(657e) Optimal Closest-Feasible-Points Non-Invariance Compensators

In 2018, we [1, 2] introduced a new system property called the closest feasible points (CFP) invariance to characterize input-constrained systems, and considered several implications of the system property. Systems that possess this invariance property include diagonal matrices and completely decentralized (decoupled) time-invariant linear systems. Also, a system that has a nonsingular characteristic (decoupling) matrix and can be made diagonal by row or column rearrangements, has this invariance property. However, a single-input single-output system may not possess this property. This system property has implications and applications in process control [1, 2], where actuator saturation is common. For example, in a plant that does not have this invariance property and is controlled with an analytical (non-model predictive) controller, the closed-loop performance may degrade considerably when an actuator saturates.

When a control signal (controller output), w, calculated by a controller that is not a solution to a constrained optimal control problem, is sent to actuators, the actuators implement the control signal as it is, only if the control signal is within the lower and upper limits of the actuators. Otherwise, at least one actuator saturates; that is, it clips the control signal component before applying the signal to the plant. In this case, the control system performance may degrade significantly due to two phenomena: (a) integral windup when the controller is dynamical; and (b) CFP non-invariance (CFPN) when closest feasible points are not invariant under the plant that is subjected to control. The former phenomenon is caused by the state variables of the dynamical controller not being properly informed of the actual controller action applied to the plant under control [3, 4]. The latter phenomenon is due to the output response of the plant to sat(w) not being closet to the output response of the plant to w. To decrease the former control performance degradation, many different anti-windup compensators have been proposed to properly inform the states of a controller, of the actual controller action that is applied to the plant under control [3, 4].

In this paper, the definition of the CFP invariance is used to derive optimal CFPN compensators that gracefully decrease the control performance degradation caused by the CFPN of plants. Two classes (discrete-time and continuous-time) nonlinear dynamical systems that do not have the CFP invariance property are considered. Several plant examples are simulated numerically to show the application and performance of the optimal CFPN compensators.

References

[1] Soroush, M., “Closest Feasible Point Invariance: a New System Property to Characterize Input-Constrained Systems,” AIChE Annual Meeting, Pittsburgh, PA (2018)

[2] Soroush, M., “Closest Feasible Points Invariance: a System Property to Characterize Input-Constrained Systems,” IEEE Transactions on Auto. Control, under review (2018)

[3] Sajjadi-Kia, S., and Faryar Jabbari, “Multi-stage Anti-windup Compensation for Open-loop Stable Plants,” IEEE Transactions on Auto. Control, 56(9), 2166–2172 (2011)

[4] Kapoor, N., and P. Daoutidis, “An Observer-based Anti-windup Scheme for Non-linear Systems with Input Constraints,” International Journal of Control, 72(1),18–29 (1999)