Differential flatness properties and multivariable adaptive control of ovarian system dynamics

Flatness-based adaptive neurofuzzy control of induction generators using output feedback

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 article proposes flatness-based control and non-linear Kalman Filtering for power generation units comprising synchronous generators connected to steam turbines. It is shown that the dynamic model of this power system is a differentially flat one. This property signifies that all its state variables and its control inputs can be expressed as differential functions of selected state variables of it, the latter standing for the flat outputs of the system. This allows for its transformation into a linear canonical form in which the design of a feedback controller and the solution of the state estimation problem become possible. Moreover, by introducing a Kalman Filter-based disturbance observer it becomes possible to identify in real time the perturbation terms that affect the power system's model. Through the proposed flatness-based controller, fast and accurate tracking of all reference setpoints is achieved by the state vector elements of the power unit. Moreover, through the proposed Kalman Filter-based disturbances estimator the control loop is given additional robustness against modelling errors and external perturbations.

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.

The article proposes a method for nonlinear control of the dynamical system that is formed by a DC–DC converter and a DC motor, making use of differential flatness theory. First it is proven that the aforementioned system is differentially flat which means that all its state vector elements and its control inputs can be expressed as differential functions of primary state variables which are defined to be the system’s flat outputs. By exploiting the differential flatness properties of the model its transformation to a linearized canonical (Brunovsky) form becomes possible. For the latter description of the system one can design a stabilizing feedback controller. Moreover, estimation of the nonmeasurable state vector elements of the system is achieved by applying a new nonlinear Filtering method which is known as Derivative-free nonlinear Kalman Filter. This filter consists of the Kalman Filter recursion applied on the linearized equivalent model of the system and of an inverse transformation that is based on differential flatness theory and which enables to obtain estimates of the initial nonlinear state-space model. Moreover, to compensate for parametric uncertainties and external perturbations the filter is redesigned as a disturbance observer. By estimating the perturbation inputs that affect the joint model of the DC–DC converter and of the DC motor their compensation becomes possible. The efficiency of the proposed control scheme is further confirmed through simulation experiments.

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

The paper studies differential flatness properties and an input–output linearization procedure for doubly fed induction generators (DFIGs). By defining flat outputs which are associated with the rotor's turn angle and the magnetic flux of the stator, an equivalent DFIG description in the Brunovksy (canonical) form is obtained. For the linearized canonical model of the generator, a feedback controller is designed. Moreover, a comparison of the differential flatness theory-based control method against Lie algebra-based control is provided. At the second stage, a novel Kalman Filtering method (Derivative-free nonlinear Kalman Filtering) is introduced. The proposed Kalman Filter is redesigned as disturbance observer for estimating additive input disturbances to the DFIG model. These estimated disturbance terms are finally used by a feedback controller that enables the generator's state variables to track desirable setpoints. The efficiency of the proposed state estimation-based control scheme is tested through simulation experiments.

This paper proposes a new state-space controller for the robust trajectory tracking control problem of multi-mass resonant systems. It is designed in state-space by combining Differential Flatness (DF) and a higher order Disturbance Observer (DOb). The former systematically generates the control input and state references of a state feed-back controller so that not only regulation but also trajectory tracking control can be performed in state-space. They are obtained in terms of differentially flat output variable in DF. The latter estimates disturbances and their successive time derivatives. Its order depends on the resonant modes of the system. The proposed controller has a two-degrees-of-freedom motion control structure. Its performance and robustness can be independently adjusted by tuning the state feed-back controller and DOb, respectively. It is designed in two steps. In the first one, the state feed-back controller is tuned by only considering the nominal plant model of the system. The nominal performance is adjusted by using pole placement; e.g., all poles of the nominal system are placed on the real axis so as to suppress the vibration at tip. The state and control input references of the state feed-back controller are obtained by using DF. In the second step, the robustness of the system is achieved by reconstructing the state vector via the estimations of disturbances and their successive time derivatives. Thanks to the proposed new state variables, mismatched disturbances are automatically cancelled by the state feed-back controller. The matched disturbances are cancelled by feeding-back their estimations through control input. Hence, the robustness of the multi-mass resonant systems is achieved. Without losing generality, the proposal is verified by considering three-mass resonant systems. Its simulation results are given to validate the proposal.

The automatic pouring of molten metal from a ladle into a mold by a robot requires an high-precision tracking control of the flow rate. Due to the hydrodynamics of the liquid metal, the model of the pouring process is highly nonlinear which has to be considered for the model-based design of the tracking control. The two design approaches studied in the paper combine a feedforward and a feedback control in order to obtain a precise and robust tracking performance. Thereby, the property of differential flatness is used for the design of the tracking control, and the unmeasured states are reconstructed for the feedback control by an unscented Kalman filter. Two proposed control schemes are compared by simulations of the automatic pouring robot together with the pouring process.

In this paper, we propose an adaptive fuzzy output feedback control method for trajectory tracking control problem for robotic systems. Using Lyapunov method, we first develop a stable adaptive fuzzy state feedback control algorithm by assuming that the systems output and its derivatives are available for feedback control design. The algorithm combines fuzzy systems with robust adaptive controller. The fuzzy system approximates the certainty equivalent (CE)-based optimal controller while robustifying adaptive control term is used to cope with uncertainties that appeared from the effect of external disturbance, fuzzy approximation errors and other modeling errors. Then, an output feedback form of the position-velocity (state feedback) controller is proposed where unknown velocity signal is replaced by the output of model-free linear estimator. We show via asymptotic analysis that the tracking performance of the output feedback design can recover the performance achieved under the state feedback control design. Finally, the proposed method is implemented and evaluated on a 2-DOF robotic system to demonstrate the theoretical development for the real-time applications.

ABSTRACTThe problem of feedback control of bandwidth allocation for Internet links is analysed in this article. It is shown that the procedure for numerical solution of a fluid flow partial differential equation (PDE) which describes the bandwidth allocation on a link results into a set of ordinary differential equations (ODEs) and into an associated state-space equations model. For the local subsystems, into which the flow PDE is decomposed, it becomes possible to apply boundary-based feedback control. The solution of the control problem proceeds by showing that the state-space model of the flow PDE stands for a differentially flat system. Next, for each subsystem which is related to an ODE, a virtual control input is computed, which can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the data-flow PDE is found. This control input contains recursively all virtual control inputs which were computed for the individual ODE subsystems associated with the previous rows of the state-space equation. Thus, by tracing the rows of the state-space model backwards, at each iteration of the control algorithm, one can finally obtain the control input that should be applied to the data flow PDE of the communication network so as to assure that all its state variables will converge to the desirable set points.

Flatness-based control implemented in successive loops is proposed for modifying the business cycles of interacting finance agents. The dynamics of the business cycle of each individual agent is described as a nonlinear oscillator, while interaction terms exist between such oscillators denoting transactions in the form of goods trade or services hiring. A state-space description for the dynamics of the interacting business cycles is obtained. The differential flatness properties of this model are proven. It is demonstrated that each row of the state-space model is a differentially flat subsystem and that a stabilizing feedback controller can be designed for it. Actually, local control is achieved by considering specific state variables of the model as virtual control inputs. From the last row of the model, the control input that is actually exerted on the system is found and this is shown to contain recursively the virtual control inputs associated with the individual rows of the state-space description. The global stability properties of the control method are confirmed through Lyapunov analysis. The method comes to confirm that exogenous investements appearing as control inputs to the system can provide the impetus for growth to the interconnected finance agents. Consequently, changes of the business cycles of the individual agents (reflecting growth or recession) can be centrally controlled.

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

Flatness-based trajectory planning for electromagnetic spacecraft proximity operations in elliptical orbits

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).