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Unformatted text preview: Chapter 8 Internal Model Principle and Repetitive Control In this chapter, we look at disturbance rejection and reference tracking in which the disturbances or reference signals have known structure - namely they can be thought of as being generated by an exo-system or the signal satisfies a known linear differential or difference equation. In this scenario, the controller can reject these disturbance or track the references by incorporating the model of the disturbance or the reference signal within itself. This approach is known as internal model principle first championed by Francis and Wonham (1976). We shall develop IMP first in the continuous time transfer function mode, and then in the state-space formulation. Finally, for the disturbance and reference are periodic, a special type of IMP controller known as repetitive control (see work by Tsao, Tomizuka and co-workers) will be discussed. 8.1 Disturbance and reference Signal model If the reference signal, or disturbance d ( t ) satisfy some differential equation: e.g. d n d dt n d d ( t ) + n d- 1 d n d- 1 dt n d- 1 d ( t ) + ... 1 d dt d ( t ) + d ( t ) = 0 then, taking Laplace transform, bracketleftbig s n d + n d- 1 s n d- 1 + ... + bracketrightbig bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright d ( s ) D ( s ) = f (0 ,s ) where f (0 ,s ) is a polynomial in s arises because of initial conditions, d (0), d (0), d (0) etc. We call d ( s ) the disturbance generating polynomial. Example : d ( t ) = sin ( t ): d ( s ) = ( s 2 + 2 ). d ( t ) = d a constant: d ( s ) = s d ( t ) = e at : d ( s ) = ( s a ). d ( t ) = d + d 1 e at : d ( s ) = s ( s a ). From the last example, we see that we can form disturbance generating polynomials for d ( t ) = ( t ) + ( t ) by combining (multiplying) the disturbance generating polynomials for ( t ) and ( t ). 131 132 c circlecopyrt Perry Y.Li 8.2 Internal Model Principle The internal model principle says that if the input disturbance, d i ( t ) , the output disturbance, d o ( t ) , or a reference r ( t ) has d ( s ) as the generating polynomial, then using a controller of the form: C ( s ) = P ( s ) d ( s ) L ( s ) (8.1) in the standard one degree-of-freedom control architecture can asymptotically reject the effect of the disturbance and cause the output to track the reference. Note that only the generating polynomial is needed. The magnitude of the disturbances or of the reference is not needed. To see why IMP works, let the plant model be G o ( s ) = B o ( s ) A o ( s ) = b n- 1 s n- 1 + b n- 2 s n- 2 + ... + b s n + a n- 1 s n- 1 + a n- 2 s n- 2 + ... + a ....
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This note was uploaded on 02/07/2012 for the course ME 8281 taught by Professor Staff during the Fall '08 term at Minnesota.
- Fall '08