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0066_frame_C31.fm Page 6 Wednesday, January 9, 2002 7:49 PM

r(t) Controller u(t) Plant y(t)
controller unknown
reference input
parameters parameters

parameter Adaptive Update
estimates Mechanism

FIGURE 31.2 The self-tuning control architecture.

principle and both direct and indirect parameter update procedures can be adopted within the MRAC
framework. Much of the work in this area deals with continuous time systems.

Self-Tuning Controller (STC)

In contrast to MRAC, there is no reference model in the STC design. A schematic sketch is shown in
Fig. 31.2. In this formulation, the controller parameters of the plant parameters are estimated in real
time, depending on whether it is a direct or indirect approach. These estimates are then used as if they
are equal to the true parameters (certainty equivalence design). Parameter estimation involves finding
the best-fit set of parameters based on the plant input–output data. This is different from the MRAC
parameter adaptation scheme, where the parameter estimates are updated in such a way to achieve
asymptotic tracking between the tracking error between the plant and the reference model. In several
STC estimation schemes, it is also possible to quantify a measure of the quality of the parameter estimates,
which can be used in the design of the controller. Many different combinations of the estimation methods
can be adopted and can be applied to both continuous time and discrete time plants. Due to the
“separation” between parameter estimation and control in STC, there is greater fiexibility in design.
However, stability and convergence are difficult to prove and stronger conditions on input signals are
required (persistent excitation) to guarantee parameter convergence. Historically speaking, STC designs
arose in the study of the stochastic regulation problem and much of the literature is devoted to discrete
time plants using an indirect approach. In spite of the seeming difference between MRAC and STC, a
direct correspondence exists between problems from both the areas. 8

31.5 Nonlinear Adaptive Control Systems

For the most general case of nonlinear systems, there exists very limited theory in the field of adaptive
control. Even though there is great interest in this area due to potential applications in a wide variety of
complex machanical systems, theoretical diffculties exist because of the lack of general analysis tools.
However, some important special cases are well understood by now, and we summarize the conditions
that these classes of systems satisfy:
1. The unknown parameters within the nonlinear plant are linearly parameterized.
2. The complete state vector is measured.
3. When the unknown parameters are assumed known, the control input can cancel all the nonlin-
earities in a feedback-linearization sense and any remaining internal dynamics should be stable.
The adaptive design is then accomplished by certainty equivalence.
We now show a typical nonlinear MRAC methodology to deal with a situation in which the nonlinear
plant model has unknown parameters. Consider the nonlinear system

x ˙ = q fx() + u (31.5)

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