Online Identification of Electrically Stimulated Muscle Models Fengmin Le, Ivan Markovsky, Christopher Freeman and Eric Rogers* Abstract— Online identification of electrically stimulated muscle under isometric conditions, modeled as a Hammerstein structure, is investigated in this paper. Motivated by the significant time-varying properties of muscle, a novel recur- sive algorithm for Hammerstein structure is developed. The linear and nonlinear parameters are separated and estimated recursively in a parallel manner, with each updating algorithm using the most up-to-date estimation produced by the other algorithm at each time instant. Hence the procedure is termed the Alternately Recursive Least Square (ARLS) algorithm. When compared with the Recursive Least Squares (RLS) algorithm applied to the over-parametric representations of the Hammerstein structure, ARLS exhibits superior performance on experimental data from electrically stimulated muscles and a faster computational time for a single updating step. Performance is further augmented through use of two separate forgetting factors. I. I NTRODUCTION As a result of the tradeoff between the complexity of general nonlinear systems identification and the interpretabil- ity of linear dynamical systems, Hammerstein structures have received considerable attention, and have been used in various areas to, for example, model chemical [24], bio- logical [10] and electrical [26] processes. The Hammerstein structure consists of a memoryless nonlinear block followed by a linear dynamic system, and the difficulty is that the inner signal is not measurable, that is, only input-output data measurements can be used to separate the nonlinear component from the linear one. There are many identification methods applicable to Hammerstein models and in general they can be roughly classified into two categories: iterative, for example, [21] and [10] with application to stretch reflex electromyogram, and non-iterative methods, for example, an equation-error parameter estimation method in [6], an optimal two-stage algorithm in [1], and decoupling methods in [2]. However, after reviewing the existing techniques, limitations were encountered when identifying an input- output model of electrically stimulated muscles with incom- plete paralysis. Consequently [20] developed two iterative algorithms for the identification of electrically stimulated muscles, and their efficacy was demonstrated through ap- plication to experimentally measured data. The algorithms developed in [20] represent significant progress in the identification of electrically stimulated mus- cles, but the models were only verified over a short time interval (20 sec duration). However, when applied to stroke rehabilitation, stimulation must be applied during intensive, *School of Electronics and Computer Science, University of Southamp- ton, Southampton, SO17 1BJ, UK goal orientated practice tasks in order to maximise improve- ment in motor control [23]. In clinical trials this translates to sustained application of stimulation during each treatment session of between 30 minutes and 1 hour duration [9]. In this case, slowly time-varying properties of the muscle system arise due to fatigue, changing physiological conditions or spasticity [14]. Motivated by this, online identification will be considered in this paper where in this approach, the model parameters are updated once new data is available. Only a few of the existing identification methods are recursive, and can be divided into three categories. The first category is the recently developed recursive subspace identification method [4]. Firstly, the Markov parameters of the system are estimated by least squares support vector machines (LS-SVM) regression and over- parameterizations. This is followed by recursive estimation of state-space model matrices by a propagator-based subspace identification method. This procedure does not have sparsity due to the LS-SVM model, and the resulting computational load makes it unsuitable for real-time implementation. The second category comprises stochastic approxima- tion [15] where algorithms with expanding truncations are developed for recursive identification of Hammerstein sys- tems. Two major issues with this method are the rather slow rates of convergence, and the lack of information on how to select the optional parameters in the algorithm for problems from different areas. The third category is Recursive Least Squares (RLS) or Extended Recursive Least Squares (ERLS). The RLS algorithm is a well-known method for recursive identification of linear-in-parameter models and if the data is generated by correlated noise, the parameters describing the model of the correlation can be estimated by ERLS. Here, a typical way to use these two algorithms is to treat each of the cross-product terms in the Hammerstein system equations as an unknown parameter. This procedure, which results in an increased number of unknowns, is usually referred to as the over-parameterization method [1] and [6]. After this step, the RLS or ERLS method can be applied [5]. The limitations of current algorithms are stated next and used to justify some of the critical choices necessary for this work to progress • The first two categories have only been applied in simu- lation and the stochastic approximation has not considered time-varying linear dynamics. This, together with the draw- backs described above, is the reason for not considering them further for the application treated in this paper. The third category is the most promising as it has already been applied to electrically stimulated muscle in [8] and [22]. 2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 978-1-4577-0079-8/11/$26.00 ©2011 AACC 90