error dynamics.
The equivalent control is the expected mean control signal on the sliding surface, while in reality the control signal is switching very fast. The equivalent control can be used to predict the mean control signal. In this analysis, measurement noise is neglected. So the term de/dt in the equivalent control is not a problem, as long as xd(t) is two times differentiable.
The problem of this control concept is, that chattering (fast oscillations of the control variable) will occur in time discrete implementations. This is not a flaw of your design, but a property of sliding mode control.
perhaps you have a look at: K. David Young, Vadim I. Utkin and ĻUmit ĻOzgĻuner,"A Control Engineer’s Guide to Sliding Mode Control", IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 7, NO. 3, MAY 1999, p.328-342
I hope this helps a bit.
Jan
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