Optimal control problems with a continuous inequality constraint on the state and the control

We consider an optimal control problem with a nonlinear continuous inequality constraint. Both the state and the control are allowed to appear explicitly in this constraint. By discretizing the control space and applying a novel transformation, a corresponding class of semi-infinite programming problems is derived. A solution of each problem in this class furnishes a suboptimal control for the original problem. Furthermore, we show that such a solution can be computed efficiently using a penalty function method. On the basis of these two ideas, an algorithm that computes a sequence of suboptimal controls for the original problem is proposed. Our main result shows that the cost of these suboptimal controls converges to the minimum cost. For illustration, an example problem is solved.     

Stability margins of nonlinear receding-horizon control via inverse optimality

Using the nonlinear analog of the Fake Riccati equation developed for linear systems, we derive an inverse optimality result for several receding-horizon control schemes. This inverse optimality result unifies stability proofs and shows that receding-horizon control possesses the stability margins of optimal control laws.     

Optimal sampling schedule for parameter estimation of linear models with unknown but bounded measurement errors

The problem of optimal sampling design for parameter estimation when data are generated by linear models is addressed. The measurements are assumed to be corrupted by an unknown but bounded additive noise. The sampling design assumes that the number of samples is unconstrained and no replication is allowed. Two main results are shown: 1) for particular classes of linear models, the optimal number of measurements is equal to the number of parameters, as in the statistical context; 2) the uncertainty intervals of the parameter estimates are bounded from above by quantities that can be computer a priori, knowing only the model and the error structure.     

有界随机测量时滞的网络控制系统的最优估计

在假设测量没有丢包的情况下, 研究了带有随机测量时滞的网络控制系统的最优估计问题. 利用已知的时 滞分布概率, 建立新的模型来描述随机时滞测量. 进一步将带有时滞的测量等价成每个通道是单时滞的多通道测 量, 从而利用新息重组方法, 通过求解黎卡提方程求解最优估计器. 最后给出仿真实例验证了该算法的有效性.     

Input to state stability of min-max MPC controllers for nonlinear systems with bounded uncertainties

Min-max model predictive control (MPC) is one of the control techniques capable of robustly stabilize uncertain nonlinear systems subject to constraints. In this paper we extend existing results on robust stability of min-max MPC to the case of systems with uncertainties which depend on the state and the input and not necessarily decaying, i.e. state and input dependent bounded uncertainties. This allows us to consider both plant uncertainties and external disturbances in a less conservative way.It is shown that the input-to-state practical stability (ISpS) notion is suitable to analyze the stability of worst-case based controllers. Thus, we provide Lyapunov-like sufficient conditions for ISpS. Based on this, it is proved that if the terminal cost is an ISpS-Lyapunov function then the optimal cost is also an ISpS-Lyapunov function for the system controlled by the min-max MPC and hence, the controlled system is ISpS. Moreover, we show that if the system controlled by the terminal control law locally admits certain stability margin, then the system controlled by the min-max MPC retains the stability margin in the feasibility region.     

Discrete minimum time tracking problem for a chain of three integrators with bounded input

Based on the theoretical results from optimal control theory and sliding mode control, this paper proposes a discrete time nonlinear control law that guarantees the minimum time global stabilization of a chain of three integrators with bounded control input. Moreover, the proposed control law allows the minimum time trajectory tracking for a wide class of reference signals (such as step, ramp, saw tooth, etc.) and it can be used as nonlinear filter for the trajectories generation.     

Optimal linear estimation for systems with multiple packet dropouts

This paper is concerned with the optimal linear estimation problem for linear discrete-time stochastic systems with multiple packet dropouts. Based on a packet dropout model, the optimal linear estimators including filter, predictor and smoother are developed via an innovation analysis approach. The estimators are computed recursively in terms of the solution of a Riccati difference equation of dimension equal to the order of the system state plus that of the measurement output. The steady-state estimators are also investigated. A sufficient condition for the convergence of the optimal linear estimators is given. Simulation results show the effectiveness of the proposed optimal linear estimators.     

Optimal linear estimation for networked systems with communication constraints

This paper is concerned with the optimal linear estimation problem for discrete time-varying networked systems with communication constraints. The communication constraint considered is that only one network node is allowed to gain access to a shared communication channel, then the various network nodes of the networked systems are scheduled to transmit data according to a specified media access control protocol, and a remote estimator performs the estimation task with only partially available measurements. The channel accessing processes of those network nodes are modeled by Bernoulli processes, and optimal linear filters are designed by using the orthogonal projection principle and the innovation analysis approach. It is shown that the optimal estimation performances critically depend on the channel accessing probabilities of the network nodes and the packet loss probability, and the optimal filters can be obtained by solving recursive Lyapunov and Riccati equations. An illustrative example is finally given to show the effectiveness of the proposed filters.     

Hybrid Solution Method for Dynamic Programming Equations for MDOF Stochastic Systems

An optimal control problem is considered for a multi-degree-of-freedom (MDOF) system, excited by a white-noise random force. The problem is to minimize the expected response energy by a given time instantT by applying a vector control force with given bounds on magnitudes of its components. This problem is governed by the Hamilton-Jacobi-Bellman, or HJB, partial differential equation. This equation has been studied previously [1] for the case of a single-degree-of-freedom system by developing a hybrid solution. Specifically, an exact analitycal solution has been obtained within a certain outer domain of the phase plane, which provides necessary boundary conditions for numerical solution within a bounded in velocity inner domain, thereby alleviating problem of numerical analysis for an unbounded domain. This hybrid approach is extended here to MDOF systems using common transformation to modal coordinates. The multidimensional HJB equation is solved explicitly for the corresponding outer domain, thereby reducing the problem to a set of numerical solutions within bounded inner domains. Thus, the problem of bounded optimal control is solved completely as long as the necessary modal control forces can be implemented in the actuators. If, however, the control forces can be applied to the original generalized coordinates only, the resulting optimal control law may become unfeasible. The reason is the nonlinearity in maximization operation for modal control forces, which may lead to violation of some constraints after inverse transformation to original coordinates. A semioptimal control law is illustrated for this case, based on projecting boundary points of the domain of the admissible transformed control forces onto boundaries of the domain of the original control forces. Case of a single control force is considered also, and similar solution to the HJB equation is derived.     

Advanced control with parameter estimation of batch transesterification reactor

The objective of this work is to enhance the economic performance of a batch transesterification reactor producing biodiesel by implementing advanced, model based control strategies. To achieve this goal, a dynamic model of the batch reactor system is first developed by considering reaction kinetics, mass balances and heat balances. The possible plant-model mismatch due to inaccurate or uncertain model parameter values can adversely affect model based control strategies. Therefore, an evolutionary algorithm to estimate the uncertain parameters is proposed. It is shown that the system is not observable with the available measurements, and hence a closed loop model predictive control cannot be implemented on a real system. Therefore, the productivity of the reactor is increased by first solving an open-loop optimal control problem. The objective function for this purpose optimizes the concentration of biodiesel, the batch time and the heating and cooling rates to the reactor. Subsequently, a closed-loop nonlinear model predictive control strategy is presented in order to take disturbances and model uncertainties into account. The controller, designed with a reduced model, tracks an offline determined set-point reactor temperature trajectory by manipulating the heating and cooling mass flows to the reactor. Several operational scenarios are simulated and the results are discussed in view of a real application. With the proposed optimization and control strategy and no parameter mismatch, a revenue of 2.76 $ min⁻¹ can be achieved from the batch reactor. Even with a minor parameter mismatch, the revenue is still 2.01 $ min⁻¹. While these values are comparable to those reported in the literature, this work also accounts for the cost of energy. Moreover, this approach results in a control strategy that can be implemented on a real system with limited online measurements.     

A radial basis collocation method for Hamilton–Jacobi–Bellman equations

In this paper we propose a semi-meshless discretization method for the approximation of viscosity solutions to a first order Hamilton–Jacobi–Bellman (HJB) equation governing a class of nonlinear optimal feedback control problems. In this method, the spatial discretization is based on a collocation scheme using the global radial basis functions (RBFs) and the time variable is discretized by a standard two-level time-stepping scheme with a splitting parameter θ. A stability analysis is performed, showing that even for the explicit scheme that θ=0, the method is stable in time. Since the time discretization is consistent, the method is also convergent in time. Numerical results, performed to verify the usefulness of the method, demonstrate that the method gives accurate approximations to both of the control and state variables.     

Parameter identification for models with bounded errors in all variables

In this paper, the problem of parameter identification for models with bounded measurement errors both on the input and on the output is addressed and some corrections to previously published results are presented. In particular, it is shown that only parameter overbounds can in general be computed for systems of the form y = (φ + δφ)θ + δy when the bounded measurement errors δφ and δy are correlated. Since ARMAX and bilinear systems can be represented in this form, it turns out that tight parameter bounds are in general not available for these systems. Finally, we show that it is possible to check a posteriori whether the obtained bounds are tight or not.     

Optimal transition maneuvers for a class of V/STOL aircraft

This paper focuses on the problem of computing optimal transition maneuvers for a particular class of tail-sitter aircraft able to switch their flight configuration from hover to level flight and vice versa. Both minimum-time and minimum-energy optimal transition problems are formulated and solved numerically in order to compute reference maneuvers to be employed by the onboard flight control system to change the current flight condition. In order to guide the numerical computation and to validate its results, in a first stage approximated solutions are obtained as a combination of a finite number of motion primitives corresponding to analytical trajectories of approximated dynamic models. The approximated solution is then employed to generate an initial guess for the numerical computation applied to a more accurate dynamic model. Numerical trajectories computed for a small scale prototype of tail-sitter aircraft are finally presented, showing the effectiveness of the proposed methodology to deal with the complex dynamics governing this kind of systems.     

Optimal tracking control for nonlinear large-scale systems with persistent disturbances

This paper studies the optimal control with zero steady-state error problem for nonlinear large-scale systems affected by external persistent disturbances. The nonlinear large-scale system is transformed into N nonlinear subsystems with interconnect terms. Based on the internal model principle, a disturbance compensator is constructed such that the ith subsystem with external persistent disturbances is transformed into an augmented subsystem without disturbances. According to the sensitivity approach, the optimal tracking control law for the ith nonlinear subsystem can be obtained. The optimal tracking control law for the nonlinear large-scale systems can be obtained. A numerical simulation shows that the method is effective.     

带有持续扰动的时滞非线性大系统的最优跟踪控制

研究具有外界持续扰动的时滞非线性大系统的无静差最优跟踪控制问题.将时滞非线性大系统分解为带有互联项的N个时滞非线性子系统,基于内模原理对子系统构造扰动补偿器,将带有外部持续扰动的子系统化为无扰动的增广系统.通过灵敏度法求解不含时滞的两点边值问题,得到子系统的最优跟踪控制律,截取最优跟踪控制律的前N项作为次优控制律来近似系统的最优控制律.仿真实例表明了该设计方法的有效性.     

Optimal control of a fractional diffusion equation with state constraints

This paper is concerned with the state constrained optimal control problems of a fractional diffusion equation in a bounded domain. The fractional time derivative is considered in the Riemann-Liouville sense. Under a Slater type condition we prove the existence a Lagrange multiplier and a decoupled optimality system.     

考虑交易费的融资与分红最优控制模型

在实际金融市场中股份公司在红利分配和再融资过程中都需要支付固定交易费和比例交易费 ,而如何确定交易费对公司财务决策的影响还没有进行过讨论 .本文利用随机脉冲控制理论研究了在收取固定和比例交易费的市场环境下 ,公司如何制定其最优的财务策略 .首先给出了最优控制问题对应的Hamilton_Jacobi_Bellman方程 ,接着构造出了它的连续可微解 .利用解的性质和推广的It^o公式 ,构造出了最优的再融资及分红策略 .最后对模型的应用做了经济学上的解释 ,并与已有模型做了比较     

随机运动目标搜索问题的最优控制模型

提出了Rn空间中做布朗运动的随机运动目标的搜索问题的最优控制模型.采用分析的方法来研究随机运动目标的最优搜索问题,并将原问题转化为由一个二阶偏微分方程(HJB方程)所表示的确定性分布参数系统的等价问题,推导出随机运动目标的最优搜索问题的HJB方程,并证明了该方程的解即是所寻求的最优搜索策略.由此给出了一个计算最优搜索策略的算法和一个实例.