Economic plant-wide control design with backoff estimations using internal model control

Economic optimal operation typically involves operating as close as possible to the active constraints. However, in the presence of disturbances it is necessary to back-off from the constraints in order to avoid violating them. The backoff approach aims at selecting the control structure that minimizes the economic loss associated with the required constraint backoffs. This paper revisits the backoff approach and proposes a framework for estimating the constraint backoffs based on well-known elements of internal model control (IMC) theory, such as an automatic procedure for tuning the IMC low-pass filters, a stability condition, and an uncertainty representation based on diagonal input multiplicative uncertainty. Since the constraint backoffs are estimated using a linear dynamic model, the inclusion of input multiplicative uncertainty allows introducing conservatism in the estimation of the backoffs, which is required in order to avoid constraint violations. A forced-circulation evaporator benchmark problem is used to illustrate the approach.     

Delta算子系统具有极点约束的鲁棒非脆弱方差控制

研究Delta算子描述的线性不确定系统具有圆形区域极点约束的鲁棒非脆弱方差控制问题. 针对系统参数范数有界不确定性和控制器的乘性增益不确定性, 运用线性矩阵不等式(LMI)方法, 提出具有圆形区域极点约束的状态反馈鲁棒非脆弱方差控制器存在的条件和设计方法. 数值算例表明设计方法的可行性.     

Robust Stabilization Over Communication Channels in the Presence of Unstructured Uncertainty

This technical note is concerned with the problem of controlling plants over communication channels, where the plant is subject to two types of unstructured uncertainty: additive uncertainty and stable coprime factor uncertainty. Necessary lower bounds on the rate of transmission (or channel capacity) C, for robust stabilization, are computed explicitly. In particular, it is shown that the lower bound in the additive uncertainty case corresponds to a fixed point of a particular function. In the stable coprime factor uncertainty case, the derivation relies on linear fractional transformation concepts. The results are important in determining the minimum channel capacity needed in order to stabilize plants subject to unstructured uncertainty over communication channels. For instance, the bounds obtained can be used to analyze the effect of uncertainty on the channel capacity. An illustrative example is provided.     

Robust H∞ control of uncertain nonlinear systemvia state feedback

This technical note presents a solution of robust H_∞ control problem for an affine nonlinear system with gain bounded uncertainty. It is shown that the L₂-induced norm of a closed-loop system with the uncertainty is less than one if an extended nonlinear system with no uncertainty is dissipative with respect to a supply rate. In consequence of this result, a state feedback law such that the closed-loop system has H_∞ robust performance is derived based on the Hamilton-Jacobi inequality. It is also shown that the existing results for linear systems are special cases of the presented results     

Probabilistic tubes in linear stochastic model predictive control

This paper considers constrained control of linear systems with additive and multiplicative stochastic uncertainty and linear input/state constraints. Both hard and soft constraints are considered, and bounds are imposed on the probability of soft constraint violation. Assuming the plant parameters to be finitely supported, a method of constraint handling is proposed in which a sequence of tubes, corresponding to a sequence of confidence levels on the predicted future plant state, is constructed online around nominal state trajectories. A set of linear constraints is derived by imposing bounds on the probability of constraint violation at each point on an infinite prediction horizon through constraints on one-step-ahead predictions. A guarantee of the recursive feasibility of the online optimization ensures that the closed loop system trajectories satisfy both the hard and probabilistic soft constraints. The approach is illustrated by a numerical example.     

An Asymptotically Exact Approach to Robust Semidefinite Programming Problems with Function Variables

This technical note provides an approximate approach to a semidefinite programming problem with a parameter-dependent constraint and a function variable. This problem covers a variety of control problems including a robust stability/performance analysis with a parameter-dependent Lyapunov function. In the proposed approach, the original problem is approximated by a standard semidefinite programming problem through two steps: first, the function variable is approximated by a finite-dimensional variable; second, the parameter-dependent constraint is approximated by a finite number of parameter-independent constraints. Both steps produce approximation error. On the sum of these approximation errors, this technical note provides an upper bound. This bound enables quantitative analysis of the approach and gives an efficient way to reduction of the approximation error. Moreover, this technical note discusses how to verify that an optimal solution of the approximate problem is actually optimal also for the original problem.      

New Approach to Robust$ cal D$-Stability Analysis of Linear Time-Invariant Systems With Polytope-Bounded Uncertainty

This note presents a new approach to robust$ cal D$-stability analysis of linear time-invariant systems with polytope-bounded uncertainty. The proposed approach combines sufficient conditions for robust$ cal D$-stability in terms of feasibility problems with linear matrix inequalities (LMI) constraints and a new polytope partition technique. If the initial polytope does not attain the robust$ cal D$-stability sufficient condition, the polytope is successively subdivided until all subpolytopes attain the sufficient condition, in the case of robustly$ cal D$-stable uncertain system, or it is found a subpolytope vertex that does not attain the regional pole-placement constraints, in the case of an uncertain system that is not robustly$ cal D$-stable. It is presented a new general format polytope partition technique that allows the implementation of the proposed approach. The efficiency of the proposed approach is verified by means of illustrative examples and three different LMI-based analysis formulations.     

Local Control Lyapunov Functions for Constrained Linear Discrete-Time Systems: The Minkowski Algebra Approach

This technical note utilizes Minkowski algebra of convex sets to characterize a family of local control Lyapunov functions for constrained linear discrete-time systems. Local control Lyapunov functions are induced by parametrized contractive invariant sets. Underlying contractive invariant sets belong to a family of Minkowski decomposable convex sets and are, in fact, parametrized by a basic shape set and linear transformations of system matrices and a set of design matrices. Corresponding local control Lyapunov functions can be detected by solving a single, tractable, convex optimization problem which in case of polyhedral constraints reduces to a single linear program. The a priori complexity estimate of the characterized local control Lyapunov function is provided for some practically relevant cases. An illustrative example and relevant numerical experience are also reported.      

On the use of mixed‐integer linear programming for predictive control with avoidance constraints

This technical note concerns the predictive control of discrete‐time linear models subject to state, input and avoidance polyhedral constraints. Owing to the presence of avoidance constraints, the optimization associated with the predictive control law is non‐convex, even though the constraints themselves are convex. The inclusion of the avoidance constraints in the predictive control law is achieved by the use of a modified version of a mixed‐integer programming approach previously derived in the literature. The proposed modification consists of adding constraints to ensure that linear segments of the system trajectories between consecutive sampling times do not cross existing obstacles. This avoids the significant extra computation that would be incurred if the sampling time was reduced to prevent these crossings. Simulation results show that the inclusion of these additional constraints successfully prevents obstacle collisions that would otherwise occur. Copyright © 2008 John Wiley & Sons, Ltd.     

Comments on “Model predictive control for systems with stochastic multiplicative uncertainty and probabilistic constraints” [Automatica 45 (2009) 167–172]

In a recent paper (Cannon, Kouvaritakis, & Wu, 2009), an MPC algorithm for systems with stochastic multiplicative uncertainty was proposed. It offers potentially significant computational advantages and it is less conservative. In this note, a modification is suggested to strengthen the original result.     

Predictive control of systems with Markovian jumps under constraints and its application to the investment portfolio optimization

In the paper, we study a problem of control with a predictive model for discrete systems with Markovian jumps and multiplicative noises. A strategy to control with regard for explicit constraints on control variables is defined. The results are applied to control an investment portfolio under constraints on investment amounts.     

Methodology for robust multi-parametric control in linear continuous-time systems

This paper presents an extension of the recent multi-parametric (mp-)NCO-tracking methodology by Sun et al. [Comput. Chem. Eng. 92 (2016) 64–77] for the design of robust multi-parametric controllers for constrained continuous-time linear systems in the presence of uncertainty. We propose a robust-counterpart formulation and solution of multi-parametric dynamic optimization (mp-DO), whereby the constraints are backed-off based on a worst-case propagation of the uncertainty using either interval analysis or ellipsoidal calculus and an ancillary linear state feedback. We address the case of additive uncertainty, and we discuss approaches to dealing with multiplicative uncertainty that retain tractability of the mp-NCO-tracking design problem, subject to extra conservativeness. In order to assist with the implementation of these controllers, we also investigate the use of data classifiers based on deep learning for approximating the critical regions in continuous-time mp-DO problems, and subsequently searching for a critical region during on-line execution. We illustrate these developments with the case studies of a fluid catalytic cracking (FCC) unit and a chemical reactor cascade.     

基于多层概率集的随机预测控制算法设计

考虑具有乘型不确定性的离散随机系统约束控制问题, 设计了一种基于多层概率集的随机预测控制算法. 多层概率集描述了状态在多步反馈控制律下的一系列不同概率的分布区域, 因此能够同时保证多个不同概率要求的软约束. 通过动态优化多步反馈律, 算法具有较大的可行范围. 之后设计的简化算法在降低计算负担的同时保证了算法的可行范围.     

Analysis and synthesis of nested feedback systems

The H/sub /spl infin//-loopshaping framework is extended to the design of nested (or cascaded) feedback systems consisting of an inner and an outer loop. Three robust stability margins are defined for such systems and an inequality relating them obtained for the case when the inner loop is implemented in observer form. This motivates the introduction of two synthesis algorithms which are compared with a sequential approach in the design of a cascade-limiter for the Quanser Helicopter experiment. By establishing an equivalence between normalized coprime factor uncertainty and a combination of output multiplicative and input inverse multiplicative uncertainty, it is possible to demonstrate in what sense the new compensators are superior to the one obtained by the sequential method.     

Model Predictive Control Tuning by Controller Matching

The effectiveness of model predictive control (MPC) in dealing with input and state constraints during transient operations is well known. However, in contrast with several linear control techniques, closed-loop frequency-domain properties such as sensitivities and robustness to small perturbations are usually not taken into account in the MPC design. This technical note considers the problem of tuning an MPC controller that behaves as a given linear controller when the constraints are not active (e.g., for perturbations around the equilibrium that remain within the given input and state bounds), therefore inheriting the small-signal properties of the linear control design, and that still optimally deals with constraints during transients. We provide two methods for selecting the MPC weight matrices so that the resulting MPC controller behaves as the given linear controller, therefore solving the posed inverse problem of controller matching, and is globally asymptotically stable.      

Simulation-Based Discrete Optimization of Stochastic Discrete Event Systems Subject to Non Closed-Form Constraints

This technical note addresses the discrete optimization of stochastic discrete event systems for which both the performance function and the constraint function are not known but can be evaluated by simulation and the solution space is either finite or unbounded. Our method is based on random search in a neighborhood structure called the most promising area proposed in and a moving observation area. The simulation budget is allocated dynamically to promising solutions. Simulation-based constraints are taken into account in an augmented performance function via an increasing penalty factor. We prove that under some assumptions, the algorithm converges with probability 1 to a set of true local optimal solutions. These assumptions are restrictive and difficult to verify but we hope that the encouraging numerical results would motivate future research exploiting ideas of this technical note.      

Smith predictor design for robust performance

A method is outlined for designing Smith predictor controllers that provide robust performance despite real parameter uncertainties in the process model. Insight into the design process is gained by viewing the Smith predictor from the perspective of internal model control. Performance requirements are written in terms of a frequency-domain weight restricting the magnitude of the closed-loop sensitivity function. A general method for approximating multiple parameter uncertainties by a single multiplicative uncertainty is developed—an exact bound is derived for the magnitude of multiplicative uncertainty used to approximate simultaneous uncertainties in process gain, time-constant, and time-delay. Three different tuning methods are demonstrated; each is applied to a wide range of parameter uncertainties in a first-order with time-delay model. The first tuning method locates loop transfer-function uncertainty regions to test for robust performance—real parameter uncertainties are considered exactly. The second tuning method approximates real parameter uncertainties by multiplicative uncertainty and uses structured singular value analysis to guarantee robust performance. The third is a ‘quick design’ method that considers the unit magnitude crossing of the multiplicative uncertainty. Finally, the Smith predictor controller is compared with the structured-singular-value-optimal controller.     

Moment stability of discontinuous stochastic dynamical systems

In this note, we establish new Lyapunov and Lagrange stability results in the pth mean for a class of discontinuous stochastic dynamical systems. We apply these results in the qualitative analysis of a class of digital feedback control systems that are subjected to multiplicative and additive disturbances in the plants. The present results constitute natural extensions of our earlier results for discontinuous deterministic dynamical systems     

Dynamic Output Feedback H∞ Control Problem for Linear Neutral Systems

This note presents the solvability conditions of the dynamic output feedback H_infin control problem for linear neutral systems with time- varying state delays in the delay-dependent case, where the delay is assumed to be time-varying continuous or differentiable uniformly bounded function, but no restriction on the derivative of the delay is needed, which means that the delay may be the fast time-varying function. An improved delay-dependent bounded real lemma (BRL) for a closed-loop system is established. Based on the obtained BRL, the dynamic output feedback H_infin controller is designed in terms of the linear matrix inequalities with inverse constraints. An iterative algorithm involving convex optimization methods is used to satisfy these constraints. The proposed results are illustrated in the examples.