Discrete-time adaptive learning control for parametric uncertainties with unknown periods

In this paper, we approach the problem of unknown periods for a class of discrete-time parametric nonlinear systems with nonlinearities which do not necessarily satisfy the sector-bounded condition. The unknown periods hide in the parametric uncertainties, which is difficult to estimate. By incorporating a logic-based switching mechanism, we estimate the period and bound of unknown parameter simultaneously under Lyapunov-based analysis. Rigorous proof is given to demonstrate that a finite number of switchings can guarantee the asymptotic regulation of the nonlinear system considered. The simulation result also shows the efficacy of the proposed switching periodic adaptive control method.Peer reviewe

Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization

We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.Comment: 25 pages; Added new Section 7.2; Added references; Corrected typos; Submitted to SIAD

On the adaptive and learning control design for systems with repetitiveness

Ph.DDOCTOR OF PHILOSOPH

Current-cycle iterative learning control for high-precision position tracking of piezoelectric actuator system via active disturbance rejection control for hysteresis compensation

As a typical smart structure, piezoelectric actuator (PEA) is an essential constituent component in piezoelectric-driven positioning stages. Nevertheless, the positioning precision is severely degraded by its innate rate-dependent hysteretic nonlinearity. In this paper, an innovative control method which combines active disturbance rejection control (ADRC) and current-cycle iterative learning control (CILC) is proposed by constructing PEA as a second-order disturbance-based (SODB) structure to handle both hysteretic nonlinearities and dynamic uncertainties of PEA. The proposed method differs from the prevalent model-inverse solution in hysteresis compensation, where the control performance of the latter extremely relies on the accurateness of the hysteretic model while the former does not require a mathematical model of hysteresis since it is considered as a general disturbance and eliminated. Compared with the existing hysteresis compensation via pure ADRC method, the proposed method has improved robustness by incorporating an additional ILC loop to ADRC. Comparative experimentations are executed on a PEA system and results imply that the proposed approach has better control performance than pure proportionalintegral (PI) control and ADRC

Iterative learning of human partner's desired trajectory for proactive human-robot collaboration

A period-varying iterative learning control scheme is proposed for a robotic manipulator to learn a target trajectory that is planned by a human partner but unknown to the robot, which is a typical scenario in many applications. The proposed method updates the robot’s reference trajectory in an iterative manner to minimize the interaction force applied by the human. Although a repetitive human–robot collaboration task is considered, the task period is subject to uncertainty introduced by the human. To address this issue, a novel learning mechanism is proposed to achieve the control objective. Theoretical analysis is performed to prove the performance of the learning algorithm and robot controller. Selective simulations and experiments on a robotic arm are carried out to show the effectiveness of the proposed method in human–robot collaboration

Controlling fluid flows with positive polynomials

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation oflong-time averages of key flow quantities is presented. The key idea, first outlined in Ref. [1], is that the difficulties of treatingand optimising long-time averages are relaxed by shifting the analysis to upper/lower bounds for minimisation/maximisationproblems, respectively. In this setting, control design reduces to finding the polynomial-type state-feedback controller thatoptimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controlleritself and a tunable polynomial function. A numerically tractable approach, based on Sum-of-Squares of polynomials techniquesand semidefinite programming, is proposed. As a prototypical example of control of separated flows, the mitigation of thefluctuation kinetic energy in the unsteady two-dimensional wake past a circular cylinder at a Reynolds number equal to 100,via controlled angular motions of the surface, is investigated. A compact control-oriented reduced-order model, resolving thelong-term behaviour of the fluid flow and the effects of actuation, is first derived using Proper Orthogonal Decomposition andGalerkin projection. In a full-information setting, linear state-feedback controllers are then designed to reduce the long-timeaverage of the resolved kinetic energy associated to the limit cycle of the system. Controller performance is then assessed indirect numerical simulations

Sum-of-Squares approach to feedback control of laminar wake flows

A novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenko et al. 2014, Philos. T. Roy. Soc. 372(2020), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable approach to the solution of such optimisation problems, based on Sum-of-Squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at Re=100, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is derived using Proper Orthogonal Decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, energy efficiency, and physical control mechanisms identified are analysed. Key elements, implications and future work are discussed