Sequential games are those in which players apply their
strategies following a certain predefined order. The sequential
game is a natural framework to address some real problems, such as
the "Action-Reaction-Counteraction" paradigm used in military
intelligence and advertising campaigns strategies of several
competing firms in economics. This dissertation focuses on
stability, adaptivity and disturbance attenuation analysis of state
feedback Nash strategies for a class of noncooperative dynamic
Linear Quadratic Sequential Games (LQSGs).
For LQSGs with finite planning horizons, state feedback Nash
strategies are provided and their existence and uniqueness within
the class of state feedback strategies are proved. When the
planning horizon approaches infinity, it is proved that the
feedback systems with the state feedback Nash strategies are
uniformly asymptotically stable, given that the associated coupled
discrete-time algebraic Riccati equations (DAREs)have a positive
definite solution. Moreover, it is also proved that at least one
positive definite solution to the coupled DAREs of a scalar LQSG
exists.
When the parameters of the objective functions are not shared
among the decision makers, an on-line adaptive scheme is provided
for each player to estimate the actual control gain used by the
other player. The convergence to Nash strategies is proved with the
condition that the associated coupled DAREs have a unique positive
definite solution. The requirement of Persistency of Excitation
(PE) is satisfied by two methods: reference-signal tracking and
system uncertainties represented by small white noises.
The disturbance attenuation analysis of state feedback Nash
strategies is based on a H∞-optimal control problem, which
is converted into a zero-sum game. In this approach, the
uncertainties are controlled by a virtual player which maximizes
the cost functions that are being minimized by the real decision
makers. We first derive, for finite-horizon LQSGs, state feedback
Nash strategies with the optimal attenuation level
√λ̂>0. We extend the approach to
infinite-horizon LQSGs. A lower bound
λ̂∞ is provided in the sense that for
every
λ < λ̂∞ the values of the LQSGs
are unbounded. We also prove that the feedback system with state
feedback Nash strategies is Bounded Input Bounded Output (BIBO)
stable with respect to the disturbances.