A computational approach to approximate input/state feedback linearization

The problem of feedback linearization (FL) of continuous and discrete nonlinear MIMO systems is considered. The idea of FL method consists in converting the original nonlinear system into a linear one by means of feedback. Then, the methods of control theory for linear systems are used for system design. A widespread approach to FL design is based on the method of normal form, that uses a nonsingular transformation of system state variables z = T(x). In order to obtain a normal form of the nonlinear system in the neighbor of a some point, it is necessary to determine a special function - the system virtual output, for which a relative degree (in the case of single input) or a vector relative degree (in the case of multiple input) is determined. Applicability of the normal form method for FL is provided by the conditions of controllability and involutivity for the considered nonlinear system, which are not always true. Moreover, when developing a linearizing control law, the main difficulty lies in the transition from transformed variables z to state variables x of the original system. In this paper, we propose another approach, based on representing the original nonlinear system into a state-dependent coefficient form and applying the canonical similarity transformation z = T(x)x, that allow getting the system to canonical form, that considerably simplifies the FL problem. Such similarity transformation allow accomplishing linearization of system without determining of the virtual system output. Another advantage of the proposed method is that the technique of the transition from the transformed variables z to the state variables x of the original system is simpler. The results are illustrated by examples for continuous and discrete nonlinear systems.

This paper presents the real time control of Magnetic Levitation System using feedback linearization and online learning using Radial Basis function neural networks. It starts with the comparison between adaptive linear controller and neurocontroller used here. Both the controllers uses feedback linearization method and Lyapunov's stability theory to derive update laws to control the nonlinear system. Simulation results justify the use of neurocontroller over adaptive linear controller. Magnetic Levitation System which is open loop unstable is stabilized using feedback linearization approach which relies on canceling the nonlinearities, whereas the online learning is implemented via function approximation in which the radial basis function neural network is used as a function approximator. The combination of feedback linearizing control law and adaptive laws ensures that the closed-loop system is stable.

This article presents a feedback linearization (FL)-based direct torque control (DTC) for an asymmetrical six-phase permanent magnet synchronous motor (PMSM). The decoupled linear model of PMSM is yielded using the FL approach. In this way, the advantages of classical hysteresis comparator-based DTC, such as fast dynamic response, are preserved. The output of the FL, namely, the desired vector, is synthesized by four active vectors for the sake of harmonics mitigation in the secondary subspace. The existing work usually adopts the synthetic vectors with zero voltage in the secondary subspace to suppress the harmonic currents, while this practice ignores the back electromotive force (EMF) harmonics. Thus, the four-vector-based space vector pulsewidth modulation (SVPWM) with and without considering the back EMF harmonics is investigated. The effectiveness of the proposed method is validated through experimental results. Compared with the classical switching table (ST)-based DTC, the proposed FL-DTC method achieves less torque and flux ripple in the primary subspace and less harmonic currents in the secondary subspace.

This paper proposes a shifting feedback linearization controller for nonlinear systems under input saturation. The suggested approach for determining a linear parameter‐varying (LPV) state‐feedback controller is based on employing a linear matrix inequality (LMI)‐based methodology, the input‐output feedback linearization approach, and the shifting paradigm concept. The resultant controller deals with a full linearized version of the nonlinear system under state‐dependent input constraints, and it modifies the closed‐loop convergence speed of the system in accordance with changes in the system's region of linearity. The proposed approach is validated by means of an experimental example.

This paper presents a sensorless feedback linearization direct torque control (FBL-DTC) for an induction motor drive with five-level cascaded H-bridge (CHB) inverter. The control of CHB multilevel inverter is realized by multicarrier based third harmonic injection PWM (THIPWM) technique. The α and β components of reference voltage vector are obtained from the feedback linearization approach that achieves a decoupled torque and flux control. To estimate the rotor speed and the stator resistance simultaneously, an extended Kalman filter (EKF) based observer is developed. An adaptive scheme that uses the estimated states and parameters is established to increase the robustness of feedback linearization control against model and measurement uncertainties. The effectiveness of proposed drive system is tested under some challenging operating conditions and verified by the simulation results.

This paper focuses on the design and implementation of a genetic algorithm (GA) tuned modified simple neuro-fuzzy control (NFC) for optimal performance of induction motor (IM) drive employing feedback linearization (FBL) approach. The intuitive FBL with the proposed simplified NFC with GA (SNFC-GA) substantially reduces the torque chattering and improves the speed response of the IM drive. This new technique also has the main advantage of improved computational efficiency by reducing computational burden over classical NFC and thus, fit for real-time industrial applications. Moreover, the GA searches the optimal parameters of the simplified NFC in order to ensure the global convergence of error. The efficacy of the proposed controller using FBL IM drive is investigated in simulation as well as in experiment, and it is evident that the robust and optimal dynamic performance of the system is achieved in terms of parameter variations and external load disturbance.

On Measures of Non-Integrability of Pfaffian Systems

On Feedback Linearization for Robust Tracking Control of Flexible Joint Robots

The magnetic levitation system's control problem is of considerable scientific interest because the system is open-loop unstable and highly non-linear, and the system's parameters are uncertain. In this paper, the system is formulated by a third-order non-linear differential equation. Based on this third-order non-linear system, the system is first linearized by applying the feedback linearization method. Then, the controller is designed based on a global sliding mode technique, which is able to stabilize a linearized system. Experimental results show that, compared with the classic proportional—integral—derivative (PID) controller, the proposed controller provides excellent transient response performance and the system is robust against the non-linear parameter perturbation. Further experiments show that the sampling period is an important parameter to the system robustness. The reason is explained theoretically, and the analysis shows that in the practical global sliding mode control (GSMC) system the smaller the sampling period is, the more robust the controller will be.

In the case of Autonomous Underwater Vehicle (AUV) navigating with low speed near water surface, a new method for design of roll motion controller is proposed in order to restrain wave disturbance effectively and improve roll stabilizing performance. Robust control is applied, which is based on uncertain nonlinear horizontal motion model of AUV and the principle of zero speed fin stabilizer. Feedback linearization approach is used to transform the complex nonlinear system into a comparatively simple linear system. For parameter uncertainty of motion model, the controller is designed with mixed-sensitivity method based on H-infinity robust control theory. Simulation results show better robustness improved by this control method for roll stabilizing of AUV navigating near water surface.

This work presents a solution to the position‐heading control problem of a small quadrotor. The main idea of this study is to exploit the advantages of the linear parameter varying (LPV) control approach together with a feedback linearisation approach for designing a cascade control structure without the need to perform simplifications with respect to the reference dynamic model used in the literature. The use of a LPV representation of the attitude subsystem allows to consider it as independent of the position one, allowing the use of a cascade control structure. For the inner attitude control loop, the proposed quasi‐LPV–linear‐quadratic regulator (LQR) controller and proportional–integral observer (PIO) are solved in an optimal manner by solving a set of linear matrix inequalities (LMIs). While for the outer position control loop, a feedback linearisation approach is developed before designing the corresponding controller and PIO also minimising the LQR problem by means of LMIs. Furthermore, another outer loop, interchangeable with the position one, such that controls the velocities in the body frame is proposed. This velocities controller is also solved in an optimal manner using a combination of feedback linearisation and LPV techniques. The proposed solutions are applied to an AscTec Hummingbird unmanned aerial vehicle in simulation.

This work presents a novel fuzzy adaptive sliding mode-based feedback linearization controller for trajectory tracking of a flexible robot manipulator. To reach this goal, after deriving the dynamical equations of the robot, the feedback linearization approach is utilized to change the nonlinear dynamics to a linear one and find the control law. Then, the sliding mode control strategy is implemented to design a stabilizer for trajectory tracking of the flexible robot. In order to adaptively tune the parameters of the designed controller, the gradient descent approach and the chain derivative rule are employed. Moreover, the Takagi–Sugeno–Kang fuzzy system is applied to regulate the controller gains. Finally, a multiobjective particle swarm optimization algorithm is used to find the optimum fuzzy rules. The conflicting objective functions considered as the integrals of the absolute values of the state error and the control effort should be minimized, simultaneously. The simulation results illustrate the effectiveness and capability of the introduced scenario in comparison with other methods.

The tracking and almost disturbance decoupling problem of multi-input multi-output nonlinear systems based on the feedback linearisation approach are studied. The main contribution of this study is to construct a controller, under appropriate conditions, such that the resulting closed-loop system is valid for any initial condition and bounded tracking signal with the following characteristics: input-to-state stability with respect to disturbance inputs and almost disturbance decoupling, that is, the influence of disturbances on the L2 norm of the output tracking error can be arbitrarily attenuated by changing some adjustable parameters. One example, which cannot be solved by the first paper of the almost disturbance decoupling problem on account of requiring some sufficient conditions that the nonlinearities multiplying the disturbances satisfy structural triangular conditions, is proposed to exploit the fact that the tracking and the almost disturbance decoupling performances are easily achieved by the proposed approach. To demonstrate the practical applicability, a famous half-car active suspension system has been investigated.

One of the most important applications of cable robots is load carrying along a specific path. Control procedure of cable robots is more challenging compared to linkage robots since cables can't afford pressure. Meanwhile carrying the heaviest possible payload for this kind of robots is desired. In this paper a nonlinear optimal control is proposed in order to control the end-effector within a predefined trajectory while the highest Dynamic Load Carrying Capacity (DLCC) can be carried. This purpose is met by applying optimum torque distribution among the motors with acceptable tracking accuracy. Besides, other algorithms are applied to make sure that the allowable workspace constraint is also satisfied. Since the dynamics of the robot is nonlinear, feedback linearization approach is employed in order to control the end-effector on its desirable path in a closed loop way while Linear Quadratic Regulator (LOR) method is used in order to optimize its controlling gains since the state space is linearized by the feedback linearization. The proposed algorithm is supported by doing some simulation studies on a two Degrees of Freedom (DOF) constrained planar cable robot as well as a six DOFs under constrained cable suspended robot and their DLCCs are calculated by satisfying the motor torque, tracking error and allowable workspace constraints. The results including the angular velocity, motors' torque, actual tracking of the end-effector and the DLCC of the robot are calculated and verified using experimental tests done on the cable robot. Comparison of the results of open loop simulation results, closed loop simulation results and experimental tests, shows that the results are improved by applying the proposed algorithm. This is the result of tuning the motors' torque and accuracy in a way that the highest DLCC can be achieved.

This paper presents a theoretical framework for linearization and stabilization of second-order nonholonomic chained systems. By giving a new criterion for the problem of nonsmooth nonregular feedback linearization, it is proved that second-order chained systems are nonregular static state feedback linearizable. A discontinuous control law is obtained based on linear system theory and the inversion technique. Simulation studies are carried out to show the effectiveness of the approach.