System sensitivity for optimal problems with singular control

Extensions of the differential performance index sensitivity to small parameter variations are presented in this paper. The system considered is assumed to be linear with free end time and initial and final target manifolds.

Results, previously reported for systems with bang.bang type implementation or control law resulting in from a minimum-fuel type performance index implementation. were all subject to the normality condition constraint. In this paper the normality requirements were removed from the system considered, thus extending the results to the singular control case. A universal expression thus presented which relates the variations of the performance index to that of the initial and final manifolds and a term that represents a physical constant, related to the Hamiltonian and which is independent of the way by which the optimal control was implemented.     

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.     

Optimal control of batteries with fully and partially available rechargeability

Motivated by the increasing dependence of many systems on battery energy, we study the problem of optimally controlling how to discharge and recharge a non-ideal battery so as to maximize the work it can perform over a given time period and still maintain a desired final energy level. Modeling a battery as a dynamic system, we adopt a Kinetic Battery Model (KBM) and formulate a finite-horizon optimal control problem when recharging is always feasible under the constraint that discharging and recharging cannot occur at the same time. The solution is shown to be of bang–bang type with the property that the battery is always in recharging mode during the last part of the interval. When the length of the time horizon exceeds a critical value, we also show that the optimal policy includes chattering. Numerical results are included to illustrate our analysis. We then extend the problem to settings where recharging is only occasionally feasible and show that it can be reduced to a nonlinear optimization problem which can be solved at least numerically.     

Modified bang‐bang controller for maximal and minimal time optimal control problems

Recently, there have been a series of results regarding two time optimal control problems for a class of linear and nonlinear systems ‐ one is to keep the system states within certain bound for the longest time during feedback disruption and the other is to derive the system states to near the origin as fast as possible after feedback recovery, both under bounded control inputs. These are called maximal and minimal time optimal control problems, respectively. In the existing results, a bang‐bang controller has been commonly suggested as the actual implementation of the optimal controller. In this paper, we suggest a modified version of the bang‐bang controller which can also serve as an approximate optimal controller. Our proposed controller provides the (near) optimal performance with (i) possible reduction of a number of switchings; (ii) possible reduction of control input magnitude.     

Time optimal control of second-order systems with transport lag†

The time optimal control of particular second–order systems with constant transport lags is determined by application of Pontryagin's Maximum Principle. A system with a delayed control and a system with a delayed state are considered. The optimal control for these systems is compared with the optimal control of a similar system which does not contain transport lag. It is found that the optimal control is essentially bang–bang but that it is non–unique under certain conditions.     

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.     

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.     

Enhancement of genuine multipartite entanglement and purity of three qubits under decoherence via bang–bang pulses with finite period

We propose a scheme to control the dynamics of genuine multipartite entanglement and purity of qubits within spatially separated thermal baths using the bang–bang pulses with finite period. The qubits are initially entangled and have no direct interactions. The genuine multipartite entanglement of the system is measured by an entanglement monotone based on a generalization of the Peres–Horodecki criterion to multipartite systems. We first derive a master equation to describe the non-Markovian dynamics of an arbitrary number of qubits within their baths with decoherence and dynamical decoupling. Then, we calculate the entanglement monotone and purity of three qubits in super-Ohmic, sub-Ohmic, and Ohmic baths numerically. The effects of the period of pulses on the non-Markovian dynamics of qubits are discussed. We show the genuine multipartite entanglement and purity can be simultaneously improved by applying the bang–bang pulses with finite period. In particular, the bang–bang pulses with finite period are more efficient when the qubits are put into the sub-Ohmic or Ohmic baths than the case of the super-Ohmic bath.     

Bang–bang control model with optimistic value criterion for uncertain switched systems

This paper studies an optimal control problem for uncertain switched linear systems with subsystems perturbed by uncertainty. A model for this problem is investigated with optimistic value criterion. The goal is to jointly design a deterministic switching law and a continuous feedback to optimize an uncertain objective function. A two-stage algorithm is applied to handle such model. In the first stage, the maximum value of the objective function and the bang–bang control are obtained under fixed switching instants, and in the second stage, GA and PSO algorithm are used to get the optimal switching instants, respectively. An example is shown to validate the method.     

Optimal strategies for biomass productivity maximization in a photobioreactor using natural light

We address the question of the optimization of the microalgal biomass long term productivity in the framework of production in photobioreactors under the influence of day/night cycles. For that, we propose a simple bioreactor model accounting for light attenuation in the reactor due to biomass density and we obtain the control law that optimizes productivity over a single day through the application of Pontryagin’s maximum principle. The dilution rate is the main control, the input concentration being only used as the secondary control to maintain the substrate concentration high. An important constraint on the obtained solution is that the biomass in the reactor should be at the same level at the beginning and at the end of the day so that the same control can be applied everyday and optimizes some form of long term productivity. Several scenarios are possible depending on the microalgae’s strain parameters and the maximal admissible value of the dilution rate: bang–bang or bang–singular–bang control or, if the growth rate of the algae is very strong in the presence of light, constant maximal dilution. A bifurcation diagram is presented to illustrate for which values of the parameters these different behaviors occur. Finally, a simple sub-optimal bang–bang strategy is proposed that numerically achieves productivity levels that almost match those of the optimal strategy.     

Synthesis of continuous control of a mechanical system with an unknown inertia matrix

The problem of the synthesis of a bounded control for a Lagrange scleronomic system is considered. We assume that the kinetic energy matrix of the system is known with some accuracy and that the system is subjected to uncontrolled bounded perturbations. The feedback control law, which allows us to steer the system to the given state at rest in a finite time, is constructed. The sufficient conditions of this behavior are formulated. The approach applied was proposed earlier for the case of a system with a known kinetic energy matrix. It is based upon the methods of the stability theory of motion. In order to construct the control law and justify it, we use an implicit Lyapunov function. The effectiveness of the control law is demonstrated with the help of numerical modeling of controlled motions of a two-link manipulator, which holds a weight of unknown mass in its hand.     

Bang–bang property for an uncertain saddle point problem

In this paper, we propose a bang–bang control model for a saddle point problem using the optimistic value criterion. By using equation of optimality in uncertain optimal control, a bang–bang control problem is investigated. And then, an example is given to illustrate our results.     

基于无源性分析的鲁棒控制系统设计

给出不确定非线性系统鲁棒控制的一种设计方法.经校正网络将一微分算子传递给 对象后,采用bang-bang控制使余下的部分无源化,并引入适当的符号跟随系统,使直输回路 严格无源.于是全反馈控制系统具有很强的鲁棒性.同时讨论了系统稳定、不稳定、逆不稳定 的各种情况的设计.     

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.     

Advanced automotive thermal management – Nonlinear radiator fan matrix control

Advanced automotive cooling systems for gasoline and diesel engines can improve the powertrain performance. The replacement of the mechanical driven coolant pump and radiator fans with computer controlled servo-motor actuators, and update of the wax-based thermostat valve with a 3-way variable position smart valve, allow the coolant flow rate and proportion directed through the radiator to be carefully adjusted. A smart thermal management system approach can regulate the forced convection heat transfer process to match the engine׳s cooling needs. This paper presents a Lyapunov based nonlinear control strategy to solely operate the radiator fan matrix for transient engine temperature tracking. A reduced order mathematical model serves as the basis for the closed-loop feedback system. An adaptive backstepping method was implemented to derive the control law. An experimental test bench with multiple radiator fans, heat exchanger, wind tunnel, coolant pump, three way valve, and engine thermal load has been fabricated. Representative numerical and experimental tests demonstrate that the advanced control strategy can regulate the engine temperature tracking error within 0.12 °C and compensate the unknown heat load. The nonlinear controller provided superior performance in terms of power consumption and temperature tracking as evident by the reduced magnitude when compared to a classical PI with lookup table based controller and a bang bang controller.     

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。     

Fastest recovery from feedback loss: Bounded overshoot

The problem of designing robust optimal controllers to reduce in minimal time operating errors that had accumulated during a period of feedback loss is revisited, with the objective of imposing a constraint on the maximal overshoot of the controlled system. It is shown that robust optimal controllers that satisfy this constraint exist under rather broad conditions. It is also shown that optimal performance can be closely approximated by bang–bang controllers – controllers that are relatively easy to design and implement.     

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.     

Efficient time-optimal two-corner trajectory planning algorithm for differential-driven wheeled mobile robots with bounded motor control inputs

We propose an efficient time-optimal trajectory planning algorithm for differential-driven wheeled mobile robots with bounded motor control inputs in the environment with cranked road where two corners with arbitrary angles exist. Based on dynamics for differential-driven wheeled mobile robots including actuating motors as well as on the bang–bang principle for time-optimality, we plan the time-optimal free path trajectory which unifies path planning and trajectory generation. Thus it improves total time significantly while handling obstacle avoidance. We divide our trajectory into three sections and then divide each section into an appropriate number of subsections to make five subsections in total. We introduce the concept of transition angle to solve the problem with formulating search loops efficiently. Numerical results are provided to validate the effectiveness of the proposed algorithm.     

Output feedback stabilization of an unstable wave equation with general corrupted boundary observation

We consider boundary output feedback stabilization for an unstable wave equation with boundary observation subject to a general disturbance. We adopt for the first time the active disturbance rejection control approach to stabilization for a system described by the partial differential equation with corrupted output feedback. By the approach, the disturbance is first estimated by a relatively independent estimator; it is then canceled in the feedback loop. As a result, the control law can be designed almost as that for the system without disturbance. We show that with a time varying gain properly designed, the observer driven by the disturbance estimator is convergent, and that all subsystems in the closed-loop are asymptotically stable in the energy state space. We also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying gain estimator on peaking value reduction.