Suboptimal Feedback Control Research Articles

Feedback controller design for linear and a class of nonlinear optimal control problems

SUMMARYThis paper presents a feedback controller designing approach for a large class of finite‐time optimal control problems. This approach involves a piecewise truncated variational iteration method (PTVIM) for solving the nonlinear Hamilton–Jacobi–Bellman equation. By using the finite iterations of PTVIM, an analytic approximate solution for value function and suboptimal feedback control law is obtained. Some illustrative examples are employed to demonstrate the accuracy and efficiency of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.

Dynamic state feedback and its application to linear optimal control

We investigate the linear optimal control problems in the context of dynamic state feedback configuration. The dynamic state feedback is a dual structure of the dynamic observer. In conjunction with the well known arguments on linear matrix differential and Lyapunov equations, we elicit the fact that the quadratic performance index is always computable with this configuration. Based on this property, we suggest a nonlinear optimization programming method to get suboptimal or near optimal time-invariant dynamic state feedback controls. One can also evaluate the efficacy of pre-designed dynamic state feedback controllers utilizing this property. Two illustrative examples are given to show the effectiveness of the proposed approach.

A suboptimal feedback control for nonlinear time-varying systems with continuous inequality constraints

A feedback control law, which is constructed as a first-order approximation to the optimal control, is proposed for nonlinear time-varying systems subject to continuous inequality constraints on the control and state. This control law is effective under small state perturbations caused by changes on initial conditions and/or modeling uncertainty.

Optimal Pulse Strategies for Relative Orbit Transfer Along a Circular Orbit

This paper is concernedwith a fuel-optimal relative orbit transfer problem forHill–Clohessy–Wiltshire equations. The input is of the pulse type and the performance index is the total velocity change required for a transfer. For the in-plane motion, the existence of optimal three-pulse strategies is shown and the minimum total velocity change is explicitly given under the condition that the difference of orbit size is relatively larger than that of drift velocity. For the out-of-plane motion, the existence of optimal single-pulse strategies is shown. The orbit transfer problem covers formation reconfiguration problem as a special case. Midpoints of the optimal pulses coincide with those points at which the velocity along the radial direction is zero. Using this information and null controllability with vanishing energy of the Hill–Clohessy–Wiltshire equations, suboptimal feedback controllers are designed. Simulation results with optimal open-loop strategies and feedback controls are given for two formation reconfiguration problems.

Suboptimal feedback control of flow over a sphere

Abstract In this study, we control unsteady motion in the wake behind a sphere using a suboptimal feedback control method based on the sensing of surface pressure. The cost function to be reduced is the square of the difference between the potential-flow and real pressures at the sphere surface. The actuation velocity (blowing/suction) is obtained from the suboptimal feedback control procedure. The sensing and actuation on the whole surface of the sphere is considered. This is an ideal case but provides a clear understanding of the effect of suboptimal feedback control on the present flow. We choose four different Reynolds numbers, Re = 100, 250, 300, and 425, covering four different flow regimes (steady axisymmetric, steady planar–symmetric, unsteady planar–symmetric, and unsteady asymmetric flows, respectively). With the present control, the vortex shedding disappears for Re = 300 and 425 and the drag is significantly reduced for all the Reynolds numbers considered.

Periodic Orbits of Nonlinear Relative Dynamics and Satellite Formation

In this paper, leader―follower formation and reconfiguration problems based on the periodic orbits of the nonlinear relative dynamics along a circular orbit are considered. First, initial conditions of coplanar and noncoplanar relative orbits are characterized by the initial true anomaly, mean motion, semimajor axis, eccentricity, and inclination angle of the follower's inertial orbit. Based on the property of null controllability with vanishing energy of the Hill―Clohessy―Wiltshire equations, L 1 suboptimal feedback controllers are designed via the linear quadratic regulator theory. Simulation results for three examples are given. A comparison with the reconfiguration problem based on the periodic orbits of the Hill―Clohessy―Wiltshire equations shows that replacement by nonlinear periodic orbits does not increase the L 1 -norm of the feedback control.

Rotational Motion Control by Feedback with Minimum L1-Norm

Rotational Motion Control by Feedback with Minimum L1-Norm

APPROXIMATE NONLINEAR OPTIMAL SDRE TRACKING CONTROL

The intrinsic nonlinearities of aircraft dynamics become exceptionally significant when there are demands for large maneuvers, causing aircraft to operate near the stability limits of the system, where series problems can be encountered, since the real issue becomes dealing with nonlinearities that dominate system behavior. This has eminent consequences in aerospace applications where catastrophic failure in high-performance aircraft must be prevented. Recently, a novel algorithm has been derived for nonlinear suboptimal tracking control, which can be realized for real-time computer implementation. The algorithm is based on the State-Dependent Riccati Equation (SDRE) strategy that has become well-known within the control community. This paper focuses on illustrating the application, computational advantage and validity of the proposed tracking control methodology on a realistic simulation example of a ducted fan engine model for high-performance thrust-vectored aircraft. The proposed method delivers a computationally simple, yet effective, algorithm for constructively synthesizing nonlinear suboptimal feedback controls for trajectory tracking problems.

Suboptimal Solution of a Cheap Control Problem for Linear Systems with Multiple State Delays

A finite-horizon linear-quadratic optimal control problem with multiple pointwise and distributed state delays in the dynamics is considered. The control cost in the performance index is small with respect to the state cost. An asymptotic solution of the singularly perturbed set of Riccati-type functional-differential equations, associated with the original control problem by the optimality conditions, is constructed and justified. Based on this solution, two types of suboptimal feedback control for the original problem are derived. These suboptimal controls are justified for two classes of the initial condition for the state variable.

Suboptimal feedback control of TCP flows in computer network using random early discard (RED) mechanism

We consider a dynamic model that simulates the interaction of TCP sources with active queue management system (AQM). We propose a modified version of an earlier dynamic model called RED. This is governed by a system of stochastic differential equations driven by a doubly stochastic point process with intensity as the control. The feedback control law proposed observes the router (queue) status and controls the intensity by sending congestion signals (warnings) to the sources for adjustment of their transmission rates. The (feedback) control laws used are of polynomial type (including linear) with adjustable coefficients. They are optimized by use of genetic algorithm (GA) and random recursive search (RRS) technique. The numerical results demonstrate that the proposed model and the method can improve the system performance significantly.

Suboptimal feedback control of turbulent flow over a backward-facing step

The objective of the present numerical study is to increase mixing in turbulent flow behind a backward-facing step using a systematic feedback control method. Spatially and temporally varying blowing and suction with zero-net mass flow rate are provided at the step edge, based on the sensing of the spanwise distribution of the wall pressure fluctuations at a downstream location. The cost functional to be increased is the root-mean-square spanwise pressure-gradient fluctuations at the sensing location, which may be associated with mixing behind the backward-facing step. Given the cost functional, the actuation at the step edge is determined through the suboptimal feedback control procedure of Choi et al. (1993). Large-eddy simulations of turbulent flow are conducted at a Reynolds number of 5100 based on the step height and free-stream velocity. The results of suboptimal feedback controls are compared with those of non-feedback single-frequency actuations. In case of the suboptimal control, velocity and vorticity fluctuations substantially increase downstream of the backward-facing step as well as in the recirculation zone. As a result, the reattachment length is significantly reduced, as compared to those of uncontrolled flow and flow with single-frequency actuations. A simple open-loop control method is devised from the suboptimal feedback control result, producing nearly the same mixing enhancement as the feedback control.

Fuzzy controller for a system with uncertain load

In many applications of motion control, problems associated with imprecisely measured or changing load (a mass or a moment of inertia) can be a serious obstacle in the formation of satisfactory controlling systems. This barrier compels the designer to include various kinds of uncertainties in engineering solutions. The present paper deals with the time-optimal control for mechanical systems with uncertain load. A fuzzy approach is used in the design of suboptimal feedback controllers, robust with respect to the load. The methodology proposed in this work may be easily adapted to other modeling uncertainties of mechanical systems, e.g. parameters of drive or motion resistance.

Optimal operation of CVT-based powertrains

Continuously Variable Transmissions (CVTs) are considered by many as a promising contribution to the improvement of the fuel economy of passenger cars. This paper discusses the fuel-optimal operation of a CVT-equipped powertrain in stationary and in transient conditions. The well-known optimal solution in the stationary case will be given only a short review discussing the importance of a correct CVT efficiency model. For transient vehicle operation so far only heuristic control strategies are known. This work aims to fill this gap. After detailed modelling of the complete powertrain and the introduction of a meaningful cost function, the resulting nonlinear optimal control problem is solved. The optimization is carried out for two different engines: a conventional SI engine and a downsized supercharged (DSC) SI engine. Each can be considered as a typical representative of the corresponding engine class. In both cases the optimal solutions are shown to be superior to standard CVT control algorithms, yielding larger gains in fuel economy for DSC engines. The optimal feedforward controls are then approximated by suboptimal feedback controllers that realize almost the same benefits in fuel economy at a substantially reduced computational effort. Finally, it is shown how the well-known ‘single track modified’ control approach can be changed such that it produces the same results as the suboptimal controller. Copyright © 2001 John Wiley & Sons, Ltd.

Optimal operation of CVT-based powertrains

Continuously Variable Transmissions (CVTs) are considered by many as a promising contribution to the improvement of the fuel economy of passenger cars. This paper discusses the fuel-optimal operation of a CVT-equipped powertrain in stationary and in transient conditions. The well-known optimal solution in the stationary case will be given only a short review discussing the importance of a correct CVT efficiency model. For transient vehicle operation so far only heuristic control strategies are known. This work aims to fill this gap. After detailed modelling of the complete powertrain and the introduction of a meaningful cost function, the resulting nonlinear optimal control problem is solved. The optimization is carried out for two different engines: a conventional SI engine and a downsized supercharged (DSC) SI engine. Each can be considered as a typical representative of the corresponding engine class. In both cases the optimal solutions are shown to be superior to standard CVT control algorithms, yielding larger gains in fuel economy for DSC engines. The optimal feedforward controls are then approximated by suboptimal feedback controllers that realize almost the same benefits in fuel economy at a substantially reduced computational effort. Finally, it is shown how the well-known ‘single track modified’ control approach can be changed such that it produces the same results as the suboptimal controller. Copyright © 2001 John Wiley & Sons, Ltd.

Fuzzy controller for mechanical systems

In many applications of motion control, the occurrence of nonlinear friction constitutes a fundamental obstacle in the design of satisfactory controlling systems. Since it is seldom possible to obtain a relatively accurate model of resistance to motion, a solution more and more often applied in practice is to introduce approaches that incorporate the inevitable imprecisions of the model in the form of uncertainties. This paper deals with the time-optimal (minimum-time) control for mechanical systems with a discontinuous and uncertain model of resistance to motion, A fuzzy approach is used in the design of suboptimal feedback controllers, convenient in practice thanks to their many advantages, especially in respect to robustness.

Suboptimal feedback control of vortex shedding at low Reynolds numbers

The objective of this study is to develop a method of controlling vortex shedding behind a bluff body using control theory. A suboptimal feedback control procedure for local sensing and local actuation is developed and applied to the flow behind a circular cylinder. The location of sensors for feedback is limited to the cylinder surface and the control input from actuators is the blowing and suction on the cylinder surface. Three different cost functionals to be minimized (J1 and J2) or maximized (J3) are investigated: J1 is proportional to the pressure drag of the cylinder, J2 is the square of the difference between the target pressure (inviscid flow pressure) and real flow pressure on the cylinder surface, and J3 is the square of the pressure gradient on the cylinder surface, respectively. Given the cost functionals, the flow variable to be measured by the sensors and the control input from the actuators are determined from the suboptimal feedback control procedure. Several cases for each cost functional have been numerically simulated at Re = 100 and 160 to investigate the performance of the control algorithm. For all actuations, vortex shedding becomes weak or disappears, and the mean drag and drag/lift fluctuations significantly decrease. For a given magnitude of the blowing/suction, reducing J2 provides the largest drag reduction among the three cost functionals.

Robust controller for fuzzy mechanical systems

The paper deals with the time-optimal (minimum-time) control for single-degree-of-freedom mechanical systems with a discontinuous model of resistance to motion. A fuzzy approach is used in the design of suboptimal feedback controllers, convenient in practice thanks to their many advantages, especially in respect to robustness.

Asymptotic Solution of a Cheap Control Problem with State Delay

A linear-quadratic optimal control problem with point and distributed delays in state variables and a small control cost in the performance index is considered. An asymptotic solution of the singularly perturbed set of Riccati-type functional-differential equations, associated with this problem, is constructed and justified. Based on this solution, the suboptimal feedback control law is derived. An accuracy of the corresponding value of the performance index is established.

A uniqueness result for the isaacs equation corresponding to nonlinear H∞ control

The dynamic programming equation (DPE) corresponding to nonlinear H∞ control is considered. When the cost grows quadratically in the state, it is well known that there may be an infinite number of viscosity solutions to the DPE. In fact, there may be more than one classical solution when a classical solution exists. For the case of fixed feedback control, it is shown that there exists a unique viscosity solution in the class of solutions meeting a certain growth condition, and a representation in terms of available storage is obtained. For the active control case, where the H∞ problem is represented by a differential game, a similar representation result is obtained under the assumption of existence of a suboptimal feedback control.

Suboptimal local feedback control for a class of constrained discrete time nonlinear control problems

In this paper, we consider a class of constrained discrete time optimal control problems involving general nonlinear dynamics with fixed terminal time. A method to solve the feedback control problem for a class of unconstrained continuous time nonlinear systems has been proposed previously. In that work, the solution is based on synthesizing an approximate suboptimal feedback controller locally in the neighbourhood of a certain nominal optimal trajectory. This paper expands on the same theme by considering problems involving discrete time systems. Taking advantage of the nature of discrete time systems, a further reduction on the computational effort of synthesising the feedback controller is made possible. Also, this paper extends the applicability of the method to constrained systems. For illustration, a numerical example is solved using the proposed method.