Flatness-Based Control of Drilling Vibrations

The paper proposes derivative-free nonlinear Kalman Filtering for MIMO nonlinear dynamical systems. The considered nonlinear filtering scheme which is based on differential flatness theory extends the class of systems to which Kalman Filtering can be applied without the need for calculation of Jacobian matrices. To deduce if a dynamical system is differentially flat, the following should be examined: (i) the existence of the flat output, which is a variable that can be written as a function of the system's state variables (ii) the system's state variables and the input can be written as functions of the flat output and its derivatives. Nonlinear systems satisfying the differential flatness property can be written in the Brunovsky form via a transformation of their state variables and control inputs. After transforming the nonlinear system to the canonical form it is straightforward to apply the standard Kalman Filter recursion. The performance of the proposed derivative-free nonlinear filtering scheme is tested through simulation experiments on benchmark nonlinear multi-input multi-output dynamical systems, such as robotic manipulators.

: The Commanding General, Marine Corps Combat Development Command (MCCDC) asked the Center for Naval Analyses to assess the general applicability of the new science to land warfare. New Sciences is a catch-all phrase that refers to the tools and methodologies used in nonlinear dynamics and complex systems theory to study physical dynamical systems exhibiting a complicated dynamics. This report concludes that the concepts, ideas, theories, tools and general methodologies of nonlinear dynamics and complex systems theory show enormous, almost unlimited, potential for not just providing better solutions for certain existing problems of land combat, but for fundamentally altering our general understanding of the basic processes of war, at all levels. Indeed, the new sciences' greatest legacy may, in the end, prove to be not just a set of creative answers to old questions but and entirely new set of questions to be asked of what really happens on the battlefield The central thesis of this paper is that land combat is a complex adaptive system. That is to say, that land combat is essentially a nonlinear dynamical system composed of many interacting semi-autonomous and hierarchically organized agents continuously adapting to a changing environment.

Introduction A new entry guidance concept is developed for an unpowered lifting vehicle. The two-dimensional longitudinal plane entry guidance scheme used by the U.S. Space Shuttle Orbiter is extended to a threedimensional entry guidance scheme. The guidance task is posed as planning and tracking a reference trajectory. This paper concerns the design and analysis of the tracking law. In the new guidance scheme, the complete ground-track is planned as a function of energy and tracked by commanding both the angle of attack and angle of bank. This extension to the Shuttle entry guidance scheme is carried out within the theoretical framework of differential flatness and feedback linearization. Approximately flat outputs for the system dynamics are identified which are then used to construct a state-variable transformation which together with a static state feedback control transformation renders the system approximately input-output, as well as input-state linear. In the transformed state and control variables, the system dynamics can be represented by two approximately decoupled linear systems. A feedback tracking law is then designed by taking advantage of the linear structure of the system dynamics in the transformed coordinates. This tracking law is shown to achieve bounded tracking of the flat outputs. Simulation results indicate that, if provided with an appropriate reference trajectory, the tracking law guides the vehicle to within one nautical mile of the target in downrange and cross-range during the hypersonic entry phase. 'Graduate Student, Student Member AIAA. t Research Assistant, AIAA member, presently at The Aerospace Corporation, 2350 E. El Segundo Blvd., El Segundo, CA 90245. ^Associate Professor, Associate Fellow AIAA. § Copyright ©by Kenneth D. Mease, Published by AIAA with permission Current efforts to develop a reusable launch vehicle (RLV) and a space station crew return vehicle provide particular motivation for improving the entry guidance capability for a lifting unpowered vehicle. In this paper we develop a natural extension to the longitudinal entry guidance scheme used by the U.S. Space Shuttle Orbiter. The new entry guidance scheme follows from the property of differential flatness of a A controlled system is differentially flat if there exist output functions, termed flat outputs, equal in number to the control inputs, such that all the state and control functions can be determined from the output functions and their derivatives through purely algebraic relations. A differentially flat system offers the possibility of reduced-order trajectory planning in terms of the flat outputs. Moreover, the flat outputs often permit feedback linearization of the controlled system dynamics, i.e., they lead to state and control transformations that globally linearize the dynamics. Thus guidance law design and analysis may become greatly simplified for a differentially flat system. Following this paradigm, the guidance task is divided naturally into a trajectory planning subtask of determining the reference output functions consistent with a trajectory that meets all the guidance objectives, and a trajectory tracking subtask of determining the control commands to track the reference output functions. The longitudinal-plane entry guidance logic used for the Shuttle can be viewed as a product of the approach just described. Drag is the flat output used in trajectory planning and the magnitude of the bank angle is commanded to track the reference drag trajectory. The reference drag trajectory is planned as a function of velocity such that it achieves the desired down-range and satisfies the vehicle thermal, normal load and dynamic pressure constraints, as well as the controllability constraint imposed by the

Two notions from the theory of non-linear systems without time delays are generalized to non-linear delay systems: flatness and quasi-static state feedback. Depending on the generality of the relations considered, flatness properties of increasing generality are obtained: flatness, δ-flatness, π-flatness, and difference-differential flatness. It is then shown that difference-differentially flat delay systems are linearizable by a certain class of “predictive” quasi-static state feedback, i.e., that they are equivalent by such feedback to a linear controllable system. The mathematical framework is difference-differential algebra.

We address the problem of cooperative transportation of a cable-suspended payload by multiple quadrotors. In previous work, quasi-static models have been used to study this problem. However, these approaches are severely limited because they ignore the payload dynamics, and do not explicitly model the underactuation in the control problem. Thus, there are no guarantees on the payload trajectory or the cable tensions, which must be non negative. In this paper, we develop a complete dynamic model for the case when payload is a point load and for the case when the payload is a rigid body. We show in both cases the resulting system is differentially flat when the cable tensions are strictly positive. We also consider the case where the tensions are non negative (including the case with zero tensions) and establish that these systems are differentially flat hybrid systems by considering the switching dynamics induced by the unilateral tension constraints. We use the differential flatness property to find dynamically feasible trajectories for the payload+quadrotors system. We show using numerical and experimental methods that these trajectories are superior to those obtained by quasi-static models.

The ovarian system exhibits nonlinear dynamics which is modeled by a set of coupled nonlinear differential equations. The paper proposes adaptive fuzzy control based on differential flatness theory for the complex dynamics of the ovarian system. It is proven that the dynamic model of the ovarian system, having as state variables the LH and the FSH hormones and their derivatives, is a differentially flat one. This means that all its state variables and its control inputs can be described as differential functions of the flat output. By exploiting differential flatness properties the system’s dynamic model is written in the multivariable linear canonical (Brunovsky) form, for which the design of a state feedback controller becomes possible. After this transformation, the new control inputs of the system contain unknown nonlinear parts, which are identified with the use of neurofuzzy approximators. The learning procedure for these estimators is determined by the requirement the first derivative of the closed-loop’s Lyapunov function to be a negative one. Moreover, Lyapunov stability analysis shows that H-infinity tracking performance is succeeded for the feedback control loop and this assures improved robustness to the aforementioned model uncertainty as well as to external perturbations. The efficiency of the proposed adaptive fuzzy control scheme is confirmed through simulation experiments.

Presenting an important potential in the representation of nonlinear systems, the multimodel approach remains an attractive axis for research. One of the important problems in the multimodel structure concerns the validity calculation which is a fundamental point especially when the process is corrupted with noise and/or its parameters are of high variations. A new approach based on the use of both two type of validity is proposed. A developed specification of the need of each one is explained by an optimization procedure. The conduct of this approach requires, first, the classification of the numerical data into a set of clusters. The frequency-sensitive competitive learning (FSCL) algorithm is used to select the number of models and the fuzzy k-means algorithm identify the operating clusters. From the satisfactory results in terms of precision and robustness obtained on theoretical examples, we are incited to confirm our contribution to real process reactor. The results obtained are compared to the classical approaches showing its ability to represent adequately the nonlinear process with a superior precision and accuracy and from this the classic strategy of multimodel representation is oriented towards a multifaceted approach.

This paper presents a formulation of differential flatness---a concept originally introduced by Fliess, Levine, Martin, and Rouchon---in terms of absolute equivalence between exterior differential systems. Systems that are differentially flat have several useful properties that can be exploited to generate effective control strategies for nonlinear systems. The original definition of flatness was given in the context of differential algebra and required that all mappings be meromorphic functions. The formulation of flatness presented here does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems. In particular, it is shown that, under regularity assumptions and in the case of single input control systems (i.e., codimension 2 Pfaffian systems), a system is differentially flat if and only if it is feedback linearizable via static state feedback. In higher codimensions our approach does not allow one to prove that feedback linearizability about an equilibrium point and flatness are equivalent: one must be careful with the role of time as well as the use of prolongations that may not be realizable as dynamic feedback in a control setting. Applications of differential flatness to nonlinear control systems and open questions are also discussed.

Theory of nonlinear systems

Differential Flatness-Based Approach for Sensors and Actuators Fault Diagnosis of a Multirotor UAV

A new discretization method for the calculation of a sampled-data representation of nonlinear continuous-time system is proposed. The suggested method is based on the well-known Taylor-series expansion and zero-order hold (ZOH) assumption. The mathematical structure of the new discretization method is analyzed. On the basis of this structure the sampled-data representation of nonlinear system with time-delayed multi-input is derived. First the new approach is applied to nonlinear systems with two inputs. And then the delayed multi-input general equation has been derived. In particular, the effect of the time-discretization method on key properties of nonlinear control systems, such as equilibrium properties and asymptotic stability, is examined. And ’hybrid’ discretization schemes that result from a combination of the ’scaling and squaring’ technique with the Taylor method are also proposed, especially under conditions of very low sampling rates. Practical issues associated with the selection of the method’s parameters to meet CPU time and accuracy requirements, are examined as well. A performance of the proposed method is evaluated using a nonlinear system with time-delay: maneuvering an automobile.KeywordsNonlinear SystemDiscretization MethodNonlinear Control SystemTaylor MethodFast Time ConstantThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This paper examines the health-conscious optimal control of lithium-ion batteries. The paper focuses specifically on the optimizing battery control policies to minimize solid electrolyte interphase (SEI) layer growth. However, its approach is applicable to other degradation mechanisms. The paper exploits the fact, established by the mathematical control literature1–4, that Fick’s second law of diffusion is differential flat for both linear and nonlinear diffusion problems. Differential flatness only applies to each diffusion medium in the battery separately. In the single particle model (SPM), for instance, each electrode’s diffusion sub-model alone is differential flat. Ensuring differential flatness for the entire battery is a two-step process. First, one must truncate redundant integrators from the two electrodes’ diffusion sub-models. Second, one must then relate the truncated integral state variable to the remaining state variables through an affine transformation. The differential flatness of the resulting battery model makes it possible to solve the health-conscious battery management problem efficiently using pseudospectral methods. This exploitation of the structure (i.e., differential flatness) of lithium-ion battery dynamics for more computationally efficient trajectory optimization is novel and significant. The existing literature already examines the problem of health-conscious optimal battery charging and discharging5,6. However, the optimal control tools used in this literature are not tailored to exploit the structure (e.g., differential flatness) of lithium-ion battery models. This can lead to higher computational burdens compared to the novel and more efficient approach we present here. The approach has the added advantage that it can be used for both offline trajectory optimization and online model-predictive control.The paper demonstrates the above ideas using the SPM. The optimization problem is formulated by equation [1-6] and the SPM with film growth model. The cost function in Equation [1] aims to track the reference state of charge (SOC) and at the same time minimize the film growth rate, which is defined by equation [7]-[14] in Ramadass’s paper7. The weight β represents a tradeoff/balance between aggressive charging and battery degradation. Constraints [2-6] place limits on battery charge/discharge current, reflecting battery management hardware capabilities. Furthermore, these constraints limit battery SOC to prevent over-charging and over-discharging. Finally, these constraints also place bounds on battery state variables contributing to damage phenomena such as lithium plating and mechanical degradation. For instance, we bound the concentration gradients in the battery cell and the overpotentials driving the lithium plating side reaction8–10. The optimization approach presented in this paper transforms these equations and constraints from a dynamic programming problem to a nonlinear programming (NLP) problem. Solving this NLP problem using traditional optimization methods leads to a health-conscious battery management policy. The value of this paper lies not in this policy, but rather in the computational efficiency with which it is obtained. Specifically, to the best of the authors’ knowledge, this paper represents the first attempt to exploit lithium-ion battery model structure for more efficient solution of the health-conscious optimal management problem.ACKNOWLEDGMENTSThe research was funded by ARPA-E AMPED program grant # 0675-1565. The authors gratefully acknowledge this support.REFERENCES1. M. FLIESS, J. LÉVINE, P. MARTIN, and P. ROUCHON, Int. J. Control, 61, 1327–1361 (1995).2. B. Laroche, P. Martin, and P. Rouchon, Int. J. Robust Nonlinear Control, 10, 629–643 (2000).3. M. Fliess and R. Marquez, Int. J. Control, 73, 606–623 (2000).4. T. Meurer, Automatica, 47, 935–949 (2011).5. R. Klein and N. Chaturvedi, Am. Control … (2011).6. V. Boovaragavan and V. R. Subramanian, J. Power Sources, 173, 1006–1011 (2007).7. P. Ramadass, B. Haran, P. M. Gomadam, R. White, and B. N. Popov, J. Electrochem. Soc., 151, A196 (2004).8. J. Christensen and J. Newman, J. Solid State Electrochem., 10, 293–319 (2006).9. X. Zhang, A. M. Sastry, and W. Shyy, J. Electrochem. Soc., 155, A542 (2008).10. N. Chaturvedi, R. Klein, J. Christensen, J. Ahmed, and A. Kojic, IEEE Control Syst. Mag., 30, 49–68 (2010).

This manuscript presents a universal control law for a class of complex dynamic systems, such as mobile robots. The trajectory tracking problem is among the major problems in the field of robotics. The proposed universal control law solves the trajectory tracking problem of mobile robots. Despite a large body of research on developing robot's control laws conducted in the literature, the dynamic effects of robots are often not taken into consideration while deriving control laws that define their trajectories. In most cases, the robot's trajectory tracking and/or stabilization problems are addressed based on its kinematic model due to simplicity. Therefore, the need for a universal control law based on the robot's dynamic and/or kinematic model is significant. Even though the feedback control theory is well-established in the field of robotics, the control laws for dynamic systems are typically more complex than system models themselves. Furthermore, control laws are required to be adapted depending on models of different mobile robot systems. Here, the development of universal control law is underscored, i.e., the paper is aimed at developing universal dynamic tracking control law that solves the trajectory tracking problem of a class of mobile robots. The controller is tested through a set of computer simulations using a differential drive mobile robot operating in an indoor planar environment.

Differential flatness has been investigated in the context of mobile vehicles for planning and control of their motions. In these models, the wheels are considered to be non-slipping, i.e., the system dynamics is subject to non-holonomic constraints. If a manipulator arm is mounted on such a mobile vehicle, the dynamics becomes highly nonlinear due to the nonlinear coupling between the motions of the mobile vehicle and the manipulator arm. A challenging question is how to perform point-to-point motions of such a system in the state space of the mobile manipulator. If some of the actuators are absent in the mechanical arm, the mobile manipulator becomes under-actuated and consequently even harder to plan and control. This paper presents a methodology for design of mobile vehicles, mounted with under-actuated manipulators operating in a horizontal plane, such that the combined system is differentially flat. In this paper, we show that by appropriate inertia distribution of the links and addition of torsion springs at the joints, a range of under-actuated designs are possible where the underactuated mobile manipulator system is differentially flat. The differential flatness property allows to efficiently solve the problem of trajectory planning and feedback controller design for point to point motions of the system. The proposed method is illustrated by the example of a mobile vehicle with under-actuated three-link manipulator.

Under-actuated systems are unavoidable in certain applications. For example, a biped can not have an actuator between the foot and the ground. For industrial robots, underactuation is preferable due to cost considerations. A fully actuated system can execute any joint trajectory. However, if the system is under-actuated, not all joint trajectories are attainable. For such systems, it is difficult to characterize attainable joint trajectories analytically and numerical methods are generally used to characterize them. This paper investigates the property of differential flatness for under-actuated planar open chain robots and study its dependence on inertia distribution within the system. Once this property is established, trajectory between any two points in its differentially flat output space is feasible and can be shown to be consistent with the dynamics of the under-actuated system. It is shown that certain choices of inertia distributions make an under-actuated open-chain planar robot with revolute joints feedback linearizable, i.e., also differentially flat. Hence, both cyclic and point to point trajectories can be guaranteed with these under-actuated systems. The methodology proposed is demonstrated with an under-actuated three degree-of-freedom planar robot.