  This paper investigates closed-loop stability of a linear discrete-time plant
subject to bounded disturbances when controlled according to packetized
predictive control (PPC) policies. In the considered feedback loop, the
controller is connected to the actuator via a digital communication channel
imposing bounded dropouts. Two PPC strategies are taken into account. In both
cases, the control packets are generated by solving sparsity-promoting
optimization problems. One is based upon an l2-constrained l0 optimization
problem. Such problem is relaxed by an l1-l2 optimization problem in the other
sparse PPC setting. We utilize effective solving methods for the latter
optimization problems. Moreover, we show that in the presence of plant
disturbances, the l2-constrained l0 sparse PPC and unconstrained l1-l2 sparse
PPC guarantee practical stability for the system if certain conditions are met.
More precisely, in each case, we can derive an upper bound on system state if
the design parameters satisfy certain conditions. The bounds we derive are
increasing with respect to the disturbance magnitude. We show via simulation
that in both cases of proposed sparse PPC strategies, larger disturbances bring
about performance degradation with no effect on system practical stability.
