LEARN - Project Funded by an Advanced Grant from the European Research Council

The Project

LEARN is an acrynom for

Limitations, Estimation, Adaptivity, Reinforcement and Networks in System Identification

It is a five-year project that started January 1, 2011 and will be finished December 31, 2015 and it is supported by a grant of 2.5 million Euros from the European Research Council, ERC. The project is carried out by a joint team from Division of Automatic control at Linköping Univerisity (LiU) and from the Division of Automatic control at the Royal Institute of Technology in Stockholm (KTH).

An outline of ideas in LEARN, along with some results from the first year is given in

EJC paper: Lennart Ljung, Håkan Hjalmarsson, and Henrik Ohlsson: Four encounters with system identification. European Journal of Control, 2011, Nr 5-6, pp 449-471.

The Principal Investigator is Lennart Ljung, LiU, and Håkan Hjalmarsson, KTH, is co-PI. The project is organized into five themes:

Theme I: Encounters with Convex Programming Techniques
A convex optimization problem has no local minima other than the global one, and can be solved by efficient algorithms. It is therefore very useful to formulate any task as a convex problem.
Theme II: Fundamental Limitations
The Cramer-Rao lower limit for variance of unbiased estimates represents a fundamental limitation to what can be achieved by estimation and identification. It is an important problem to fully investigate the consequences of this.
Theme III: Experiment Design and Reinforcement Techniques
Careful design of experiments may lead to essential improvement of the model accuracy. At the same time it is important to take the experinment cost (in a wide sense) into account.
Theme IV: Potentials of Non-parametric Models of Dynamical Systems
The dominating approach in ``main stream'' system identification is to use parametric models and methods. An important reason for that is that control design methods are predominantly now model based and analytical. Also parametric models and methods have a strong statistical historical background ever since Gauss. The non-parametric approaches in system identification have mostly been confined to spectral analysis techniques for estimating frequency functions.
Theme V: Managing Structural Constraints
The increasing use of physical modeling of integrated and complex systems in e.g.the automotive, aerospace and process industry, results in highly structured models. Another source of motivation for this research theme stems from the many emerging networked and decentralized engineering applications, e.g. networked control systems and wireless sensor networks.

Theme I: Encounters with Convex Programming Techniques

State of the art

The development of convex and semidefinite programming has been booming in recent years, and has played a major role in several research communities. Convexification of estimation problems has been a very visible theme in the statistics community. However, such activities have not been particularly pronounced in the System Identification community which has largely been sticking to a maximum likelihood (or related) framework. It is perhaps symptomatic that some very recent and interesting applications of semidefinite programming techniques to system identification, have their origins in optimization rather than identification research groups. To be fair, it must be said that also research on subspace identification methods and attempts to work with predictors that are linear in the parameters, like LS Support Vector Machines and kernel-like techniques, could be seen as a convexification trend. There is thus a clear link to the research Theme IV. Another area that belongs to the state-of-the-art in this context is model reduction. Model reduction is closely related to System Identification, by its inherent system approximation feature. It is therefore interesting to follow convexification attempts for model reduction problems, and see if they have implications on the identification problem.

Objectives

The goal is to understand the potential of convexification and to utilize modern convex programming techniques for system identification problems. This may lead to more efficient estimation algorithms, getting grips with the constant peril of getting caught in local minima for grey-box models and non-linear black-box models. The links to machine learning and Theme IV are also important to understand

Results

Convexity Issues in System Identification.
A survey of convexity issues in system identification was presented as a plenary presentation at the ICCA conference in Hangzhou in June 2013: The paper and the slides.
What can regularization offer for estimation of dynamical systems?
A summary of the possibilties and usefulness of regularization in system identification was presented in a plenary at ALCOSP 2013 in Caen, France: The paper and the slides.

 

 

Boxplot

Theme II: Fundamental Limitations

State of the art

Using fundamental system limitations to establish benchmarks regarding what can be realistically achieved with given resources is an often used concept in engineering. The importance of understanding limitations in engineering systems imposed by data based modeling is accentuated as complexity and structural constraints grow. To this end the Cramér-Rao lower bound (CRLB) is fundamental. It translates into performance bounds for model based applications employing identified models, e.g. it implies a lower bound on the regulatory precision in control applications.

Objectives

While the basic expression for the CRLB is well known, understanding how it depends on system and model complexity as well as experimental conditions during the data collection has been subject to rather intense research for a range of problem settings. Recently it has been recognized that the variance of estimated frequency functions is subject to a water-bed effect, reminiscent to Bode's sensitivity integral.

Results

Convergence of the prediction error method: Existence of local minima of the cost function can severely hamper the efficiency of the prediction error method. In the contribution below we explore the fact that the input can be used to shape the cost function so that this problem is alleviated.

D. Eckhard and A.S. Bazanella and C.R. Rojas and H. Hjalmarsson Input design as a tool to improve the convergence of PEM Automatica Vol 49, pp 3282-3291.

Theme III: Experiment Design and Reinforcement Techniques

State of the art

The limitations discussed in Theme II) above imply that it is impossible to identify a complex system accurately with only a modest amount of (noisy) sensor information. The key to circumvent this curse of complexity is the observation that an application often requires only a modest amount of system properties to be accurately modelled.

Objectives

Thus carefully designed, but still ``inexpensive'', experiments may provide sufficient information for the intended application. Consider for example the very simple case of optimizing the yield of a product with respect to one parameter. The model of the yield can be very poor far away from the optimum (as long as it does not predict better yield than the real optimum); it is only close to the optimum that the modeling accuracy becomes important. It is immediate from this example that obtaining such models requires access to actuation abilities: Measurements should be concentrated to the vicinity of the optimal parameter value. There are two well-known issues that hamper the implementation of this objective:
(i) Computational restrictions:
Many optimal experiment design problems, e.g. those involving non-linear dynamical systems, correspond to highly non-convex optimization problems.
(ii) The chicken and egg problem:
Good experiments typically depend on the to be identified system itself. In the example above, the optimum, where the measurements should be taken, is unknown.
Convexification has proved to be a viable route to cope with Issue i) for linear dynamical systems. Corresponding methods for non-linear dynamical systems are still lacking.

Results

Adaptive experiment design:

We have shown that, for a general family of model structures for linear systems, adaptive experiment design asymptotically has the same accuracy as if complete knowledge of the true system was used in the experiment design.

Paper: http://arxiv.org/abs/1310.3973

Experiment design for biased models Most experiment design methods assume that the true system is in the model set. It is well known that the total error may be reduced by using simpler models, trading variance for bias. In the contribution below we study experiment design for such models.

D. Eckhard, H. Hjalmarsson, C.R. Rojas and M. Gevers Mean-squared error experiment design for linear regression models 16th IFAC Symposium on System Identification Brussels, Belgium, pp 1629-1634, 2012.

Experiment design for non-linear systems A graph theoretic approach is taken to optimal experiment design for non-linear systems in the contribution below.

P.E. Valenzuela, C.R. Rojas and H. Hjalmarsson Optimal input design for non-linear dynamic systems: a graph theory approach Proceedings 51st IEEE Conference on Decision and Control, Florence, Italy, 2013.

Theme IV: Potentials of Non-parametric Models of Dynamical Systems

State of the art

Various non-parametric methods have been important tools in statistics since long. Nearest neighbour, kernel, and local approximation techniques are successfully used to estimate surfaces in regressor spaces. We have ourselves experience in developing, analysing and testing such local approaches (Direct Weight Optimisation, DWO). Also Support Vector Machines, SVM,can be understood as such kernel methods, although the formal treatment is via parameter estimation. The convergence of Machine Learning towards Statistical Learning (or the other way around) has stressed the role of kernel approximations. Gaussian Process regression (originally conceived in the 1950's) for example has become a widely used tool for function approximation in Machine Learning also for applications to dynamical systems. Some very recent contributions, have shown that conventional parametric methods can be successfully combined with learning techniques even for estimating standard linear models. The concept and use of manifold learning techniques is related to this area. These are methods to identify areas (manifolds) in the regressor space that are of special interest for a given application. By focusing on the system's behaviour on that manifold, simpler and more effective models can be constructed.

Objectives

The goal of the research in this area is to provide methodology and algorithms for estimating complex (nonlinear) dynamical systems in reliable and effective manners. We strongly feel that non-parametric methods are underutilised in the system identification research community and that powerful results can be obtained by adjoining manifold learning and machine learning strategies to conventional identification methods.

Results

Encounters with System Identification:
In the plenary at the 50th CDC/ECC in Orlando, the potentials of Gaussian processes and Bayesian thinking for system identification were outlined: The presentation, the paper and the video recording.
On the estimation of transfer functions:
An extensive analysis of the use of Gaussian Processes techniques for system identification has been presented in Automatica paper: T. Chen, H. Ohlsson and L. Ljung On the estimation of transfer functions, regulariztions amd Gaussian processes - Revisited Automatica, Vol 48, pp 1525-1535, 2012.

Theme V: Managing Structural Constraints

State of the art

Recently identifiability of systems composed of cascade, feed-forward, feedback and multiplicative connections of linear dynamic and zero memory nonlinear elements have been studied. In even more recent work it ha discussed how to include structural information in subspace identification of ARMAX models with unequal polynomial orders. It has been shown quite surprisingly, there are certain configurations of cascade systems where some of the sensors do not contribute with information regarding system blocks further ``upstreams''. Distributed signal processing very much targets the emerging area of wireless sensor networks (WSN). There is also a flood of results concerning networked control but there exists few results on identification of distributed and networked systems with control as application.

Objectives

The overall objective is to develop structure preserving identification methods suitable for networked and decentralized systems. This includes experiment design, identification (e.g. subspace) and model reduction methods that respect structural constraints. It also includes design methods for both communication strategies (between nodes) and sensor/actuator placement in decentralized identification with the objective to maximize modeling accuracy. This requires understanding how structural information and decentralized data processing influence a model's accuracy.

Results

Rational model structures

Parameter estimation in standard model structures, such as Box-Jenkins, can be seen as impulse response estimation with a rank constraint. By a convex relaxation of the constraint a convex problem can be obtained. In the contribution below, the nuclear norm is used as surrogate for the rank constraint.

H. Hjalmarsson, C.R. Rojas and J. Welsh Identification of Box-Jenkins models using structured {ARX} models and nuclear norm relaxation 16th IFAC Symposium on System Identification Brussels, Belgium, pp 322-327, 2012.

Sparse estimation l-1 methods such as LASSO depend on a regularization parameter which typically is selected using cross-validation. In the first contribution below, the statistical properties of the estimate are used to directly determine this parameter, leading to improved computational efficiency. In the second contribution, this technique is applied to sparse estimation of rational model structures.

C.R. Rojas and H. Hjalmarsson Sparse estimation based on a validation criterion Proceedings 50th IEEE Conference on Decision and Control Orlando, FA, USA, 2011

C.R. Rojas, B. Wahlberg and H. Hjalmarsson A Sparse Estimation Technique for General Model Structures European Control Conference, Zurich, Switzerland, 2013.

Variance analysis of block structured systems From a system identification perspective, understanding how where to place sensors in a block-structured system requires understanding of how sensors influence the accuracy of the estimate of a specific block. Below we report on results for cascaded systems.

N. Everitt, H. Hjalmarsson and C.R. Rojas A Geometric Approach to Variance Analysis of Cascaded Systems Proceedings 51st IEEE Conference on Decision and Control, Florence, Italy, 2013.


Last modified: Wed Jan 29 16:02:40 CET 2014