Two-degree-of-freedom output feedback controllers for discrete-time nonlinear systems

A discrete-time, model-based output feedback control structure for nonlinear processes is developed in the present work. The structure makes use of a closed-loop observer, while at the same time it guarantees that the overall feedback controller possesses integral action. An algebraic transformation is applied on the observer states to insure that the input/output gain of the observer matches the model upon which the static state feedback control law is based. The resulting control algorithm is a two-degree-of-freedom control law, in the sense that the output and the set point are processed in different ways. The control structure is shown not only to have the same properties as the standard model-state feedback structure, but also that it emerges from a model algorithmic control framework. Finally, a simulation example using an exothermic CSTR operating at an open-loop unstable steady state is used to evaluate the closed-loop performance of the proposed method.     

Two-degree-of-freedom output feedback controllers for nonlinear processes

In this work, a nonlinear output feedback control algorithm is proposed, in the spirit of model-state feedback control. The structure provides state estimates using a process model, the measured output, and the residual between the model output and the measured output. These estimates will track the process states at a rate determined by a set of tunable parameters. An algebraic transformation of the state estimates is incorporated in the control structure to ensure that the input/output gain of the observer matches the model upon which the static state feedback control law is based. The transformed states are then used in the control law. This leads to a controller of minimal order possessing integral action. The control structure is shown to have the same properties as the standard model-state feedback structure. The resulting algorithm is a two-degree of freedom control law, in the sense that the control action is not a function of the error only, but the output and the set point are processed in different ways. Finally, a simulation example using an exothermic CSTR operating at an open-loop unstable steady state is used to demonstrate the closed-loop performance of the proposed method.     

NONLINEAR MODEL PREDICTIVE CONTROL

Nonlinear Model Predictive Control (NMPC), a strategy for constrained, feedback control of nonlinear processes, has been developed. The algorithm uses a simultaneous solution and optimization approach to determine the open-loop optimal manipulated variable trajectory at each sampling instant. Feedback is incorporated via an estimator, which uses process measurements to infer unmeasured state and disturbance values. These are used by the controller to determine the future optimal control policy. This scheme can be used to control processes described by different kinds of models, such as nonlinear ordinary differential/algebraic equations, partial differential/algebraic equations, integra-differential equations and delay equations. The advantages of the proposed NMPC scheme are demonstrated with the start-up of a non-isothermal, non-adiabatic CSTR with an irreversible, first-order reaction. The set-point corresponds to an open-loop unstable steady state. Comparisons have been made with controllers designed using (1) nonlinear variable transformations, (2) a linear controller tuned using the internal model control approach, and (3) open-loop optimal control. NMPC was able to bring the controlled variable to its set-point quickly and smoothly from a wide variety of initial conditions. Unlike the other controllers, NMPC dealt with constraints in an explicit manner without any degradation in the quality of control. NMPC also demonstrated superior performance in the presence of a moderate amount of error in the model parameters, and the process was brought to its set-point without steady-state offset.     

Multiplicity of steady states in fluidized bed reactors—V: The effect of catalyst decay

A simplified two-phase model is used to investigate the behaviour of non-isothermal fluidized bed reactors experiencing catalyst decay. The investigation shows that for highly exothermic reaction it is almost always desirable to operate the reactor at the middle unstable steady state, since it gives higher accumulative yield than both the high and the low temperature steady states. A simple feedback control scheme with time varying set point is suggested to stabilize the middle steady state. The dynamic behaviour and stability of the system is investigated for the open-loop reactor (uncontrolled) and the closed loop reactor (controlled).     

A DIRECT NONLINEAR ADAPTIVE CONTROL OF STATE FEEDBACK LINEARIZABLE SYSTEMS

A direct nonlinear adaptive control of state feedback linearizable single-input single-output systems is proposed in the case when parametric uncertainties are represented linearly in the unknown parameters. The main feature of the proposed nonlinear adaptive control system is that the linearizing coordinate transformation and the state feedback are updated by parametric adaptive law, derived using the second method of Lyapunov. The proposed adaptive control scheme is relatively straightforward and simple in the sense that it does not use the concept of augmented error. This adaptive control scheme is numerically applied to an exothermic chemical reactor system and is compared with the nonadaptive stale feedback linearization which has an integral action. The simulation shows that the proposed adaptive control scheme can be applied effectively to highly nonlinear, uncertain chemical systems.     

Synthesis of PID controller for unstable and integrating processes

Properly designed controllers provide stable closed-loop response for open-loop unstable processes. Internal model controller equivalent PID tuning rules for low order unstable plus dead time systems are synthesized in this work. The controller is approximated near the vicinity of zero (origin). Controller parameters are derived by equating the closed-loop response to a control-signature (desired closed-loop response) involving a user defined tuning parameter, λ. Simulations are carried out to show the performance of the proposed tuning scheme for both set point and disturbance rejection cases.     

Design of multi-parametric NCO tracking controllers for linear dynamic systems

A methodology for combining multi-parametric programming and NCO tracking is presented in the case of linear dynamic systems. The resulting parametric controllers consist of (potentially nonlinear) feedback laws for tracking optimality conditions by exploiting the underlying optimal control switching structure. Compared to the classical multi-parametric MPC controller, this approach leads to a reduction in the number of critical regions. It calls for the solution of more difficult parametric optimization problems with linear differential equations embedded, whose critical regions are potentially nonconvex. Examples of constrained linear quadratic optimal control problems with parametric uncertainty are presented to illustrate the approach.     

Control of periodically operated reactors

Control of periodically operated reactors has in common with control of reactors operating at steady state the objective of minimizing the effect of disturbances on an objective function such as the cost of a product or the deviation of an outlet concentration of a pollutant from a statuary target. Simple feedback control employing feedback PID regulators, however, is not “adequate for most disturbances because of problems with tracking a time-varying output and the necessarily non-linear character of the reactors with respect to controlled as well as uncontrolled inputs. This contribution is a review of the literature and a discussion of research needs. The literature on the control of periodically operated reactors is not voluminous. Nevertheless this literature clearly indicates that model based predictive controllers can be used for this type of reactor”. Further research on the limitations, maintenance and implementation costs of model based controllers in this application would be worthwhile. Experimental studies are wholly absent. Unique regulators for other periodic operations, such as adaptive control or forcing the output toward a reference trajectory using an open loop model based control strategy, certainly warrant study of their application to periodically operated reactors. Additionally, proper design of the reactor may lead to configurations that are simpler to control and that may not require complex control strategies for efficient operation.     

不确定离散线性切换系统的鲁棒控制

利用共同李亚普诺夫函数方法和线性矩阵不等式处理方法,研究了一类含有结构参数不确定性的离散时间切换系统的鲁棒渐近稳定问题。基于凸组合条件,通过各个子系统的状态反馈控制器和相应切换规律的设计,在有限控制器之间进行切换使得该系统是全局渐近稳定的。文中用数学方法证明了该设计方法的有效性,所得结果为线性切换系统的分析提供了更深的视角,有助于控制系统其它性能实现。     

强制不稳态环状反应器网络的特性

The multi-reactors network, a closed sequence of two or more catalytic fixed bed reactors with periodical change of the feed position, was studied by means of numerical simulations. Two advantages of such a reactor configuration, the possibility of exploiting the thermal storage capacity of the catalyst and the optimal temperature profile for exothermic equilibrium-limited reactions, were analyzed. The former feature acting as a regenerative heat exchanger was simulated in the case of the combustion of lean volatile organic compound (VOC) mixtures, with the possibility of multiple stability windows found when rich mixtures are fed. The latter was demonstrated using the methanol synthesis, with the enhancement of the conversion and product selectivity predicted. The influence of the number of the reactors in the network was pointed out. Some results obtained in the reverse-flow reactors were also presented for comparison.     

Recuperative coupling of exothermic and endothermic reactions

Coupling energy intensive endothermic reaction systems with suitable exothermic reactions improves the thermal efficiency of processes and reduces the size of the reactors. One type of reactor suitable for such a type of coupling is the heat exchanger reactor. In this work, a one-dimensional pseudo-homogeneous plug flow model is used to analyze and compare the performance of co-current and counter-current heat exchanger reactors. A parametric analysis is carried out to address the vital issues, such as the exit conversion of the endothermic reaction, the temperature peak (hot spot) of the exothermic reaction and the reactor volumetric productivity. The measures to reduce the hot spot by different catalyst profiling techniques are also addressed. Some features of the dynamic behavior exhibited by these reactors, which are important from design, operational and control point of view, are presented.     

Globally stable control systems for processes with input multiplicities

A nonlinear process with input multiplicity has two or more input values for a given output at the steady state, and the process steady state gain changes its sign as the operating point changes. A control system with integral action will be unstable when both signs of the process gain and the controller integral gain are different, and its stability region will be limited to the boundary where the process steady state gain is zero. Unlike processes with output multiplicities, feedback controllers cannot be used to correct the sign changes of process gain. To remove such stability limitation, a simple control system with parallel compensator is proposed. The parallel compensator can be easily designed based on the process steady state gain information and tuned in the field. Using the two time scale method, the stability of proposed control systems for processes with input multiplicities can be checked.     

General rules for optimal tuning the PIλDµ controllers with application to first‐order plus time delay processes

For two main reasons optimal tuning the PI^λD^µ controllers is a challenging task: First, the search space is very large in dealing with such controllers, and second, there is not any generally applicable method for stability testing of the linear feedback systems containing both time delay and fractional‐order controllers. Hence, easy‐to‐use and effective rules for optimal tuning such controllers are highly demanded. In this paper, explicit formulas for optimal tuning the parameters of the PI^λD^µ controller, when it is applied in series with a first‐order plus time‐delay process in a standard output‐feedback system, are proposed. © 2012 Canadian Society for Chemical Engineering     

Control of chemical reactors in the subspace of reaction and control variants

Control of stirred tank reactors is studied. Possibilities of utilizing variant and invariant properties of a general reactor model for designing control systems are investigated. The studied optimal feedback controllers are based on linearized reactor models and are computed from the reaction and control variant part of the system only. One advantage of this approach, compared to state feedback, is that the dimension of the reaction and control variant space usually is smaller than the dimension of the state space of the reactor.Another advantage is that the reaction and control variants only have to be measured or estimated. This advantage is, however, not revealed by a straightforward implementation since the reaction and control variants are combinations of all states. In the paper it is shown how the control law can be modified so that just as many states as there are reaction and control variants have to be fed back, thus reducing the number of sensors needed.The methods discussed in the paper are illustrated through simulation of linear-quadratic control of a nonlinear reactor with heat exchange.     

Variable Cascade Control Structure for Tubular Reactors

The application of a variable control structure for tubular reactors, based on multiple temperature measurements, is explored. The proposed structure allows the controller to adapt to temperature error variations along the tubular reactor. Controllers can be designed from simple input‐output first‐order dynamical models obtained from step responses and are coupled with a dynamic law for assigning a variable weight to conventional feedback cascade control loops. The simulation results show that the variable structure based on linear proportional‐integral controllers enhances the performance and robustness properties of conventional cascade schemes for controlling the outlet reactor concentration. Major advantages are shown in the face of a variety of concentration and temperature feed disturbances.     

Dynamics and control of autothermal reactors for the production of hydrogen

Autothermal reactors, coupling endothermic and exothermic reactions in parallel channels, represent one of the most promising technologies for hydrogen production. The bulk of existing research work concerning their operation refers, however, to steady state conditions. In the present work, the dynamic behavior of autothermal reactors is analyzed. It is demonstrated that such systems are modeled by systems of equations that are stiff, their dynamics consequently featuring two time scales. Within the framework of singular perturbations, reduced-order, nonstiff models are derived for the transient evolution in each time scale. Furthermore, the challenges posed by the transient operation of autothermal reactors are identified, along with demonstrating the implementation of feedback control in order to improve transient performance and to avoid severe issues (such as reactor extinction) that can arise in the course of operating the reactor. All theoretical concepts are illustrated with numerical simulations performed using the model of a hydrogen production reactor.     

A method for robustness analysis of controlled nonlinear systems

This paper presents an approach to analyzing robustness properties of nonlinear systems under feedback control. The core idea is to apply numerical bifurcation analysis to the closed-loop process, using the controller/observer tuning parameters, the set points, and parameters describing model uncertainty (parametric as well as unmodeled dynamics) as bifurcation parameters. By analyzing the Hopf bifurcation and saddle-node bifurcation loci with respect to these parameters, bounds on the controller tuning are identified which can serve as a measure for the robustness of the controlled system. These bounds depend upon the type as well as the degree of mismatch that exists between the plant and the model used for controller design.The method is illustrated by analyzing three control systems which are applied to a continuously operated stirred tank reactor: a state feedback linearizing controller and two output feedback linearizing controllers. While model uncertainty has only a minor effect on the tuning of the state feedback linearizing controller, this does not represent a very realistic scenario. However, when an observer is implemented in addition to the controller and an output feedback linearizing scheme is investigated, it is found that the plant-model mismatch has a much more profound impact on the tuning of the observer than it has on the controller tuning. In addition, two observer designs with different level of complexity are investigated and it is found that a scheme which makes use of additional knowledge about the system will not necessarily result in better stability properties as the level of uncertainty in the model increases. These investigations are carried out using the robustness analysis scheme introduced in this paper.     

Feedback control of unstable states in a laboratory reactor

The principal objective of the experimental work reported in this paper was to employ automatic feedback control to stabilize those steady states in a nonadiabatic CSTR which were naturally unstable (that is, unstable in open-loop operation). The exothermic liquid-phase reaction between sodium thiosulfate and hydrogen peroxide in aqueous solution was used in all experiments, and the unstable states of interest were the so-called intermediate states (saddle points in the phase plane). Described here are experimental results which show successful stabilization of the unstable states. The reactor temperature and the coolant flow rates were the controlled and manipulated variables respectively. The controller used was of the conventional proportional—integral type. This paper also contains a discussion of the theoretical behavior of the controlled system as well as comparisons of the predictions of a mathematical model with experimental results.The experimental attainment of smooth stabilized reactor operation at the intermediate states was found to be relatively easy, although some care in the planning of start-up procedures was demanded in certain cases. Further, the agreement between theoretical and experimental behavior was generally good. Apparently this is the first published report of an experimental investigation into controlled operation at the unstable intermediate states of an exothermic system. Theoretical studies of the problem were first reported in the 1950's.     

Nonlinear feedback control for operating a nonisothermal CSTR near an unstable steady state

Application of relay feedback controllers to maintain process states near unstable steady states is extended to a second order system, the CSTR with an irreversible, exothermic reaction. The control scheme is successful provided the bandwith (the reactor temperature deviation which causes relay switching) is not chosen too large relative to the drive levels (the deviation in coolant temperature caused by relay switching). In this context Tsypkin's analytical methods for approximate solution of this and higher order problems are elaborated and tested. Tsypkin's technique proves superior to Describing Function analysis in accuracy and in bounds on the maximum permissible bandwidth.