Stability of Large Space Structure Control Systems Using Positivity Concepts

A robust stability test and associated design procedure based on the positivity of operators is proposed. The test does not rely on modal truncation or high-order truth models of the structure and is independent of the numerical values of the modal data. The stability criterion is applied to the plant (structure) and the controller individually, assuring global stability when the loop is closed by negative feedback. Therefore, design/stability evaluations need only iterate on the low-order controller part of the loop. The method can be extended to nonlinear systems. ODAL truncation in the dynamic models of large space structures (LSS) can cause well-known stability problemsM when these models are used as the design basis for active structure control. Therefore, most active structure control methods proposed so far demand an extensive stability evaluation after completion of the control system design. This is done by using very high-order "truth models" of the structure, accounting for as many residual modes (modes previously neglected in the controller design) as is computationally practical. When the system can justifiably be assumed linear, solution of a large eigenvalue problem will suffice to establish stability; however, if the system contains significant nonlinearities in the actuators and sensors, ex- tensive simulations of a very high-order system would be required. Aside from the fact that in either case such stability evaluations are time-consuming and costly, they are usually based on erroneous mathematical models of the structure, which casts doubt on the validity of the results obtained. It is well known from on-orbit experience (e.g., Skylab) that modal models of large structures obtained by finite-elemen t methods (e.g., NASTRAN) must be expected to contain errors greater than 10% in the determined modal frequencies and mode shapes. An additional source of error is that even in a high-order truth model, modal truncation must occur at some place. For the above reasons there exists an urgent need for the following. 1) Robust stability tests independent of the number and exact numerical values of modal frequencies and mode shapes, and tests that can be extended to nonlinear (and potentially even time-varying) systems. 2) Control laws which can be assured to remain stable while a system is undergoing preoperational on-orbit tests to identify system parameters for subsequent controller tuning and performance improvement. 3) Design procedures which can assure stability a priori, or at least require iteration of the design/stability evaluations on no more than the low-order controller part of the loop. It is clear that for this purpose design methods and stability tests broader than "poles in the left-half plane" are required. We are proposing such a method based on the positivity of operators. Physically, the method has the interpretation of assuring that the closed-loop system will be energy dissipative. The proposed method makes extensive use of functional analysis techniques and is an outgrowth of Popov's 5'6 work on absolute and hyperstability, which was subsequently extended by many researchers (for example, Refs. 7-12). The basic underlying theory is general and not restricted to linear and time-invariant plants. In fact, the method has in the past been associated with the stability of nonlinear systems (Lur'e problem13) where both the nonlinear and linear parts of the system must meet certain positivity conditions. These con- ditions are only sufficient conditions for stability and therefore tend to be conservative at times. Since well-known necessary and sufficient conditions exist to establish the stability of linear systems, it is apparent why positivity techniques have generally not been applied to linear systems in the past. The stability problem in the control of large space struc- tures (LSS) differs from more conventional linear systems, because the plant to be controlled is theoretically of infinite order (and practically of very high order), while the con- troller/estimator must, of necessity, be of relatively low order. It appears desirable, therefore, to use tests which do not require explicit evaluation of the closed-loop system, but which impose conditions individually on the plant and the controller, assuring stability when the loop is closed by negative feedback. The stability theory based on the positivity of operators is a powerful method having just this charac- teristic.