8_feedback_lin - Control of Nonlinear Dynamic Systems:...

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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 133 8 ` Feedback Linearization Feedback linearization is an approach to nonlinear control design that has attracted lots of research in recent years. The central idea is to algebraically transform nonlinear systems dynamics into (fully or partly) linear ones , so that linear control techniques can be applied. This differs entirely from conventional (Jacobian) linearization , because feedback linearization is achieved by exact state transformation and feedback, rather than by linear approximations of the dynamics . Key points Feedback linearization = ways of transforming original system models into equivalent models of a simpler form. Completely different from conventional (Jacobian) linearization, because feedback linearization is achieved by exact state transformation and feedback, rather than by linear approximations of the dynamics. Input-Output, Input-State Internal dynamics, zero dynamics, linearized zero dynamics Jacobi’s identity, the theorem of Frobenius MIMO feedback linearization is also possible.
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 134 The basic idea of simplifying the form of a system by choosing a different state representation is not completely unfamiliar; rather it is similar to the choice of reference frames or coordinate systems in mechanics. Feedback linearization = ways of transforming original system models into equivalent models of a simpler form. Applications: helicopters, high-performance aircraft, industrial robots, biomedical devices, vehicle control. Warning: there are a number of shortcomings and limitations associated with the feedback linearization approach. These problems are very much topics of current research. References: Sastry, Slotine and Li, Isidori, Nijmeijer and van der Schaft Terminology Feedback Linearization A “catch-all” term which refers to control techniques where the input is used to linearize all or part of the system’s differential equations. Input/Output Linearization A control technique where the output y of the dynamic system is differentiated until the physical input u appears in the r th derivative of y. Then u is chosen to yield a transfer function from the “synthetic input”, v, to the output y which is: r s s V s Y 1 ) ( ) ( = If r, the relative degree, is less than n, the order of the system, then there will be internal dynamics. If r = n, then I/O and I/S linearizations are the same. Input/State Linearization A control technique where some new output y new = h new (x) is chosen so that with respect to y new , the relative degree of the system is n. Then the design procedure using this new output y new is the same as for I/O linearization.
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Control of Nonlinear Dynamic Systems: Theory and Applications J. K. Hedrick and A. Girard © 2005 135 Input/Output Linearization Input/State Linearization I/O and I/S coincide Natural output r < n Where the natural output y has relative degree n Choose new output y
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This note was uploaded on 08/01/2008 for the course ME 237 taught by Professor Hedrick during the Spring '08 term at University of California, Berkeley.

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8_feedback_lin - Control of Nonlinear Dynamic Systems:...

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