A review of convex approaches for control, observation and safety of linear parameter varying and Takagi-Sugeno systems

This paper provides a review about the concept of convex systems based on Takagi-Sugeno, linear parameter varying (LPV) and quasi-LPV modeling. These paradigms are capable of hiding the nonlinearities by means of an equivalent description which uses a set of linear models interpolated by appropriately defined weighing functions. Convex systems have become very popular since they allow applying extended linear techniques based on linear matrix inequalities (LMIs) to complex nonlinear systems. This survey aims at providing the reader with a significant overview of the existing LMI-based techniques for convex systems in the fields of control, observation and safety. Firstly, a detailed review of stability, feedback, tracking and model predictive control (MPC) convex controllers is considered. Secondly, the problem of state estimation is addressed through the design of proportional, proportional-integral, unknown input and descriptor observers. Finally, safety of convex systems is discussed by describing popular techniques for fault diagnosis and fault tolerant control (FTC).Peer ReviewedPostprint (published version

A Linear Parameter-Varying Control Method for Inline Wheel Systems

The design of the bicycle and other single-track systems are continually evolving and have become a key tool for people and goods transportation worldwide [1],[2]. The form factor, carrying capacity, maneuverability, and cost of single-track vehicles makes them advantageous in a variety of circumstances and justifies their use case in the 21st Century [2] [3],[4]. As autonomous double track vehicles arrive on public roads, it is natural that single-track autonomous systems will be developed as well; however, the unstable and non-minimum phase dynamics of single-track vehicles make their control have an additional layer of complexity compared to double track vehicles. Although many researchers have provided commentary on the stability and tracking of a riderless bicycle, relatively few bodies of work have validated their analysis through experimental testing. This work successfully demonstrates that, through gain scheduling, a PID-type controller can balance a riderless single-track vehicle by using a linear actuator to implement front-fork steering control. This control system is novel in the way in which the front fork is actuated. The manual PID tuning process outlined in this body of work is also unique, as well as the specifics of the control law (although PID controllers have been used by other authors). The works of other authors on this topic is briefly summarized and a second-order dynamics system model is derived. Then controller analysis is simulated and then validated experimentally. Suggestions are also made on next steps that can be taken to build upon the work outlined in this thesis.MSEElectrical Engineering, College of Engineering & Computer ScienceUniversity of Michigan-Dearbornhttp://deepblue.lib.umich.edu/bitstream/2027.42/169157/1/Ronald Smith Final Thesis.pd

LMI-based design of state-feedback controllers for pole clustering of LPV systems in a union of -regions

This paper introduces an approach for the design of a state-feedback controller that achieves pole clustering in a union of DR-regions for linear parameter varying systems. The design conditions, obtained using a partial pole placement theorem, are eventually expressed in terms of linear matrix inequalities. In addition, it is shown that the approach can be modified in a shifting sense. Hence, the controller gain is computed such that different values of the varying parameters imply different regions of the complex plane where the closed-loop poles are situated. This approach enables the online modification of the closed-loop performance. The effectiveness of the proposed method is demonstrated by means of simulations.Peer ReviewedPostprint (author's final draft

Design of state-feedback controllers for linear parameter varying systems subject to time-varying input saturation

All real-world systems are affected by the saturation phenomenon due to inherent physical limitations of actuators. These limitations should be taken into account in the controller’s design to prevent a possibly severe deterioration of the system’s performance, and may even lead to instability of the closed-loop system. Contrarily to most of the control strategies, which assume that the saturation limits are constant in time, this paper considers the problem of designing a state-feedback controller for a system affected by time-varying saturation limits with the objective to improve the performance. In order to tie variations of the saturation function to changes in the performance of the closed-loop system, the shifting paradigm is used, that is, some parameters scheduled by the time-varying saturations are introduced to schedule the performance criterion, which is considered to be the instantaneous guaranteed decay rate. The design conditions are obtained within the framework of linear parameter varying (LPV) systems using quadratic Lyapunov functions with constant Lyapunov matrices and they consist in a linear matrix inequality (LMI)-based feasibility problem, which can be solved efficiently using available solvers. Simulation results obtained using an illustrative example demonstrate the validity and the main characteristics of the proposed approach.Peer ReviewedPostprint (published version

Development of a Test Platform for Experimental Testing of Bicycle Models

Electrical Engineerin

Performance enhancement of multivariable model reference optimal adaptive motor speed controller using error-dependent hyperbolic gain functions

The main contribution of this paper is to formulate a robust-adaptive and stable state-space speed control strategy for DC motors. The linear-quadratic-integral (LQI) controller is utilized as the baseline controller for optimal speed-regulation, accurate reference-tracking and elimination of steady-state fluctuations in the motor’s response. To reject the influence of modelling errors, the LQI controller is augmented with a Lyapunov-based model reference adaptation system (MRAS) that adaptively modulates the controller gains while maintaining the asymptotic stability of the controller. To further enhance the system’s robustness against parametric uncertainties, the adaptation gains of MRAS online gain-adjustment law are dynamically adjusted, after every sampling interval, using smooth hyperbolic functions of motor’s speed-error. This modification significantly improves the system’s response-speed and damping against oscillations, while ensuring its stability under all operating conditions. It dynamically re-configures the control-input trajectory to enhance the system’s immunity against the detrimental effects of random faults occurring in practical motorized systems such as bounded impulsive-disturbances, modelling errors, and abrupt load–torque variations. The efficacy of the proposed control strategy is validated by conducting credible hardware-in-the-loop experiments on QNET 2.0 DC Motor Board. The experimental results successfully validate the superior tracking accuracy and disturbance-rejection capability of the proposed control strategy as compared to other controller variants benchmarked in this article

LMI-based design of state-feedback controllers for pole clustering of LPV systems in a union of DR-regions

This paper introduces an approach for the design of a state-feedback controller that achieves pole clustering in a union of DR-regions for linear parameter varying systems. The design conditions, obtained using a partial pole placement theorem, are eventually expressed in terms of linear matrix inequalities. In addition, it is shown that the approach can be modified in a shifting sense. Hence, the controller gain is computed such that different values of the varying parameters imply different regions of the complex plane where the closed-loop poles are situated. This approach enables the online modification of the closed-loop performance. The effectiveness of the proposed method is demonstrated by means of simulations.acceptedVersio

Automatic Stabilization of a Riderless Bicycle using the Active Disturbance Rejection Control Approach

[ES] Este trabajo propone una estrategia de Control por Rechazo Activo de Perturbaciones (ADRC), usando observadores extendidos de perturbación, para estabilizar una bicicleta en movimiento, sin conductor y con una velocidad de avance variable. Aunque la bicicleta tiene una dinámica inestable y no lineal alrededor de su posición vertical, que puede modelarse como un sistema Lineal de Parámetros Variantes (LPV) dependientes de la velocidad, el diseño del controlador usa un modelo simplificado de parámetros concentrados invariantes en el tiempo y una velocidad nominal constante. El esquema ADRC agrupa las discrepancias entre el modelo simplificado y la planta, junto con las perturbaciones externas en una señal aditiva unificada, que es estimada a través del observador y realimentada mediante una ley de control lineal para rechazarla. La efectividad de la estrategia es validada mediante una co-simulación entre ADAMS y MATLAB, la cual exhibe un alto desempeño y robustez sobre un modelo dinámico virtual de la bicicleta, sometida a perturbaciones externas severas y variaciones de parámetros.[EN] This work proposes an ADRC (Active Disturbance Rejection Control) strategy by disturbance extended observers to stabilize a moving riderless bicycle with a variant forward speed. Although the bicycle has an unstable and non-linear dynamics when in its upright position, which can be modeled as a LPV (Linear-Parameter-Varying) system that depends on the forward speed, a simplified time-invariant and lumped-parameter model, with an nominal constant forward speed is used in the controller design. ADRC scheme groups discrepancies between the simplified model and the plant, with external disturbances into an equivalent additive unified disturbance signal at input, which is estimated via the observer and rejected through a linear control law. The effectiveness of this strategy is validated by a co-simulation between ADAMS and MATLAB, which exhibits a high performance and robustness in a virtual dynamic model of the bicycle, submitted to severe external disturbances and parameter variations. Baquero-Suárez, M.; Cortes-Romero, J.; Arcos-Legarda, J.; Coral-Enriquez, H. (2017). Estabilización Automática de una Bicicleta sin Conductor mediante el Enfoque de Control por Rechazo Activo de Perturbaciones. Revista Iberoamericana de Automática e Informática industrial. 15(1):86-100. https://doi.org/10.4995/riai.2017.8832OJS86100151Ai-Buraiki, O., Thabit, M. B., Jun 2014. Model Predictive Control Design Approach for Autonomous Bicycle Kinematics Stabilization. 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Control Systems, IEEE 30 (5), 23-32. https://doi.org/10.1109/MCS.2010.937745Cortés Romero, J., Luviano Juárez, A., Álvarez Salas, R., Sira Ramírez, H., Aug 2010. Fast Identification and Control of an Uncertain Brushless DC Motor Using Algebraic Methods. In: 12th International Power Electronics Congress (CIEP). pp. 9-14. https://doi.org/10.1109/CIEP.2010.5598844Cortés Romero, J., Ramos, G., Coral Enriquez, H., Aug 2014. Generalized Proportional Integral Control for Periodic Signals under Active Disturbance Rejection Approach. ISA Transactions 53 (6), 1901-1909. https://doi.org/10.1016/j.isatra.2014.07.001Emaru, T., Tsuchiya, T., Dec 2005. Research on Estimating Smoothed Value and Differential Value by using Sliding Mode System. IEEE Transactions on Robotics and Automation 21 (6), 391-402.Gao, B., Junpeng Shao, Xiaodong Yang, Nov 2014. A Compound Control Strategy Combining Velocity Compensation with ADRC of Electroydraulic Position Servo Control System. 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Science 332 (6027), 339-342. https://doi.org/10.1126/science.1201959Kooijman, J. D. G., Schwab, A. L., Meijaard, J. P., May 2008. Experimental Validation of a Model for the Motion of an Uncontrolled Bicycle. Multibody System Dynamics 19 (1), 115-132. https://doi.org/10.1007/s11044-007-9050-xLam, P. Y., Sep 2011. Gyroscopic Stabilization of a Kid-Size Bicycle. In: 2011 IEEE 5th International Conference on Cybernetics and Intelligent Systems (CIS). pp. 247-252. https://doi.org/10.1109/ICCIS.2011.6070336Lewis, F. L., Popa, L. X. D., 2008. Optimal and Robust Estimation, with an Introduction to Stochastic Control Theory, 2nd Edition. CRC Press, Ch. 3, pp. 151-204.Limebeer, D. J. N., Sharp, R. S., Oct 2006. Bicycles, Motorcycles, and Models. IEEE Control Systems 26 (5), 34-61. https://doi.org/10.1109/MCS.2006.1700044Meijaard, J., Papadopoulos, J. M., Ruina, A., Schwab, A., 2007. Linearized Dynamics Equations for the Balance and Steer of a Bicycle: a Benchmark and Review. 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Path Following and Stabilization of an Autonomous Bicycle

In this thesis we investigate the problem of designing a control system for a modern bicycle so that the bicycle is stable and follows a path. We propose a multi-loop control architecture, where each loop is systematically designed using linear control techniques. The proposed strategy guarantees that the bicycle asymptotically converges to paths of constant curvature. A key advantage of our approach is that by using linear techniques analysis and controller design are relatively simple. We base our control design on the nonlinear (corrected) Whipple model, which has been previously verified for correctness and experimentally validated. The equations of motion for the nonlinear model are very complicated, and would take many pages to explicitly state. They also have no known closed form solution. To enable analysis of the model we linearize it about a trajectory such that the bicycle is upright and travelling straight ahead. This linearization allows us to arrive at a parameterized linear time-invariant state-space representation of the bicycle dynamics, suitable for analysis and control design. The inner-loop control consists of a forward-speed controller as well as a lean and steer controller. To keep the bicycle at a constant forward speed, we develop a high-bandwidth proportional controller that uses a torque along the axis of the rear wheel of the bicycle to keep the angular velocity of the rear wheel at a constant setpoint. To stabilize the bicycle at this forward speed, lean torque and steer torque are treated as the control signals. We design a state-feedback controller and augment integrators to the output feedback of the lean angle and steer angle to provide perfect steady-state tracking. To arrive at the gains for state feedback, linear-quadratic regulator methods are used. When following a constant-curvature path, a vehicle has a constant yaw rate. Using this knowledge, we begin designing the outer-loop path-following control by finding a map that converts a yaw rate into appropriate lean angle and steer angle references for the inner loop. After the map is completed, system identification is performed by applying a yaw-rate reference to the map and analyzing the response of the bicycle. Using the linear approximation obtained, a classical feedback controller for yaw-rate tracking is designed. In addition to yaw-rate control, to track a path the yaw angle of the bicycle must match that of the path and the bicycle must physically be on the path. To analyze these conditions a linear approximation for the distance between the bicycle to the path is found, enabling construction of a linear approximation of the entire system. We then find that by passing the signal for the difference in yaw rate and the distance through separate controllers, summing their output, and subtracting from the reference yaw rate of the path, the bicycle converges to the path. After developing the general design procedure, the final part of the thesis shows a step by step design example and demonstrates the results of applying the proposed control architecture to the nonlinear bicycle model. We highlight some problems that can arise when the bicycle is started far from the path. To overcome these problems we develop the concept of a virtual path, which is a path that when followed returns the bicycle to the actual path. We also recognize that, in practice, typical paths do not have constant curvature, so we construct more practical paths by joining straight line segments and circular arc segments, representing a practical path similar to a path that would be encountered when biking through a series of rural roads. Finally, we finish the design example by demonstrating the performance of the control architecture on such a path. From these simulations we show that using the suggested controller design that the bicycle will converge to a constant curvature path. Additionally with using the controllers we develop that in the absence of disturbance the bicycle will stay within the intended traffic lane when travelling on a typical rural road