One-shot data-driven design of fractional-order PID controller considering closed-loop stability: fictitious reference signal approach

A one-shot data-driven tuning method for a fractional-order proportional-integral-derivative (FOPID) controller is proposed. The proposed method tunes the FOPID controller in the model-reference control formulation. A loss function is defined to evaluate the match between a given reference model and the closed-loop response while explicitly considering the closed-loop stability. A loss function value is based on the fictitious reference signal computed using the input/output data. Model matching is achieved via loss function minimization. The proposed method is simple and practical: it needs only one-shot input/output data of a plant (no plant model required), considers the bounded-input bounded-output stability of the closed-loop system, and automatically determines the appropriate parameter value via optimization. Numerical simulations show that the proposed approach facilitates good control performance, and destabilization can be avoided even if perfect model matching is unachievable

H∞ Controller Design for Spectral MIMO Models by Convex Optimization

A new method for robust fixed-order H∞ controller design by convex optimization for multivariable systems is investigated. Linear Time-Invariant Multi-Input Multi- Output (LTI-MIMO) systems represented by a set of complex values in the frequency domain are considered. It is shown that the Generalized Nyquist Stability criterion can be approximated by a set of convex constraints with respect to the parameters of a multivariable linearly parameterized controller in the Nyquist diagram. The diagonal elements of the controller are tuned to satisfy the desired performances, while simultaneously, the off-diagonal elements are designed to decouple the system. Multimodel uncertainty can be directly considered in the proposed approach by increasing the number of constraints. A simulation example illustrates the effectiveness of the proposed approach. by a simulation example on an unstable system

Design of nonlinear controllers through the virtual reference method and regularization

This work proposes a new extension for the nonlinear formulation of the data-driven control method known as the Nonlinear Virtual Reference Feedback Tuning. When the process to be controlled contains a significant quantity of noise, the standard Nonlinear VRFT approach – that uses the Least Squares method – yield estimates with poor statistical properties. These properties may lead the control system to undesirable closed loop performances and even instability. With the intention to improve these statistical properties and controller sparsity and hence, the system’s closed loop performance, this work proposes the use of ℓ1 regularization on the nonlinear formulation of the VRFT method. Regularization is a component that has been extensively employed and researched in the Machine Learning and System Identification communities lately. Furthermore, this technique is appropriate to reduce the variance in the estimates. A detailed analysis of the noise effect on the estimate is made for the Nonlinear VRFT method. Finally, three different regularization methods, the third one proposed in this work, are compared to the standard Nonlinear VRFT.Este trabalho propõe uma nova extensão para a formulação não linear do método de controle orientado por dados conhecido como Método da Referência Virtual Não Linear, ou Nonlinear Virtual Reference Feedback Tuning – denominado aqui somente como VRFT. Quando o processo a ser controlado contém uma quantidade significativa de ruído, a abordagem padrão do VRFT – que usa o método dos Mínimos Quadrados – fornece estimativas com propriedades estatísticas pobres. Essas propriedades podem levar o sistema de controle a desempenhos indesejáveis em malha fechada. Com a intenção de melhorar essas propriedades estatística, identificar um controlador simples em quantidade de parâmetros e melhorar o desempenho em malha fechada do sistema, este trabalho propõe o uso da regularização ℓ1 na formulação não linear do método VRFT. A regularização é uma técnica que tem sido amplamente empregada e pesquisada nas comunidades de Aprendizagem de Máquina e Identificação de Sistemas ultimamente. Além disso, esta técnica é apropriada para reduzir a variância das estimativas. Uma análise detalhada do efeito do ruído na estimativa é feita para o método VRFT não linear. Finalmente, três diferentes métodos de regularização, o terceiro proposto neste trabalho, são comparados com o VRFT

Data-driven controller tuning using the correlation approach

The essential ingredients of control design procedures include the acquisition of process knowledge and its efficient integration into the controller. In many practical control applications, a reliable mathematical description of the plant is difficult or impossible to obtain, and the controller has to be designed on the basis of measurements. This thesis proposes a new datadriven method labeled Correlation-based Tuning (CbT). The underlying idea is inspired by the well-known correlation approach in system identification. The controller parameters are tuned iteratively either to decorrelate the closed-loop output error between designed and achieved closed-loop systems with the external reference signal (decorrelation procedure) or to reduce this correlation (correlation reduction). Ideally, the resulting closedloop output error contains only the contribution of the noise and perfect model-following can be achieved. By the very nature of the control design criterion, the controller parameters are asymptotically insensitive to noise. Both theoretical and implementation aspects of CbT are treated. For the decorrelation procedure, a correlation equation is solved using the stochastic approximation method. The iterative procedure converges to the solution of the correlation equation even in the case when an approximate gradient of the closed-loop output error with respect to controller The asymptotic distribution of the resulting controller parameter estimates is analyzed. When perfect decorrelation is not possible, the correlation reduction method can be used. That is, instead of solving the correlation equation, the norm of a cross-correlation function is minimized. A frequency domain analysis of the criterion shows that the algorithm minimizes the two-norm of the difference between the achieved and designed closed-loop systems.With the correlation reduction method, an unbiased estimate of the gradient of the closed-loop output error is necessary to guarantee convergence of the algorithm to a local minimum of the criterion. Furthermore, this criterion can be generalized to allow handling the mixed sensitivity specifications. An extension of this method for the tuning of linear time-invariant multivariable controllers is proposed for both procedures. CbT allows tuning some of the elements of the controller transfer function matrix to satisfy the desired closed-loop performance, while the other elements are tuned to mutually decouple the closed-loop outputs. The tuning of all decouplers and controllers can be made by performing only one experiment per iteration regardless of the number of inputs and outputs since all reference signals can be excited simultaneously. However, due to the fact that decoupling is imposed as a design criterion, simultaneous excitation of all references brings a negative impact on the variance of the estimated controller parameters. In fact, one must choose between low experimental cost (simultaneous excitation) and better accuracy of the estimated parameters (sequential excitation). The CbT algorithm has been tested on numerous simulation examples and implemented experimentally on a magnetic suspension system and the active suspension system benchmark problem proposed for a special issue of European Journal of Control on the design and optimization of restricted-complexity controllers

Fixed-Order Robust Controller Design by Convex Optimization Using Spectral Models

This thesis proposes a new method to design fixed-order controllers in frequency domain using convex optimization. The method is based on the shaping of open-loop transfer function in the Nyquist diagram with infinity norm constraints on weighted closed-loop transfer functions. A parametric model is not required in this method as it directly uses frequency-domain data. Furthermore, systems with multi-model uncertainty as well as systems with frequency-domain uncertainties can be considered. Fixed-order linearly parameterized controllers are designed with the proposed method for single-input single-output (SISO) linear time-invariant plants. The shaping of the open-loop transfer function is performed based on the minimization of the difference with a desired open-loop transfer function under H∞ constraints on the closed-loop sensitivity functions. Since these constraints represent a nonconvex set in the space of the controller parameters, an inner convex approximation of this set is proposed using the desired open-loop transfer function. This approximation makes the problem of robust fixed-order controller design a convex optimization problem. An extension of the method is proposed to design two-degree-of-freedom (2DOF) controllers for SISO plants. The method is also extended to tune fixed-order linearly parameterized multivariable controllers for multiple-input multiple-output (MIMO) linear time-invariant plants where the stability of the closed-loop system is guaranteed using Gershgorin bands. The control problem is solved only using a finite number of frequency-domain samples. However, the stability and performance conditions between frequency samples are also verified if a frequency-domain uncertainty is considered. It is shown that this adds some conservatism to the solution. The proposed frequency-domain method has been tested on many simulation examples. The method has been applied to a flexible transmission benchmark for robust controller design giving extremely good results. Additionally, the method has also been implemented on an experimental high-precision double-axis positioning system. These results show the effectiveness of the proposed methods

Non-Iterative Data-Driven Model Reference Control

In model reference control, the objective is to design a controller such that the closed-loop system resembles a reference model. In the standard model-based solution, a plant model replaces the unknown plant in the design phase. The norm of the error between the controlled plant model and the reference model is minimized. The order of the resulting controller depends on the order of the plant model. Furthermore, since the plant model is not exact, the achieved closed-loop performance is limited by the quality of the model. In recent years, several data-driven techniques have been proposed as an alternative to this model-based approach. In these approaches, the order of the controller can be fixed. Since no model is used, the problem of undermodeling is avoided. However, closed-loop stability cannot, in general, be guaranteed. Furthermore, these techniques are sensitive to measurement noise. This thesis treats non-iterative data-driven controller tuning. This controller tuning approach leads to an identification problem where the input is affected by noise, and not the output as in standard identification problems. A straightforward data-driven tuning scheme is proposed, and the correlation approach is used to deal with measurement noise. For linearly parameterized controllers, this leads to a convex optimization problem. The accuracy of the correlation approach is compared to that of several solutions proposed in the literature. It is shown that, if the order of the controller is fixed, both the correlation approach and a specific errors-in-variables approach can be used. The model reference controller-tuning problem is extended with a constraint that ensures closed-loop stability. This constraint is derived from stability conditions based on the small-gain theorem. For linearly parameterized controllers, the resulting optimization problem is convex. The proposed constraint for stability is conservative. As an alternative, a non-conservative a posteriori stability test is developed based on similar stability conditions. The proposed methods are applied to several numerical and experimental examples

Data-driven methods for tracking improvement

The tracking precision required by modern industrial applications is continuously increasing. Feedback control alone is often no longer capable of giving the necessary tracking accuracy and so the use of two-degree-of-freedom controllers, which include a feedforward term, has become commonplace. Traditionally the feedforward term is a filter based on the inverse of an identified model of the system. It is, however, not possible to obtain very high precision tracking with this approach because the identified model will always suffer from model uncertainty. In this thesis, data-driven methods are investigated. These methods derive the feedforward control directly from measured data and thus avoid the system identification step, which is where the model uncertainty is introduced. They are, therefore, capable of producing higher precision tracking than the traditional methods. For the general tracking problem, a precompensator controller is considered as the feedforward term. This controller filters the desired output signal before it is applied as an input to the system. The precompensator's parameters are tuned directly using measured data. These data are affected by stochastic disturbances, such as measurement noise. The effect of these disturbances on the calculated parameters is studied and the correlation approach is used to reduce it. For the specific problem where the tracking task is repetitive, a situation frequently encountered in industrial applications, Iterative Learning Control is proposed. Iterative Learning Control uses measurements from previous repetitions to adjust the system's input for the current repetition in a manner that improves the tracking. As measurements are used, the calculated input is sensitive to the stochastic disturbances. The effect of these disturbances on the learning procedure is examined and algorithms, which are less sensitive to their presence, are developed. Extensions of the methods are also made for linear parameter varying systems in which the system's dynamics change as a function of a scheduling parameter. The developed methods are successfully applied to an industrial linear motor positioning system

Extensão de método baseado em dados para projeto de controladores através da abordagem de correlação para rejeição de perturbação na entrada de controle

Métodos diretos de controle baseado em dados ainda estão crescendo em popularidade mesmo mais de duas décadas após terem sido introduzidos. Esses métodos usam dados coletados do processo para identificar os parâmetros de um controlador ótimo usando bem pouca informação sobre o próprio processo, sendo este o seu principal diferencial em relação ao controle baseado em modelo. A literatura mostra que muitos casos podem se beneficiar dessas características, principalmente quando o processo é complexo ou difícil de modelar. Porém, a literatura cobre mais o problema de seguimento de referência, enquanto que há evidência de que muitos problemas encontrados na vida real são de rejeição ou atenuação de distúrbios. Ademais, a maior parte dos trabalhos lida com controladores parametrizados linearmente, o que equivale a fixar os polos da função de transferência do controlador. Embora a identificação dos polos seja possível, como indicado por alguns trabalhos, houve pouco esforço para apresentar uma solução baseada em dados para esse problema. Por conta disso, o presente trabalho propõe uma abordagem baseada em dados capaz de ajustar os parâmetros de controladores monovariáveis com parametrização do denominador e também os parâmetros de controladores multivariáveis com vistas à rejeição de distúrbios de carga. Em particular, essa abordagem combina a abordagem de correlação com o erro de predição de um modelo do controlador ótimo obtido dos sinais virtuais propostos por outro método baseado em dados com o mesmo objetivo. Ou seja, desejase que a resposta a perturbação do sistema seja similar à resposta de um determinado modelo de referência. Entretanto, o ajuste pelo modelo de referência pode levar a um baixo desempenho ou mesmo instabilidade quando este é muito distante do que pode ser atingido com a estrutura de controle disponível. Um meio de lidar com esse problema é utilizar um modelo de referência flexível, i.e. identificar o melhor modelo de referência juntamente com o controlador. Como isso não é suficiente para garantir a estabilidade, uma técnica de certificação baseada na condição de Vinnicombe também foi proposta para o caso de controladores monovariáveis. Por fim, o método de síntese proposto foi comparado ao outro método da literatura para rejeição de distúrbios através de um exemplo onde a abordagem de correlação mostrou-se mais imune ao ruído do que a abordagem de mínimos quadrados e variáveis instrumentais da literatura. Além disso, o método proposto também foi avaliado através de alguns estudos de caso e apresentou resultados satisfatórios. Já o algoritmo de certificação foi comparado com outros dois métodos de certificação baseados em dados e apresentou vantagens como baixa complexidade em relação a um e menor conservadorismo em relação ao outro.Data-driven direct methods are still growing in popularity more than two decades after they were introduced. These methods use data collected from the process to identify an optimal controller’s parameters with little knowledge about the process itself, and this is the main characteristic that sets them apart from model-based control. The literature shows that many cases may benefit from this characteristic, mainly when the process is complex and difficult to be modelled. However, the literature covers more the reference tracking problem, whereas there is evidence that many of the problems faced in real-life are of disturbance rejection or attenuation. Also, the vastly majority of those works deals with linearly parametrized controllers, which amounts to fixing the poles of the controller’s transfer function. Although the identification of the controller’s poles is not prohibitive, as hinted by some of the papers, there is little effort on presenting a data-driven solution capable of doing so. With all that in mind, this work proposes a data-driven approach which is able to adjust the parameters of monovariable controllers with parameters in the denominator and the parameters of multivariable controllers aiming at the load disturbance rejection. In particular, this approach combines the correlation approach with the prediction error of some model of the optimal controller obtained from the virtual signals proposed by another data-driven method with the same goal. That is, the goal is that the closed loop response be made as close as possible to some determined reference model’s response. However, employing a reference model leads to poor performance and even instability when the reference model is much different from what may be achieved with the controller structure available. One way to deal with that problem is by using a flexible reference model, i.e. identify the best reference model along with the controller. Because that is not enough to guarantee the closed loop stability, a controller certification technique based on the Vinnicombe’s condition is also proposed for the monovariable case. Finally, the proposed synthesis method was compared against the other literature method for load disturbance rejection through an example, where the correlation approach presented itself more imune to noise than the least squares and the instrumental variable approach from the literature. Besides, the proposed method was also evaluated through some case studies and presented satisfactory results. The proposed algorithm for certification was compared with two other data-driven certification methods from the literature and presented advantages such as low complexity against one of them and less conservatism against the other

Correlation-based tuning of decoupling multivariable controllers

The iterative method labelled correlation-based tuning (CbT) is considered in this paper for tuning linear time-invariant multivariable controllers. The approach allows one to tune some elements of the controller transfer function matrix to satisfy the desired closed-loop performance. while the other elements are tuned to mutually decouple the closed-loop outputs. Decoupling is achieved by decorrelating a given reference with the non-corresponding outputs. The controller parameters are calculated either by solving a correlation equation (decorrelation procedure) or by minimizing a cross-correlation function (correlation reduction). In addition, the preferred way of exciting a 2 x 2 system for CbT is investigated via the accuracy of the estimated controller parameters. It is shown that simultaneous excitation of both reference signals does not improve the accuracy of the estimated controller parameters compared to the case of sequential excitation. In fact, one must choose between low experimental cost (simultaneous excitation) and better accuracy of the estimated parameters (sequential excitation). The theoretical results are illustrated via three simulation Studies. (C) 2007 Elsevier Ltd. All rights reserved