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Wikipedia / Feedback linearization

Feedback linearization is a common strategy employed in nonlinear control to control nonlinear systems. Feedback linearization techniques may be applied to nonlinear control systems of the form

take the form of a linear, controllable control system,

An outer-loop control strategy for the resulting linear control system can then be applied to achieve the control objective.

Typically, feedback linearization is applied to unconstrained systems, as the required nonlinear transformations can make simple state constraints significantly more complicated in the new coordinate system.

Feedback linearization of SISO systems

that transforms the system (1) into the so-called normal form which will reveal a feedback law of the form that will render a linear input. To ensure that the transformed system is an equivalent representation of the original system, the transformation must be a diffeomorphism. That is, the transformation must not only be invertible (i.e., bijective), but both the transformation and its inverse must be smooth so that differentiability in the original coordinate system is preserved in the new coordinate system. In practice, the transformation can be only locally diffeomorphic and the linearization results only hold in this smaller region.

Several tools are required to solve this problem.

Lie derivative

derivatives. To understand the structure of this target system, we use the Lie derivative. Consider the time derivative of (2), which can be computed using the chain rule,

Now we can define the Lie derivative as,

as,

as,

Note that the notation of Lie derivatives is convenient when we take multiple derivatives with respect to either the same vector field, or a different one. For example,

and

Relative degree

if,

  1. in a neighbourhood

appears explicitly. In an linear time-invariant (LTI) system, the relative degree is the difference between the degree of the transfer function's denominator polynomial (i.e., number of poless) and the degree of its numerator polynomial (i.e., number of zeross).

Linearization by feedback

times we have,

th derivatives.

derivatives. In particular,

coordinate system. So long as this transformation is a diffeomorphism trajectories will be described by the new system,

Hence, the feedback control law

renders a linear input. The resulting linearized system

may be chosen using standard linear system methodology. In particular, a state-feedback control law of

derivatives, results in the LTI system

with,

, we can arbitrarily place the closed-loop poles of the linearized system.

Unstable zero dynamics

. However, the normal form of the system will include zero dynamics (i.e., states that are not observable from the output of the system) that may be unstable. In practice, unstable dynamics may have deleterious effects on the system (e.g., it may be dangerous for internal states of the system to grow unbounded). These unobservable states may be controllable or at least stable, and so measures can be taken to ensure these states do not cause problems in practice. Minimum phase systems provide some insight on zero dynamics.

Feedback linearization of MIMO systems

, as is shown below.

, as is shown in Eq. 5.

'th output. Analogously, this gives

such that , as is shown below.

See also

Further reading

  • A. Isidori, Nonlinear Control Systems, third edition, Springer Verlag, London, 1995.

  • H. K. Khalil, Nonlinear Systems, third edition, Prentice Hall, Upper Saddle River, New Jersey, 2002.

  • M. Vidyasagar, Nonlinear Systems Analysis, second edition, Prentice Hall, Englewood Cliffs, New Jersey, 1993.

  • B. Friedland, Advanced Control System Design, facsimile edition, Prentice Hall, Upper Saddle river, New Jersey, 1996.

References