Adaptive Control Based on Retrospective Cost Optimization

U NLIKE robust control, which chooses control gains based on a prior, fixed level of modeling uncertainty, adaptive control algorithms tune the feedback gains in response to the true plant and exogenous signals: that is, commands and disturbances. Generally speaking, adaptive controllers require less prior modeling information than robust controllers and thus can be viewed as highly parameter-robust control laws. The price paid for the ability of adaptive control laws to operate with limited prior modeling information is the complexity of analyzing and quantifying the stability and performance of the closed-loop system, especially in light of the fact that adaptive control laws, even for linear plants, are nonlinear. Stability and performance analysis of adaptive control laws often entails assumptions on the dynamics of the plant. For example, a widely invoked assumption in adaptive control is passivity [1], which is restrictive and difficult to verify in practice. A related assumption is that the plant is minimum phase [2,3], which may entail the same difficulties. In fact, sampling may give rise to non-minimum-phase zeros whether or not the continuous-time system is minimum phase [4], which must ultimately be accounted for by any adaptive control algorithm implemented digitally in a sampled-data control system. Beyond these assumptions, adaptive control laws are known to be sensitive to unmodeled dynamics and sensor noise [5,6], which necessitates robust adaptive control laws [7]. In addition to these basic issues, adaptive control laws may entail unacceptable transients during adaptation, which may be exacerbated by actuator limitations [8–10]. In fact, adaptive control under extremely limited modeling information, such as uncertainty in the signof thehigh-frequencygain [11,12],mayyielda transient response that exceeds the practical limits of the plant. Therefore, the type and quality of the available modeling information as well as the speed of adaptation must be considered in the analysis and implementation of adaptive control laws. These issues are stressed in [13]. Adaptive control laws have been developed in both continuoustime and discrete-time settings. In the present paper, we consider discrete-time adaptive control laws, since these control laws can be implemented directly in embedded code for sampled-data control systems without requiring an intermediate discretization step that may entail loss of stability margins. References on discrete-time adaptive control include [2,3,14–24]. In [2], a discrete-time adaptivecontrol lawwith guaranteed stability is developed under a minimum-phase assumption. Extensions given in [3] based on internal model control [25] and Lyapunov analysis also invoke this assumption. To circumvent the minimum-phase assumption, the zero annihilation periodic control law [23] uses lifting to move all of the plant zeros to the origin. The drawback of lifting, however, is the need for open-loop operation during alternating data windows. An alternative approach, developed in [14,15,17,18], is to exploit knowledge of the non-minimum-phase zeros. In [14], knowledge of the non-minimum-phase zeros is used to allow matching of a desired closed-loop transfer function, recognizing that minimum-phase zeros can be canceled but not moved, whereas non-minimum-phase zeros can neither be canceled nor moved. In [15,18], knowledge of a diagonal matrix that contains the non-minimum-phase zeros is used within a multi-input/multioutput (MIMO)direct adaptivecontrol algorithm.Finally, knowledge of the unstable zeros of a rapidly sampled continuous-time singleinput/single-output (SISO) system with a real non-minimum-phase zero is used in [17]. Motivated by the adaptive control laws given in [3,24], the goal of the present paper is to develop a discrete-time adaptive control law that is effective for non-minimum-phase systems. In particular, we present an adaptive control algorithm that extends the retrospective cost optimization approach used in [24]. This extension is based on a retrospective cost that includes controlweighting aswell as a learning rate, which can be used to adjust the rate of controller convergence and thus the transient behavior of the closed-loop system. Unlike [24], which uses a gradient update, the present paper uses a Newtonlike update for the controller gains, as the closed-form solution to a quadratic optimization problem.Nooffline calculations are needed to implement the algorithm. A key aspect of this extension is the fact that the required modeling information is the relative degree, the first nonzero Markov parameter, and non-minimum-phase zeros, if any. Exceptwhen theplant hasnon-minimum-phase zeroswhoseabsolute value is less than the plant’s spectral radius, we show that the required zero information can be approximated by a sufficient number of Markov parameters from the control inputs to the performance variables. No matching conditions are required on either the plant uncertainty or disturbances. The goal of the present paper is to develop the retrospective correction filter (RCF) adaptive control algorithm and demonstrate its effectiveness for handling non-minimum-phase zeros. To this end, we consider a sequence of examples of increasing complexity, ranging fromSISOminimum-phase plants toMIMOnon-minimumphase plants, including stable and unstable cases. We then revisit these plants under offnominal conditions: that is, with uncertainty in the required plant modeling data, unknown latency, sensor noise, and Received 15 August 2009; revision received 7 November 2009; accepted for publication 16 November 2009. Copyright © 2009 by Dennis S. Bernstein. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/10 and $10.00 in correspondence with the CCC. ∗National Defense Science and Engineering Graduate Fellow, Aerospace Engineering Department, 1320 Beal Avenue. Member AIAA. Professor, Aerospace Engineering Department, 1320 Beal Avenue. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 33, No. 2, March–April 2010