Q&A regarding L1 adaptive control

This page contains questions and answers regarding the contribution of L1 adaptive control theory to the general field of control theory and engineering. We will continue the postings, as we get more questions in various forums and by e-mail.


Q1: What is the main contribution of L1 adaptive control theory?

A1: L1 adaptive control theory decouples estimation (adaptation) from control (robustness), and with that allows for arbitrary  FAST adaptation with bounded away from zero time-delay margin. The architectures of this theory, with a single set of control design parameters and without any redesign or retuning, lead to uniform (apriori PREDICTABLE) performance bounds throughout the entire operation of the system, including both transient and steady-state phases.


Q2: Why is it called L1 adaptive control theory?

A2: The proofs of stability and performance use the L1-norm of a cascaded system for determining the bandwidth of the filter, used in the design. Because the L1-norm is the induced L_infty norm of the input/output signals, the sufficient condition for stability, written in terms of L1-norm, leads to UNIFORM performance bounds for input/output signals.


Q3: How does L1 adaptive control theory compare to robust control?

A3: The architectures of L1 adaptive control theory aim for (partial) compensation of uncertainties within the bandwidth of the control channel. The benefit of L1 architecture is that it does not invert the desired system dynamics, which is unavoidable in several LTI architectures of robust control. This consequently allows for application of L1 adaptive controllers to a broader class of systems in the presence of various nonlinearities. This paper clarifies the relationship.


Q4: How broad is the class of systems, for which L1 adaptive control theory offers a solution?

A4: In state-feedback case, multi-input multi-output systems with both matched and unmatched uncertainties have been presented in (Xargay et al., American Control Conference 2010). Exactly this solution has been applied to control of NASA’s subscale generic transport aircraft model in (Gregory et al., AIAA Conference on Guidance, Navigation and Control, 2010). The flight tests included post-stall flight regimes, with the speed of the aircraft dropping from 80 knots to 40 knots. The L1 adaptive controller was able to guarantee safe operation of the vehicle during the entire flight and the pilot was able to satisfactorily fly the specified tasks, consistently with the theoretical predictions. The L1 adaptive flight control system tested was the main controller and was not the augmentation of any baseline controller. It used a single set of control parameters and did not rely on control reconfiguration, gain-scheduling, or persistent excitation to achieve desired transient and steady-state response. In output feedback, L1 adaptive control theory has solutions for both strictly positive real (SPR) reference systems and non-SPR reference systems. In these solutions, however, the verification of the L1-norm sufficient condition for stability is not straightforward. Obviously, there are classes of systems, for which the design of L1 adaptive control solution in output feedback is not straightforward, and/or would have very conservative performance bounds.


Q5: What are the open questions in this theory?

A5: Optimality is the open question at this point. How to choose the structure of the filter and how to select its parameters so that to achieve the desired trade-off between performance and robustness? It is a non-convex optimization problem due to the L1-norm condition, used in the performance bounds. Nevertheless, this problem can be addressed using tools from linear and robust control.


Q6: What are the opportunities availed by this theory in a broader context?

A6: The architectures of L1 adaptive control theory offer decoupling not only between adaptation and robustness, but also between design of various modules in networked control theory, like quantization, event-triggering, scheduling, etc. The main benefit offered by these architectures is that the performance of uncertain cyber-physical systems can be predicted apriori; these performance bounds can be tuned by increasing the adaptation rate, reducing the event threshold and increasing the bandwidth of the low-pass filters locally.


Q7: What benefits does L1 adaptive control offer on the level of classical control, as compared to the standard PI controller?

A7: For a linear time-invariant system with known parameters and unknown disturbance, the structure of L1 controller can be derived from a PI controller, by considering two SIMULTANEOUS modifications: i) the error for the integrator needs to be replaced by the prediction error, and ii) a low-pass filter needs to be inserted at the output of the control channel. The prediction error is the error between the state of the system and its predictor, which mimics the system’s structure, replacing the unknown disturbance by its estimate. If for the PI controller the trade-off between performance and robustness is governed by the gain of the integrator, for the L1-type PI controller the trade-off between performance and robustness is resolved by means of selection of the low-pass filter. Meantime, the integrator gain can be selected as high as the CPU permits, as the integration (which is the adaptation in this case) is decoupled from robustness. This fast adaptation is the MAIN benefit of the L1-type PI controller as compared to the standard PI controller, as it allows for compensation of rapid variation of uncertainties with uniform performance.