With growing complexity of power grid interconnections, power systems may become increasingly vulnerable to low frequency oscillations (especially inter-area oscillations) and dependent on stabilizing controls using either local signals or wide-area signals to provide adequate damping. In recent years, the ability and potential to use wide-area signals for control purposes has increased since a significant investment has been made in the U. S. in deploying synchrophasor measurement technology. Fast and reliable communication systems are essential to enable the use of wide-area signals in controls. If wide-area signals find increased applicability in controls the security and reliability of power systems could be vulnerable to disruptions in communication systems. Even though numerous modern techniques have been developed to lower the probability of communication errors, communication networks cannot be designed to be always reliable. Given this background the motivation of this work is to build resiliency in the power grid controls to respond to failures in the communication network when wide-area control signals are used. In addition, this work also deals with the delay uncertainty associated with the wide-area signal transmission. In order to counteract the negative impact of communication failures on control effectiveness, two approaches are proposed and both approaches are motivated by considering the use of a robustly designed supplementary damping control (SDC) framework associated with a static VAr compensator (SVC). When there is no communication failure, the designed controller guarantees enhanced improvement in damping performance. When the wide-area signal in use is lost due to a communication failure, however, the resilient control provides the required damping of the inter-area oscillations by either utilizing another wide-area measurement through a healthy communication route or by simply utilizing an appropriate local control signal. Simulation results prove that with either of the proposed controls included, the system is stabilized regardless of communication failures, and thereby the reliability and sustainability of power systems is improved. The proposed approaches can be extended without loss of generality to the design of any resilient controller in cyber-physical engineering systems.