The use of general tensegrity systems that incorporate rigid bodies beyond axially loaded members has garnered increasing attention in practical applications. Recent preliminary studies have been conducted on the analysis and form design of general tensegrity systems with disconnecting rigid bodies. However, existing methods cannot account for connections between different rigid bodies. In practical applications, general tensegrity systems may have interconnected rigid bodies, rendering the analysis method proposed in previous studies inapplicable. To address this issue, this work proposes a comprehensive and unified analysis method for general tensegrity systems. The proposed formulation allows for the incorporation of connections between rigid bodies and general tensegrity systems with supports into the developed framework, enabling uniform analysis. Equilibrium and compatibility equations are derived through an energy approach combined with the Lagrange multiplier method. Self-stress states and mechanism modes are then computed based on these formulations. The stiffness of the mechanism mode is analyzed and validated using both the product force method and the reduced geometric stiffness matrix method. Furthermore, a self-stress design approach based on semi-definite programming (SDP) is proposed to determine feasible member forces that can stabilize general tensegrity systems. Illustrative examples are presented to verify the effectiveness of the proposed approach. This study expands the scope of the analysis theory for tensegrity systems and provides a fundamental and unified analysis approach that can be applied to any type of tensegrity system.