Adaptive H/sub ∞/ control with application to systems with structural flexibility

Abstract

In this paper we develop a robust adaptive control algorithm using a combination of H 1 design and system identification. We derive frequency dependent bounds for the tolerated unmodeled dynamics and show that the approach gives a closed-loop system with bounded`1 and`2 gain when the model mismatch is small in the frequency range where the control gain is large. Our application focus is systems with structural flexibility. We present a parameter estimation algorithm that uses constrained least squares with prefiltering to overcome the problem of identifying lightly damped antiresonances (a common problem in identification of flexible systems). The estimation and control design are executed at a low frequency and only when parameter updating is needed. This allows us to apply computationally expensive control design and signal processing algorithms. It also eliminates many of the problems of earlier adaptive controllers (such as bursting, parameter drift, etc.) by turning off the estimator. We show results from the application of adaptive H1 control to a high-fidelity model of the Martin Marietta flexible beam testbed.

Stability of supervised adaptive control

“…Assumption (2) constrains the types of external disturbances and model/plant mismatch handled by the algorithm. Effectively, the second assumption states that the plant disturbances are bounded, resulting in un-modeled dynamics with bounded H ∞ norm [13], [14]. We are now ready to state our result.…”

“…The Lyapunov result for the supervisor case (always updated) is exactly as the previous result for V C (t) (Lemma 2). In order to incorporate the Lyapunov result for the control model we need to incorporate the indicator function, ∆ C (t), defined in (13).…”

“…First we establish the relationship between the estimate of the disturbance variance r C (t) and the comparison 46th IEEE CDC, New Orleans, USA, Dec. [12][13][14]2007 WePI19.9 signal x(t). Second we relate the comparison sequence x(t) and the normalization sequence n C (t) to the control model prediction error e C (t).…”

Adaptive control with selective memory

“…What the inequality states is that the external disturbances should be bounded by a constant and that the model-mismatch should be 'small' relative to the comparison sequence aðtÞ: The second inequality is satisfied if we use a controller that stabilizes the estimated model at each time instant. Sufficient conditions for the existence of such constants using one step ahead control is given in Reference [6] and for H 1 control in Reference [14].…”

“…The constants K g ; k g ; K f ; k f of Assumption 1 depend on filtering as well as choice of control law as pointed out by [6,14]. 3.…”