Prof. Dr. Diego Eckhard

The Prediction Error Method is related to a non-convex optimization problem. It is usual to apply iterative algorithms to solve this optimization problem. However, iterative algorithms can get stuck at a local minimum of the cost function or converge to the border of the searching space. An analysis of the cost function and sufficient conditions to ensure the convergence of the iterative algorithms to the global minimum are presented in this work. It is observed that this conditions depend on the spectrum of the input signal used in the experiment. This work presents tools to improve the convergence of the algorithms to the global minimum, which are based on the manipulation of the input spectrum.

Data-based control design methods consist of adjusting the parameters of the controller directly from batches of input-output data of the process; no process model is used. The adjustment is done by solving an optimization problem, which searches the argument that minimizes a specific cost function. Iterative algorithms based on the gradient are applied to solve the optimization problem, like the steepest descent algorithm, Newton algorithm and some variations. The only information utilized for the steepest descent algorithm is the gradient of the cost function, while the others need more information like the hessian. Longer and more complex experiments are used to obtain more informations, that turns the application more complicated. For this reason, the steepest descent method was chosen to be studied in this work. The convergence of the steepest descent algorithm to the global minimum is not fully studied in the literature. This convergence depends on the initial conditions of the algorithm and on the step size. The initial conditions must be inside a specific domain of attraction, and how to enlarge this domain is treated by the methodology Cost Function Shaping. The main contribution of this work is a method to compute efficiently the step size, to ensure convergence to the global minimum. Some informations about the process are utilized, and this work presents how to estimate these informations. Simulations and experiments demonstrate how the methods work.

This work presents a comparison among three different control strategies for multivariable processes. The techniques were implemented in a pilot plant with coupled control loops, where all steps used to design the controllers were described allowing to establish a trade-off between algorithm complexity, information needed from the process and achieved performance. Two data-driven control techniques are used: multivariable ultimate point method to design a decentralized PID controller and virtual reference feedback tuning to design a centralized PID controller. A mathematical model of the process is obtained and used to design a model-based generalized predictive controller. Experimental results allow us to evaluate the performance achieved for each method, as well as to infer on their advantages and disadvantages.

Abstract This paper deals with Data-Driven (DD) control design in a Model Reference (MR) framework. We present a new \{DD\} method for tuning the parameters of a controller with a fixed structure. Because the method originates from embedding the control design problem in the Prediction Error identification of an optimal controller, it is baptized as Optimal Controller Identification (OCI). Incorporating different levels of prior information about the optimal controller leads to different design choices, which allows to shape the bias and variance errors in its estimation. It is shown that the limit case where all available prior information is incorporated is tantamount to model-based design. Thus, this methodology also provides a framework in which model-based design and \{DD\} design can be fairly and objectively compared. This comparison reveals that \{DD\} design essentially outperforms model-based design by providing better bias shaping, except in the full order controller case, in which there is no bias and model-based design provides smaller variance. The practical effectiveness of the design methodology is illustrated with experimental results.

Abstract Identification of an output error model using the prediction error method leads to an optimization problem built on input/output data collected from the system to be identified. It is often hard to find the global solution of this optimization problem because in most cases both the corresponding objective function and the search space are nonconvex. The difficulty in solving the optimization problem depends mainly on the experimental conditions, more specifically on the spectra of the input/output data collected from the system. It is therefore possible to improve the convergence of the algorithms by properly choosing the data prefilters; in this paper we show how to perform this choice. We present the application of the proposed approach to case studies where the standard algorithms tend to fail to converge to the global minimum.

Abstract Continuous process industries usually have hundreds to thousands of control loops, most of which are coupled, i.e. one control loop affects the behavior of another control loop. In order to properly design the controllers and reduce the interactions between loops it is necessary to consider the multivariable structure of the process. Usually {MIMO} (multiple-input, multiple-output) controllers are designed using {MIMO} models of the process, but obtaining these models is a task very demanding and time consuming. Virtual Reference Feedback Tuning ({VRFT}) is a data-driven technique to design controllers which do not use a model of the process; all the needed information is collected from input/output data from an experiment. The method is well established for {SISO} (single-input, single-output) systems and there are some extensions to {MIMO} process which assume that all the outputs should have the same closed-loop performance. In this paper we develop a complete framework to {MIMO} {VRFT} which provides unbiased estimates to the optimal {MIMO} controller (when it is possible) even when the closed-loop performances are distinct to each loop. When it is not possible to obtain the optimal controller because the controller class is too restrictive (for example {PID} controllers) then we propose the use of a filter to reduce the bias on the estimates. Also, when the data is corrupted by noise, the use of instrumental variables to eliminate the bias on the estimate should be considered. The article presents simulation examples and a practical experiment on a tree tank system where the goal is to control the level of two tanks.

This paper presents the identifiability analysis of a nonlinear model for a batch bioreactor and the estimation of the identifiable parameters within the prediction error framework. The output data of the experiment are the measurements of the methane gas generated by the process, during 37 days, and knowledge of the initial conditions is limited to the initial quantity of chemical oxygen demand. It is shown by the identifiability analysis that only three out of the eight model parameters can be identified with the available measurements and that identification of the remaining parameters would require further knowledge of the initial conditions. A prediction error algorithm is implemented for the estimation of the identifiable parameters. This algorithm is iterative, relies on the gradient of the prediction error, whose calculation is implemented recursively, and consists of a combination of two classic optimization methods: the conjugated gradient method and the Gauss?Newton method.

In the present work a systematic methodology for computing dynamic output stabilizing feedback control laws for nonlinear systems subject to saturating inputs is presented. In particular, the class of Lur'e type nonlinear systems is considered. Based on absolute stability tools and a modified sector condition to take into account input saturation effects, an \{LMI\} framework is proposed to design the controller. Asymptotic as well as input-to-state and input-to-output (in a L2 sense) stabilization problems are addressed both in regional (local) and global contexts. The controller structure is composed of a linear part, an anti-windup loop and a term associated to the output of the dynamic nonlinearity. Convex optimization problems are proposed to compute the controller considering different optimization criteria. A numerical example illustrates the potentialities of the methodology.

The Prediction Error Method (PEM) is related to an optimization problem built on input/output data collected from the system to be identified. It is often hard to find the global solution of this optimization problem because the corresponding objective function presents local minima and/or the search space is constrained to a nonconvex set. The shape of the cost function, and hence the difficulty in solving the optimization problem, depends directly on the experimental conditions, more specifically on the spectrum of the input/output data collected from the system. Therefore, it seems plausible to improve the convergence to the global minimum by properly choosing the spectrum of the input; in this paper, we address this problem. We present a condition for convergence to the global minimum of the cost function and propose its inclusion in the input design. We present the application of the proposed approach to case studies where the algorithms tend to get trapped in nonglobal minima.

Data-based control design methods most often consist of iterative adjustment of the controller?s parameters towards the parameter values which minimize an Formula performance criterion. Typically, batches of input-output data collected from the system are used to feed directly a gradient descent optimization algorithm ? no process model is used. Two topics are important regarding this algorithm: the convergence rate and the convergence to the global minimum. This paper discusses these issues and provides a method for choosing the step size to ensure convergence with high convergence rate, as well as a test to verify at each step whether or not the algorithm is converging to the global minimum.

Iterative data-based controller tuning consists of iterative adjustment of the controller parameters towards the parameter values which minimise an {H2} performance criterion. The convergence to the global minimum of the performance criterion depends on the initial controller parameters and on the step size of each iteration. This article presents convergence properties of iterative algorithms when they are affected by disturbances.

Model reference control design methods fail when the plant has one or more non-minimum phase zeros that are not included in the reference model, leading possibly to an unstable closed loop. This is a very serious problem for data-based control design methods, where the plant is typically unknown. In this paper, we extend the Virtual Reference Feedback Tuning method to non-minimum phase plants. This extension is based on the idea proposed in Lecchini and Gevers (2002) for Iterative Feedback Tuning. We present a simple two-step procedure that can cope with the situation where the unknown plant may or may not have non-minimum phase zeros.

This study addresses the control of linear systems subject to both sensor and actuator saturations and additive L2-bounded disturbances. Supposing that only the output of the linear plant is measurable, the synthesis of stabilising output feedback dynamic controllers, allowing to ensure the internal closed-loop stability and the finite L2-gain stabilisation, is considered. In this case, it is shown that the closed-loop system presents a nested saturation term. Therefore, based on the use of some modified sector conditions and appropriate variable changes, synthesis conditions in a quasi-linear matrix inequality (LMI) form are stated in both regional (local) as well as global stability contexts. Different LMI-based optimisation problems for computing a controller in order to maximise the disturbance tolerance, the disturbance rejection or the region of stability of the closed-loop system are proposed.

A method for computing time-varying dynamic output feedback controllers for discrete-time linear systems subject to input saturation is proposed. The method is based on a locally valid polytopic representation of the saturation term. From this representation, it is shown that, at each sampling time, the matrices of the stabilising time-varying controller can be computed from the current system output and from constant matrices obtained as a solution of some matrix inequalities. Linear matrix inequality-based optimisation problems are therefore proposed in order to compute the controller aiming at the maximisation of the basin attraction of the closed-loop system, as well as aiming at ensuring a level of {L2} disturbance tolerance and rejection.

This paper proposes the use of regularization on the multivariable formulation of the Virtual Reference Feedback Tuning (VRFT). When the process to be controlled has a significant amount of noise, the standard VRFT approach, that uses the instrumental variable technique, provides estimates with very poor statistical properties. To cope with that, this paper considers the use of regularization on the estimation procedure, reducing the covariance error at the cost of inserting a small bias. Also, this paper explains different types of regularization matrices and presents the methodology to tune these matrices. In order to demonstrate the benefits of the proposed formulation, a numerical example is presented.

Virtual Reference Feedback Tuning (VRFT) is a data-driven technique used to design controllers without the need of a process model, only input-output data is utilized. When the process has non-minimum phase (NMP) zeros, the original method usually presents poor performance, because scarcely the reference model has the same NMP zeros as the process. To overcome this problem, a flexible criterion has been proposed to the VRFT method, in a way that both the controller parameters and the NMP zeros of the process are estimated together. In this paper we present the application of the VRFT method with flexible criterion to the level control of a MIMO pilot plant. We show that a sequential controller design may incorporate NMP behavior to the process. We then use the VRFT method with flexible criterion to design the controller using only closed-loop data from the process.

This paper considers the design and implementation of a discrete-time fast tracking controller for quadrotor vehicles subject to perturbations. The proposed controller consists of a model-based disturbance observer and a Composite Nonlinear Feedback (CNF) controller. The CNF control law introduces nonlinear damping to the system so that it possesses a fast rise time without overshoot. The least square identification method is applied to develop a model based disturbance observer, thus decoupling the problems of track following and disturbance rejection. Experimental results are provided in order to validate the proposed approach.

In this work a modified Resonant Controller is proposed to deal with the tracking/rejection problem of periodic signals robust to period variations and parametric uncertainties in the plant. The control strategy is based on a resonant structure in series with a notch filter, which will be responsible to improve the robustness to period variation. A robust state feedback controller is designed by solving a linear matrix inequality (LMI) optimization problem guaranteeing the robust stability of the closed loop system. A numerical example is presented to illustrate the method.

Iterative Feedback Tuning (IFT) is a data-driven method used to tune parameters of feedback controllers minimising an H2 criterion. The method uses data from experiments to estimate the gradient of the criterion, and uses iterative quasinewton algorithms to adjust the controllers. When the method is used in cascade systems, usually the inner loop is firstly adjusted, and after the outer loop. In this article we describe an extension to the IFT method that adjusts both inner and outer loop at the same time using only data from closed-loop experiments at each iteration.

Quadcopter is a type of Unmanned Aerial Vehicle which is lifted and propelled by four rotors. The vehicle has a complex non-linear dynamic which makes the tuning of the roll and pitch controllers difficult. Usually the control design is based on a mathematical model which is strongly related to physical components of vehicle: mass, moment of inertia and aerodynamic. When a tool is attached to the vehicle, a new model must be computed to redesign the controllers. In this article we will adjust the controllers of a real experimental quadcopter using the Cascade Iterative Feedback Tuning method. The method is data-driven, so it does not uses a model for the vehicle; all it uses is input-output data collect from the closed-loop system. The method minimizes the \{H2\} error between the desired response and the actual response of the vehicle angle using the Newton-Raphson algorithm. The method achieves the desired performance without the need of the vehicle model, with low cost and low complexity.

Wireless networks are gaining space in industrial environments due to the low installation costs and low maintenance. Robustness is also one of the main requirements for these systems to be adopted, and, in this context, WirelessHART (WH), ISA SP100.11a, and WIA-PA protocols met these characteristics. In order to provide low maintenance, these protocols must provide reliable radio links while keeping low power consumption to allow battery powered devices. Unfortunately, the standards of these protocols do not impose any RF power modulation technique, which is a form to increase even more the battery endurance of a wireless field device. Instead, RF power levels are fixed and selected by commissioning, and must allow the longest link per device. In this case, devices in closer ranges waste energy during transmissions, as they could save energy by modulating the RF power. This paper presents a RF power modulation technique that employs a proportional-integral controller and allows reduction of energy consumption while keeping the robustness of RF links. A proof of concept of the power modulation technique is implemented and verified showing good results and proving that the proposed controller is feasible. The proposal has the advantage to be fully compatible with the standard.