Gergely Takács

Adaptive Model Predictive Vibration Control of a Cantilever Beam with Real-Time Parameter Estimation

This paper presents an adaptive-predictive vibration control system using extended Kalman filtering for the joint estimation of system states and model parameters. A fixed-free cantilever beam equipped with piezoceramic actuators serves as a test platform to validate the proposed control strategy. Deflection readings taken at the end of the beam have been used to reconstruct the position and velocity information for a second-order state-space model. In addition to the states, the dynamic system has been augmented by the unknown model parameters: stiffness, damping constant, and a voltage/force conversion constant, characterizing the actuating effect of the piezoceramic transducers. The states and parameters of this augmented system have been estimated in real time, using the hybrid extended Kalman filter. The estimated model parameters have been applied to define the continuous state-space model of the vibrating system, which in turn is discretized for the predictive controller. The model predictive control algorithm generates state predictions and dual-mode quadratic cost prediction matrices based on the updated discrete state-space models. The resulting cost function is then minimized using quadratic programming to find the sequence of optimal but constrained control inputs. The proposed active vibration control system is implemented and evaluated experimentally to investigate the viability of the control method.

Model predictive control algorithms for active vibration control: a study on timing, performance and implementation properties

The dynamic properties of vibration control systems pose unique requirements and challenges on the implementation of model predictive control (MPC) algorithms with stability and feasibility guarantees. This article presents a comprehensive experimental comparison of computation timing and damping performance for various MPC methods; analyzing their offline and online properties in active vibration control and their impact on practical implementability. Optimal and sub-optimal MPC algorithms providing guaranteed stability and constraint feasibility have been applied to the real-time active vibration attenuation of a lightly damped mechanical test structure. Based on the experiments presented in this paper, the standard and sequential quadratic programming-based, optimal and sub-optimal minimum time multi-parametric programming-based and the sub-optimal Newton–Raphson’s algorithm-based MPC methods demonstrate closely comparable vibration attenuation performance. The offline and online timing analysis indicates that the underlying difference between the investigated MPC algorithms lies mainly in practical implementability difficulties caused by inherent algorithm efficiency, rendering certain variants of MPC more suitable for vibration control than others.

Basics of Vibration Dynamics

This chapter intends to introduce the reader to the theoretical basics of vibration dynamics analysis. Every well designed active vibration control (AVC) system requires at least a fundamental understanding of the underlying vibration phenomenon. The simple point mass oscillator is used as an example to build more and more complicated systems gradually. After the free response of undamped and damped one degree of freedom systems is discussed, forced response from a harmonic source is considered as well. The basics of engineering vibration analysis of lumped mass multiple degree of freedom systems is investigated with a concise account on the eigenvalue problem and modal decomposition. The transversal vibration of a clamped-free cantilever beam is used as an example to show, how exact solutions for distributed parameter systems may be developed. The chapter is finished with a section discussing modeling techniques used in vibration control, such as first principle transfer function models, state-space models, FEM based models and experimental identification. The aim of this chapter is to introduce the mathematical description of vibration phenomena briefly, in order to characterize the nature of the mechanical systems to be controlled by the model predictive control (MPC) strategy presented in the upcoming chapters of this book.

Efficient Embedded Model Predictive Vibration Control via Convex Lifting

This paper presents an efficient real-time implementation of embedded model predictive control, adopted in the context of active vibration control with the objective of minimizing the tip deflection of lightly damped cantilever beams. In particular, we focus on memory and time-efficient explicit solutions of the associated constrained optimal control problems that are easily implementable on low-end embedded hardware. To this end, we exploit the concept of convex lifting and show how it can be used to devise low-complexity, regionless piecewise affine controllers without any loss of optimality and performance. The efficiency of this constructive procedure is quantified via an extensive complexity analysis, evidenced by a successful practical deployment and optimal vibration control performance using a family of 32-bit ARM Cortex-M-based microcontroller platforms.

Efficiency and performance of embedded model predictive control for active vibration attenuation

The development of efficient solution approaches and technological advances facilitate the use of predictive control on embedded systems, even for fast systems or on computationally limited hardware platforms. The practical implementation of predictive control is, however, still often very time consuming and demands insight into the formulation and solution strategies for predictive control. Tools for automatic code generation tailored for deployment on embedded systems promise to overcome these problems and thus enable the fast and reliable implementation of predictive control. This paper exploits the efficiency and performance of automatic code generation for linear model predictive control for the embedded active vibration control of a flexible mechanical structure. μAO-MPC is used to automatically generate C code, based on a high level problem description. The resulting code is used for real-time implementation on an embedded platform. The embedded model predictive controller efficiently computes the voltage input for a piezoceramic actuator based on state estimates from a Kalman filter. The performance, computational efficiency, memory requirements, task execution timing and other properties of practical interest are examined via experiments on the embedded controller for active vibration control.

Adaptive predictive control of transient vibrations of cantilevers with changing weight

Abstract A method to control the transient vibrations in cantilever beam structures with variable weight is presented. The proposed adaptive approach is based on the hybrid extended Kalman filter (EKF) for the joint estimation of dynamic states and the weight parameter, in combination with dual-mode infinite horizon model predictive control (MPC). This adaptive predictive method is compared to nominal linear quadratic (LQ) control and positive position feedback (PPF) in experiment. Experiments are performed on an active cantilever beam with piezoelectric actuation, subjected to transient vibration effects and weight variations. The results presented in this article suggest that the proposed algorithm outperforms its traditional counterparts, while requiring less inputs to the actuated structure.

Online Structural Health Monitoring and Parameter Estimation for Vibrating Active Cantilever Beams Using Low-Priced Microcontrollers

This paper presents a structural health monitoring and parameter estimation system for vibrating active cantilever beams using low-cost embedded computing hardware. The actuator input and the measured position are used in an augmented nonlinear model to observe the dynamic states and parameters of the beam by the continuous-discrete extended Kalman filter (EKF). The presence of undesirable structural change is detected by variations of the first resonance estimate computed from the observed equivalent mass, stiffness, damping, and voltage-force conversion coefficients. A fault signal is generated upon its departure from a predetermined nominal tolerance band. The algorithm is implemented using automatically generated and deployed machine code on an electronics prototyping platform, featuring an economically feasible 8-bit microcontroller unit (MCU). The validation experiments demonstrate the viability of the proposed system to detect sudden or gradual mechanical changes in real-time, while the functionality on low-cost miniaturized hardware suggests a strong potential for mass-production and structural integration. The modest computing power of the microcontroller and automated code generation designates the proposed system only for very flexible structures, with a first dominant resonant frequency under 4 Hz; however, a code-optimized version certainly allows much stiffer structures or more complicated models on the same hardware.

Algorithms in Active Vibration Control

With the advent of active vibration control (AVC) systems and their gradual transfer to commercial products, building a solid knowledge base on feedback systems and their components has become increasingly important for the vibration engineering community. In addition to the actuating elements that transfer the necessary dynamic changes to vibrating mechanical systems and sensors that provide feedback on vibration levels, the control strategy itself is also an essential component of the feedback system. This chapter introduces the reader to some control strategies that are routinely implemented in vibration attenuation systems. In addition to a brief theoretical primer on the control theory standing behind these strategies, examples of their use in AVC applications are given. The chapter is meant to provide a review of strategies alternative to the model predictive control (MPC) approach that is at the center of attention of this book. First, classical control strategies are introduced which are based on position or velocity feedback and use a fixed gain to compute control input. After a short discussion on the ever-so-popular proportional integral derivative (PID) controller, the focus is shifted to the essentials of optimization-based algorithms. The linear quadratic (LQ) controller is in close relationship with MPC and it is utilized both often and very effectively in vibration control. The underlying idea behind another optimization based strategy, the \({{\fancyscript{H}}}_{\infty}\) (H-infinity) controller is reviewed as well. The chapter is finished by a section on some of the more exotic control approaches, which due to their potential to tackle hysteresis and non-linearity can be very valuable for AVC. These soft computing approaches are genetic algorithms, neural networks and fuzzy control.

Applications of Model Predictive Vibration Control

This chapter will briefly review some of the existing applications of model predictive control for vibration attenuation or its closely related fields. The application of model predictive control as a vibration reduction strategy is not common, and there are only a handful of available publications related to this field.

Stability and Feasibility of MPC

Stability is perhaps the most important property of controllers. A stable controller performs its task reliably in any circumstance; however, an unstable controller may cause system failure with dramatic effects. Unstable controllers in an active vibration control (AVC) system may even cause structural failure. Stability can be guaranteed and studied in traditional control systems by inspecting the location of the closed-loop poles of the transfer function, or equivalently the magnitude of the eigenvalues of the state matrix. Unfortunately, the constrained model predictive control (MPC) law is nonlinear and it cannot be expressed in a closed form. Considering the cost function as a Lyapunov function is the key to understand the stability of the constrained MPC strategy. As long as the cost function is monotonically decreasing, the control process remains stable. It can be proved that the condition for this is to ensure the feasibility of the input sequence from the current time up to infinity. Fortunately, this seemingly formidable task can be implemented with a finite number of optimization variables. An additional constraint checking horizon in infinite horizon cost dual-mode constrained MPC ensures stability and feasibility. The series of additional constraints to ensure stability also allocate a special region of state-space called the region of attraction and its subset, the target set. For an optimal MPC strategy with maximum performance, these have a complex polyhedral shape. In the interest decreasing complexity and increasing computational speeds, these sets can be simplified to low complexity hypercubes or a hyperellipsoids. This chapter reviews the stability properties of the constrained MPC control law, how it can be guaranteed at all times, introduces the reader to maximal invariant target sets and some of its simplified approximations.