Abstracts for Poster Session at Multi-Agent Control Summer School in Kranjska Gora

**Parametrical Modelling of Nonlinear Systems**

Gregor Gregorcic and Gordon Lightbody

University College Cork

This poster will give an overview of some of the key issues associated with parametrical modelling of nonlinear systems. This will include:

- comparison of the various modelling paradigms,
- local vs. global modelling and the need for carefully designed data,
- off--equilibrium behavior,
- switching and interpolation between the models.

Global modelling difficulties such as:

- selection of structure of the model,
- complex modelling algorithm,
- curse of dimensionality,
- lack of transparency of the global model and
- bias problem, will also be highlighted.

**Hierarchical Gaussian Process Mixtures for Regression
**

Shi, J. Q., Murray-Smith, R. and Titterington, D. M.

Glasgow University

As a result of their good performance in practice and their desirable analytical properties, Gaussian process regression models are becoming increasingly interesting in engineering and other fields. However, there are two major problems when the model is applied to a large data-set with repeated measurement. One is the heterogeneity among the different replications, and the other is the requirement to invert a covariance matrix which is involved in the implementation of the model. The dimension of this matrix equals the sample size of the training data-set. This clearly becomes time-consuming for large data-sets. In this paper, a mixture regression model of Gaussian processes is proposed, and a hybrid Markov chain Monte Carlo (MCMC) algorithm is developed for the implementation. Using the model and the algorithm, the computational burden decreases dramatically. A real application is used to illustrate the mixture model and its implementation.

Gaussian Process Priors with Correlated Noise Models

Agathe Girard (Ph. D. student, supervised by Roderick Murray-Smith)

Glasgow University

The Gaussian Process prior is a probabilistic model in a Bayesian setting. Rather than placing a prior over the unknown parameters of the model, we put directly a prior on the space of functions. The strength of the GP modeling is its flexibility (non parametric) and its interpretability, in view of the covariance function. The covariance function of the targets (noise corrupted versions of the model outputs) is made of two parts: one associated to the modelling function and one for the noise model. Here, we propose to assume parametric correlated noise models: the auto regressive (AR) and moving average (MA) models. The improvement in performance is illustrated on some simulation examples of data generated by nonlinear static functions corrupted with additive ARMA noise.

**Hidden coupling in blended local PID control schemes
**

Q. Rong, S. McLoone and G. Irwin

Queen's University Belfast

Analysis of conventional blended local model network based PID controller designs reveals the existence of a hidden coupling term which imposes a slow variation constraint on the blending functions. By developing a PID controller scheme within a velocity-based local model network framework this hidden coupling term is eliminated leading to a global controller whose dynamics reflect more closely those of the local controllers. Simulation studies on a continuous stirred tank reactor are used to support the analysis.

**Force Feedback in FES Assisted Standing-Up in Paraplegia
**

Roman Kamnik*, Jian Qing Shi, Roderick Murray-Smith, Tadej Bajd*

* University of Ljubljana

Glasgow University

After the spinal cord injury, the paraplegic patients stand up from the sitting to the standing position primarily by the use of arm support. In addition, the manoeuvre can be facilitated with the functional electrical stimulation (FES) artificially invoking muscle contractions of the paralyzed limbs. The traditional approach employs a minimum of two channels of FES delivered to both knee extensors through two pairs of large surface electrodes. The stimulation patterns are constant and voluntary on/off triggered by the patient. Due to drawbacks of open loop control approach, the closed loop approaches to the FES control in a standing-up in paraplegia are under development.

The poster presents an analysis of the standing-up manoeuver in paraplegia considering the supportive forces as a potential feedback source in the task of determining the human body state throughout the rising. The paraplegic’s body center of mass (COM) trajectory was used as a target body state trajectory. The analysis was accomplished on standing-up data acquired in a group of eight paraplegic patients. In the measurements, the body motion kinematics and the reaction force vectors under the feet, chair, and arm handles were accessed. For the modelling, the conventional neural network model, Gaussian process prior and mixture models were compared. It was shown that Gaussian process prior has a superior performances in comparison to the neural network model.

RVM methods for regression and classification

Joaquin Quinero Candela - Ph.D. Student

Technical University of Denmark

The Relevance Vector Machine (RVM), proposed by Tipping, is a probabilistic model similar to the widely established Support Vector Machine (SVM). The RVM relies on a bayesian scheme where maximum likelihood II estimation of the hyperparameters leads to sparsity. In the poster presented RVM methods are applied to both regression and classification in well-known examples. Results, which show a comparable performance to SVM's with a much more sparse architecture, and implementation details are exposed. Furthermore, some related research issues are arised.

Artificial Control of Electrically Stimulated Muscle in Paraplegia

Henrik Gollee

Glasgow University

For muscle which has lost nervous control, artificial electrical stimulation can be used as a technique to produce functionally useful movement. This work is concerned with the design and experimental evaluation of feedback control techniques for such systems. As muscle is a

non-linear dynamic plant which is subject to significant uncertainties and disturbances such as fatigue and spasticity, we develop design approaches which are robust enough to deal with these uncertainties while being simple to set up and easy to implement. Results of experimental evaluation in a system which provides unsupported standing for a paraplegic subject are presented.

On the Infinity-Norm and the Asymptotic Stability of a Class of Second Order Time-Varying Systems

Kai Wulff

National University of Ireland, Maynooth

In this paper we present a sufficient condition for global asymptotic stability of certain classes of second order time-varying systems. While the proof is based upon geometrical and non-quadratic Lyapunov arguments, the stability conditions are expressed as simple restrictions on a matrix pencil. These restrictions possess a simple geometrical interpretation and may generalize to higher order systems. The conditions are invariant under similarity transformations, and are readily verifiable via root-locus checking. Further, the conditions establish a direct connection between a matrix pencil and the generalized infinity norm, and can therefore be viewed as extending the link between known matrix pencil conditions and quadratic stability criteria.