Controlling chaos in El Niño

Many weather and climate phenomena are chaotic in nature; indeed for many people this is the canonical example of a chaotic system. However, because of this, it is at least theoretically possible to have significant influence over these systems with extremely small control inputs. This potential is explored using the Cane-Zebiak 33 000-state model of the El-Niño/Southern Oscillation (ENSO). The model dynamics are nonlinear and chaotic, and the optimal control input can be found through iteration using the adjoint simulation. The performance of this optimal control (which implicitly assumes perfect model and state information) is compared with a simple SISO linear feedback. Significant reductions in ENSO amplitude are (theoretically) possible with very small control inputs, illustrating that it is possible to have significant influence over large-scale climatic phenomena without correspondingly large control effort

Pinning control of spatiotemporal chaos

Linear control theory is used to develop an improved localized control scheme for spatially extended chaotic systems, which is applied to a coupled map lattice as an example. The optimal arrangement of the control sites is shown to depend on the symmetry properties of the system, while their minimal density depends on the strength of noise in the system. The method is shown to work in any region of parameter space and requires a significantly smaller number of controllers compared to the method proposed earlier by Hu and Qu [Phys. Rev. Lett. 72, 68 (1994)]. A nonlinear generalization of the method for a 1D lattice is also presented

Typicalness of chaotic fractal behaviour of integral vortexes in Hamiltonian systems with discontinuous right hand side

We consider a linear-quadratic deterministic optimal control problem where the control takes values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely the chaotic behaviour of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighbourhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us the possibility to calculate the entropy and the Hausdorff dimension of the non-wandering set which appears to have a Cantor-like structure as in Smale's Horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.Comment: 113 pages, 22 figure

Chaos control via Mathieu-Van der Pol system and Linear Optimal Control design with a non-ideal excitation and parametric uncertainties

The hyper-chaotic Mathieu-Van der Pol system is an important autonomous system model, with four state variables and four Lyapunov exponents, three of which are positive. Also called the Tri-Chaos, the Mathieu-Van der Pol system was mathematically modeled via linear coupling of Mathieu and Van der Pol non-linear and non-autonomous systems. In this work, a non-ideal system based on the Mathieu-Van der Pol System is modeled considering its parameters as uncertain, which makes it closer to reality. Numerical simulations are presented demonstrating that the system has a chaotic behavior with three positive Lyapunov exponents. Given such unstable and unpredictable behavior, the linear optimal control design is proposed to reduce the chaotic movement of this system to a fixed point. The simulation results show that the identification by Linear Optimal Control is very effective.Peer Reviewe

Nonlinear Control and Estimation with General Performance Criteria

This dissertation is concerned with nonlinear systems control and estimation with general performance criteria. The purpose of this work is to propose general design methods to provide systematic and effective design frameworks for nonlinear system control and estimation problems. First, novel State Dependent Linear Matrix Inequality control approach is proposed, which is optimally robust for model uncertainties and resilient against control feedback gain perturbations in achieving general performance criteria to secure quadratic optimality with inherent asymptotic stability property together with quadratic dissipative type of disturbance reduction. By solving a state dependent linear matrix inequality at each time step, the sufficient condition for the control solution can be found which satisfies the general performance criteria. The results of this dissertation unify existing results on nonlinear quadratic regulator, Hinfinity and positive real control. Secondly, an H2-Hinfinity State Dependent Riccati Equation controller is proposed in this dissertation. By solving the generalized State Dependent Riccati Equation, the optimal control solution not only achieves the optimal quadratic regulation performance, but also has the capability of external disturbance reduction. Numerically efficient algorithms are developed to facilitate effective computation. Thirdly, a robust multi-criteria optimal fuzzy control of nonlinear systems is proposed. To improve the optimality and robustness, optimal fuzzy control is proposed for nonlinear systems with general performance criteria. The Takagi-Sugeno fuzzy model is used as an effective tool to control nonlinear systems through fuzzy rule models. General performance criteria have been used to design the controller and the relative weighting matrices of these criteria can be achieved by choosing different coefficient matrices. The optimal control can be achieved by solving the LMI at each time step. Lastly, since any type of controller and observer is subject to actuator failures and sensors failures respectively, novel robust and resilient controllers and estimators are also proposed for nonlinear stochastic systems to address these failure problems. The effectiveness of the proposed control and estimation techniques are demonstrated by simulations of nonlinear systems: the inverted pendulum on a cart and the Lorenz chaotic system, respectively