Robust Adaptive Control

This chapter presents several techniques for improved robustness of model-reference adaptive control. These techniques, called robust modification, achieve increased robustness through two general principles: (1) limiting adaptive parameters and (2) adding damping mechanisms in the adaptive laws to bound adaptive parameters. The dead-zone method and the projection method are two common robust modification schemes based on the principle of limiting adaptive parameters. The dead-zone method prevents the adaptation when the tracking error norm falls below a certain threshold. This method prevents adaptive systems from adapting to noise which can lead to a parameter drift. The projection method is widely used in practical adaptive control applications. The method requires the knowledge of a priority bounds on system parameters. Once the bounds are given, a convex set is established. The projection method then permits the normal adaptation mechanism of model-reference adaptive control as long as the adaptive parameters remain inside the convex set. If the adaptive parameters reach the boundary of the convex set, the projection method changes the adaptation mechanism to bring the adaptive parameters back into the set. The \(\sigma \) modification and e modification are two well-known robust modification techniques based on the principle of adding damping mechanisms to bound adaptive parameters. These two methods are discussed, and the Lyapunov stability proofs are provided. The optimal control modification and the adaptive loop recovery modification are two recent robust modification methods that also add damping mechanisms to model-reference adaptive control. The optimal control modification is developed from the optimal control theory. The principle of the optimal control modification is to explicitly seek a bounded tracking as opposed to the asymptotic tracking with model-reference adaptive control. The bounded tracking is formulated as a minimization of the tracking error norm bounded from an unknown lower bound. A trade-off between bounded tracking and stability robustness can therefore be achieved. The damping term in the optimal control modification is related to the persistent excitation condition. The optimal control modification exhibits a linear asymptotic property under fast adaptation. For linear uncertain systems, the optimal control modification causes the closed-loop systems to tend to linear systems in the limit. This property can be leveraged for the design and analysis of adaptive control systems using many existing well-known linear control techniques. The adaptive loop recovery modification is designed to minimize the nonlinearity in a closed-loop plant so that the stability margin of a linear reference model could be preserved. This results in a damping term proportional to the square of the derivative of the input function. As the damping term increases, in theory the nonlinearity of a closed-loop system decreases so that the closed-loop plant can follow a linear reference model which possesses all the required stability margin properties. The \(\mathscr {L}_{1}\) adaptive control has gained a lot of attention in the recent years due to its ability to achieve robustness with fast adaptation for a given a priori bound on the uncertainty. The underlying principle of the \(\mathscr {L}_{1}\) adaptive control is the use of fast adaptation for improved transient or tracking performance coupled with a low-pass filter to suppress high-frequency responses for improved robustness. As a result, the \(\mathscr {L}_{1}\) adaptive control can be designed to achieve stability margins under fast adaptation for a given a priori bound on the uncertainty. The basic working concept of the \(\mathscr {L}_{1}\) adaptive control is presented. The bi-objective optimal control modification is an extension of the optimal control modification designed to achieved improved performance and robustness of systems with input uncertainty. The adaptation mechanism relies on two sources of errors: the normal tracking error and the predictor error. A predictor model of a plant is constructed to estimate the open-loop response of the plant. The predictor error is formed as the difference between the plant and the predictor model. This error signal is then added to the optimal control modification adaptive law to enable the input uncertainty to be estimated. Model-reference adaptive control of singularly perturbed systems is presented to address slow actuator dynamics. The singular perturbation method is used to decouple the slow and fast dynamics of a plant and its actuator. The asymptotic outer solution of the singularly perturbed system is then used in the design of model-reference adaptive control. This modification effectively modifies an adaptive control signal to account for slow actuator dynamics by scaling the adaptive law to achieve tracking. Adaptive control of linear uncertain systems using the linear asymptotic property of the optimal control modification method is presented for non-strictly positive real (SPR) systems and non-minimum phase systems. The non-SPR plant is modeled as a first-order SISO system with a second-order unmodeled actuator dynamics. The plant has relative degree 3 while the first-order reference model is SPR with relative degree 1. By invoking the linear asymptotic property, the optimal control modification can be designed to guarantee a specified phase margin of the asymptotic linear closed-loop system. For non-minimum phase systems, the standard model-reference adaptive control is known to be unstable due to the unstable pole-zero cancellation as a result of the ideal property of asymptotic tracking. The optimal control modification is applied as an output feedback adaptive control design that prevents the unstable pole-zero cancellation by achieving bounded tracking. The resulting output feedback adaptive control design, while preventing instability, can produce poor tracking performance. A Luenberger observer state feedback adaptive control method is developed to improve the tracking performance. The standard model-reference adaptive control still suffers the same issue with the lack of robustness if the non-minimum phase plant is required to track a minimum phase reference model. On the other hand, the optimal control modification can produce good tracking performance for both minimum phase and non-minimum phase reference models.