Complementarity Constraints for Nonlinear Systems

This paper develops complementarity constraints for nonlinear feedback control systems with additive output disturbances and sensor noise, and for nonlin-ear ltering in the presence of process noise and additive measurement noise. In the control case, the complementarity constraint is the nonlinear counterpart of the well known fact in linear feedback control theory that the sum of the sensitivity operator, S(s), and complementary sensitivity operator, T (s), is the identity operator. Performance limitations induced by this nonlinear complementarity constraint in the case of open loop unstable and non-minimum phase plants are explored. 1 COMPLEMENTARITY CONSTRAINTS AND PERFORMANCE LIMITS: FROM LINEAR TO NONLINEAR When designing a controller or a lter for a certain process, it is desirable to recognize, a pri-ori, the limitations in achievable performance that exist irrespective of the speciic design procedure employed. These limitations are intrinsic to the particular structure of the control or ltering loop and are imposed by the non-minimum phase zeros and unstable poles of the open loop system. In linear feedback control theory, the structure of the classical feedback connguration leads to the key observation that the sum of the sensitivity operator, S(s), and the complementary sensitivity operator, T (s), is the identity operator. The sensitivity operator maps output disturbances to the system output, whilst the complementary sensitivity operator maps sensor noise to the system output. Thus, a fundamental trade-oo exists when shaping any of these mappings during design. The complementarity constraint S(s) + T (s) = I represents an algebraic trade-oo, since it constrains the properties of the closed loop system at each frequency. These restrictions are called interpolation constraints. However, the comple-mentarity relation manifests itself quantitatively in linear control design in the form of integral constraints. These constraints condition the overall frequency responses of the closed loop system and are aaected by open loop zeros and poles. Work on these trade-oos in the control literature can be traced back to the seminal work of Bode in the 1940's (Bode 1945). In his analysis, it was shown that, if the open loop plant is stable and has relative degree of at least two, then, provided the closed loop system is stable, the sensitivity function must satisfy the following integral relation: Z 1 0 logjS(j!)jd! = 0: This result demonstrates that there is a compromise between sensitivity properties in diierent frequency ranges. Related results on performance limitations were developed over the past decade in the …