A Feasible Lookahead Control for Systems with Finite Control Set Sherif Abdelwahed sherif@isis.vanderbilt.edu Rong Su rongsu@control.toronto.edu Sandeep Neema sandeep@isis.vanderbilt.edu Abstract— In this paper we present a feasible control ap- proach for a general class of systems with finite control set. The objective of the controller is to direct the system towards a desired set-point state and then maintain the system within a known neighborhood of this state.The feasibility ofthe online control policy is formulated as a joint containability and attraction problem. A novel computational procedure based on nonlinear programming is presented to compute a containable region,in which each trajectory from inside cannot move out under a single-step-lookahead control policy.The proposed controlapproach and the developed feasibility technique is demonstrated on a two tank system example. I. I NTRODUCTION This paperpresents a feasible control approach for a general class ofnonlinear systems with finite control set undera limited lookahead control policy.The underlying model, referred to as switching system model, can describe the dynamics of many practical real-life systems, including real-timecomputation systems, embedded systems, and command and control systems. Owing to its nonlinear dynamicsand thefact thatthe underlying control set is finite,traditional optimalcontroltechniques cannot be applied directly to switching systems, and in general a closed expression for a feedback control map cannot be established. The situation isfurthercomplicated by the factthatin many practical systems the model is usually imprecise and system parameters may change during run- time owing to the component’s failure or aging. In addition, in certain computation and engineering applications, the controller is required to adapt in run-time to changes in operational equipments of the systems. To address the above challenges, we proposed in [1] a limited-lookahead online control policy forhandling set- pointspecifications. In the onlinecontrolapproach the controller searches a limited set of all possible future states from the current state and chooses the next step that drives the system closer to the desired state based on a given distancefunction.This controlstrategy isconceptually similar to the model predictive control [19], [24], [25], [16] in which a limited time forecast of the process behavior at each state is optimized according to a given criterion over the set of inputs.Similar receding horizon approach This work is sponsored by the DARPA/IXO Model-Based Integration of Embedded Software program, under contract F33615-02-C-4037 with the Air Force Research Laboratory Information Directorate, Wright Patterson Air Force Base. SherifAbdelwahed and Sandeep Neema are with the Electrical Eng. and Computer Science Dept., Vanderbilt University, Nashville, TN, USA Rong Su is with the Department of Electrical and Computer Engineer- ing,University of Toronto, Toronto, Ontario, Canada has been investigated in [23] for linear quantized systems proposed in . Also related to our work is the online limited lookahead supervision of discrete event systems [9]. The limited lookahead control policy has been applied successfully to several engineering and real-time compu- tation systems including: QoS management for distributed real-time computation systems (a signal processing appli- cation)[1], managing the powerconsumed by acom- puter processing a time-varying workload comprising HTTP and e-commerce related requests [14], [15],and resource management of the waterrecovery and air evaporation subsystems of a simulated NASA advanced life support system that is a component of an Advanced Life Support system for extended duration human space missions [2]. This paperaddresses the feasibilityof the limited- lookahead control approach forthe classof switching systems. The online controller is said to be feasible for a given set-point, tolerance domain containing the set-point, and operating region containing the tolerance domain if it can drive the system (in finite time) from any initial state in the operation region to the tolerance domain and maintain the system within this domain. The feasibility of the proposed online control approach is formulated as a joint containability and attraction problem. A novel compu- tational procedure based on nonlinear programming (NLP) is presented to compute a containable region, in which each trajectory from inside cannot move out under a single-step- lookahead control policy. The paper then discusses how to decide whether the obtained containable region is finitely reachable from a point outside the region. The proposed online control and the underlying stability analysis approach is demonstrated on a two tank system. Because of space limitation, proofs for the main propositions in this papers have be omitted. They can be found in [26]. II. L IMITED LOOKAHEAD CONTROL OF SWITCHING SYSTEMS This paper considers a class of nonlinear systems with finite control set. This class will be referred to a switching systems and is defined formally as follows. x(k + 1) = f (x(k), u(k)) (1) where u(k) ∈ U ⊂ R m and f :R n × U → R n is differentiable over x for each fixed u ∈ U . The control action set U is finite. The above model belongs to the class of switched systems [7], [3], [4], where each control input o U corresponds to a potential transition between two modes. In this paperwe consider a set-point controlproblem Proceedings of the 2005 IEEE Conference on Control Applications Toronto, Canada, August 28-31, 2005 TA3.5 0-7803-9354-6/05/$20.00 ©2005 IEEE 663