Backstepping controller synthesis and characterizations of incremental stability

Incremental stability is a property of dynamical and control systems, requiring the uniform asymptotic stability of every trajectory, rather than that of an equilibrium point or a particular time-varying trajectory. Similarly to stability, Lyapunov functions and contraction metrics play important roles in the study of incremental stability. In this paper, we provide characterizations and descriptions of incremental stability in terms of existence of coordinate-invariant notions of incremental Lyapunov functions and contraction metrics, respectively. Most design techniques providing controllers rendering control systems incrementally stable have two main drawbacks: they can only be applied to control systems in either parametric-strict-feedback or strict-feedback form, and they require these control systems to be smooth. In this paper, we propose a design technique that is applicable to larger classes of (not necessarily smooth) control systems. Moreover, we propose a recursive way of constructing contraction metrics (for smooth control systems) and incremental Lyapunov functions which have been identified as a key tool enabling the construction of finite abstractions of nonlinear control systems, the approximation of stochastic hybrid systems, source-code model checking for nonlinear dynamical systems and so on. The effectiveness of the proposed results in this paper is illustrated by synthesizing a controller rendering a non-smooth control system incrementally stable as well as constructing its finite abstraction, using the computed incremental Lyapunov function.Comment: 23 pages, 2 figure

Two-layer on-line parameter estimation for adaptive incremental backstepping flight control for a transport aircraft in uncertain conditions

Presence of uncertainties caused by unforeseen malfunctions of the actuator or changes in aircraft behavior could lead to aircraft loss of control during flight. The paper presents two-layer parameter estimation procedure augmenting Incremental Backstepping (IBKS) control algorithm designed for a large transport aircraft. IBKS uses angular accelerations and current control deflections to reduce the dependency on the aircraft model. However, it requires knowledge of the control effectiveness. The proposed identification technique is capable to detect possible problems such as a failure or presence of unknown actuator dynamics even in case of redundancy of control actuation. At the first layer, the system performs monitoring of possible failures. If a problem in one of the control direction is detected the algorithm initiates the second-layer identification determining the individual effectiveness of the each control surface involved in this control direction. Analysis revealed a high robustness of the IBKS to actuator failures. However, in severe conditions with a combination of multiple failures and presence of unmodelled actuator dynamics IBKS could lost stability. Meanwhile, proposed control derivative estimation procedure augmenting the IBKS control helps to sustain stability

Closed-loop analysis with incremental backstepping controller considering measurement bias

In this paper, closed loop system characteristics with an incremental backstepping controller are investigated through theoretical analysis when both measurement biases and model uncertainties exist. Incremental backstepping algorithm is proposed in previous studies to reduce model dependency of classical backstepping algorithm with additional measurements about state derivatives and control surface deflection angles. This research enables to have following critical understandings especially about the effects of biases on these additional measurements to system characteristics with incremental backstepping method. First, these biases do not affect a characteristic equation, so they do not have any influence about a condition for absolute stability. Second, these biases cause a steady state error, and model uncertainty in control effectiveness information starts to have an impact to it when these biases are additionally considered

Understandings of classical and incremental backstepping controllers with model uncertainties

This paper suggests closed-loop analysis results for both classical and incremental backstepping controllers considering model uncertainties. First, transfer functions with each control algorithm under the model uncertainties, are compared with the ones for the nominal case. The effects of the model uncertainties on the closed-loop systems are critically assessed via investigations on stability conditions and performance metrics. Second, closed-loop characteristics with classical and incremental backstepping controllers under the model uncertainties are directly compared using derived common metrics from their transfer functions. This comparative study clarifies how the effects of the model uncertainties to the closed-loop system become different depending on the applied control algorithm. It also enables understandings about the effects of additional measurements in the incremental algorithm. Third, case studies are conducted assuming that the uncertainty exists only in one aerodynamic derivative estimate while the other estimates have true values. This facilitates systematic interpretations on the impacts of the uncertainty on the specific aerodynamic derivative estimate to the closed-loop system

Two-layer adaptive augmentation for incremental backstepping flight control of transport aircraft in uncertain conditions

Presence of uncertainties caused by unforeseen malfunctions in actuation system or changes in aircraft behaviour could lead to aircraft loss-of-control during flight. The paper presents Two-Layer Adaptive augmentation for Incremental Backstepping (TLA-IBKS) control algorithm designed for a large transport aircraft. IBKS uses angular accelerations and current control deflections to reduce the dependency on the aircraft model. However, it requires knowledge of control effectiveness. The proposed technique is capable to detect possible failures for an overactuated system. At the first layer, the system performs monitoring of a combined effectiveness and detects possible failures via an innovation process. If a problem is detected the algorithm initiates the second-layer algorithm for adaptation of effectiveness of individual control effectors. Filippov generalization for nonlinear differential equations with discontinuous right-hand sides is utilized to develop Lyapunov based tuning function adaptive law for the second layer adaptation and to prove uniform asymptotic stability of the resultant closed-loop system. Conducted simulation manifests that if the input-affine property of the IBKS is violated, e.g., in severe conditions with a combination of multiple failures, the IBKS can lose stability. Meanwhile, the proposed TLA-IBKS algorithm demonstrates improved stability and tracking performance