Article

Feedback linearisation of mechanical systems using data-driven models

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... Recently, Floren et al. [16] proposed a data-driven approach utilizing a model predictive control (MPC) framework for system identification based on this internal nonlinear feedback force concept. They explored the proposed method for the real-time feedback linearization on a physical nonlinear vibration system. ...
... Therefore, major challenge in fully describing the nonlinear system lies in identifying the nonlinear feedback force acting on the system at each instant of time. To address this issue, we propose to use a signal processing framework based on the model predictive control (MPC) formulation, as recently explored by Floren et al. [16], which will be presented in the following subsection. It is important to note that the objective of this data processing framework, which is carried out offline, is the characterization of nonlinear restoring forces. ...
Conference Paper
Full-text available
The design of active controllers for vibration attenuation in mechanical systems depends on an accurate identification of the system. This task becomes very challenging in nonlinear systems due to the complexity and variety of nonlinear phenomena, and where a parametric modeling approach may have difficulty in capturing the full dynamic behavior. In this paper, we investigate the use of a data-driven identification approach using a Gaussian process regression model for feedback linearization control of nonlinear dynamics systems. The proposed methodology is investigated in a numerical demonstration case of a Duffing oscillator. The advantages and limitations of the data-driven identification approach are discussed with respect to the feedback linearization control aspect.
Article
Full-text available
The algebraic formalism developed in this paper unifies the study of the accessibility problem and various notions of feedback linearizability for discrete-time nonlinear systems. The accessibility problem for nonlinear discrete-time systems is shown to be easy to tackle by means of standard linear algebraic tools, whereas this is not the case for nonlinear continuous-time systems, in which case the most suitable approach is provided by differential geometry. The feedback linearization problem for discrete-time systems is recasted through the language of differential forms. In the event that a system is not feedback linearizable, the largest feedback linearizable subsystem is characterized within the same formalism using the notion of derived flag of a Pfaffian system. A discrete-time system may be linearizable by dynamic state feedback, though it is not linearizable by static state feedback. Necessary and sufficient conditions are given for the existence of a so-called linearizing output, which in turn is a sufficient condition for dynamic state feedback linearizability.
Article
Full-text available
An offset-free control is one that drives the controlled outputs to their desired targets at steady state. In the linear model predictive control (MPC) framework, the elimination of steady-state offset may seem a little obscure, since the closed-loop optimization tends to hide the integral action. Theoretically, implementing a well-posed optimization problem and having unbiased steady-state predictions are sufficient conditions to eliminate the output offset. However, these basic conditions are not always achieved in practical applications, especially when state-space models are used to perform the output predictions. This paper presents a detailed practical analysis of the existing strategies to eliminate offset when using linear state-space models with moderated uncertainties. The effectiveness of these strategies is demonstrated by simulating three different control problems: a linear SISO system where the effect of using the estimation of the control variable is highlighted, a continuous stirred tank reactor (CSTR) with non-linear dynamics and the consequent model uncertainty and, a 2×2 system representing a distillation column that verifies the consistency of previous results and extends the conclusions to higher dimension systems.
Conference Paper
Full-text available
This paper points out the flaws in using the extended Kalman filter (EKE) and introduces an improvement, the unscented Kalman filter (UKF), proposed by Julier and Uhlman (1997). A central and vital operation performed in the Kalman filter is the propagation of a Gaussian random variable (GRV) through the system dynamics. In the EKF the state distribution is approximated by a GRV, which is then propagated analytically through the first-order linearization of the nonlinear system. This can introduce large errors in the true posterior mean and covariance of the transformed GRV, which may lead to sub-optimal performance and sometimes divergence of the filter. The UKF addresses this problem by using a deterministic sampling approach. The state distribution is again approximated by a GRV, but is now represented using a minimal set of carefully chosen sample points. These sample points completely capture the true mean and covariance of the GRV, and when propagated through the true nonlinear system, captures the posterior mean and covariance accurately to the 3rd order (Taylor series expansion) for any nonlinearity. The EKF in contrast, only achieves first-order accuracy. Remarkably, the computational complexity of the UKF is the same order as that of the EKF. Julier and Uhlman demonstrated the substantial performance gains of the UKF in the context of state-estimation for nonlinear control. Machine learning problems were not considered. We extend the use of the UKF to a broader class of nonlinear estimation problems, including nonlinear system identification, training of neural networks, and dual estimation problems. In this paper, the algorithms are further developed and illustrated with a number of additional examples
Article
Full-text available
Two noniterative subspace-based algorithms which identify linear, time-invariant MIMO (multi-input/multioutput) systems from frequency response data are presented. The algorithms are related to the recent time-domain subspace identification techniques. The first algorithm uses equidistantly, in frequency, spaced data and is strongly consistent under weak noise assumptions. The second algorithm uses arbitrary frequency spacing and is strongly consistent under more restrictive noise assumptions, promising results are obtained when the algorithms are applied to real frequency data originating from a large flexible structure
Article
Nonlinear system identification is an extremely broad topic, since every system that is not linear is nonlinear. That makes it impossible to give a full overview of all aspects of the fi eld. For this reason, the selection of topics and the organization of the discussion are strongly colored by the personal journey of the authors in this nonlinear universe.
Article
In this paper, we propose a parametric system identification approach for a class of continuous-time Lur’e-type systems. Using the Mixed-Time-Frequency (MTF) algorithm, we show that the steady-state model response and the gradient of the model response with respect to its parameters can be computed in a numerically fast and efficient way, allowing efficient use of global and local optimization methods to solve the identification problem. Furthermore, by enforcing the identified model to be inside the set of convergent models, we certify a stability property of the identified model, which allows for reliable generalized usage of the model also for other excitation signals than those used to identify the model. The effectiveness and benefits of the proposed approach are demonstrated in a simulation case study. Furthermore, we have experimentally shown that the proposed approach provides fast identification of both medical equipment and patient parameters in mechanical ventilation and, thereby, enables improved patient treatment.
Article
Linear parameter-varying (LPV) models form a powerful model class to analyze and control a (nonlinear) system of interest. Identifying an LPV model of a nonlinear system can be challenging due to the difficulty of selecting the scheduling variable(s) a priori, especially if a first principles based understanding of the system is unavailable. Converting a nonlinear model to an LPV form is also non-trivial and requires systematic methods to automate the process. Inspired by these challenges, a systematic LPV embedding approach starting from multiple-input multiple-output (MIMO) linear fractional representations with a nonlinear feedback block (NLFR) is proposed. This NLFR model class is embedded into the LPV model class by an automated factorization of the (possibly MIMO) static nonlinear block present in the model. As a result of the factorization, an LPV-LFR or an LPV state-space model with affine dependency on the scheduling is obtained. This approach facilitates the selection of the scheduling variable and the connected mapping of system variables. Such a conversion method enables to use nonlinear identification tools to estimate LPV models. The potential of the proposed approach is illustrated on a 2-DOF nonlinear mass-spring-damper example.
Conference Paper
We consider a novel control architecture for an input non-affine thermoelectric system, which is used to control the temperature of an object subject to unknown thermal disturbances. A key component in this architecture is given by an input-output linearizing feedback controller to deal with the nonlinear dynamics associated with the input. This enables us to use linear control techniques with the associated performance guarantees. Using a Lyapunov-based stability analysis we derive sufficient conditions for asymptotic stability in the nominal operating regime. To prevent instability outside the nominal operating regime, e.g. in the face of large disturbances where stability is inevitably compromised, we propose to saturate the control input by using state-dependent bounds. These bounds automatically trade-off performance and stability, thereby avoiding the need for complicated stability analysis per application, and as such, allowing the designer to focus on performance. The effectiveness of the nonlinear control approach is demonstrated through measurement results.
Article
The present paper deals with the identification of nonlinear mechanical vibrations. A grey-box, or semi-physical, nonlinear state-space representation is introduced, expressing the nonlinear basis functions using a limited number of measured output variables. This representation assumes that the observed nonlinearities are localised in physical space, which is a generic case in mechanics. A two-step identification procedure is derived for the grey-box model parameters, integrating nonlinear subspace initialisation and weighted least-squares optimisation. The complete procedure is applied to an electrical circuit mimicking the behaviour of a single-input, single-output (SISO) nonlinear mechanical system and to a single-input, multiple-output (SIMO) geometrically nonlinear beam structure.
Conference Paper
The problem of controlling a variable-speed-variable-pitch wind turbine in non conventional operating points is addressed. We aim to provide a control architecture for a general active power tracking problem for the entire operating envelope. The presented control enables to cope with system non linearities while handling state and input constraints, and avoiding singular points. Simulations are carried out based on a 600 kW turbine parameters. Montecarlo simulation shows that the proposed controller presents a certain degree of robustness with respect to the system major uncertainties.
Article
Nonlinear system identification is a vast research field, today attracting a great deal of attention in the structural dynamics community. Ten years ago, an MSSP paper reviewing the progress achieved until then [1] concluded that the identification of simple continuous structures with localised nonlinearities was within reach. The past decade witnessed a shift in emphasis, accommodating the growing industrial need for a first generation of tools capable of addressing complex nonlinearities in larger-scale structures. The objective of the present paper is to survey the key developments which arose in the field since 2006, and to illustrate state-of-the-art techniques using a real-world satellite structure. Finally, a broader perspective to nonlinear system identification is provided by discussing the central role played by experimental models in the design cycle of engineering structures.
Article
Block-oriented models are popular in nonlinear modeling because of their advantages to be quite simple to understand and easy to use. Many different identification approaches were developed over the years to estimate the parameters of a wide range of block-oriented models. One class of these approaches uses linear approximations to initialize the identification algorithm. The best linear approximation framework and the ϵ\epsilon-approximation framework, or equivalent frameworks, allow the user to extract important information about the system, guide the user in selecting good candidate model structures and orders, and they prove to be a good starting point for nonlinear system identification algorithms. This paper gives an overview of the different block-oriented models that can be modeled using linear approximations, and of the identification algorithms that have been developed in the past. A non-exhaustive overview of the most important other block-oriented system identification approaches is also provided throughout this paper.
Article
This article addresses the following problems: 1) First, a nonlinearity analysis is made looking for the presence of nonlinearities in an early phase of the identification process. The level and the nature of the nonlinearities should be retrieved without a significant increase in the amount of measured data. 2) Next it is studied if it is safe to use a linear system identification approach, even if the presence of nonlinear distortions is detected. The properties of the linear system identification approach under these conditions are studied, and the reliability of the uncertainty bounds is checked. 3) Eventually, tools are provided to check how much can be gained if a nonlinear model were identified instead of a linear model. Addressing these three questions forms the outline of this article. The possibilities and pitfalls of using a linear identification framework in the presence of nonlinear distortions will be discussed and illustrated on lab-scale and industrial examples. In this article, the focus is on nonparametric and parametric black box identification methods, however the results might also be useful for physical modeling methods. Knowing the actual nonlinear distortion level can help to choose the required level of detail that is needed in the physical model. This will strongly influence the modeling effort. Also, in this case, significant time can be saved if it is known from experiments that the system behaves almost linearly. The converse is also true. If the experiments show that some (sub-)systems are highly nonlinear, it helps to focus the physical modeling effort on these critical elements.
Article
Aluminium alloys have been the primary material for the structural parts of aircraft for more than 80 years because of their well known performance, well established design methods, manufacturing and reliable inspection techniques. Nearly for a decade composites have started to be used more widely in large commercial jet airliners for the fuselage, wing as well as other structural components in place of aluminium alloys due their high specific properties, reduced weight, fatigue performance and corrosion resistance. Although the increased use of composite materials reduced the role of aluminium up to some extent, high strength aluminium alloys remain important in airframe construction. Aluminium is a relatively low cost, light weight metal that can be heat treated and loaded to relatively high level of stresses, and it is one of the most easily produced of the high performance materials, which results in lower manufacturing and maintenance costs. There have been important recent advances in aluminium aircraft alloys that can effectively compete with modern composite materials. This study covers latest developments in enhanced mechanical properties of aluminium alloys, and high performance joining techniques. The mechanical properties on newly developed 2000, 7000 series aluminium alloys and new generation Al-Li alloys are compared with the traditional aluminium alloys. The advantages and disadvantages of the joining methods, laser beam welding and friction stir welding, are also discussed.
Conference Paper
To achieve a good performance of a boiler-turbine unit, dynamic variables such as steam pressure, water level of drum and electric output of turbine must be controlled. In this paper a nonlinear model of the boiler-turbine unit is considered in which the inputs are the valve positions of fuel flow, steam control, and feed-water flow. Using two control methods, feedback linearization and gain scheduling, a PI controller is designed. It is shown that by applying both methods, system goes from one operating point to another with an appropriate specification of time response. Results show that system with a controller designed based on gain scheduling method has a better time response from a nominal operating point to a far operating point. In addition, applying any of these control approaches guarantees robustness of the system against the uncertainties associated with dynamic model.
Conference Paper
A robust model predictive control algorithm solving the tracking and the infeasible reference problems for constrained systems subject to bounded disturbances is presented in this technical note. The proposed solution relies on three main concepts: 1) the reformulation of the system in the so-called velocity form to obtain offset-free tracking when constant disturbances are present, 2) the use of a tube-based approach to cope with non-constant but bounded disturbances, 3) the use of reference outputs as arguments of the optimization problem to cope with infeasible references. Convergence results are derived by suitably defining the auxiliary control law and the terminal set used in the problem formulation.
Article
We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback. For n = 2, we show, under the assumption of completeness of , that if the discretized system is lineariable by state coordinate change and feedback, then the continuous time affine complete analytic system is linearizable by state coordinate change only. Also, we suggest a method of proof when n ≥ 3.
Article
We give necessary and sufficient conditions for a nonlinear discrete-time system to be locally linearizable by a change of state coordinates and a feedback. The conditions are expressed in terms of controllability distributions associated to the system.
Conference Paper
In this paper, following ideas recently developed for a class of continuous systems, we introduce the notion of zero dynamics and minimum phase for discrete time nonlinear systems. On this basis, sufficient conditions are given for state feedback stabilization and full linearization via dynamic compensation. Relations with other properties of the system are also investigated.
Article
We refer to Model Predictive Control (MPC) as that family of controllers in which there is a direct use of an explicit and separately identifiable model. Control design methods based on the MPC concept have found wide acceptance in industrial applications and have been studied by academia. The reason for such popularity is the ability of MPC designs to yield high performance control systems capable of operating without expert intervention for long periods of time. In this paper the issues of importance that any control system should address are stated. MPC techniques are then reviewed in the light of these issues in order to point out their advantages in design and implementation. A number of design techniques emanating from MPC, namely Dynamic Matrix Control, Model Algorithmic Control, Inferential Control and Internal Model Control, are put in perspective with respect to each other and the relation to more traditional methods like Linear Quadratic Control is examined. The flexible constraint handling capabilities of MPC are shown to be a significant advantage in the context of the overall operating objectives of the process industries and the 1-, 2-, and ∞-norm formulations of the performance objective are discussed. The application of MPC to non-linear systems is examined and it is shown that its main attractions carry over. Finally, it is explained that though MPC is not inherently more or less robust than classical feedback, it can be adjusted more easily for robustness.
Article
In the general case of non-uniformly spaced frequency-domain data and/or arbitrarily coloured disturbing noise, the frequency-domain subspace identification algorithms described in McKelvey, Akçay, and Ljung (IEEE Trans. Automatic Control 41(7) (1996) 960) and Van Overschee and De Moor (Signal Processing 52(2) (1996) 179) are consistent only if the covariance matrix of the disturbing noise is known. This paper studies the asymptotic properties (strong convergence, convergence rate, asymptotic normality, strong consistency and loss in efficiency) of these algorithms when the true noise covariance matrix is replaced by the sample noise covariance matrix obtained from a small number of independent repeated experiments. As an additional result the strong convergence (in case of model errors), the convergence rate and the asymptotic normality of the subspace algorithms with known noise covariance matrix follows.
Article
Model Predictive Control (MPC) has a long history in the field of control engineering. It is one of the few areas that has received on-going interest from researchers in both industry and universities. It has been recognised that there are three major branches of MPC algorithms consisting of step-response model based design: Dynamic Matrix Control (DMC); transfer function model based design: Generdised Predictive Control (GPC); and a general state space model based design. The DMC and GPC algorithms can also be caste in the state space framework. Along the genera lines of state space methods, there are two mainstreunts: one solves for the optinzal control signal while the other solves for the increment of the optimal control signal. The latter can be implemented in a velocity form analogous to the implementation of a PID controller on an industrial plant. Motivated by this advantage. and that integral action is naturally embedded in the algorithm, this tutorial paper focuses on an introduction to Model Predictive Control based on the state space approach using a linear velocity-form model.
Conference Paper
In this paper we describe a new recursive linear estimator for filtering systems with nonlinear process and observation models. This method uses a new parameterisation of the mean and covariance which can be transformed directly by the system equations to give predictions of the transformed mean and covariance. We show that this technique is more accurate and far easier to implement than an extended Kalman filter. Specifically, we present empirical results for the application of the new filter to the highly nonlinear kinematics of maneuvering vehicles
Article
It is shown that feedback linearizability of a continuous-time system can be destroyed through the introduction of the usual sample-and-hold devices. A method whereby a feedback linearizable discrete-time model can be recovered is given