کنترل ردیابی سیستم های زمان پیوسته مکانیکی فشرده غیر خطی : روش یادگیری تکرار ی مبتنی بر مدل

This paper presents a nonlinear model-based iterative learning control procedure to achieve accurate tracking control for nonlinear lumped mechanical continuous-time systems. The model structure used in this iterative learning control procedure is new and combines a linear state space model and a nonlinear feature space transformation. An intuitive two-step iterative algorithm to identify the model parameters is presented. It alternates between the estimation of the linear and the nonlinear model part. It is assumed that besides the input and output signals also the full state vector of the system is available for identification. A measurement and signal processing procedure to estimate these signals for lumped mechanical systems is presented. The iterative learning control procedure relies on the calculation of the input that generates a given model output, so-called offline model inversion. A new offline nonlinear model inversion method for continuous-time, nonlinear time-invariant, state space models based on Newton's method is presented and applied to the new model structure. This model inversion method is not restricted to minimum phase models. It requires only calculation of the first order derivatives of the state space model and is applicable to multivariable models. For periodic reference signals the method yields a compact implementation in the frequency domain. Moreover it is shown that a bandwidth can be specified up to which learning is allowed when using this inversion method in the iterative learning control procedure. Experimental results for a nonlinear single-input-single-output system corresponding to a quarter car on a hydraulic test rig are presented. It is shown that the new nonlinear approach outperforms the linear iterative learning control approach which is currently used in the automotive industry on durability test rigs.

In many control applications [1] the system dynamics can be approximated by a linear model with, often additive or multiplicative, uncertainty [2]. If these linear model uncertainties are too large the performance that can be obtained with linear control techniques [2] will be moderate or insufficient. If this is the case, one has to resort to nonlinear models [3], [4] and [5], nonlinear system identification [6], [7], [8] and [5] and nonlinear control techniques [9], [10] and [11]. Usually however these nonlinear system identification methods generate models that do not satisfy the conditions and structure requirements imposed by the nonlinear control design techniques. For example most of the nonlinear control design methods assume that a physical continuous-time state space representation of the system is available [9], [10] and [11], while the existing nonlinear system identification procedures generate discrete-time, black-box, input–output models [8]. In addition, usually no direct link between these different system representations exists, as there is for linear systems. This is a consequence of the fact that the class of nonlinear systems and models is very broad, or as Stanislaw Ulam, godfather of what is now known as nonlinear science, famously remarked [12]: “Using the term ‘nonlinear science’ is like calling the bulk of zoology the study of non-elephants”. This paper presents a new nonlinear state space model structure for nonlinear time-invariant (NLTI) multiple-input-multiple-output (MIMO or multivariable) systems. It consists of a linear and a nonlinear part. The nonlinear part uses so-called features to capture the nonlinear system characteristics in a black-box setting. The features discussed in this paper correspond to sigmoidal neurons [13] and [14], which have been shown to be successful in approximating complex nonlinear functions [15]. This paper focusses on NLTI lumped mechanical systems. It is shown that these systems typically contain important, physically interpretable, linear dynamics besides the nonlinear characteristics. To capture these dynamics within the linear part of the model, an intuitive two-step iterative identification algorithm to estimate the model parameters is presented. It alternates the estimation of the parameters of the linear and the nonlinear part of the model structure at each iteration. A similar modeling concept and two-step identification procedure are presented in [16] and [17]. The modeling approach presented in this paper is, however, more general since the nonlinear part is based on sigmoidal neurons. The nonlinear part of the model structure presented in [16] and [17] are static nonlinearities and NARX models, respectively. The main drawback of the presented modeling approach is that the nonlinear part is nonlinear in the unknown parameters hence requiring nonlinear techniques to estimate them with no guarantee for global convergence. For the identification it is assumed that, besides the input and the output, also the states and their time derivatives are available. This is an important restriction for the application of the presented modeling approach. However, it is illustrated in this paper that by proper instrumentation this restriction can be overcome for lumped mechanical systems. This paper also presents a measurement and signal processing procedure to obtain accurate estimates of the states and their time derivatives for lumped mechanical systems based on sensor fusion. The combination of the new model structure with the intuitive two-step iterative identification algorithm yields a gray-box NLTI modeling approach [6]. The main advantage of this modeling approach is that the structure of the physical model of the system can be copied into the proposed model, while well established black-box estimation techniques are used to identify the nonlinear characteristics. This helps to keep the number of unknown model parameters low and yields physical state variables. Accurate tracking control can be obtained using the nonlinear internal model control (NIMC) scheme of Economou [18] and Henson [19]. The NIMC scheme assumes that a (perfect) nonlinear model of the system and an (exact) inverse of that model are available. Currently the only method to calculate the inverse model of a nonlinear state space model was derived by Hirschorn [20] but is restricted to minimum phase systems having a well defined relative degree [9]. Hirschorn's method uses feedback linearization and requires the calculation of the model's higher order Lie derivatives which assumes smooth nonlinear characteristics. Moreover the NIMC scheme assumes closed loop stability under the assumption of a perfect model. In practice a model is never perfect such that stability of the nonlinear closed loop NIMC scheme should be imposed during the design. It is well known that this is an extremely difficult task. To avoid closed loop stability problems, feedforward control can be used to obtain accurate tracking control. Iterative learning control (ILC) aims at designing a feedforward signal by repeating the same control task and updating the control signal iteratively based on the system response measured during the previous iteration. With iterative learning, accurate tracking can be obtained even if the system model is uncertain. ILC is only applicable to systems that perform the same action repeatedly over a finite time interval. The research on linear iterative learning control (LILC), that is, ILC for linear systems or based on a linear model of the system, was initiated by the work of Arimoto et al. [21]. A historical overview and general classification of LILC algorithms is presented by Moore [22]. Overviews of specific LILC procedures and their applications are given in [23], [24] and [25]. If the system is too nonlinear, the convergence of the LILC procedures can be slow, requiring many trials before a predefined tracking accuracy is obtained. Moreover, De Cuyper [26] shows that due to system nonlinearities, the LILC procedure can fail to produce the desired tracking accuracy. This paper presents a new model-based nonlinear iterative learning control (NILC) law. It is based on the NIMC scheme which is adopted to introduce the iterative learning behavior. The control signal at each iteration is based on the previous control signal and the difference between the reference output and measured output projected to the input space of the model. Because the update uses solely information about the previous iteration, it is an open loop NILC procedure [27] such that closed loop stability is not an issue. This NILC law requires at each iteration the calculation of the inverse signal of a NLTI model for the measured output, that is, the model's input signal that produces the measured output. Devasia et al. [28] developed an offline iterative inversion procedure for nonlinear systems based on the inverse of the Byrnes–Isidori normal form transformation [9]. Both the normal form and the inverse are in general hard, if not impossible, to find. Moreover, in the case of non-minimum phase systems, the procedure yields a non-causal control signal with an infinite time horizon. This paper presents a new approach to calculate this inverse signal iteratively using Newton's method [29] and [30]. The application of ILC implicitly assumes periodicity of the reference signal. Therefore this paper focusses on the tracking performance in steady state. The periodic inverse signal that generates in steady-state the given periodic output is referred to as the steady-state inverse. It is shown in this paper that using this new approach to calculate this steady-state inverse yields a compact implementation in the frequency domain. A maximum frequency can be defined up to which the steady-state inverse inverts the model. When applying this inversion method within the NILC procedure this maximal frequency corresponds to the tracking bandwidth, that is, the bandwidth up to which learning is effective. Both the model structure, identification algorithm, NILC procedure and the calculation of the steady-state inverse have been developed with the application of indoor durability tests in the automotive industry in mind. These tests correspond to a MIMO tracking control problem that requires calculation of actuator control signals to reproduce so-called target signals on indoor hydraulic test rigs until failure of the device under test occurs [26]. Such test rigs can be adequately modeled as lumped mechanical systems. Moreover the reference trajectories or target signals are reproduced repeatedly such that ILC is applicable. These target signals are assumed to be periodic and bandlimited. This paper presents experimental validation of the model structure, identification algorithm, NILC procedure and the calculation of the steady-state inverse on a nonlinear test setup which represents a scale model of a single-input-single-output (SISO) hydraulic test rig. It is shown that by applying the NILC procedure, improved tracking accuracy and faster convergence are obtained compared to the LILC procedure currently used in industry. The paper is organized as follows. Section 2 presents the developed NLTI state space model structure. The two-step iterative identification algorithm is discussed in Section 3. Section 4 presents the NILC procedure together with the calculation of the steady-state inverse using Newton's method. The discussed model structure, identification algorithm and NILC procedure are validated experimentally in Section 5 on a SISO system representing a quarter car hydraulic test rig.

This paper presents a new nonlinear model structure and a new nonlinear iterative learning control procedure to track bandlimited periodic signals. Both the model structure and the nonlinear iterative learning control procedure are applicable to multivariable time-invariant nonlinear systems, both minimum and non-minimum phase. The nonlinear model structure consists of a linear and a nonlinear model part and is therefore suited for systems with significant nonlinear dynamics. This paper also presents an intuitive two-step iterative identification algorithm that allows to incorporate prior knowledge in the model structure. The nonlinear iterative learning control procedure is a model-based iterative feedforward design approach that generates accurate tracking. The key element in the procedure is a new method to calculate the so-called steady-state inverse, that is the input vector that produces a desired, periodic, output vector, up to a maximal frequency which can be specified. This inversion method is applicable to multivariable nonlinear state space models, both minimum and non-minimum phase. The maximal frequency specifies a learning bandwidth up to which accurate tracking is obtained. The identification of the presented model structure and the nonlinear iterative learning control procedure is validated experimentally on a nonlinear system representing a scaled model of a quarter car test rig. Comparison of this nonlinear iterative learning control procedure with a linear procedure shows a significant improvement of the convergence speed and tracking accuracy.