Håkan Hjalmarsson

Nonlinear black-box modeling in system identification: a unified overview

A nonlinear black-box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics. There has been considerable recent interest in this area, with structures based on neural networks, radial basis networks, wavelet networks and hinging hyperplanes, as well as wavelet-transform-based methods and models based on fuzzy sets and fuzzy rules. This paper describes all these approaches in a common framework, from a users perspective. It focuses on what are the common features in the different approaches, the choices that have to be made and what considerations are relevant for a successful system-identification application of these techniques. It is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space. These mappings are discussed separately. The latter mapping is usually formed as a basis function expansion. The basis functions are typically formed from one simple scalar function, which is modified in terms of scale and location. The expansion from the scalar argument to the regressor space is achieved by a radial- or a ridge-type approach. Basic techniques for estimating the parameters in the structures are criterion minimization, as well as two-step procedures, where first the relevant basis functions are determined, using data, and then a linear least-squares step to determine the coordinates of the function approximation. A particular problem is to deal with the large number of potentially necessary parameters. This is handled by making the number of ‘used’ parameters considerably less than the number of ‘offered’ parameters, by regularization, shrinking, pruning or regressor selection.

Iterative feedback tuning: theory and applications

We have examined an optimization approach to iterative control design. The important ingredient is that the gradient of the design criterion is computed from measured closed loop data. The approach is thus not model-based. The scheme converges to a stationary point of the design criterion under the assumption of boundedness of the signals in the loop. From a practical viewpoint, the scheme offers several advantages. It is straightforward to apply. It is possible to control the rate of change of the controller in each iteration. The objective can be manipulated between iterations in order to tighten or loosen performance requirements. Certain frequency regions can be emphasized if desired. This direct optimal tuning algorithm is particularly well suited for the tuning of the basic control loops in the process industry, which are typically PID loops. These primary loops are often very badly tuned, making the application of more advanced (for example, multivariable) techniques rather useless. A first requirement in the successful application of advanced control techniques is that the primary loops be tuned properly. This new technique appears to be a very practical way of doing this, with an almost automatic procedure.

From experiment design to closed-loop control

The links between identification and control are examined. The main trends in this research area are summarized, with particular focus on the design of low complexity controllers from a statistical perspective. It is argued that a guiding principle should be to model as well as possible before any model or controller simplifications are made as this ensures the best statistical accuracy. This does not necessarily mean that a full-order model always is necessary as well designed experiments allow for restricted complexity models to be near-optimal. Experiment design can therefore be seen as the key to successful applications. For this reason, particular attention is given to the interaction between experimental constraints and performance specifications.

Nonlinear black-box models in system identification: mathematical foundations

We discuss several aspects of the mathematical foundations of the nonlinear black-box identification problem. We shall see that the quality of the identification procedure is always a result of a certain trade-off between the expressive power of the model we try to identify (the larger the number of parameters used to describe the model, the more flexible is the approximation), and the stochastic error (which is proportional to the number of parameters). A consequence of this trade-off is the simple fact that a good approximation technique can be the basis of a good identification algorithm. From this point of view, we consider different approximation methods, and pay special attention to spatially adaptive approximants. We introduce wavelet and ‘neuron’ approximations, and show that they are spatially adaptive. Then we apply the acquired approximation experience to estimation problems. Finally, we consider some implications of these theoretical developments for the practically implemented versions of the ‘spatially adaptive’ algorithms.

A convergent iterative restricted complexity control design scheme

In this contribution we propose an optimization approach to the design of a restricted complexity controller. The design criterion is of LQG type containing two terms. The first term is the quadratic norm of the error between the output of the true closed loop and a desired response. The second term is the quadratic norm of the input signal. It is shown that the minimization of this criterion does not require a model of the system. Closed loop experimental data can be used instead. The result is an iterative scheme of closed loop experiments and controller updates which converges to a local minimum of the design criterion under the condition of bounded signals.<<ETX>>

Neural Networks in System Identification

Neural Networks are non-linear black-box model structures, to be used with conventional parameter estimation methods. They have good general approximation capabilities for reasonable non-linear systems. When estimating the parameters in these structures, there is also good adaptability to concentrate on those parameters that have the most importance for the particular data set.

System identification of complex and structured systems

A key issue in system identification is how to cope with high system complexity. In this contribution we stress the importance of taking the application into account in order to cope with this issue. We define the concept “cost of complexity” which is a measure of the minimum required experimental effort (e.g. used input energy) as a function of the system complexity, the noise properties, and the amount, and desired quality, of the system information to be extracted from the data. This measure gives the user a handle on the trade-offs that must be considered when performing identification with a fixed experimental “budget”. Our analysis is based on the observation that the identification objective is to guarantee that the estimated model ends up within a pre-specified “level set” of the application objective. This geometric notion leads to a number of useful insights: Experiments should reveal system properties important for the application but may also conceal irrelevant properties. The latter, dual, objective can be explored to simplify model structure selection and model error assessment issues. We also discuss practical issues related to computation and implementation of optimal experiment designs. Finally, we illustrate some fundamental limitations that arise in identification of structured systems. This topic has bearings on identification in networked and decentralized systems.

The fundamental role of general orthonormal bases in system identification

The purpose of this paper is threefold. Firstly, it is to establish that contrary to what might be expected, the accuracy of well-known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure. Secondly, it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any fixed poles or zeros. This is achieved by employing certain new results on generalized Fourier series and the asymptotic properties of Toeplitz-like matrices in such a way that the new variance expressions presented here encompass pre-existing ones as special cases. Via this latter analysis a new perspective emerges on recent work pertaining to the use of orthonormal basis structures in system identification. Namely, that orthonormal bases are much more than an implementational option offering improved numerical properties. In fact, they are an intrinsic part of estimation since, as shown here, orthonormal bases quantify the asymptotic variability of the estimates whether or not they are actually employed in calculating them.

Efficient tuning of linear multivariable controllers using iterative feedback tuning

Iterative feedback tuning is a direct tuning method using closed-loop experimental data. The method is based on numerical optimization and in each iteration an unbiased gradient estimate is used. Due to these unbiased gradient estimates, the method converges to a stationary point of the control criterion provided the closed loop signals remain bounded throughout the iterations. In this contribution, it is shown how such unbiased estimates can be obtained for multivariable linear time-invariant systems. Particular attention is given to the issue of keeping the experiment time to a minimum and several efficient algorithms are presented. It is shown that, for tuning an arbitrary linear time-invariant multivariable controller with nw inputs and nu outputs, 1+nuA—nw experiments are sufficient in each iteration of the algorithm. For disturbance rejection, an alternative algorithm is proposed which requires nu+nw experiments. As an illustration, the method is applied to a simulation model of a gas turbine engine.

Brief Relay auto-tuning of PID controllers using iterative feedback tuning

In this paper, ideas from iterative feedback tuning (IFT) are incorporated into relay auto-tuning of the proportional-plus-integral-plus-derivative (PID) controller. The PID controller is auto-tuned to give specified phase margin and bandwidth. Good tuning performance according to the specified bandwidth and phase margin can be obtained and the limitation of the standard relay auto-tuning technique using a version of Ziegler-Nichols formula can be eliminated. Furthermore, by using common modelling assumptions for the relay system, some of the required derivatives in the IFT algorithm can be derived analytically. The algorithm was tested in the laboratory on a coupled tank and good tuning result was demonstrated.