Self-optimising control for a class of continuous bioreactor via variable-structure feedback

A self-optimising controller is designed for stabilisation of a class of bioreactor exploiting sliding-mode techniques. The stability analysis for the class of bioreactor, in open-loop configuration, suggests that the optimal behaviour, respect to maximal biomass production, occurs in an unstable region (structurally unstable). In this contribution, a variable-structure controller is designed, exploiting the inhibitory effect of substrate concentration under the biomass growth rate, such that the closed-loop system reaches the optimal manifold where the effect induced by the growth rate gradient is compensated (favouring the maximum growth rate). The self-optimising comprises an uncertainty estimator which computes the unknown terms for increasing the robustness issues of the sliding-mode scheme. Numerical experiments illustrate the performance and execution of the control strategy considering different parameter values for biomass growth rate. The robustness and fragility of the proposed controller are also discussed with respect to the modelling uncertainty and small changes in the controller gains, respectively.     

Cell population models for bifurcation analysis and nonlinear control of continuous yeast bioreactors

Saccharomyces cerevisiae (baker's yeast) can exhibit sustained oscillations over a wide range of operating conditions when produced in a continuous bioreactor. In this paper the bifurcations leading to these periodic solutions are investigated using an unstructured, segregated model in which the population balance equation (PBE) for the cell mass distribution is coupled to the mass balance of the rate limiting substrate. The PBE model is shown to produce periodic solutions over a range of dilution rates due to the presence of two supercritical Hopf bifurcations. The problem of oscillation attenuation using nonlinear feedback control with four candidate input/output variable pairings is investigated. The controller designs are based on a low dimensional moment representation of the PBE model. The performance of the nonlinear controllers are compared and discussed.     

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.     

Tracking of non-square nonlinear continuous time systems with piecewise constant model predictive control

The paper presents a model predictive control (MPC) algorithm for continuous-time, possibly non-square nonlinear systems. The algorithm guarantees the tracking of asymptotically constant reference signals by means of a control scheme were the integral action is directly imposed on the error variables rather than on the control moves. The plant under control, the state and control constraints and the performance index to be minimized are described in continuous time, while the manipulated variables are allowed to change at fixed and uniformly distributed sampling times. The algorithm is used to control a continuous fermenter where the manipulated variables are the dilution rate and the feed substrate concentration while the controlled variable is the biomass concentration.     

Optimization of microorganisms growth processes

Microorganisms growth processes are encountered in many biotechnological applications. For an increased economic benefit, optimizing their productivity is of great interest. Often the growth is inhibited by the presence in excess of other components. Inhibition determines the occurrence of multiple equilibrium points, which makes the optimal steady state reachable only from a small region of the system state space. Thus dynamic control is needed to drive the system from an initial state (characterized by a low concentration of microorganisms) to the optimal steady state. The strategy presented in this paper relies on the solutions of two optimization problems: the problem of optimal operation for maximum productivity in steady state (steady state optimization) and the problem of the start-up to the optimal steady state (transient optimization). Steady state optimization means determining the optimal equilibrium point (the amount of microorganisms harvested is maximum). The transient optimization is solved using the maximum principle of Pontryagin.The proposed control law, which drives the bioreactor from an initial state to the optimal steady state while maximizing the productivity, consists of switching the manipulated variable (dilution rate) from the minimum to the maximum value and then to the optimal value at well defined instants. This control law substantially increases the stability region of the optimal equilibrium point. Aside its efficiency, the strategy is also characterized by simplicity, being thus appropriate for implementation in real-life systems. Another important advantage is its generality: this technique may be applied to any microorganisms growth process which involves only one biochemical reaction. This means that the sequence of the control levels does not depend on the structure and parameters of the reaction kinetics, the values of the yield coefficients or the number of components in the bioreactor.     

Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

We consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND). The eigenstates of the measured observable are thus the open-loop stationary states: they are used to construct a closed-loop supermartingale playing the role of a strict control Lyapunov function. The parameters of this supermartingale are calculated by inverting a Metzler matrix that characterizes the impact of the control input on the Kraus operators defining the Markov process. The resulting state feedback scheme, taking into account a known constant delay, provides the almost sure convergence to the target state. This convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box, a cavity quantum electrodynamics system.     

Transfer function conditions for output feedback disturbance rejection

A recent paper [1] has derived state space conditions under which the disturbance transfer function in a linear multivariable system can be zeroed by a dynamic compensator forced by a prescribed set of measurements. The present note derives necessary conditions for this problem in terms of the orders of the open-loop control, disturbance, and measurement transfer functions. These necessary conditions are shown to be generically sufficient for solvability. Moreover, they provide additional insight into the geometric solvability conditions, are simple to check, and extend the corresponding results obtained for the state feedback case [2].     

On the optimal control of discrete-time linear systems with random parameters

This work is concerned with the optimal control of a discrete-time linear system with random parameters. It is assumed that the parameters of the system vary randomly during the process, namely, the parameters constitute sequences of random variables. These random variables are not necessarily independent. An important particular case occurs where there are unknown constant parameters in the system. The measurements of the state of the system contain additive noise. A quadratic function of the state and controller, with appropriate weighting, serves as the criterion function. The solutions for the open-loop controller and the open-loop feedback controller are presented. The method of solution is based on the dynamic programming approach which leads to functional recurrence equations.     

具有完整性的线性调节器设计

到目前为止,控制系统的完整性分析基本上局限在开环渐近稳定的系统上。本文在L.S.Shieh提出的方法的基础上,把它推广到开环不稳定系统中去,得到了系统在正常状态下具有较好动态性能的具有完整性的状态反馈控制律。整个方法已编制成FORTRAN语言,设计人员利用计算机可很快地设计出所需要的反馈控制律。     

工业重载液压力伺服系统的抗扰动控制

由于重载荒磨机负载惯性大阻尼小, 施力系统开环传递函数含有一对纯虚零点, 而且负载特性变化会明显影响开环频带宽度. 同时, 砂轮高速旋转会产生高频率大幅度的偏心力干扰, 由于系统频响较低, 单纯的输出反馈方式无法抑制干扰. 本文通过采用非线性跟踪微分器提取状态观测量并引入非线性前馈系数, 消除了纯虚零点, 消除了负载特性变化对开环频宽的影响, 显著地抑制了外部位移干扰和偏心力干扰, 从而使得产品质量大幅度提高.     

Design of control systems with random parameters†

In this paper a method for designing control systems with random parameters and random initial state is presented. The observed signal available for feedback is a function of the state and system parameters corrupted by additive white Gaussian noise with zero mean. The proposed control consists of an open-loop term, which is the optimal control for the parameters and initial state equal to their mean values, plus a feedback correction term. The feedback correction term is a linear function of the estimated values of the deviation in initial state and system parameters from their mean values. The control tends to the optimally sensitive control when the measurement noise tends to zero, and the feedback correction tends to zero when the measurement noise tends to infinity. A numerical experiment with a simplified model for nuclear reactor control shows that in spite of fairly large measurement noise, significant improvement in performance can be achieved using the proposed control compared to open-loop control.     

Stochastic Control of Linear and Nonlinear Econometric Models: Some Computational Aspects

This paper considers the optimal control of small econometric models applying the OPTCON algorithm. OPTCON determines approximate numerical solutions to optimum control problems for nonlinear stochastic systems. These optimum control problems consist in minimizing a quadratic objective function for linear and nonlinear econometric models with additive and multiplicative (parameter) uncertainties. The algorithm was programmed in C# and in MATLAB and allows for stochastic control with open-loop and passive learning (open-loop feedback) information patterns. Here we compare the results of applying the OPTCON2 version of the algorithm to two macroeconomic models for Slovenia, the nonlinear model SLOVNL and the linear model SLOVL. The results for both models are similar, with open-loop feedback controls giving better results on average and less outliers than open-loop controls. The number of outliers is higher in the nonlinear case and especially under high parameter uncertainty.     

Min-max quadratic cost control of systems described by approximate models

This paper considers the design of output regulators with the use of approximate models. The measure of the approximation between process and model outputs is represented by a bound in norm of the output error signals and it requires the computation of two numbers. The design is achieved with a rain-max approach where control and error signals are the two antagonists. The min-max solution is obtained as a linear function of the model state (open-loop solution). It is shown that no dosed-loop controls can improve the open-loop min-max performance. Conditions are given so as to preserve the min-max performance by means of proportional feedback of the system's output. In this case the min-max feedback law is obtained (closed-loop solution).     

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.     

A probabilistic likelihood approach to trajectory sensitivity

Comparative trajectory sensitivity is investigated with respect to small probabilistic perturbations in initial state and plant parameters when a neighboring feedback control rather than the nominally equivalent open-loop control is used.The uncertainty in initial state and plant parameters is modeled by jointly normally distributed random variables with known mean and known covariance matrix. The nominal control is assumed to yield a satisfactory nominal trajectory and it is desired to preserve this shape in non-nominal situations. The likelihood of the state provides a measure of insensitivity at time t.Using the system equations linearized around the nominal trajectory, the joint distributions of the incremental state are computed in both open-loop and closed-loop cases. It is established that the augmented nominal state when a neighboring optimal feedback control is used is at least as likely as that when the nominally equivalent open-loop control is used. The domain where the augmented incremental state for the optimal closed-loop control is more densely distributed than that corresponding to open-loop control is shown to be a hyper-hyperboloid. This domain is unbounded in certain directions, thus pointing out a potential disadvantage in using optimal feedback control.     

Output feedback adaptive extremum seeking control of a continuous stirred tank bioreactor with Monod's kinetics

An adaptive extremum seeking controller is presented for the optimization of the production rate of a continuous stirred tank bioreactor with Monod's kinetics. This controller is saturated outside a domain of interest and a reduced-order observer is designed to estimate the substrate concentration in the bioreactor. It is shown that once a persistence of excitation condition is satisfied, the convergence of the parameter estimates to their true values is guaranteed. Semi-global asymptotic stability for the output feedback closed-loop system is proved based on Lyapunov's theorem. Simulation results are shown to illustrate the performance of the proposed approach.     

Synthesis of optimal feedback controller by neural networks

In nonlinear optimal control problems, open-loop solutions from a fixed initial condition are much easier to compute than closed-loop solutions which do not depend on initial conditions. Two methods of using neural networks to approximate the optimal feedback controller are discussed. The indirect method uses a neural network to interpolate the whole field of extremals obtained from open-loop calculation. The direct method directly trains a neural network such that a general nonlinear optimal control performance index is minimized. The novelty of the modified backpropagation training is the requirement of the jacobian matrix of the neural network function. Simulation studies show that the closed-loop solution can be made to be arbitrarily close to the optimal open-loop solution with initial conditions chosen from a nontrivial subset of the state space.     

关联大系统的分散开环回路传递再

基于H∞理论和分散状态观测器,提出了不确定性关联大系统的分散开环回路传递再生方法.给出了满足开环再生矩阵H∞范数要求的观测器的存在性.在大系统关联项块对角奇异值分解的基础上,证明了通过解Riccati方程可以求得使开环再生差阵满足要求的分散观测器增益阵,以实现分散开环传递再生.试验结果表明,分散开环回路传递再生控制性能与分散状态反馈控制性能基本相同.     

Relations between the stabilization properties of two classes of hereditary linear systems with commensurate delays

Two classes of hereditary linear systems each subjected to two alternative controller structures (one of a free delay dynamic nature while the other one includes delays under integral forms) are analysed from a stabilizability viewpoint. The first class of plants has pure commensurate delays in both state and input while the second has its time delayed contributions to the dynamics under integral form in the differential equations. Under certain weak conditions, the four closed-loop systems resulting from combining each open-loop plant structure with each controller structure can lead to identical state/output trajectories for appropriately related controller parametrizations. This allows the derivation of stabilizability results for any of the four closed-loop combinations of open-loop system plus controller by using results from the literature, each one being originally applicable to just one of the above combinations     

Adaptive observers for nonlinearly parameterized class of nonlinear systems

In this paper, one proposes adaptive observers for a class of uniformly observable MIMO nonlinear systems with general nonlinear parameterizations. The state and the unknown parameters of the considered systems are supposed to lie in bounded domains which size can be arbitrarily large and the exponential convergence of the observers is shown to result under a well-defined persistent excitation condition. Moreover, the gain of the observers involves a design function that has to satisfy a simple condition which is given. Different expressions of such a function are proposed and it is shown that adaptive high gain like observers and adaptive sliding mode like observers can be derived by considering particular expressions of the design function. The theory is supported by simulation results related to the estimation of the biomass concentration and the Contois model parameters in a bioreactor.