Nonlinear state feedback control of second-order nonminimum-phase nonlinear systems

The present work addresses the problem of synthesizing nonlinear state feedback controllers for second-order nonminimum-phase nonlinear systems. The concept of a first-order nonlinear all-pass is first introduced. A class of static state feedback control laws is then developed that makes the closed-loop system equivalent, under an appropriate coordinate transformation, to a nonlinear first-order all-pass in series with a linear first-order lag. A particular control law from this class is calculated that results in I.%!?-optimal response. The performance of the proposed methodology in set point tracking is evaluated through numerical simulations in a CSTR example. Major contributions in the area of linear process control in the last decade have established the idea that a controller must explicitly or implicitly generate a process inverse (Garcia and Morari, 1982). When dealing with minimum-phase linear systems, such an inverse is stable and can be used for controller synthesis. When dealing with nonminimum-phase linear systems, an appropriate decomposition of the process model into a part with stable inverse and a part with unstable inverse is necessary and the con- troller must invert only the part with stable inverse. In the context of linear state feedback, the same idea arises in placing closed-loop poles at the left-half plane zeros and at the reflection of the right-half plane zeros. Such a control strategy has been shown to be ZSE-optimal for step changes. In the field of nonlinear process control, the idea of a controller that generates a process inverse is central in general synthesis methods for minimum-phase sys- tems, like the nonlinear IMC structure (Economou et al., 1986; Parrish and Brosilow, 1988), which explic- itly generates a process inverse on-line and the input/ output linearization method (Kravaris and Chung, 1987) which implicitly generates a process inverse. The latter has been shown to lead to ZSE-optimal responses for step changes (Kravaris, 1988). How- ever, the control of nonminimum-phase nonlinear systems in this vein remains a major unmet challenge. In this work, a control law for second-order non- minimum-phase nonlinear systems will be developed, that leads to an ZSE-optimal closed-loop response for changes in the set point. The proposed methodology hopes to serve as a starting point for the development of a more general framework for the control of nonminimum-phase nonlinear systems and to moti- vate further research effort in this area. In Section 2 the characterization of nonminimum- phase behavior for second-order systems will be reviewed following the approach of Bymes and Isidori (1985). In Section 3, a nonlinear analog of the linear first-order all-pass will be introduced. Section 4 will review a standard result for ZSE-optimal state feedback control of linear second-order nonmini- mum-phase systems and will motivate the develop- ment that follows. In Section 5, a class of control laws will be developed that lead to a closed-loop response of a nonlinear first-order all-pass in series with a linear first-order lag. In Section 6, a particular control law from this class will be calculated, that results in ZSE-optimal closed-loop response in the limit as the time constant of the linear lag tends to zero. Finally, Section 7 will illustrate the application of the pro- posed control methodology and evaluate its perform- ance in a CSTR example.