Reconfigurable intelligent surface (RIS) is regarded as a promising technology with great potential to boost wireless networks. Affected by the "double fading" effect, however, conventional passive RIS cannot bring considerable performance improvement when users are not close enough to RIS. Recently, active RIS is introduced to combat the double fading effect by actively amplifying incident signals with the aid of integrated reflection-type amplifiers. In order to reduce the hardware cost and energy consumption due to massive active components in the conventional fully-connected active RIS, a novel hardware-and-energy efficient sub-connected active RIS architecture has been proposed recently, in which multiple reconfigurable electromagnetic elements are driven by only one amplifier. In this paper, we first develop an improved and accurate signal model for the sub-connected active RIS architecture. Then, we investigate the joint transmit precoding and RIS reflection beamforming (i.e., the reflection phase-shift and amplification coefficients) designs in multiuser multiple-input single-output (MU-MISO) communication systems. Both sum-rate maximization and power minimization problems are solved by leveraging fractional programming (FP), block coordinate descent (BCD), second-order cone programming (SOCP), alternating direction method of multipliers (ADMM), and majorization-minimization (MM) methods. Extensive simulation results verify that compared with the conventional fully-connected structure, the proposed sub-connected active RIS can significantly reduce the hardware cost and power consumption, and achieve great performance improvement when power budget at RIS is limited.