Model Predictive Vibration Control

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.

The mechanical behavior of classical materials can be described by their elastic constant, which relates stress to strain. In advanced engineering materials that are often referred to as smart or intelligent materials, the mechanical behavior is also influenced by other fields; such as magnetic, electric charge, temperature, light and chemical composition. This is also reflected in the underlying constitutive equations, which couple two or more of these fields to describe the physical behavior of the material. These aforementioned materials have desirable properties when it comes to their use in active vibration control (AVC), since they may be readily integrated within the controlled structure and do not alter the mass of the mechanical system significantly. The aim of this chapter is thus to give a review of advanced engineering materials used in active and semi-active vibration control. The chapter covers the shape memory and superelastic property of shape memory alloys (SMA) and their current use in vibration control. Magnetostrictive and electrostrictive (MS, ES) materials are less commonly utilized in vibration control, however due to their engineering potential we will give a concise account of these materials and the underlying physical principles. The advantages of magnetorheological (MR) fluid based dampers with adjustable properties have become increasingly recognized in the engineering community, thus the discussion of magnetorheological and the related electrorheological (ER) fluids is provided here as well. Piezoelectric materials such as piezoceramics are probably the most commonly used smart materials in AVC. Here, we introduce the direct and converse piezoelectric effect, a short review on the application of transducers in vibration damping and some notes on the mathematical modeling of their dynamics. The chapter is finished by the emerging electrochemical materials or electroactive polymers (EAP).

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.

A laboratory demonstration hardware for the verification of model predictive control (MPC) algorithms in active vibration control (AVC) is introduced in detail. The laboratory device featured in the experiments comparing model predictive vibration control algorithms is a simple clamped cantilever beam with piezoelectric actuation. Despite of its elementary physical construction such a lightly damped vibrating device models the dynamics of a class of real-life applications, such as helicopter rotor beams, wing surfaces, antenna masts, manipulators and others. After a brief summary of this laboratory device, its experimental identification procedure is discussed. Some of the characteristic properties of the device are introduced as well, such as actuator linearity and noise tolerance. As finite element modeling (FEM) of vibrating structures is a valuable tool for the engineering practitioner, some of the results of the preliminary FEM analyses performed on the device are also presented. The chapter is closed with a section on hardware component details, which can be an aid to those who are unfamiliar with the components of such AVC demonstrators and are planning to build one.

Model predictive control (MPC) is a modern optimization based control strategy, which has been predominantly used for applications with slow dynamics so far. The implementation of MPC for fast systems such as active vibration attenuation (AVC) requires a firm theoretical knowledge of the method, its issues and limitations. This chapter is aimed at the reader with little or no prior contact with model predictive control. Discussing basic concepts like prediction, penalty, cost and optimization helps the reader to get up to speed using the most straightforward approach possible. Taking the basic building blocks of MPC and assembling them, one should ultimately understand how the optimal, constrained, quadratic programming-based MPC algorithm works. The chapter starts with an introduction of the fundamental idea behind model predictive control, followed by a review of the historic development of this field of control engineering. Next, generating a sequence of future states based on the current measurement and a state-space model is presented. Assembling a numerical indicator of the quality of control from the grounds up is just as important as predicting the sequence of future states, thus essentials of creating a quadratic cost are also presented. With the cost function available, one is able to create a simple MPC control law in the absence of constraints. Finally, the formulation of process constraints is described, and the resulting quadratic programming (QP) optimization problem is reviewed. The mathematical problem of quadratic programming is presented through an example, along with two basic solution algorithms for QP. The chapter is finished by the discussion of the important infinite horizon cost dual-mode MPC algorithm.

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.

Computing the control input for constrained model predictive control (MPC) is a complex and time-consuming process. The constrained problem formulation does not allow the control law to be expressed as a closed equation; instead, a numerical optimization problem has to be performed online at each sampling period. The use of formulations with a priori stability guarantees further increase the complexity of the optimization task. It is a logical requirement for every control law, that the computation time of the control moves must be strictly shorter than the sampling period. Due to the computational demands of the MPC strategy, its implementations have been mostly applied to processes with slow dynamics. The application of stabilized constrained MPC to plants with fast dynamics, such as active vibration control (AVC), is a challenge even with up to date computing platforms. To allow the implementation of stabilized MPC on high sampling rate systems, this chapter reviews the theoretical foundation of some of the existing formulations that may considerably cut back on computational loads. The core of the chapter is devoted to a method, which sacrifices the optimality of the MPC process in order to create a simple and computationally very efficient MPC strategy. This method referred to as Newton–Raphson’s MPC (NRMPC) is based on a pre-stabilized, augmented state-space formulation with ellipsoidal target sets, eventually leading to an online algorithm only requiring the computation of a polynomial root. Instead of sacrificing optimality, another major line of efficient MPC formulation uses multi-parametric programming to transfer the online computational load into the offline regime (MPMPC). In addition to reviewing the theoretical essentials of this explicit MPC approach, other efficient MPC methods are briefly revised as well.

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.

This chapter is devoted to the implementation of model predictive control (MPC) algorithms in active vibration control (AVC) applications. Even though the main area of interest is AVC, the software implementation tasks presented here are valid for any other engineering application of MPC, thus the material may be recommended to anyone interested in practical issues with MPC software deployment. Three different MPC strategies are discussed, each having its own advantages and disadvantages: the well known infinite horizon cost dual-mode quadratic programming based MPC (QPMPC), optimal and suboptimal explicit pre-computed multi-parametric programming based MPC (MPMPC) and the efficient but suboptimal Newton–Raphson MPC (NRMPC). The offline portion of the algorithms is implemented in the Matlab m-file scripting language, while the real-time controllers are realized in Simulink and subsequently transferred to the xPC Target rapid control software prototyping platform. The practical approach utilized here is focused at simplicity. An off-the shelf quadratic solver called qpOASES is used for the online implementation of QPMPC, while the MPMPC algorithm is implemented using the MPT Toolbox. As the implementation of NRMPC to physical systems is unique to this book, the most attention is devoted to the code developed for the use of Newton–Raphson MPC in the vibration damping of lightly damped structures.

This chapter presents the results of simulations performed with various computationally efficient model predictive control (MPC) strategies applied to the state space model of an active vibration control (AVC) demonstration device. A numerical study analyzing the minimal prediction horizon necessary to steer system state into equilibrium for the AVC demonstrator model suggests very long horizons, even for relatively small outside disturbances. Because infinite horizon cost dual-mode quadratic programming based MPC (QPMPC) with stability guarantees is assumed for this test, the feasibility of the online implementation with several hundred steps long horizon is unlikely due to excessive computation times. Through the computation of offline optimization times, number of regions and controller sizes, it is demonstrated here that the computational requirements of multi-parametric programming based MPC (MPMPC) also render it as an unlikely candidate for the AVC of lightly damped systems. From the viewpoint of computational complexity, Newton–Raphson MPC (NRMPC) is certainly a good choice for AVC, however, not without drawbacks. As the simulation results presented here suggest, invariance of the target set and therefore constraints can be violated due to numerical imprecision at the offline computational stage. Moreover, the suboptimality of this approach becomes troubling with increasing problem dimensionality. As the numerical tests demonstrate, the offline computational drawbacks can be partly remedied by performance bounds and proper solver settings, while the optimality is somewhat influenced by a suitable input penalty choice. The chapter is finished with a simulation comparison of the QPMPC, MPMPC and NRMPC algorithms, which shows input and output sequences in agreement with theoretical expectations.

This chapter presents the results of experiments comparing different computationally efficient model predictive control (MPC) methods applied to a laboratory device, demonstrating the active vibration control (AVC) of lightly damped mechanical structures. Because of the combination of long prediction horizons, short sampling times and large actuator-disturbance asymmetry, the implementation of the predictive control strategy on lightly damped vibrating structures is highly demanding. The vibration damping effect and online timing properties of model predictive control algorithms such as infinite horizon cost dual-mode quadratic programming based MPC (QPMPC), pre-computed explicit multi-parametric programming based MPC (MPMPC), minimum-time MPMPC and the very efficient but suboptimal Newton–Raphson based MPC (NRMPC); all with guaranteed stability and constraint feasibility are analyzed in different disturbance and loading scenarios. All MPC methods along with the baseline linear quadratic (LQ) controller decrease vibration settling to equilibrium by an order of magnitude time. The damping effect of all investigated MPC strategies is comparable with a slight decrease in performance for the suboptimal minimum-time MPMPC and NRMPC controllers. Due to the excessive online computational needs of QPMPC, it is a very unlikely candidate for lightly damped vibrating systems given currently available hardware. The online timing analysis presented here demonstrates that MPMPC provides significantly shorter online execution times, however its suboptimal minimum-time version does not bring a convincing improvement. NRMPC can provide online execution times on par with linear quadratic controllers; however, its suboptimality becomes excessive with increasing prediction model orders.