Suboptimal Feedback Control Research Articles

Stochastic Optimal Linear Control for Generalized Cost Functions With Time-Invariant Stochastic Parameters.

This study presents the design of feedback controllers for generalized cost functions to deal with stochastic optimal control problems. Target linear systems contain time-invariant stochastic parameters that describe system uncertainty. The cost functions involve nonlinear mappings and polynomial forms of the system states and inputs to express various performance metrics. Unfortunately, these properties cause difficulties in solving the problems. Conventional methods, such as the principle of optimality, are not employed to solve such problems owing to the time-invariant parameters. As opposed to the well-known quadratic functions, handling the generalized cost functions is a complicated task. This study overcomes these challenges by deriving an explicit relation between the cost function and the linear feedback gain of a controller. The derived relation enables the feedback gain to be optimized via a gradient method. A theoretical analysis ensures the convergence of the proposed gradient method. A suboptimal feedback controller is obtained to solve the problem, even for the generalized cost. Furthermore, the controller guarantees robust stability of the feedback system even with the stochastic parameters. It is demonstrated that the proposed cost function can express an expectation of a quadratic cost, risk-sensitive cost, polynomial cost, and input-to-state gain. A numerical simulation shows the effectiveness of the proposed method.

Reduced order model predictive control for parametrized parabolic partial differential equations

Model Predictive Control (MPC) is a well-established approach to solve infinite horizon optimal control problems. Since optimization over an infinite time horizon is generally infeasible, MPC determines a suboptimal feedback control by repeatedly solving finite time optimal control problems. Although MPC has been successfully used in many applications, applying MPC to large-scale systems – arising, e.g., through discretization of partial differential equations – requires the solution of high-dimensional optimization problems and thus poses immense computational effort. We consider systems governed by parametrized parabolic partial differential equations and employ the reduced basis method as a low-dimensional surrogate model for the finite time optimal control problem. The reduced order optimal control serves as feedback control for the original large-scale system. We analyze the proposed RB-MPC approach by first developing a posteriori error bounds for the errors in the optimal control and associated cost functional. These bounds can be evaluated efficiently in an offline-online computational procedure and allow us to guarantee asymptotic stability of the closed-loop system using the RB-MPC approach in several practical scenarios. We also propose an adaptive strategy to choose the prediction horizon of the finite time optimal control problem. Numerical results are presented to illustrate the theoretical properties of our approach.

State-dependent Riccati equation feedback stabilization for nonlinear PDEs

The synthesis of suboptimal feedback laws for controlling nonlinear dynamics arising from semi-discretized PDEs is studied. An approach based on the State-dependent Riccati Equation (SDRE) is presented for 2 and ∞ control problems. Depending on the nonlinearity and the dimension of the resulting problem, offline, online, and hybrid offline-online alternatives to the SDRE synthesis are proposed. The hybrid offline-online SDRE method reduces to the sequential solution of Lyapunov equations, effectively enabling the computation of suboptimal feedback controls for two-dimensional PDEs. Numerical tests for the Sine-Gordon, degenerate Zeldovich, and viscous Burgers’ PDEs are presented, providing a thorough experimental assessment of the proposed methodology.

Robust suboptimal feedback control for a fed-batch nonlinear time-delayed switched system

The optimal control strategy constructed in the form of a state feedback is effective for small state perturbations caused by changes in modeling uncertainty. In this paper, we investigate a robust suboptimal feedback control (RSPFC) problem governed by a nonlinear time-delayed switched system with uncertain time delay arising in a 1,3-propanediol (1,3-PD) microbial fed-batch process. The feedback control strategy is designed based on the radial basis function to balance the two (possibly competing) objectives: (i) the system performance (concentration of 1,3-PD at the terminal time of the fermentation) is to be optimal; and (ii) the system sensitivity (the system performance with respect to the uncertainty of the time-delay) is to be minimized. The RSPFC problem is subject to the continuous state inequality constraints. An exact penalty method and a novel time scaling transformation approach are used to transform the RSPFC problem into the one subject only to box constraints. The resulting problem is solved by a hybrid optimization algorithm based on a filled function method and a gradient-based algorithm. Numerical results are given to verify the effectiveness of the developed hybrid optimization algorithm.

Optimal production of microalgae in the presence of grazers

Zooplankton contamination represents a major constraint in large-scale microalgal cultivation systems. While zooplankton contamination cannot be avoided, their development can be controlled by regulating the dilution rate. However, it is not straightforward to find the best control strategy for the dilution rate. Low dilution rates (or long retention times) favor grazer development and high dilution rates avoid their establishment at the risk of reducing microalgal productivity. Furthermore, the presence of periodic regimes arising from the interaction predator–prey makes it unclear if the presence of grazers must be completely avoided. In this paper, we study the role of the dilution rate in the control of zooplankton populations and in the optimization of biomass productivity. We show that in the long-term operation (static optimal control problem or SOCP), the optimal constant dilution rate must ensure the eradication of the zooplankton population. In the case of time-varying dilution rate, we numerically solve an optimal control problem (OCP) over a finite interval of time. We find that the optimal solution approaches the solution for the SOCP most of the time, except when zooplankton actively avoid the pond outflow. Based on these results, we propose a simple sub-optimal feedback control that approximately matches the solution of the OCP when the initial concentration of grazers is low.

Impedance control as an optimal control problem: a novel formulation of impedance controllers as a subcase of optimal control

A formulation of impedance control for redundant manipulators is presented here as a particular case of an optimal control problem. This allows the design of an impedance controller that benefits from the stability and efficiency of an optimal controller. To circumvent the high computational costs of finding optimal controllers, a suboptimal feedback controller based on state-dependent Riccati equations (SDRE) is developed. This approach is compared with the quadratic programming (QP) formulation, commonly used to solve redundancy of robotic manipulators. Numerical simulations of a redundant planar four-DOF serial-link manipulator show that the SDRE controller offers superior performance over the QP one in terms of stability, performance and required control effort.

Low-Energy Sub-Optimal Low-Thrust Trajectories to Libration Points and Halo-Orbits

The paper considers the problem of calculating direct and low-energy, low-thrust trajectories to the libration points of the Earth–Moon system and to halo-orbits. A method for solving the problem is proposed. It consists of calculating stable manifolds of libration points or halo-orbits and calculating a low-thrust trajectory from an initial circular Earth orbit to the given point of this manifold using sub-optimal feedback control. With a fixed final mass of a spacecraft, the last stage of calculations is reduced to solving the Cauchy problem. The numerical examples are given for the calculation of direct and low-energy trajectories to libration points and to halo-orbits and the optimization of entry points to stable manifolds for low-energy trajectories.

Stabilization of a class of nonlinear control systems via a neural network scheme with convergence analysis

In this paper, the stability of a class of nonlinear control systems is analyzed. We first construct an optimal control problem by inserting a suitable performance index; this problem is referred to as an infinite horizon problem. By a suitable change of variable, the infinite horizon problem is reduced to a finite horizon problem. We then present a feedback controller designing approach for the obtained finite horizon control problem. This approach involves a neural network scheme for solving the nonlinear Hamilton Jacobi Bellman equation. By using the neural network method, an analytic approximate solution for value function and a suboptimal feedback control law are achieved. A learning algorithm based on a dynamic optimization scheme with stability and convergence properties is also provided. Some illustrative examples are employed to demonstrate the accuracy and efficiency of the proposed plan. As a real-life application in engineering, the stabilization of a micro-electromechanical system is studied.

Solving multiobjective random interval programming problems by a capable neural network framework

In this paper, the stability of a class of nonlinear control systems is analyzed. We first construct an optimal control problem by inserting a suitable performance index, which this problem is referred to as an infinite horizon problem. By a suitable change of variable, the infinite horizon problem is reduced to a finite horizon problem. We then present a feedback controller designing approach for the obtained finite horizon control problem. This approach involves a neural network scheme for solving the nonlinear Hamilton Jacobi Bellman (HJB) equation. By using the neural network method, an analytic approximate solution for value function and suboptimal feedback control law is achieved. A learning algorithm based on a dynamic optimization scheme with stability and convergence properties is also provided. Some illustrative examples are employed to demonstrate the accuracy and efficiency of the proposed plan. As a real life application in engineering, the stabilization of a micro electro mechanical system is studied.

A separation theorem for guaranteed H2 performance through matrix inequalities

The usage of convex optimisation programs that leverage linear matrix inequalities allows for numerical solutions to the design of output-feedback controllers with guaranteed H2 performance. As decreed by the classical separation theorem for the related LQG control problem, the H2 control problem admits an optimal solution in terms of those of the separate optimal state-estimation and state-feedback design problems. This work details a new and alternative proof of this separation theorem. The proof builds on techniques for (linear) matrix inequalities and shows, in particular, that feasible but sub-optimal solutions of the state-feedback and the state-estimation problem yield a sub-optimal output feedback controller with guaranteed H2 performance.

Real-time collision avoidance maneuvers for spacecraft proximity operations via discrete-time Hamilton–Jacobi theory

This study presents a sub-optimal feedback control that implements real-time collision avoidance for spacecraft in proximity operations. The penalty function for avoiding collision with obstacles is first incorporated into the performance index of a typical optimal tracking problem. Then, the infinite-horizon feedback control law is derived by employing generating functions in the framework of discrete-time Hamilton–Jacobi theory. The derived control law, which is an explicit function of the reference states and instantaneous positions of obstacles, allows active spacecraft to avoid collision in real-time. The proposed approach has advantages over conventional optimal collision avoidance approaches in that it does not require iterations with initial guesses, repetitive shooting-based process for multiple boundary conditions, and/or trajectories of obstacles to be known a priori. Numerical simulations demonstrate that the proposed algorithm with a properly designed penalty function is suitable for implementing optimal collision-free transfers in real-time.

Optimal resource allocation for bacterial growth with degradation

Understanding adaptation of bacterial growth for a changing environment is a fundamental biological problem which also has a great interest from the biotechnological point of view. This paper extends one of existing bacterial growth models by taking macromolecular degradation into account and provides new mathematical results. The related dynamic problem of maximizing biomass accumulation is stated in a specific way so that existence of optimal resource allocation strategies can be verified. A novel numerical algorithm for approximating switching curves of the chattering control in the state space is developed. Several realistic suboptimal feedback control laws are also constructed and successfully tested. The results of numerical simulations confirm validity of the new problem statement.

An Iterative Method for Nonlinear Stochastic Optimal Control Based on Path Integrals

This paper proposes a new iterative solution method for nonlinear stochastic optimal control problems based on path integral analysis. First, we provide an iteration law for solving a stochastic Hamilton-Jacobi-Bellman (SHJB) equation associated to this problem, which is a nonlinear partial differential equation (PDE) of second order. Each iteration procedure of the proposed method is represented by a Cauchy problem for a linear parabolic PDE, and its explicit solution is given by the Feynman-Kac formula. Second, we derive a suboptimal feedback controller at each iteration by using the path integral analysis. Third, the convergence property of the proposed method is investigated. Here, some conditions are provided so that the sequence of solutions for the proposed iteration converges, and the SHJB equation is satisfied. Finally, numerical simulations demonstrate the effectiveness of the proposed method.

Suboptimal Feedback Control of PDEs by Solving HJB Equations on Adaptive Sparse Grids

An approach to solve finite time horizon suboptimal feedback control problems for partial differential equations is proposed by solving dynamic programming equations on adaptive sparse grids. A semi-discrete optimal control problem is introduced and the feedback control is derived from the corresponding value function. The value function can be characterized as the solution of an evolutionary Hamilton---Jacobi Bellman (HJB) equation which is defined over a state space whose dimension is equal to the dimension of the underlying semi-discrete system. Besides a low dimensional semi-discretization it is important to solve the HJB equation efficiently to address the curse of dimensionality. We propose to apply a semi-Lagrangian scheme using spatially adaptive sparse grids. Sparse grids allow the discretization of the value functions in (higher) space dimensions since the curse of dimensionality of full grid methods arises to a much smaller extent. For additional efficiency an adaptive grid refinement procedure is explored. The approach is illustrated for the wave equation and an extension to equations of Schrodinger type is indicated. We present several numerical examples studying the effect the parameters characterizing the sparse grid have on the accuracy of the value function and the optimal trajectory.

Distributed Control of the Stochastic Burgers Equation with Random Input Data

AbstractWe discuss a control problem involving a stochastic Burgers equation with a random diffusion coefficient. Numerical schemes are developed, involving the finite element method for the spatial discretisation and the sparse grid stochastic collocation method in the random parameter space. We also use these schemes to compute closed-loop suboptimal state feedback control. Several numerical experiments are performed, to demonstrate the efficiency and plausibility of our approximation methods for the stochastic Burgers equation and the related control problem.

モデル縮約に基づく非線形偏微分方程式の最適フィードバック制御

This paper proposes a suboptimal feedback controller design for nonlinear distributed parameter systems via stable manifold method. Stable manifold method provides approximate stabilizing solutions of Hamilton-Jacobi equations in nonlinear optimal control theory. We apply this method to a reduced-order system obtained from proper orthogonal decomposition and Galerkin projection. The feasibility of the design is demonstrated by a numerical example of the feedback controls for the viscous Burgers' equation.

Suboptimal Feedback Control of Nonlinear Distributed Parameter Systems by Stable Manifold Method

Abstract This paper proposes a suboptimal feedback controller design for nonlinear distributed parameter systems via the stable manifold method. The stable manifold method provides approximate stabilizing solutions of Hamilton-Jacobi (HJ) equations in nonlinear optimal control theory. We apply this method to a reduced-order system obtained from the proper orthogonal decomposition (POD) and Galerkin projection. The feasibility of the design is demonstrated by the numerical example of feedback controls of the viscous Burgers' equation.

3A11 Vibration and Control of Active Isolator taking Dynamical Property of Elastic Load into Account(The 12th International Conference on Motion and Vibration Control)

Ultra precision apparatus like semiconductor producing equipment needs the high accuracy at manufacturing. Therefore active isolation systems to avoid vibration transmitted from a vibration source to the precision apparatus are very important. This paper deals with the control system design for active isolation table^<[1][2][3]>. It aims at controlling vibration of the installed object and isolation table simultaneously. An experimental isolation table with flexible loaded object is built. The experimental system is modeled according to the reduced order physical modeling method. The parameters of the table are experimentally identified. Using the obtained model, control simulations are carried out by using feed-back controller designed according to sub-optimal control theory, absolute velocity feed-back control theory and feed-forward controller. The sub-optimal feed-back controller achieved good vibration damping effect, while the feed-forward controller effectively suppressed vibration transmission from ground. In order to evaluate the control performance on experimental system, control experiments are carried out and compared with simulation results. They agreed well and the effectiveness of presented modeling and controller design procedure is confirmed.

Order reduction via balancing and suboptimal control of a fuel cell – Reformer system

In this paper we first perform system balancing of an eighth-order mathematical model of a polymer electrolyte membrane fuel cell (PEMFC) dynamic coupled with a tenth-order mathematical model of a hydrogen gas reformer. Based on that information we determine reduced-order mathematical models of the original eighteen-order model by eliminating state variables that have negligible contribution to the model dynamics. Having obtained the reduced-order models, we study their step and impulse responses, and compare them to those of the original full-order model. In addition, we design corresponding suboptimal feedback controllers based on the reduced-order models. Comparing the obtained suboptimal controllers (that require a reduced number (only six or even five) of feedback loops making them easy for implementation) we find that their suboptimal performances are very close to the optimal performance of the full-state optimal feedback controller. It is important to emphasize that the full-order state feedback controller requires the same number of the feedback loops as the dimension of the original full-order state space model (in this case eighteen), which makes it complex and sometimes impractical to implement.

A separation theorem for stochastic singular linear quadratic control problem with partial information

In this paper, we provide a separation theorem for the singular linear quadratic (LQ) control problem of Ito-type linear systems in the case of the state being partially observable. Above all, the Kalman-Bucy filtering of the dynamics is given by means of Girsanov transformation, by which the suboptimal feedback control of the LQ problem is determined. Furthermore, it is shown that the well-posedness of the LQ problem is equivalent to the solvability of a generalized differential Riccati equation (GDRE).