This paper presents the dynamic modeling and performance control of a Web server hosted on a private cloud. The cloud hosting Web server is a variable capacity system with two control inputs: 1) the number of virtual machines (VMs), which is indicative of the capacity of the cloud, and 2) the admission control used for regulating workload. As the workload and the hosting conditions change frequently, the linear parameter-varying (LPV) framework is well suited to derive the model. For the hosted Web server, we obtain an multiple input multiple output (MIMO) LPV model with performance metrics such as the response time and the throughput, which is then converted to polytopic LPV form using tensor product transformation. Finally, we design a gain scheduled linear quadratic regulator controller for performance guarantees with optimal cost of VMs. The identification, validation, and control experiments are demonstrated on the open source Eucalyptus cloud platform. The HTTP requests are generated using customized synthetic workload generator tool.