Under the overarching stipulation of computationally tractable control synthesis techniques, there are two specific contributions of our work. The first concerns the simultaneous inclusion of the following four different classes of constraints in control problems:• constraints on the control actions pointwise in time,• constraints on the states pointwise in time,• frequency constraints on the control trajectories, and • frequency constraints on the state trajectories.

In this article we present a geometric discretetime Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the controls in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first order necessary conditions for optimality, and leads to two-point boundary value problems that may be solved by shooting techniques to arrive at optimal trajectories. We validate our theoretical results with a numerical experiment on the attitude control of a spacecraft on the Lie group SO(3). I. IMost engineering systems are required to operate in a certain pre-defined region of the state and control spaces. For instance, since mechanical systems are inertial, mechanical actuators have natural limitations in terms of, e.g., the torque magnitudes and the operating frequencies. In the control literature these are known as control magnitude and frequency constraints, respectively. Control magnitudes must be limited, for instance, to prevent rapid movements of robotic arms for safety considerations [1]. Frequency constraints arise from a more subtle consideration. Consider, for instance, read/write operations in disk drives [2] where excitation of the actuator at flexible modes may result in erroneous read/write operations, attitude orientation manoeuvres of satellites fitted with flexible structures such as solar panels [3] may excite the natural frequencies of the flexible structures, leading to vibrations and structural damage unless the natural frequencies are avoided, etc. It is, therefore, desirable to eliminate certain frequencies from the spectra of the control functions of controlled systems at the control synthesis stage.Traditional attempts by control engineers to handle frequency constraints includes filtering of the actuating signal after control synthesis, or techniques of more recent vintage such as H ∞ control [4] that minimize a weighted combination of transfer functions in certain frequencies as part of the synthesis technique. Both these techniques suffer from their own problems: The former is ad hoc and based on the designer's intuition of the system and actuator, and the latter, though more systematic and incorporates penalties on the frequencies in an interval, still suffers from the inability to completely suppress a pre-specified set of frequencies.More importantly, none of these techniques is capable of incorporating hard bounds on the control actions and the states. Constraints on the control actions and the states are

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We present a geometric discrete-time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. This PMP gives first-order necessary conditions for optimality and leads to two-point boundary value problems that may be solved by numerical techniques to arrive at optimal trajectories. We demonstrate our theoretical results with numerical simulations on the optimal trajectory generation of a wheeled inverted pendulum and an attitude control problem of a spacecraft on the Lie group SO(3).