A nonlinear observer design for an activated sludge wastewater treatment process

This paper treats the problem of estimating simultaneously the state and the unknown inputs of a class of nonlinear discrete-time systems. An observer design method for nonlinear Lipschitz discrete-time systems is proposed. By assuming that the linear part of this class of systems is time-varying, the state estimation problem of nonlinear system is transformed into a state estimation problem for LPV system. The stability analysis is performed using a Lyapunov function that leads to the solvability of linear matrix inequalities (LMIs). Performances of the proposed observer are shown through the application to an activated sludge process model.     

Stochastic MPC with inequality stability constraints

Stochastic uncertainty is a common feature of many control engineering problems, and is also present in a wider class of applications, e.g. finance and sustainable development. Recent work proposed a constrained MPC approach that took explicit account of the distributions of uncertain model parameters but used terminal equality constraints to ensure stability. The present paper reformulates the problem in order to relax the stability constraints by invoking appropriate terminal inequalities. The application of the proposed strategy and its advantages over earlier work are illustrated by means of a numerical example.     

Min-max MPC using a tractable QP problem

Min-max model predictive controllers (MMMPC) suffer from a great computational burden that is often circumvented by using approximate solutions or upper bounds of the worst possible case of a performance index. This paper proposes a computationally efficient MMMPC control strategy in which a close approximation of the solution of the min-max problem is computed using a quadratic programming problem. The overall computational burden is much lower than that of the min-max problem and the resulting control is shown to have a guaranteed stability. A simulation example is given in the paper.     

MPC for continuous piecewise-affine systems

A large class of hybrid systems can be described by a max–min-plus-scaling (MMPS) model (i.e., using the operations maximization, minimization, addition and scalar multiplication). First, we show that continuous piecewise-affine systems are equivalent to MMPS systems. Next, we consider model predictive control (MPC) for these systems. In general, this leads to nonlinear, nonconvex optimization problems. We present a new MPC method for MMPS systems that is based on canonical forms for MMPS functions. In case the MPC constraints are linear constraints in the inputs only, this results in a sequence of linear optimization problems such that the MPC control can often be computed in a much more efficient way than by just applying nonlinear optimization as was done in previous work.     

Conditions for which linear MPC converges to the correct target

This paper considers the efficacy of disturbance models for ensuring offset-free control and the determination of the optimum feasible steady-state target within linear model predictive control (MPC). Previously proposed methods for steady-state target determination can address model error, disturbances, and output target changes when the desired steady state is feasible, but may fail to achieve a feasible target that is as close as possible to the desired steady-state target when the desired target is unreachable due to active constraints. Under certain conditions, the resulting ‘feasible steady-state target’ can converge to a point that is not as close as possible to the optimal feasible target. By considering the Karush–Kuhn–Tucker (KKT) conditions of optimality for the steady-state target optimizer, sufficient multi-variable conditions are established for which convergence to the optimal feasible target is guaranteed and, conversely, when convergence to a sub-optimal feasible target is expected.     

MPC: Current practice and challenges

Linear Model Predictive Control (MPC) continues to be the technology of choice for constrained, multivariable control applications in the process industry. Successful deployment of MPC requires “getting right” multiple aspects of the control problem. This includes the design of the underlying regulatory controls, design of the MPC(s), test design for model identification, model development, and dealing with nonlinearities. Approaches and techniques that are successfully applied in practice are described, including the challenges involved in ensuring a successful MPC application. Academic contributions are highlighted and suggestions provided for improving MPC.     

Kernel methods for subspace identification of multivariable LPV and bilinear systems

Subspace identification methods for multivariable linear parameter-varying (LPV) and bilinear state-space systems perform computations with data matrices of which the number of rows grows exponentially with the order of the system. Even for relatively low-order systems with only a few inputs and outputs, the amount of memory required to store these data matrices exceeds the limits of what is currently available on the average desktop computer. This severely limits the applicability of the methods. In this paper, we present kernel methods for subspace identification performing computations with kernel matrices that have much smaller dimensions than the data matrices used in the original LPV and bilinear subspace identification methods. We also describe the integration of regularization in these kernel methods and show the relation with least-squares support vector machines. Regularization is an important tool to balance the bias and variance errors. We compare different regularization strategies in a simulation study.     

Enlarging the domain of attraction of MPC controllers

This paper presents a method for enlarging the domain of attraction of nonlinear model predictive control (MPC). The usual way of guaranteeing stability of nonlinear MPC is to add a terminal constraint and a terminal cost to the optimization problem such that the terminal region is a positively invariant set for the system and the terminal cost is an associated Lyapunov function. The domain of attraction of the controller depends on the size of the terminal region and the control horizon. By increasing the control horizon, the domain of attraction is enlarged but at the expense of a greater computational burden, while increasing the terminal region produces an enlargement without an extra cost.In this paper, the MPC formulation with terminal cost and constraint is modified, replacing the terminal constraint by a contractive terminal constraint. This constraint is given by a sequence of sets computed off-line that is based on the positively invariant set. Each set of this sequence does not need to be an invariant set and can be computed by a procedure which provides an inner approximation to the one-step set. This property allows us to use one-step approximations with a trade off between accuracy and computational burden for the computation of the sequence. This strategy guarantees closed loop-stability ensuring the enlargement of the domain of attraction and the local optimality of the controller. Moreover, this idea can be directly translated to robust MPC.     

Subspace identification of multivariable linear parameter-varying systems

A subspace identification method is discussed that deals with multivariable linear parameter-varying state-space systems with affine parameter dependence. It is shown that a major problem with subspace methods for this kind of system is the enormous dimension of the data matrices involved. To overcome the curse of dimensionality, we suggest using only the most dominant rows of the data matrices in estimating the model. An efficient selection algorithm is discussed that does not require the formation of the complete data matrices, but processes them row by row.     

一类双线性系统的广义预测控制研究

双线性模型预测控制的研究表明,采用一般双线性模型的预测控制将涉及非线性优化问题,在线处理相当困难,而采用线性近似模型的预测控制又会带来较大的偏差.针对一类输入一输出双线性系统,提出了一种双线性系统的广义预测控制算法.该算法将基于输入-输出模型双线性系统中的双线性项和线性项合并,建立了一种类似于线性系统的ARIMA模型,并充分利用多步最优预测信息,由递推近似实现多步预测.控制律具有解析形式,避免了一般非线性寻优的复杂计算,并能适用于非最小相位双线性系统.仿真实验表明该算法具有良好的控制效果.     

Fuzzy control of the activated sludge wastewater treatment process

The activated sludge process is a commonly used method for treating sewage and waste waters. It is characterised by a lack of relevant instrumentation, control goals that are not always clearly stated, the use of qualitative information in decision making and poorly understood basic biological behaviour mechanisms. In this brief paper we examine the behaviour of an experimental fuzzy control algorithm constructed to reflect actual operational practice. We conclude that this algorithm does rather well and that a fuzzy controller would be a useful and practical way of regulating the activated sludge process.     

Inferential control of product quality attributes—application to food cooking extrusion process

Reducing product quality variability is one of the most important issues for the process industry. As product quality attributes (PQAs) are often measured in a laboratory using off-line sensors, direct feedback control is not feasible unless expensive on-line sensors are used. Inferential control provides a means to the solution of this type of problem without the cost of expensive sensors. This paper reports a successful application of control of product quality attributes in a food cooking extruder. It covers the essential steps in achieving the control objective: (1) identifying the influential variables that can be measured on-line and have significant impact on the product quality attributes; (2) estimating a dynamic model between the influential variables and the PQAs for on-line prediction of product quality; (3) on-line feedback control using the influential variables. Experimental results demonstrate that the designed control system not only maintains the quality of the products but also automatically brings the PQAs to user's specification.     

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.     

Dissolved oxygen control for activated sludge processes

Controllers for dissolved oxygen reference trajectory tracking for activated sludge processes are proposed and investigated. A nonlinear model predictive controller and a direct reference adaptive controller are investigated. Both the nutrient and the phosphorous removal from a wastewater by its biological treatment using an activated sludge technology are considered. An approach to the controller design utilises a structure of the dissolved oxygen dynamics and its two time scales: fast and slow. The predictive controllers offer good tracking performance and robustness. The direct model reference adaptive controller is much simpler to implement. However, it is more difficult to compromise between tracking accuracy and rate of change and magnitudes of the control actions. The controllers are validated by simulation using real data sets and an ASM2d model of the biological reactor.     

Exponential Stabilization of Bilinear Systems With Open‐Loop Unstable Dynamics

The quadratic control (or its modification) can stabilize a homogeneous bilinear system asymptotically (or exponentially) if its open‐loop dynamics are neutrally stable. When the open‐loop dynamics are unstable, the quadratic control can only bring the system state to a bounded region around the origin. In this paper, we propose a new nonlinear control which can drive the state of an open‐loop unstable system to the origin exponentially. This control algorithm can be easily extended to time varying systems with slight modification.     

一类输入受限不确定时滞系统的准Min-Max模型预测控制

针对一类输入受限离散不确定时滞系统,提出一种基于准Ｍｉｎ-Ｍａｘ的模型预测控制器设计方法．定义了时滞系统的鲁棒性能指标,给出了系统稳定的充分条件,通过求解ＬＭＩ凸优化获得控制器．准Ｍｉｎ-Ｍａｘ预测控制将当前控制量作为独立优化变量,与其他作为反馈控制的时域控制序列分开处理,有效地降低了算法的保守性,提高了可行性．仿真算例验证了所提出控制方法的有效性．     

An MPC approach to the design of two-layer hierarchical control systems

A methodology for the design of two-layer hierarchical control systems is presented. The high layer corresponds to a system with slow dynamics, whose control inputs must be provided by subsystems with faster dynamics placed at the low layer. Model Predictive Control laws are synthesized for both layers and overall convergence properties are established. The use of different control configurations is also considered by allowing the switching on/off of the subsystems at the low layer. A simulation example is reported to witness the potentialities of the proposed solution.     

Iterative learning control-based batch process control technique for integrated control of end product properties and transient profiles of process variables

Importance of batch processes has grown recently with the increasing economic competition that has pushed the manufacturing industries to pursue small quantity production of diverse high value-added products. Accordingly, systems engineering research on advanced control and optimization of batch processes has proliferated. In this paper, we examine the potentials of ‘iterative learning control (ILC)’ as a framework for industrial batch process control and optimization. First, various ILC rules are reviewed to provide a historical perspective. Next it is shown how the concept of ILC can be fused with model predictive control (MPC) to build an integrated end product and transient profile control technique for industrial chemical batch processes. Possible extensions and modifications of the technique are also presented along with some numerical illustrations. Finally, other related techniques are introduced to note the similarities and contemplate the opportunities for synergistic integration with the current ILC framework.     

Convergence properties of constrained linear system under MPC control law using affine disturbance feedback

This paper shows new convergence properties of constrained linear discrete time system with bounded disturbances under Model Predictive Control (MPC) law. The MPC control law is obtained using an affine disturbance feedback parametrization with an additional linear state feedback term. This parametrization has the same representative ability as some recent disturbance feedback parametrization, but its choice together with an appropriate cost function results in a different closed-loop convergence property. More exactly, the state of the closed-loop system converges to a minimal invariant set with probability one. Deterministic convergence to the same minimal invariant set is also possible if a less intuitive cost function is used. Numerical experiments are provided that validate the results.