Adaptive control of single-input, single-output linear systems
About: This article is published in Conference on Decision and Control.The article was published on 1977-01-01 and is currently open access. It has received 3 citation(s) till now. The article focuses on the topic(s): Adaptive control & Linear system.
Summary (2 min read)
HERE ARE many examples of physical processes
- Twhich require feedback control syestems capable of functioning at a number of different process operating points.
- Nevertheless, it is fair to say that the basic principles governing the design and operation of such systems are only now just beginning to be understood.
- Only under very restrictive assumptions have any actually been shown to result in stable closed-loop systems.
- The only assumptions made about the process are that it admits a transfer function model with left-half plane zeros, and that: 1) an .
- In spite of its intuitive appeal, this particular configuration is dfficult to analyze and at present is not well understood.
- There are at least three reasons for this.
- 1) Known results -[ll] characterizing the behavior of dynamic identifiers (e.g., adaptive observers) are not directly applicable, since such results usually require all identifier inputs to be bounded; in the present situation, neither the process input u nor output y can be assumed bounded a priori, since the identifier is in feedback with the process.
- 2) The relationship between process model parameters p and desired feedback gains f is typically a complicated nonlinear functionf(p).
Process Model Assumptions
- The reference model is a stable, canonical linear system with strictly proper transfer function T,(s) and bounded, piecewise-continuous, reference input r( t).
- Clearly, any reference model not respecting this constraint must involve some form of differentiation.
- Since the authors have stipulated that their adaptive controller be differentiator-free, they must require that (T,(s))O >n*.
- The process model assumptions imply that the process can be represented in an especially useful way.
p(i?,r(t))-p(Xo(t),r(t)) is globally, uniformly stable, jSo(t)
- On the other hand, it cannot be concluded from Theorem 2 (or Theorem 1) that the parameter errors F(t) and g(t) tend to 'Note that C; is nonsingular because (c,A) is observable.
- The first describes certain algebraic relationships which exist between a positive-definite matrix P and matrices Mi defined by (43) .
- This together with (71) allows us to write.
- In this paper the authors have outlined a procedure for designing a parameter-adaptive controller capable of causing the output of a process to approach and track the output of a prespecified reference model with zero steady-state error.
- The principal result of this paper-that it is actually possible to adaptively control a linear system with zero steady-state tracking error-is contrary to their earlier expectations [lo].
- What this paper shows is that this problem can be avoided by applying to the process a control signal which, in effect, is an estimate of the product of desired feedback gains and process state rather than the product of distinct estimates of desired feedback gains and process state.
- Recent simulation experiments have shown, that for such models, the overall set of differential equations describing the adaptive system can be quite sensitive to computer roundoff errors unless the initial parameter errors are small.
- Systematic techniques for selecting these gains to reduce sensitivity or to increase convergence rates are obviously needed, but so far have not been developed.
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