(79b) Broadcast Model Predictive Control of Multi-Cellular System

In multi-cellular biological systems, the behavior of skeletal muscle cells and control of these vast numbers of cells through few available actuators are important and challenging problems. Key mechanisms through which a few selected cells activate from vast number of cells to produce the aggregate desired response is still an open question in cellular biology. Besides this, smooth functioning of muscle cells with partial available information to the central nervous system and control architecture for this integrated system is not known completely. To understand this integrated system, one of the approaches among various possibilities is to apply engineering tools in order to develop theoretical models and control architectures. Several models and control architectures have been reported in the literature amongst which the idea of modeling these vast numbers of cells by taking a probabilistic approach and designing a control architecture based on broadcast feedback (Ueda et.al. [1, 2, 3, 4]) appears to be promising. Inspired by this idea, we have developed a probabilistic model for this multi-cellular system and control architecture using broadcast model predictive control. To the best of our knowledge, the idea of integrating broadcast feedback with model predictive control for this multi-cellular integrated system has never been introduced before.

This presentation introduces this new idea of integrating broadcast feedback with model predictive control applicable to various systems similar to multi-cellular integrated system. We develop a non-homogeneous Markov model to represent multi-cellular systems with the known fact from literature that muscle fibers show only ?ON? (excited) and ?OFF? (relaxed) states. We consider that muscle cells are identical and behave independently. Based on this assumption, we define a pair of transition probability for switching states of cells and show the existence of such transition probabilities for an ensemble of cells by assigning the same transition probabilities to all cells within the system. These transition probabilities are function of system error defined as the difference of the desired number of ?ON? cells and number of ?ON? cells at a particular time instant. We further show the stability of this multi-cellular system rigorously using super-martingale theory [5] and robustness against permanently dead cells within the system when the information about number of dead cells is not available to the controller.

The idea of broadcasting the same information to all the cells present in a multi-cellular system based on model predictive control is partly inspired by the functioning of our central nervous system. Whether the central nervous system makes decisions in a predictive sense is still unclear in the literature but intuitively it appears that our central nervous system may function as model predictive controller. Based on this, we design a model predictive control framework to compute transition probabilities which are then broadcast to all the cells within the multi-cellular system. We derive an expectation based model for the multi-cellular system in a predictive sense. Due to the dependence of transition probabilities on the system error, this model is non-linear. We consider a quadratic cost objective function and optimize it over the predefined prediction horizon with respect to transition probabilities. For this, we implement a strategy based on the sign of the system error. If the system error is positive then we restrict all the transitions from ?ON? to ?OFF? by setting transition probabilities of ?ON? to ?OFF? to zero over the prediction horizon. Similarly we put the restriction on transition of ?OFF? to ?ON? whenever the system error is negative. With this strategy we compute all pairs of transition probabilities over the prediction horizon. Out of all computed transition probabilities, we broadcast the first pair of transition probabilities to the multi-cellular system for the next sample time calculation. We obtain analytical expressions for transition probabilities for one step and two step predictors. Higher order predictor transition probabilities are obtained by solving non-linear coupled algebraic equations numerically. Further we design an observer for obtaining information such as time varying number of ?ON? and ?OFF? dead cells present within the multi-cellular system for better control strategy. Finally, the overall integrated multi-cellular system including observer is simulated in MatLab and future enhancement are proposed.

References

[1] Jun Ueda, Lael Odhner, and H. Harry Asada. Broadcast feedback of stochastic cellular actuators inspired by biological muscle control. International Journal of Robotics Research, Special Issue on Bio-Robotics, February 2007.

[2] Jun Ueda, Lael Odhner, and H. Harry Asada. A Broadcast-Probability Approach to the Control of Vast DOF Cellular Actuators. In Proceedings of the 2006 IEEE International Conference on Robotics and Automation, 1456-1461, May 2006.

[3] Lael Odhner, Jun Ueda, and H. Harry Asada. Stochastic Optimal Control Laws for Cellular Artificial Muscles. In Proceedings of the 2007 IEEE International Conference on Robotics and Automation, 1554-1559, April 2007.

[4] Lael Odhner, Jun Ueda, and H. Harry Asada. Feedback Control of Stochastic Cellular Actuators. Experimental Robotics, 481-490, 2008.

[5] Rick Durrett. Probability: Theory and Examples. Thomson, third edition, 2005.