Application of Wiener-Hammerstein System Identification in Electrically Stimulated Paralyzed Skeletal Muscle Modeling Er-Wei Bai, Zhijun Cai, Shauna Dudley-Javoroski, and Richard K. Shields Abstract— Electrical muscle stimulation has demonstrated potential for restoring functional movement and for preventing muscle atrophy after spinal cord injury (SCI). Control systems used to optimize delivery of electrical stimulation protocols depend upon algorithms generated using computational models of paralyzed muscle force output. The existing skeletal muscle models are either not accurate or too complicated to implement for real-time control. In this paper, we propose a Wiener- Hammerstein system, Linear-Saturation-Linear (LSL) model, to model the skeletal muscle dynamics under electrical stimulus conditions. Experimental data from the soleus muscles of an individual with SCI was used to quantify the performance of the model. We demonstrate that the proposed Wiener- Hammerstein system is comparable to, in terms of model fitting, and outperforms, in terms of prediction, the Hill Huxley model, the most advanced and accurate model previously reported. On the other hand, the proposed LSL model is much simpler in terms of the structure and involves a much smaller number of unknown coefficients. This has substantial advantages in identification algorithm analysis and implementation including computational complexity, convergence and also in real time model implementation for control purposes. I. INTRODUCTION After spinal cord injury (SCI) the loss of volitional muscle activity triggers a range of deleterious adaptations. Muscle cross-sectional area declines by as much as 45% in the first six weeks after injury, with further additional atrophy occurring for at least six months [5], Muscle atrophy impairs weight distribution over bony prominences, predisposing individuals with SCI to pressure ulcers, a potentially life- threatening secondary complication [16]. The diminution of muscular loading through the skeleton precipitates severe osteoporosis in paralyzed limbs. The lifetime fracture risk for individuals with SCI is twice the risk experienced by the non-SCI population [21]. Rehabilitation interventions to prevent post-SCI muscle atrophy and its sequelae are an urgent need. Electrical muscle stimulation after SCI is an effective method to induce muscle hypertrophy [15], [11], fiber type and metabolic enzyme adaptations [8], [1], and improvements in torque output and fatigue resistance [18], [20]. New evidence suggests that an appropriate longitudinal dose of muscular load can be an effective anti-osteoporosis countermeasure [18], [17], [14]. Electrical muscle stimula- tion also has potential utility for restoration of function in tasks such as standing, reaching, and ambulating. The myriad Er-Wei Bai and Zhijun Cai are with Department of Electrical and Computer Engineering, University of Iowa, Iowa City, IA, 52242, USA. erwei,zai@engineering.uiowa.edu Shauna Dudley-Javoroski, and Richard K. Shields are Graduate Program in Physical Therapy and Rehabilitation Science University of Iowa, Iowa City, IA 52242, USA applications for electrical stimulation after SCI have created a demand for control systems that adjust stimulus param- eters in real-time to accommodate muscle output changes (potentiation, fatigue) or inter-individual force production differences. To facilitate the refinement of control system algorithms, mathematical models of muscle torque output are continuously being developed. To most successfully adapt stimulus parameters to real-time muscle output changes, an accurate and easy-to-implement model is essential. Over the last decades, researchers have developed a num- ber of muscle models aimed at predicting muscle force output [9], [10], [3]. The Hill Huxley model [10] is the most advanced and accurate model put forward to date [4]. Compared to other models, the Hill Huxley model represents muscle dynamics well. However, its complexity undermines its usefulness for real time implementation for control. Identification of a Hill Huxley model is non-trivial because it is time-varying, high dimensional and nonlinear. Local minimum versus global minimum is always a difficult issue for identification, and the user must tune identification algorithm parameters patiently (including the initial estimate) in order to have a good result. Our goal has been to develop a model that is comparable to or outperforms the Hill Huxley model, but at a reduced complexity. We propose to use a Wiener-Hammerstein sys- tem that resembles the Hill Huxley structure but has the added advantage of greater simplicity. This approach was previously suggested by Hunt and colleagues [13] but was deemed inadequate for muscle modeling. By examining the experimental data sets and the Hammerstein system, we noted two problems. First, a linear block prior to the non- linear block was missing and secondly, the static nonlinear- ity seemed suboptimal. The proposed Wiener-Hammerstein model overcomes these two deficiencies, enjoys a high degree of accuracy, and is comparable to or outperforms the Hill Huxley model. Most importantly, the proposed model is much simpler not only in the structure but also in the number of unknown coefficients. The purpose of this report is to describe a Wiener-Hammerstein system for modeling paralyzed skeletal muscle dynamics under electrical stim- ulation. By using actual soleus force data from a subject with SCI, we demonstrate that this model’s advantages over previous models are theoretically justified and numerically verified. Equally important is a demonstration of the useful- ness of block oriented nonlinear systems together with their identification algorithms. Identification of block oriented nonlinear systems including Wiener-Hammerstein systems has been extensively investigated recently in the control and Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico, Dec. 9-11, 2008 WeC05.1 978-1-4244-3124-3/08/$25.00 ©2008 IEEE 3305