Data-Driven Controller Design

In this introductory chapter, a conceptual presentation of the data-driven control design problem is given, along with a delimitation of the class of systems treated—namely, linear time invariant, discrete-time, single-input-single-output systems. General definitions and notation are also defined here.

The formal statement of the

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design problem is given in Chap. 2 along with its basic properties. The different

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control objectives are presented: minimum variance control, model reference control, economy of control effort and their combinations. A central theoretical concept is introduced: the ideal controller regarding each control objective. The resulting similarities with the theory of system identification are discussed. Then the designer’s choices in choosing her/his performance criterion are examined under the light of these concepts, deriving guidelines for these choices.

Once the designer has chosen the performance criterion, it must be minimized, which in a data-driven control design will be done using only input-output data collected from the system. It is possible in many situations to perform this minimization in only “one-shot”, that is, with only one batch of data collected in only one operating condition. These “one-shot” solutions, which are the most convenient, are the subject of Chap. 3. The

virtual reference feedback tuning

method (VRFT, for short) is presented and its statistical properties—consistency, variance—are demonstrated. Its extension to nonminimum phase processes is also presented. A number of simulation studies illustrate the properties of VRFT.

In many practical cases the theoretical conditions required by the “one-shot” solutions are not met. Moreover, operational constraints of the process often require that only small adjustments to existing parameter settings can be made, which precludes the application of VRFT in these situations. In such cases the data-driven control design must be performed through iterative procedures in which each iteration requires collecting more data, each time with a different controller in the loop. In Chap. 4 a general review of basic optimization theory is given, setting the stage for the chapters to follow. The basic convergence properties of the basic optimization algorithms—steepest descent and Newton-Raphson in particular—are analyzed. Some robustness properties, that is, convergence under imprecise information, of these algorithms are also demonstrated.

Starting in Chap. 5, the particularities of the

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cost functions that are minimized in data-driven control design are explored. Convergence to the globally optimal controller is sought in a data-driven control design, so the properties of the cost function are analyzed with respect to this goal. A number of properties of these particular cost functions and of some basic optimization algorithms when applied to them are presented in this chapter. From these properties, guidelines for the optimization are provided, involving the choice of the optimization algorithm and the automatic selection of step sizes. Step size policies that speed up the convergence to the global minimum are presented, and a general choice of optimization algorithm is advocated: start with the steepest descent then switch to the Newton-Raphson method when sufficiently close to the global optimum. A case study is presented to illustrate the convergence properties of the various algorithmic choices. The theoretical results also set the stage for the synthetic procedures to be presented in Chap.

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Chapter 6 deals with the

cost function shaping

concept. Cost function shaping is the name that we have given to a set of procedures and maneuvers that change the cost function so that it is more amenable to optimization, yet maintaining its original design objective. Cost function shaping involves mainly choosing the input signal in the experiments (aka input design) and changing stepwise the reference model (aka cautious control). Proper data windowing can also be very helpful in shaping the cost function so that it becomes “well-behaved” as desired. Each one of these procedures and maneuvers is presented, and theoretically justified, in this chapter. It is shown that it is always possible to reshape the cost function so that there are no local minima in the whole parameter space. This implies that convergence to the global minimum is “easily” obtained starting from any stabilizing controller. Several simulation examples are given along this chapter to illustrate the concepts and to show how to perform cost function shaping in real control design.

Performing the optimization requires calculation of the cost function’s derivative(s), and this must be done only with the data collected from the system—there is not an analytical expression of the cost function available. Each particular data-driven “brand” (like IFT, FDT and CbT) is characterized by a particular way of estimating the function’s derivatives from data. This computing aspect, namely the calculation of these quantities, is the subject of Chap. 7. Three different methods are described in some detail and interpreted under the light of the theory presented in the previous chapters: IFT, FDT and CbT.

In Chap. 8 the practical application of data-driven control design with the tools and concepts presented along this book is explored. Numerous experimental results showing the data-driven design of controllers for processes of different natures are presented: a thermal process, a motor speed control process and a liquid flow process. For each process various different control designs are performed, with different control objectives. In each design the problem is approached under the light of the theory presented in the previous chapters of this book. The designer’s tools provided in this book, regarding the choice of the cost function, the selection of the optimization algorithm and its parameters, cost function shaping, etc are all explored in detail. These designs illustrate how the theory translates into the real world, showing what are the designer’s choices, and how he/she should make these choices taking into account the theoretical concepts presented in this book in order to obtain the best result from a real data-driven control design.