Approximate Solutions to Nonlinearly-Constrained Optimal Control Problems Philip S. Harvey and Henri P. Gavin Abstract— The use of variational methods in optimal control problems involves solving a two-point boundary-value problem (for states and costates) and satisfying an optimality condition. For problems with quadratic integral cost that have linear state dynamics and unconstrained controls, the co-state equations are also linear. Adjoining control constraints to the objective function introduces non linearity to the costate equation, and iterative numerical methods are required to converge upon the optimal control trajectory. The nonlinear costate terms arise at times in which the control constraints are active. In the numerical methodology proposed in this paper, an approxi- mately optimal solution is converged upon from a feasible sub- optimal initial control trajectory. In each iteration the control trajectory moves toward the unconstrained optimum solution while remaining feasible. Importantly, the state and costate equations are linear and the method is applied to a multi- input system designed to minimize the response of a vibration isolation system by adjusting only the damping characteristics of a variable damping device. I. INTRODUCTION In the time domain, the solution to nonlinear optimal control problems involves solving a two-point boundary value problem posed by the Euler-Lagrange equations [1]. For most nonlinear problems, solutions are iteratively con- verged upon. Problems involving inequality constraints on the controls and states are representative of a wide range of applications but remain open problems. For a quadratic performance index and constraints on states and controls, the solution consists of trajectories that are sometimes con- strained and are sometimes within their feasible space [2]. Various methods have been proposed to find the optimal solution for linearly constrained problems [3],[4],[5],[6]. One such method uses integral penalty functions to approximately enforce the constraint [7]. This method requires a choice of weighting constant and penalty function. If the weighting constant is too small the actual constraint may not be enforced. This paper presents a numerical method exhibiting monotonic convergence to the optimal control trajectories for a linear system with nonlinear sector bound constraints on the controls and states. Semi-active control is a class of control systems in which a small amount of external power is required to modulate mechanical properties of the actuators (i.e., stiffness and damping) [8],[9],[10],[11]. The circle criterion guarantees This work was not supported by any organization P.S. Harvey is with Department of Civil and Environmental Engineering, Duke Univ., Durham, NC 27708-0287, USA Philip.Harvey@duke.edu H.P. Gavin is with the Faculty of Department of Civil and En- vironmental Engineering, Duke Univ., Durham, NC 27708-0287, USA Henri.Gavin@duke.edu the absolute stability of semi-active control systems; the plant dynamics are typically linear with strictly positive-real transfer functions and the control forces are sector-bounded (within quadrants I and III in the velocity-force plane) [12]. Implementation of semi-active control involves controls act- ing through actuators that exhibit saturation limits. Therefore, the controls are sector-bounded. In this study, trajectories for optimal damping rates are calculated for equipment isolation systems that operate on the principle of a rolling pendulum. The isolated components are supported by large ball bearings (2 cm in diameter) that roll on rigid dish-shaped bowls with a quadratic pro- file. The period of motion is determined by the curvature of the dish, independent of the mass. Damping force is modulated in the isolation system in order to minimize a quadratic performance functional that weights total response accelerations and control efforts in order to improve the isolation system transmissibility at high frequencies while simultaneously suppressing resonant behavior. This method requires a priori knowledge of the disturbance and cannot be implemented in non-autonomous systems. However, from the optimal control trajectories, parameterized feedback control laws may be deduced. The economic impact of earthquakes depends not only on the performance of primary structural components, but also the performance of non-structural building contents. The serviceability of many important facilities (e.g., hospitals, emergency-response centers, computer centers etc.) depend on the functionality of non-structural components following an earthquake even when the facility’s structural system remains operational [13],[14],[15],[16]. Therefore, seismic hazards analyses of structures must include the effects dam- age to critical equipment [16],[17]. Failure of vibration- sensitive equipment is caused not only by overturning or toppling, but also by excessive displacements and/or large absolute equipment accelerations. For this reason vibration isolation systems are installed to mitigate the seismic risk posed to mission-critical equipment. Isolation systems can considerably reduce the base accel- eration transmitted to objects by mechanically decoupling the isolator from the ground [8],[17]. Seismic equipment isola- tion systems are typically of two types—friction-pendulum or rolling-pendulum [18],[19],[20]—with natural periods be- tween 2 and 4 s [13],[17]. During low-level seismic events, passive equipment isolation systems perform extremely well [9],[21],[22],[23]. Whereas, when subjected to earthquakes with high-amplitude near-fault ground motions, considerable amplification will produce excessive isolator displacements 2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 978-1-4577-0079-8/11/$26.00 ©2011 AACC 3122