Minimizing control variation in discrete-time optimal control problems

• We present a gradient-based method for discrete-time optimal control problems.
• The cost function is the sum of terminal cost and the variation of control signal.
• A smooth transformation and the constraint transcription technique are used.
• Two examples are provided to demonstrate the feasibility of the method.

For a real practical system, a large fluctuation in the control signal is highly undesirable. To address this undesirable situation, we investigate a discrete-time optimal control problem subject to terminal state and all-time-step constraints on the state and control, where the cost function is the sum of terminal cost and the variation of the control signal. The variation of the control signal is expressed in terms of absolute value functions and hence is non-smooth. By a novel smooth transformation and the constraint transcription technique, this problem is approximated by a constrained discrete-time optimal control with the new cost function involves only smooth functions. A gradient-based computational method is then derived, which is supported by rigorous convergence analysis. Two examples are provided to demonstrate the effectiveness and advantages of the proposed method.