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1. Using local reachability sets in guiding mobile actuator and its orbiting sensors to approximate state feedback kernels in output control of PDEs

   https://doi.org/10.1109/CDC51059.2022.9993034  

   Demetriou, Michael A. ( December 2022 , 2022 IEEE 61st Conference on Decision and Control (CDC))

   This paper considers a class of distributed parameter systems that can be controlled by an actuator onboard a mobile platform. In order to avoid computational costs and control architecture complexity associated with a joint optimization of actuator guidance and control law, a suboptimal policy is proposed that significantly reduces the computational costs. By utilizing a continuous-discrete optimal control design, a mobile actuator moves to a new position at the beginning of a new time interval and resides for a prescribed time. Using the cost to go with variable lower limit, the optimization simplifies to solving algebraic Riccati equations instead of differential Riccati equations. Adding a hardware feature whereby the mobile sensors are constrained to stay within the proximity of the mobile actuator, a feedback kernel decomposition scheme is proposed to approximate a full state feedback controller by the weighted sum of sensor measurements. 

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2. Linear Optimal Regulation to Zero Dynamics

   https://doi.org/10.1109/TAC.2024.3389736  

   Ludeke, Taylor ; Iwasaki, Tetsuya ( October 2024 , IEEE Transactions on Automatic Control)

   A general problem encompassing output regulation and pattern generation can be formulated as the design of controllers to achieve convergence to a persistent trajectory within the zero dynamics on which an output vanishes. We develop an optimal control theory for such design by adding the requirement to minimize the H2 norm of a closed-loop transfer function. Within the framework of eigenstructure assignment, the optimal control is proven identical to the standard H2 control in form. However, the solution to the Riccati equation for the linear quadratic regulator is not stabilizing. Instead it partially stabilizes the closed-loop dynamics excluding the zero dynamics. The optimal control architecture is shown to have the feedback of the deviation from the subspace of the zero dynamics and the feedforward of the control input to remain in the subspace. 

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3. Linear-Quadratic Optimal Controls for Stochastic Volterra Integral Equations: Causal State Feedback and Path-Dependent Riccati Equations

   https://doi.org/10.1137/22M1492696  

   Wang, Hanxiao ; Yong, Jiongmin ; Zhou, Chao ( August 2023 , SIAM Journal on Control and Optimization)

   A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a type-II backward SVIE (BSVIE). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a type-III BSVIE is introduced whose adapted solution can be used to represent the adapted M-solution of the corresponding type-II BSVIE. Under certain conditions, it is proved that the path-dependent Riccati equation admits a unique solution, which means that the decoupling field for the optimality system is found. Therefore, a causal state feedback representation of the open-loop optimal control is constructed. An additional interesting finding is that when the control only appears in the diffusion term, not in the drift term of the state system, the causal state feedback reduces to a Markovian state feedback. 

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4. A Value Iteration Approach to Adaptive Optimal Control of Linear Time-Delay Systems

   https://doi.org/10.1016/j.ifacol.2023.10.524  

   Cui, Leilei ; Pang, Bo ; Jiang, Zhong-Ping ( October 2023 , IFAC-PapersOnLine)

   This paper presents a novel learning-based adaptive optimal controller design for linear time-delay systems described by delay differential equations (DDEs). A key strategy is to exploit the value iteration (VI) approach to solve the linear quadratic optimal control problem for time-delay systems. However, previous learning-based control methods are all exclusively devoted to discrete-time time-delay systems. In this article, we aim to fill in the gap by developing a learning-based VI approach to solve the infinite-dimensional algebraic Riccati equation (ARE) for continuous-time time-delay systems. One nice feature of the proposed VI approach is that an initial admissible controller is not required to start the algorithm. The efficacy of the proposed methodology is demonstrated by the example of autonomous driving. 

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5. Optimal Control Strategies for Systems with Input Delay using the PIE Framework

   https://doi.org/10.1016/j.ifacol.2022.11.339  

   Peet, Matthew M. ( January 2022 , IFAC-PapersOnLine)

   The Partial Integral Equation (PIE) framework provides a unified algebraic representation for use in analysis, control, and estimation of infinite-dimensional systems. However, the presence of input delays results in a PIE representation with dependence on the derivative of the control input, u˙. This dependence complicates the problem of optimal state-feedback control for systems with input delay – resulting in a bilinear optimization problem. In this paper, we present two strategies for convexification of the H∞-optimal state-feedback control problem for systems with input delay. In the first strategy, we use a generalization of Young's inequality to formulate a convex optimization problem, albeit with some conservatism. In the second strategy, we filter the actuator signal – introducing additional dynamics, but resulting in a convex optimization problem without conservatism. We compare these two optimal control strategies on four example problems, solving the optimization problem using the latest release of the PIETOOLS software package for analysis, control and simulation of PIEs. 

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