Time‐ and State‐Dependent Input Delay‐Compensated Bang‐Bang Control of a Screw Extruder for 3D Printing

In this paper, a delay‐compensated bang‐bang control design methodology for the control of the nozzle output flow rate of screw extruder‐based three‐dimensional printing processes is developed. A geometrical decomposition of the screw extruder in a partially and a fully filled regions allows to describe the material convection in the extruder chamber by a one‐dimensional hyperbolic partial differential equation (PDE) coupled with an ordinary differential equation. After solving the hyperbolic PDE by the method of characteristics, the coupled PDE–ordinary differential equation's system is transformed into a nonlinear state‐dependent input delay system. The aforementioned delay system is extended to the non‐isothermal case with the consideration of periodic fluctuations acting on the material's convection speed, which represent the process variabilities due to temperature changes in the extruder chamber, resulting to a nonlinear system with an input delay that simultaneously depends on the state and the time variable. Global exponential stability of the nonlinear delay‐free plant is established under a piecewise exponential feedback controller that is designed. By combining the nominal, piecewise exponential feedback controller with nonlinear predictor feedback, the compensation of the time‐dependent and state‐dependent input delay of the extruder model is achieved. Global asymptotic stability of the closed‐loop system under the bang‐bang predictor feedback control law is established when certain conditions related to the extruder design and the material properties, as well as to the magnitude and frequency of the materials transport speed variations, are satisfied. Simulations results are presented to illustrate the effectiveness of the proposed control design. Copyright © 2017 John Wiley & Sons, Ltd.     

Robust Optimal Control of Nonlinear Systems With System Disturbance During Feedback Disruption

In this paper, a robust optimal control problem of nonlinear systems with system disturbance during feedback disruption is considered. This is an extended work of previous time‐delay optimal control results, by adding external disturbance in the considered system. It is shown that there exists an optimal input signal which keeps the performance error within the specified bound for the longest time. Then, it is shown that such an optimal input signal can be approximated by an implementable bang‐bang input signal in terms of control performance. Two examples are given for illustration.     

The Bang‐bang principle of time optimal controls for the Kuramoto–Sivashinsky–KdV equation with internal control

This paper is concerned with the time optimal control problem governed by the internal controlled Kuramoto–Sivashinsky–Korteweg‐de Vries equation, which describes many physical processes in motion of turbulence and other unstable process systems. We prove the existence of optimal controls with the help of the Carleman inequality, which has been widely used to obtain the local controllability or null controllability of parabolic differential systems. More precisely, with the help of the Carleman inequality, we obtain a relationship between the null controllability and time optimal control problem. Moreover, we give the bang‐bang principle for an optimal control of our original problem by using the one of approximate problems. This method is new for time optimal control problems. The bang‐bang principle established here seems also to be new for fourth‐order parabolic differential equations. Copyright © 2015 John Wiley & Sons, Ltd.     

An Improved Parameterized Controller Reduction Technique via New Frequency Weighted Model Reduction Formulation

In this paper, an improved parameterized controller reduction technique via a new frequency weighted model reduction formulation is developed for the feedback control of MIMO discrete time systems particularly for non‐unity feedback control system configurations which have the controller located in the feedback path. New frequency weights which are a function of a free parameter matrix are derived based on a set of equivalent block diagrams and this leads to a generalized double sided frequency weighted model reduction formulation. Solving this generalized double sided frequency weighted model reduction problem for various values of the free parameter results in obtaining controllers which correspond to each value of the free parameter. It is shown that the proposed formulation has a useful characteristic such that selecting a controller which corresponds to a large value of the free parameter results in obtaining an optimal reduced order controller and using this optimal reduced order controller in a closed loop system results in significant reduction in the infinity norm of the approximation error between the original closed loop system and the closed loop system which uses an optimal reduced order controller (when compared to existing frequency weighted model reduction methods).     

Fuzzy bang–bang relay control of a single-axis active magnetic bearing system

Active magnetic bearings (AMB) are presently being utilized in various classes of rotating machinery. Although the rotor-AMB systems are open loop unstable, they are easily stabilized using feedback control schemes of which the PID controller is the most commonly used. The PID controller is however only effective at the vicinity of the rotor’s equilibrium position where the dynamics of the rotor-AMB system is linearized. Significant deviation of the rotor’s motion from this equilibrium position may occur due to large imbalance forces. In this situation, the nonlinearity in AMBs, which arises from the relationship between the electromagnetic force, coil current and air gap, may render the PID controller ineffective. For the control of nonlinear systems, artificial intelligence techniques such as fuzzy and hybrid techniques are effective. In this paper, a new fuzzy controller is proposed for the control of a single-axis AMB system. This controller is based on the bang–bang scheme, which is an old but effective technique to control nonlinear systems in optimal time. The performance of the proposed integrated fuzzy bang–bang relay controller (FBBRC) was found to be superior to that of the optimized PD controller and the conventional fuzzy logic controller. Comparison of the FBBRC with the fuzzy logic controller cascaded with a hard limiter (FBBC) relay revealed almost equal performance. High frequency chattering was however observed in the steady-state response of the FBBC. Such chattering is known to cause instability and distortion in the amplifiers that are used to supply current to the magnetic bearing actuators.     

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.     

Fastest recovery after feedback disruption

The problem of quickly reducing operating errors during recovery from a feedback disruption is considered. The objective is to design controllers that reduce operating errors as quickly as possible, once feedback has been restored. It is shown that robust optimal feedback controllers that achieve this objective do exist. Furthermore, it is shown that the performance of optimal controllers can be approximated as closely as desired by controllers that generate bang–bang input signals for the controlled system. Controllers that generate bang–bang signals are relatively easy to derive and implement, since bang–bang signals are characterised by a finite list of scalars – their switching times.     

Two‐stage stability control for strict‐feedback systems with input delays and input nonlinear uncertainties

In this paper, a two‐stage control procedure is proposed for stabilization of a class of strict‐feedback systems with unknown constant time delays and nonlinear uncertainties in the input. A nominal controller is first designed to compensate input time delays without considering input nonlinear uncertainties. Extended from backstepping algorithm, input delay compensation is realized by means of predicted states that are computed through integration of cascaded system dynamics, making the nominal closed‐loop system asymptotically stable. Based on the nominal controller presented for the input delay system, a multi‐timescale system is subsequently developed to estimate the unknown input nonlinearity and make the estimate approach the nominal control input as fast as possible. It is proved that the proposed control scheme can make states of the strict‐feedback systems converge to zero and all the signals of the closed‐loop systems are guaranteed to be bounded in the presence of input time delays and nonlinear uncertainties. Simulation verification is carried out to illuminate the effectiveness of the proposed control approach.     

Prespecifiable fixed‐time control for a class of uncertain nonlinear systems in strict‐feedback form

In this paper, we investigate the prespecifiable fixed‐time control problem for a class of uncertain nonlinear systems in strict‐feedback form, where the settling (convergence) time is not only bounded but also user‐assignable in advance. One of the salient features of the proposed method lies in the fact that it makes it possible to achieve any practically allowable settling time by using a simple and effective control parameter selection recipe. Both fixed‐time stabilization and fixed‐time tracking are considered for uncertain strict‐feedback systems. Firstly, by adding exponential state feedback and using fractional power integration as Lyapunov function candidate, a global stabilizing control strategy is developed. It is proved that all the system states converge to zero within prespecified fixed‐time with continuous and bounded control action. Secondly, under more general uncertain nonlinearities and external disturbances, an adaptive fixed‐time controller is derived such that the tracking error converges to a small neighborhood of zero within preassigned time. Theoretical results are also illustrated and supported by simulation studies.     

Optimality and stability in a class of bang–bang controlled biochemical reaction systems

This paper deals with the optimization of biochemical reaction systems of rank one. Two optimization problems are solved: the problem of optimal operation for maximum productivity in steady state and the problem of the start-up to the optimal steady state. Application of Pontryagin's maximum principle shows that the controller is of the bang–bang type, with no singular intervals. The determination of the optimal switching surface involves the solution of a two point boundary value problem. Solving such a difficult problem is avoided by choosing candidate switching surfaces on a heuristic basis. This study shows that switching on the stability boundary of the nominal operating point corresponding to the maximum dilution rate is the best choice. Here the value of the cost index is minimum amongst the various switching surfaces considered and the stability boundary satisfies the conditions imposed on a candidate switching surface for proper operation. Simple, robust algorithms are formulated for accurately estimating the system's stability boundary. The obtained results display the influence of feedback control on the stability of the set point. The bang–bang controller substantially increases the set point's region of attraction in state space as compared to the uncontrolled bioreactor.     

Global stabilization of uncertain stochastic nonlinear time‐delay systems by output feedback

Constructive control techniques have been proposed for controlling strict feedback (lower triangular form) stochastic nonlinear systems with a time‐varying time delay in the state. The uncertain nonlinearities are assumed to be bounded by polynomial functions of the outputs multiplied by unmeasured states or delayed states. The delay‐independent output feedback controller making the closed‐loop system globally asymptotically stable is explicitly constructed by using a linear dynamic high‐gain observer in combination with a linear dynamic high‐gain controller. A simulation example is given to demonstrate the effectiveness of the proposed design procedure. Copyright © 2007 John Wiley & Sons, Ltd.     

On the convergence of nonlinear optimal control using pseudospectral methods for feedback linearizable systems

We consider the optimal control of feedback linearizable dynamic systems subject to mixed state and control constraints. In contrast to the existing results, the optimal controller addressed in this paper is allowed to be discontinuous. This generalization requires a substantial modification to the existing convergence analysis in terms of both the framework as well as the notion of convergence around points of discontinuity. Although the nonlinear system is assumed to be feedback linearizable, the optimal control does not necessarily linearize the dynamics. Such problems frequently arise in astronautical applications where stringent performance requirements demand optimality over feedback linearizing controls. We prove that a sequence of solutions obtained using the Legendre pseudospectral method converges to the optimal solution of the continuous‐time problem under mild conditions. Published in 2007 by John Wiley & Sons, Ltd.     

Energy dispatch controllers for a photovoltaic system

In this paper two energy dispatch controllers for use in a grid-independent photovoltaic (PV) system are presented. The first, an optimal energy dispatch controller, is based on a class of Adaptive Critic Designs (ACDs) called Action Dependent Heuristic Dynamic Programming (ADHDP). This class of ACDs uses two neural networks to evolve an optimal control strategy over time. The first neural network or “Action” network dispenses the actual control signals while the second network or “Critic” network uses these control signals along with the system states to provide feedback to the action network, measuring performance using a utility function. This feedback loop allows the action network to improve behavior over time. The optimal energy dispatcher places emphasis on always meeting the critical load, followed by keeping the charge of the battery as high as possible so as to be able to power the critical load in cases of extended low output from the PV array, and lastly to power the non-critical load in so far as to not interfere with the first two objectives. The second energy dispatch controller is a smart energy dispatch controller and is built using knowledge from an expert, codified into a series of static rules. This smart energy dispatch controller is called the “PV-priority 2” controller. These energy dispatchers are compared with a static scheme called the “PV-priority 1”. The PV-priority 1 controller represents the standard control strategy. Results show that the ADHDP-based optimal energy dispatcher (or controller) outperforms the standard PV-priority 1 energy dispatcher in meeting the stated objectives, but trails the PV-priority 2 energy dispatcher. However, the major advantage of the ADHDP controller is that no expert is required for designing the controller, whereas for a rule-based controller such as the PV-priority 2 controller, an expert is always required.     

正则/非正则最优控制方法在最优趋同控制的应用

趋同控制已得到广泛研究并取得重要进展,但具有最优指标约束的趋同控制仍然面临挑战.基于此,研究多智能体系统最优趋同控制问题,基于正则/非正则最优控制方法提出新型分布式控制协议,使得半正定加权矩阵性能指标最小化的同时系统能够达到趋同.特别地,所设计的控制协议由两部分组成:一部分是个体状态的反馈,目的是最小化性能指标;另一部分是相邻个体之间相对状态的反馈,目的是保证趋同.     

Robust optimal control during feedback disruption for nonlinear systems controlled by an observer-based output feedback controller

In this paper, a robust optimal control problem during feedback disruption is considered for a class of nonlinear systems which have been controlled by an observer-based output feedback controller. It is shown that during feedback disruption, there exists an optimal control input which keeps both system states and observer errors within a specified bound for the longest time. Then, it is shown that such an optimal control input can be practically implemented by using a bang-bang control input in terms of control performance. One numerical and one practical examples are given for clear illustration.     

分段变结构Bang-Bang控制器在气动脉宽调制位置伺服系统中的研究

本文提出了一种分段变结构Bang-Bang控制方法,并应用于气动脉宽调制高速开关阀 控气缸位置伺服系统,使得控制性能得到了明显改善,有实际应用价值。     

Minimax control in substate space of a system with process uncertainty

The control of a single input-single output plant with parameter uncertainty has been approached using a minimax technique. A linear time invariant controller results which requires only partial state feedback and is optimal in the sense that it minimizes a quadratic criterion involving tracking error, control, and parameter uncertainty. Bounded input-bounded output stability is guaranteed provided the transfer function has only left half plane zeros. If uncertainties are bounded, it is always possible to stabilize the system when sufficient control amplitude is available. The number of states required to generate the control is equal to the system order less the number of zeros.     

Continuous output‐feedback finite‐time control for a class of second‐order nonlinear systems with disturbances

In this paper, a solution to the continuous output‐feedback finite‐time control problem is proposed for a class of second‐order MIMO nonlinear systems with disturbances. First, a continuous finite‐time controller is designed to stabilize system states at equilibrium points in finite time, which is proven correct by a constructive Lyapunov function. Next, because only the measured output is available for feedback, a continuous nonlinear observer is presented to reconstruct the total states in finite time and estimate the unknown disturbances. Then, a continuous output‐feedback finite‐time controller is proposed to track the desired trajectory accurately or alternatively converge to an arbitrarily small region in finite time. Finally, proposed methods are applied to robotic manipulators, and simulations are given to illustrate the applicability of the proposed control approach. Copyright © 2015 John Wiley & Sons, Ltd.     

Fast terminal sliding‐mode finite‐time tracking control with differential evolution optimization algorithm using integral chain differentiator in uncertain nonlinear systems

This paper presents a fast terminal sliding‐mode tracking control for a class of uncertain nonlinear systems with unknown parameters and system states combined with time‐varying disturbances. Fast terminal sliding‐mode finite‐time tracking systems based on differential evolution algorithms incorporate an integral chain differentiator (ICD) to feedback systems for the estimation of the unknown system states. The differential evolution optimization algorithm using ICD is also applied to a tracking controller, which provides unknown parametric estimation in the limitation of unknown system states for trajectory tracking. The ICD in the tracking systems strengthens the tracking controller robustness for the disturbances by filtering noises. As a powerful finite‐time control effort, the fast terminal sliding‐mode tracking control guarantees that all tracking errors rapidly converge to the origin. The effectiveness of the proposed approach is verified via simulations, and the results exhibit high‐precision output tracking performance in uncertain nonlinear systems.     

输入随机的不可靠柔性制造系统的反馈控制

对于具有随机输入和随机需求的一类不可靠柔性制造系统,利用转移率一致化技术和随机动态规划方法,给出了输入率和服务率分配的最优反馈控制策略,指出系统的最优控制具有bang-bang形式的天关结构,数值例子验证了文中的结果。