Optimal control of an advection-dominated catalytic fixed-bed reactor with catalyst deactivation

The paper focuses on the linear-quadratic control problem for a time-varying partial differential equation model of a catalytic fixed-bed reactor. The classical Riccati equation approach, for time-varying infinite-dimensional systems, is extended to cover the two-time scale property of the fixed-bed reactor. Dynamical properties of the linearized model are analyzed using the concept of evolution systems. An optimal LQ-feedback is computed via the solution of a matrix Riccati partial differential equation. Numerical simulations are performed to evaluate the closed loop performance of the designed controller on the fixed-bed reactor. The performance of the proposed controller is compared to performance of an infinite dimensional controller formulated by ignoring the catalyst deactivation. Simulation results show that the performance of the proposed controller is better compared to the controller ignoring the catalyst deactivation when the deactivation time is close to the resident time of the reactor.     

Optimal control design for time-varying catalytic reactors: a Riccati equation-based approach

The linear quadratic (LQ) optimal control problem is studied for a partial differential equation model of a time-varying catalytic reactor. First, the dynamical properties of the linearised model are studied. Next, an LQ-control feedback is computed by using the corresponding operator Riccati differential equation, whose solution can be obtained via a related matrix Riccati partial differential equation. Finally, the designed controller is applied to the non-linear reactor system and tested numerically.     

Asymptotic stability of infinite-dimensional semilinear systems: Application to a nonisothermal reactor

The concept of asymptotic stability is studied for a class of infinite-dimensional semilinear Banach state space (distributed parameter) systems. Asymptotic stability criteria are established, which are based on the concept of strictly m-dissipative operator. These theoretical results are applied to a nonisothermal plug flow tubular reactor model, which is described by semilinear partial differential equations, derived from mass and energy balances. In particular it is shown that, under suitable conditions on the model parameters, some equilibrium profiles are asymptotically stable equilibriums of such model.     

Identification and control of a nonlinear discrete-time system based on its linearization: a unified framework

This paper presents a unified theoretical framework for the identification and control of a nonlinear discrete-time dynamical system, in which the nonlinear system is represented explicitly as a sum of its linearized component and the residual nonlinear component referred to as a "higher order function." This representation substantially simplifies the procedure of applying the implicit function theorem to derive local properties of the nonlinear system, and reveals the role played by the linearized system in a more transparent form. Under the assumption that the linearized system is controllable and observable, it is shown that: 1) the nonlinear system is also controllable and observable in a local domain; 2) a feedback law exists to stabilize the nonlinear system locally; and 3) the nonlinear system can exactly track a constant or a periodic sequence locally, if its linearized system can do so. With some additional assumptions, the nonlinear system is shown to have a well-defined relative degree (delay) and zero-dynamics. If the zero-dynamics of the linearized system is asymptotically stable, so is that of the nonlinear one, and in such a case, a control law exists for the nonlinear system to asymptotically track an arbitrary reference signal exactly, in a neighborhood of the equilibrium state. The tracking can be achieved by using the state vector for feedback, or by using only the input and the output, in which case the nonlinear autoregressive moving-average (NARMA) model is established and utilized. These results are important for understanding the use of neural networks as identifiers and controllers for general nonlinear discrete-time dynamical systems.     

Discontinuous feedback stabilization of minimum-phase semilinear infinite-dimensional systems with application to chemical tubular reactor

This paper develops discontinuous control methods for minimum-phase semilinear infinite-dimensional systems driven in a Hilbert space. The control algorithms presented ensure asymptotic stability, global or local accordingly, as state feedback or output feedback is available, as well as robustness of the closed-loop system against external disturbances with the a priori known norm bounds. The theory is applied to stabilization of chemical processes around prespecified steady-state temperature and concentration profiles corresponding to a desired coolant temperature. Two specific cases, a plug flow reactor and an axial dispersion reactor, governed by hyperbolic and parabolic partial differential equations of first and second order, respectively, are under consideration. To achieve a regional temperature feedback stabilization around the desired profiles, with the region of attraction, containing a prescribed set of interest, a component concentration observer is constructed and included into the closed-loop system so that there is no need for measuring the process component concentration which is normally unavailable in practice. Performance issues of the discontinuous feedback design are illustrated in a simulation study of the plug flow reactor.     

有界扰动下约束非线性系统鲁棒经济模型预测控制EI北大核心CSCD

针对未知但有界扰动下约束非线性系统,提出一种新的鲁棒经济模型预测控制(Economic model predictive control,EMPC)策略,保证闭环系统对扰动输入具有输入到状态稳定性(Input-to-state stability,ISS).基于微分对策原理,分别优化经济目标函数和关于最优经济平衡点的鲁棒稳定性目标函数,其中经济最优性与鲁棒稳定性是具有冲突的两个控制目标.利用鲁棒稳定性目标最优值函数构造EMPC优化的隐式收缩约束,建立鲁棒EMPC的递推可行性和闭环系统关于最优经济平衡点相对于有界扰动输入到状态稳定性结果.最后以连续搅拌反应器为例,对比仿真验证本文策略的有效性.     

约束非线性系统稳定经济模型预测控制

考虑约束非线性系统,提出一种新的具有稳定性保证的经济模型预测控制（Economic model predictive control,EMPC）策略.由于经济性能指标的非凸性和非正定性,引入关于经济最优平衡点的正定辅助函数.利用辅助函数的最优值函数定义原始EMPC优化问题的稳定性约束.应用终端约束集、终端代价函数和局部控制器三要素,建立闭环系统关于经济最优平衡点的渐近稳定性和渐近平均性能.进一步,结合多目标理想点概念,将提出的控制策略用于多个经济性能指标的优化控制,得到稳定多目标EMPC策略.最后,以连续搅拌反应器为例,比较仿真结果验证本文策略的有效性.     

On the gain margin of nonlinear and optimal regulators

The robustness of nonlinear regulators for nonlinear systems with respect to variations in gain is investigated. It is shown that there exist regulators that produce asymptotically stable closed-loop systems, but do not tolerate any variation in gain without instability. However, if the linearized closed-loop system is also asymptotically stable, then there is always some gain margin. For a wide class of optimal regulators, it is shown that the gain margin is infinite with respect to increases in gain and that decreases down to 0.5 can be tolerated. The robustness properties of linear quadratic control laws are thus generalized.     

Differential games,finite-time partial-state stabilisation of nonlinear dynamical systems,and optimal robust control

Two-player zero-sum differential games are addressed within the framework of state-feedback finite-time partial-state stabilisation of nonlinear dynamical systems. Specifically, finite-time partial-state stability of the closed-loop system is guaranteed by means of a Lyapunov function, which we prove to be the value of the game. This Lyapunov function verifies a partial differential equation that corresponds to a steady-state form of the Hamilton–Jacobi–Isaacs equation, and hence guarantees both finite-time stability with respect to part of the system state and the existence of a saddle point for the system's performance measure. Connections to optimal regulation for nonlinear dynamical systems with nonlinear-nonquadratic cost functionals in the presence of exogenous disturbances and parameter uncertainties are also provided. Furthermore, we develop feedback controllers for affine nonlinear systems extending an inverse optimality framework tailored to the finite-time partial-state stabilisation problem. Finally, two illustrative numerical examples show the applicability of the results proven.     

Optimization of a non-linear distributed parameter system using periodic boundary control†

Conditions for loeal optimality are worked out using simple calculus of variations. To find the optimum control, a two-point boundary value problem in space and time has to be solved, which involves the solution of the adjoint differential equation together with the prooess equation. The method is applied to the optimization of a periodic process, consisting of a tubular reactor where a second-order homogeneous reaction takes place and a periodicity oondition of the state is satisfied everywhere along the reactor. Tho plug flow and the diffusion model are assumed. In the first case an exact solution is carried out. The improvements in yield compared with steady-state conditions are obtained and shown in graphs.     

Non-linear feedback control of parabolic partial differential difference equation systems

This paper proposes a general method for the synthesis of non-linear output feedback controllers for single-input singleoutput quasi-linear parabolic partial differential difference equation (PDDE) systems, for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Initially, a non-linear model reduction scheme which is based on combination of Galerkin's method with the concept of approximate inertial manifold is employed for the derivation of differential difference equation (DDE) systems that describe the dominant dynamics of the PDDE system. Then, these DDE systems are used as the basis for the explicit construction of non-linear output feedback controllers through combination of geometric and Lyapunov techniques. The controllers guarantee stability and enforce output tracking in the closed-loop parabolic PDDE system independently of the size of the state delay, provided that the separation of the slow and fast eigenvalues of the spatial differential operator is sufficiently large and an appropriate matrix is positive definite. The methodology is successfully employed to stabilize the temperature profile of a tubular reactor with recycle at a spatially non-uniform unstable steadystate.     

Dissipativity-based observer and feedback control design for a class of chemical reactors

The problem of controlling a (possibly open-loop unstable) continuous exothermic reactor with non-monotonic kinetics, temperature measurements, reactant and heat exchange rates as manipulated inputs, and operation at maximum production rate is addressed within a robustness-oriented practical (input-to-state nonlocal) stability framework, according to the related detectability and relative degree structure. The result is a dynamical output-feedback (OF) controller made by the combination of a passive state-feedback (SF) nonlinear controller with a dissipative nonlinear observer. The proposed design methodology has: (i) solvability conditions with physical meaning, and (ii) a closed-loop stability criterion coupled with simple tuning guidelines. The proposed approach is tested with a representative case example through numerical simulations.     

An integrated state space partition and optimal control method of multi-model for nonlinear systems based on hybrid systems

Multilinear model approach turns out to be an ideal candidate for dealing with nonlinear systems control problem. However, how to identify the optimal active state subspace of each linear subsystem is an open problem due to that the closed-loop performance of nonlinear systems interacts with these subspaces ranges. In this paper, a new systematic method of integrated state space partition and optimal control of multi-model for nonlinear systems based on hybrid systems is initially proposed, which can deal with the state space partition and associated optimal control simultaneously and guarantee an overall performance of nonlinear systems consequently. The proposed method is based on the framework of hybrid systems which synthesizes the multilinear model, produced by nonlinear systems, in a unified criterion and poses a two-level structure. At the upper level, the active state subspace of each linear subsystem is determined under the optimal control index of a hybrid system over infinite horizon, which is executed off-line. At the low level, the optimal control is implemented online via solving the optimal control of hybrid system over finite horizon. The finite horizon optimal control problem is numerically computed by simultaneous method for speeding up computation. Meanwhile, the model mismatch produced by simultaneous method is avoided by using the strategy of receding-horizon. Simulations on CSTR (Continuous Stirred Tank Reactor) confirm that a superior performance can be obtained by using the presented method.     

基于确定学习的机器人任务空间自适应神经网络控制

针对产生回归轨迹的连续非线性动态系统, 确定学习可实现未知闭环系统动态的局部准确逼近. 基于确定学习理论, 本文使用径向基函数(Radial basis function, RBF)神经网络为机器人任务空间跟踪控制设计了一种新的自适应神经网络控制算法, 不仅实现了闭环系统所有信号的最终一致有界, 而且在稳定的控制过程中, 沿着回归跟踪轨迹实现了部分神经网络权值收敛到最优值以及未知闭环系统动态的局部准确逼近. 学过的知识以时不变且空间分布的方式表达、以常值神经网络权值的方式存储, 可以用来改进系统的控制性能, 也可以应用到后续相同或相似的控制任务中, 节约时间和能量. 最后, 用仿真说明了所设计控制算法的正确性和有效性.     

A new Lyapunov design approach for nonlinear systems based on Zubov's method

The present research work proposes a new nonlinear controller synthesis approach that is based on the methodological principles of Lyapunov design. In particular, it relies on a short-horizon model-based prediction and optimization of the rate of “energy dissipation” of the system, as it is realized through the time derivative of an appropriately selected Lyapunov function. The latter is computed by solving Zubov's partial differential equation based on the system's drift vector field. A nonlinear state feedback control law with two adjustable parameters is derived as the solution of an optimization problem that is formulated on the basis of the aforementioned Lyapunov function and closed-loop performance characteristics. A set of system-theoretic properties of the proposed control law are examined as well. Finally, the proposed Lyapunov design method is evaluated in a chemical reactor example which exhibits nonminimum-phase behaviour.     

Structural stability of linear dynamically varying (LDV) controllers

LDV systems are linear systems with parameters varying according to a nonlinear dynamical system. This paper examines the robust stability of such systems in the face of perturbations of the nonlinear system. Three classes of perturbations are examined: differentiable functions, Lipschitz continuous functions and continuous functions. It is found that in the first two cases the system remains stable. Whereas, if the perturbations are among continuous functions, the closed-loop may not be asymptotically stable, but, instead, is asymptotically bounded with the diameter of the residual set bounded by a function that is continuous in the size of the perturbation. It is also shown that in the case of differential perturbations, the resulting optimal LDV controller is continuous in the size of the perturbation. An example is presented that illustrates the continuity of the variation of the controller in the case of a nonstructurally stable dynamical system.