Marwan A. Simaan

On the Stackelberg strategy in nonzero-sum games

The properties of the Stackelberg solution in static and dynamic nonzero-sum two-player games are investigated, and necessary and sufficient conditions for its existence are derived. Several game problems, such as games where one of the two players does not know the others performance criterion or games with different speeds in computing the strategies, are best modeled and solved within this solution concept. In the case of dynamic games, linear-quadratic problems are formulated and solved in a Hilbert space setting. As a special case, nonzero-sum linear-quadratic differential games are treated in detail, and the open-loop Stackelberg solution is obtained in terms of Riccati-like matrix differential equations. The results are applied to a simple nonzero-sum pursuit-evasion problem.

Multiple target angle tracking using sensor array outputs

The use of the output of an array of sensors to track multiple independently moving targets is reported. The output of each sensor in the array is the sum of signals received from each of the targets. The results of direction-of-arrival estimation by eigenvalue analysis are extended to derive a recursive procedure based on a matrix quadratic equation. The solution of this matrix quadratic equation is used to provide updated target positions. A linear approximation method for estimating the solution of the matrix equation is presented. The algorithm is demonstrated by the simulated tracking of two targets. The main advantage of the algorithm is that a closed-form solution for updating the target angle estimates has been obtained. Also, its application is straightforward, and the data association problem due to uncertainty in the origin of the measurements is avoided. However, it requires the inversion of an N*N as well as other linear operations, so that the computational burden becomes substantial as N becomes very large. >

Game-theoretic modeling and control of a military air operation

An attrition-type discrete-time dynamic model is formulated for two opposing forces, labeled Blue and Red, engaged in a military air operation. The Blue force consists of combat air units and its objective is to destroy a fixed target, such as an airport or a bridge, which Is being defended by the Red force. The Red force consists of ground troops and air defense units. We model the objective functions for each side and identify the associated constraints on the control and state variables. We employ a two-level hierarchy of command and control for each force. An example scenario illustrating the implementation of this approach using concepts from non-zero sum dynamic game theory is presented.

Brief paper: Game theory applied to dynamic duopoly problems with production constraints

In this paper an application of differential game theory in the area of microeconomics is presented. The problem considered is that of a dynamic duopoly where two firms each limited by a maximum capacity of production, share the same market, and try simultaneously but independently to maximize their profits over a certain planning horizon. While the static duopoly theory does not address itself to the question of the process by which changes in the price are brought about, but only compares the prices before and after the change takes place, the dynamic market theory, considered in this paper, allows for an analysis of how the price changes with time and what trajectory it follows. Necessary conditions for the existence of a Nash equilibrium solution in the general case are discussed and more specific results for the special case of linear demand and quadratic cost functions are developed.

A Dynamical State Space Representation and Performance Analysis of a Feedback-Controlled Rotary Left Ventricular Assist Device

The left ventricular assist device (LVAD) is a mechanical device that can assist an ailing heart in performing its functions. The latest generation of such devices is comprised of rotary pumps which are generally much smaller, lighter, and quieter than the conventional pulsatile pumps. The rotary pumps are controlled by varying the rotor (impeller) speed. If the patient is in a health care facility, the pump speed can be adjusted manually by a trained clinician to meet the patients blood needs. However, an important challenge facing the increased use of these LVADs is the desire to allow the patient to return home. The development of an appropriate feedback controller for the pump speed is therefore crucial to meet this challenge. In addition to being able to adapt to changes in the patients daily activities by automatically regulating the pump speed, the controller must also be able to prevent the occurrence of excessive pumping (known as suction) which may cause collapse of the ventricle. In this paper we will discuss some theoretical and practical issues associated with the development of such a controller. As a first step, we present and validate a state-space mathematical model, based on a nonlinear equivalent circuit flow model, which represents the interaction of the pump with the left ventricle of the heart. The associated model is a six-dimensional vector of time varying nonlinear differential equations. The time variation occurs over four consecutive intervals representing the contraction, ejection, relaxation, and filling phases of the left ventricle. The pump in the model is represented by a nonlinear differential equation which relates the pump rotational speed and the pump flow to the pressure difference across the pump. Using this model, we discuss a feedback controller which adjusts the pump speed based on the slope of the minimum pump flow signal, which is one of the model state variables that can be measured. The objective of the controller is to increase the speed until the envelope of the minimum pump flow signal reaches an extreme point and maintain it afterwards. Simulation results using the model equipped with this feedback controller are presented for two different scenarios of patient activities. Performance of the controller when measurement noise is added to the pump flow signal is also investigated.

Moving horizon Nash strategies for a military air operation

Dynamic game theory has recently received considerable attention as a possible technology for formulating control actions for decision makers in an extended complex enterprise that involves an adversary. Examples of such enterprises are very common in military operations. Enterprises of this type are typically modeled by a highly nonlinear discrete time dynamic system whose state is controlled by two teams of decision makers each with a different objective function and possibly with a different hierarchy of decision making. Because of the complexity of such systems, the traditional solutions from dynamic game theory that involve optimizing objective functions over the entire time horizon of the system are computationally extremely difficult, if not impossible, to derive. We discuss a solution approach where at each step the controllers limit the computation of their actions to a short time horizon that may involve only the next few time steps. This moving horizon solution, although suboptimal in the global sense, is very useful in taking into account the possible near-term control actions of the adversary. To illustrate this solution methodology, we consider an example of an extended military enterprise that involves two opposing forces engaged in a battle.

A passivity-based method for induction motor control

The control of an induction motor is a difficult problem, since the dynamics of the induction motor are nonlinear, the rotor electrical state variables (i.e., rotor fluxes or currents) are usually unavailable for measurement, and the motor parameters can vary significantly from their nominal values. The main purpose of this paper is to develop a control algorithm that forces the induction motor to track time-varying speed, position, and flux trajectories without knowledge of the rotor electrical state variables. To achieve this, a passivity-based method is developed. The key point with this method is the identification of terms, known as workless forces, which appear in the dynamic equations of the induction motor but do not have any effect on the energy balance equation of the induction motor. These terms do not influence the stability properties of the induction motor and, hence, there is no need to cancel them with feedback control. This leads to a simpler control structure and enhances the robustness of the control system. Experimental results show that the passivity-based method provides close tracking of time-varying speed, position, and flux trajectories without knowledge of the rotor electrical state variables.

An efficient algorithm for tracking the angles of arrival of moving targets

A novel technique for tracking the angles of arrival of moving targets is presented. The targets are modeled as signal sources that continuously emit narrowband signals which impinge on an array of sensors. Estimates of target angles are obtained by minimizing the norm of an error matrix function involving the covariance of the sensor outputs. The algorithm yields estimates that are automatically correctly associated with previous estimates. Consequently, the data association problem does not arise, and this results in a much more efficient scheme in comparison to existing methods involving search over N factorial possible measurement/target associations (where N is the number of targets). Performance of the algorithm is illustrated by a simulation example. >

Estimation of systemic vascular bed parameters for artificial heart control

An extended Kalman filter estimator for the identification of systemic circulation model parameters during cardiac ejection and cardiac filling is described. The estimator has been developed for use in the control of a cardiac ventricular assist device. A lumped element circuit with a time-varying capacitor was used to represent the systemic circulation and the left ventricle. Since the haemodynamic variables that are measurable in patients with impaired cardiac function vary dramatically as the patients move through different levels of care, the estimator was designed so that it can be used with different sets of blood pressure and flow measurements. Preliminary evaluation of the performance of the estimator using data from a computer simulation and from a patient during open-heart surgery is presented. The robustness of the estimator to variations in parameter initialization is also described.

A Stackelberg solution for games with many players

The concept of Stackelberg solution is widened to include games with many leaders and many followers. Necessary conditions for the existence of an open-loop Stackelberg solution in differential games where each player is using a Nash strategy within his group are also derived.