Terminal constrained robust hybrid iterative learning model predictive control for complex time-delayed batch processes

This work mainly addresses terminal constrained robust hybrid iterative learning model predictive control against time delay and uncertainties in a class of complex batch processes with input and output constraints. In this work, an equivalently novel extended two-dimensional switched system is first constructed to represent the process model by introducing state difference, output error and new relaxation variable information. Then, a hybrid predictive updating controller is proposed and an optimal performance index function including terminal constraints is designed. Under the condition that the switching signal meets certain conditions, the solvable problem of model predictive control is realized by Lyapunov stability theory. Meanwhile, the design scheme of controller parameters is also given. In addition, the robust constraint set is adopted to overcome the disadvantage that the traditional asymptotic stability cannot converge to the origin when it involves disturbances, such that the system state converges to the constraint set and meets its expected value. Finally, the effectiveness of the proposed algorithm is verified by controlling the speed and pressure parameters of the injection molding process.     

Model predictive control for switched systems with a novel mixed time/event-triggering mechanism

This paper investigates observer-based model predictive control (MPC) for switched systems with a mixed time/event-triggering mechanism. The problem of predictive control that can achieve receding horizon optimization is considered and solved by minimizing an upper bound of the quadratic cost function. Since the system state may not be fully measured in practice, state observers are employed to estimate. A mixed mechanism including adaptive event-triggering and time-triggering is proposed, which can be switched determined by a threshold describing system performance to better balance system resource utilization and performance requirements. Then, a closed-loop switched system subject to networked-time-delay is modeled. Piecewise Lyapunov function technique and average dwell time approach are utilized to ensure asymptotical stability. Afterwards, MPC controller construction problem is turned into a LMIs feasibility problem. A new solving method of sufficient conditions for co-design of the state observers, feedback controllers and mixed triggering mechanism is derived. Lastly, simulation examples illustrate the correctness and advantages of research content.     

Delay-dependent nonfragile guaranteed cost control for nonlinear time-delay systems

This paper concerns the nonfragile guaranteed cost control problem for a class of nonlinear dynamic systems with multiple time delays and controller gain perturbations. Guaranteed cost control law is designed under two classes of perturbations, namely, additive form and multiplicative form. The problem is to design a memoryless state feedback control law such that the closed-loop system is asymptotically stable and the closed-loop cost function value is not more than a specified upper bound for all admissible uncertainties. Based on the linear matrix inequality (LMI) approach, some delay-dependent conditions for the existence of such controller are derived. A numerical example is given to illustrate the proposed method.     

Characterization and computation of control invariant sets for linear impulsive control systems

Impulsive control systems are suitable to describe and control a venue of real-life problems, going from disease treatment to aerospace guidance. The main characteristic of such systems is that they evolve freely in-between impulsive actions, which makes it difficult to guarantee its permanence in a given state-space region. In this work, we develop a method for characterizing and computing approximations to the maximal control invariant sets for linear impulsive control systems, which can be explicitly used to formulate a set-based model predictive controller. We approach this task using a tractable and non-conservative characterization of the admissible state sets, namely the states whose free response remains within given constraints, emerging from a spectrahedron representation of such sets for systems with rational eigenvalues. The so-obtained impulsive control invariant set is then explicitly used as a terminal set of a predictive controller, which guarantees the feasibly asymptotic convergence to a target set containing the invariant set. Necessary conditions under which an arbitrary target set contains an impulsive control invariant set (and moreover, an impulsive control equilibrium set) are also provided, while the controller performance are tested by means of two simulation examples.     

Guaranteed cost control of time-delay chaotic systems

This article studies a guaranteed cost control problem for a class of time-delay chaotic systems. Attention is focused on the design of memory state feedback controllers such that the resulting closed-loop system is asymptotically stable and an adequate level of performance is also guaranteed. Using the Lyapunov method and LMI (linear matrix inequality) framework, two criteria for the existence of the controller are derived in terms of LMIs. A numerical example is given to illustrate the proposed method.     

Robust Transferable Sets of Linear Transferring Strategies

The problem of transferring a controlled linear system with a disturbance to a prescribed target set (a hyperplane at a prescribed time instant) is considered. The control and disturbance are subject to geometrical constraints. The original transfer problem is reduced to a scalar one. Then, for a given transferring linear feedback strategy, the respective transferable set is studied, i.e. the set of all the initial positions in the time/state plane, from which the target set is reached, respecting the control constraints, against any admissible disturbance. Necessary and sufficient conditions for the existence of the transferable set are derived, its set-theoretical properties are established, the algorithm of its boundary construction is proposed, the boundary smoothness is studied. Illustrative examples are presented.     

Optimal control of continuity equations

An optimal control problem for the continuity equation is considered. The aim of a “controller” is to maximize the total mass within a target set at a given time moment. The existence of optimal controls is established. For a particular case of the problem, where an initial distribution is absolutely continuous with smooth density and the target set has certain regularity properties, a necessary optimality condition is derived. It is shown that for the general problem one may construct a perturbed problem that satisfies all the assumptions of the necessary optimality condition, and any optimal control for the perturbed problem, is nearly optimal for the original one.     

Robust output feedback time optimal decomposed controllers for linear systems via moving horizon estimation

This paper studies the robust output feedback time optimal control (TOC) problem for linear discrete-time systems with state and input constraints. Bounded state disturbances are assumed. The moving horizon estimation (MHE) technique combined with a Luenberger observer is used to design a state estimator with which the state estimation error converges to and remains in some disturbance invariant set. A novel approach is proposed to reduce the computational complexity of TOC, in which the terminal controller comprises several predetermined local linear feedback laws, resulting in a large terminal set. Starting from this relatively large terminal set, a large domain of attraction of the proposed TOC controller can be obtained by using a short horizon, which consequently leads to a low on-line computational effort. A correction term, the output of the observer subtracted from the output of the plant and then multiplied by a design matrix, is added to the TOC controller, which aims at further correcting estimates of the state based on the present estimation error. Furthermore, by formulating a suitable cost function, as time evolves the TOC controller reaches the desired controller to obtain a good asymptotical behavior. A case study is used to illustrate the proposed approach.     

基于参数李雅普诺夫函数的鲁棒预测控制器

针对多包描述的不确定系统,提出一种新的鲁棒约束预测控制器.离线设计时引入参数Lyapunov函数以减少单一Lyapunov函数设计时的保守性,得到多包系统Worst-case情况下性能最优的不变集,在线求解多包系统无穷时域性能指标的min-max优化问题.设计采用了时变的终端约束集,扩大了初始可行域,而且能够获得较优的控制性能.仿真结果验证了该方法的有效性.     

Switching design for suboptimal guaranteed cost control of 2-D nonlinear switched systems in the Roesser model

This paper deals with the problem of switching design for guaranteed cost control of discrete-time two-dimensional (2-D) nonlinear switched systems described by the Roesser model. The switching signal, which determines the active mode of the system, is subject to a state-dependent process whose values belong to a finite index set. By using 2-D common Lyapunov function approach, a sufficient condition expressed in terms of tractable matrix inequalities is first established to design a min-projection switching rule that makes the 2-D switched system asymptotically stable. The obtained result on stability analysis is then utilized to synthesize a suboptimal state feedback controller that minimizes the upper bound of a given infinite-horizon cost function. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design method.     

Robust memory state feedback model predictive control for discrete-time uncertain state delayed systems

In this paper, we propose a memory state feedback model predictive control (MPC) law for a discrete-time uncertain state delayed system with input constraints. The model uncertainty is assumed to be polytopic, and the delay is assumed to be unknown, but with a known upper bound. We derive a sufficient condition for cost monotonicity in terms of LMI, which can be easily solved by an efficient convex optimization algorithm. A delayed state dependent quadratic function with an estimated delay index is considered for incorporating MPC problem formulation. The MPC problem is formulated to minimize the upper bound of infinite horizon cost that satisfies the sufficient conditions. Therefore, a less conservative sufficient conditions in terms of linear matrix inequality (LMI) can be derived to design a more robust MPC algorithm. A numerical example is included to illustrate the effectiveness of the proposed method.     

The problem of package guidance by a given time for a linear control system with delay

The problem of guaranteed closed-loop guidance by a given time under incomplete information on the initial state is studied for a dynamical control system with delay by means of the method of open-loop control packages. A solvability criterion is proved for this problem in the case of a finite set of admissible initial states. The proposed technique is illustrated by a specific linear control system of differential equations with delay.     

Multi-model predictive control of Hammerstein-Wiener systems based on balanced multi-model partition

ABSTRACT

Model analysis of Hammerstein-Wiener systems has been made, and it is found that the included angle is applicable to such systems to measure the non-linearity. Then, a dichotomy gridding algorithm is proposed based on the included angle. Supporting by the gridding algorithm, a balanced multi-model partition method is put forward to partition a Hammerstein-Wiener system into a set of local linear models. For each linear model, a linear model predictive controller (MPC) is designed. After that, a multi-MPC is composed of the linear MPCs via soft switching. Thus, a complex non-linear control problem is transformed into a set of linear control problems, which simplifies the original control problem and improves the control performance. Two non-linear systems are built into Hammerstein-Wiener models and investigated using the proposed methods. Simulations demonstrate that the proposed gridding and partition methods are effective, and the resulted multi-MPC controller has satisfactory performance in both set-point tracking and disturbance rejection control.     

ON/OFF strategy based minimum-time control of continuous Petri nets

This paper focuses on the target marking control problem of timed continuous Petri nets (TCPN), aiming to drive the system from an initial state to a desired final one. This problem is similar to the set-point control problem in a general continuous-state system. In a previous work, a simple and efficient ON/OFF controller was proposed for Choice-Free nets, and it was proved to be minimum-time (Wang, 2010). However, for general TCPN the ON/OFF controller may bring the system to “blocking” situations due to its “greedy” firing strategy, and the convergence to the final state is not ensured. In this work the ON/OFF controller is extended to general TCPN by adding more “fair” strategies to solve conflicts in the system: the ON/OFF+ controller is obtained by forcing proportional firings of conflicting transitions. Nevertheless, such kind of controller might highly slow down the system when transitions have flows of different orders of magnitude, therefore a balancing process is introduced, leading to the B-ON/OFF controller. A third approach introduced here is the MPC-ON/OFF controller, a combination of Model Predictive Control (MPC) and the ON/OFF strategy; it may achieve a smaller number of time steps for reaching the final states, but usually requires more CPU time for computing the control laws. All the proposed extensions are heuristic methods for the minimum-time control and their convergences are proved. Finally, an application example of a manufacturing cell is considered to illustrate the methods. It is shown that by using the proposed controllers, reasonable numbers of time steps for reaching the final state can be obtained with low computational complexity.     

A Schwarz domain decomposition method with gradient projection for optimal control governed by elliptic partial differential equations

A domain decomposition method (DDM) is presented to solve the distributed optimal control problem. The optimal control problem essentially couples an elliptic partial differential equation with respect to the state variable and a variational inequality with respect to the constrained control variable. The proposed algorithm, called SA-GP algorithm, consists of two iterative stages. In the inner loops, the Schwarz alternating method (SA) is applied to solve the state and co-state variables, and in the outer loops the gradient projection algorithm (GP) is adopted to obtain the control variable. Convergence of iterations depends on both the outer and the inner loops, which are coupled and affected by each other. In the classical iteration algorithms, a given tolerance would be reached after sufficiently many iteration steps, but more iterations lead to huge computational cost. For solving constrained optimal control problems, most of the computational cost is used to solve PDEs. In this paper, a proposed iterative number independent of the tolerance is used in the inner loops so as to save a lot of computational cost. The convergence rate of L²-error of control variable is derived. Also the analysis on how to choose the proposed iteration number in the inner loops is given. Some numerical experiments are performed to verify the theoretical results.     

An efficient model predictive control for Markovian jump systems with all unstable modes

This paper is concerned with the efficient model predictive control (EMPC) problem for a class of Markovian jump systems (MJSs) with unstable modes under polytopic uncertainties and hard constraints. The transition probability matrix and a dual-mode control strategy in the framework of EMPC are co-designed. To achieve a nice tradeoff among the computation burden, the initial feasible region, and the control performance, the EMPC is proposed, whose main idea is two-fold: (1) the terminal constraint set, the corresponding feedback gain, and proper switching rules (i.e. the transition probability) are designed simultaneously by solving an off-line “min–max” problem related to subsystem modes; and (2) a fairly large initial feasible region is obtained off-line by adjusting the dimension of the control perturbation sequence, meanwhile such a perturbation sequence is designed online to steer the system state belonging to initial feasible region into the terminal constraint set within the pre-determined steps. Furthermore, sufficient conditions are presented to rigidly guarantee the feasibility of the proposed EMPC algorithm and the mean-square stability of the underlying MJS. Finally, an illustrative example regarding the economic system is provided to verify the feasibility and effectiveness of the developed algorithm.     

Guidance problem for a distributed system with incomplete information on the state coordinates and an unknown initial state

We study the problem of guaranteed positional guidance of a linear partially observable control system with distributed parameters to a convex target set at a given time. The problem is considered under incomplete information. More precisely, we assume that the system is subjected to an unknown disturbance; in addition, the initial state is assumed to be unknown as well. Further, the sets of admissible disturbances and the set of admissible initial states, which is assumed to be finite, are known. An algorithm for solving the problem is suggested.     

A neural fuzzy control system with structure and parameter learning

A general connectionist model, called neural fuzzy control network (NFCN), is proposed for the realization of a fuzzy logic control system. The proposed NFCN is a feedforward multilayered network which integrates the basic elements and functions of a traditional fuzzy logic controller into a connectionist structure which has distributed learning abilities. The NFCN can be constructed from supervised training examples by machine learning techniques, and the connectionist structure can be trained to develop fuzzy logic rules and find membership functions. Associated with the NFCN is a two-phase hybrid learning algorithm which utilizes unsupervised learning schemes for structure learning and the backpropagation learning scheme for parameter learning. By combining both unsupervised and supervised learning schemes, the learning speed converges much faster than the original backpropagation algorithm. The two-phase hybrid learning algorithm requires exact supervised training data for learning. In some real-time applications, exact training data may be expensive or even impossible to obtain. To solve this problem, a reinforcement neural fuzzy control network (RNFCN) is further proposed. The RNFCN is constructed by integrating two NFCNs, one functioning as a fuzzy predictor and the other as a fuzzy controller. By combining a proposed on-line supervised structure-parameter learning technique, the temporal difference prediction method, and the stochastic exploratory algorithm, a reinforcement learning algorithm is proposed, which can construct a RNFCN automatically and dynamically through a reward-penalty signal (i.e., “good” or “bad” signal). Two examples are presented to illustrate the performance and applicability of the proposed models and learning algorithms.     

Minimax control of systems with uncertain state measurements

A system is controlled with the aid of uncertain state measurements, the errors of which are known to lie within a given, compact set. Control of the system incurs a cost of which it is desired that the largest value consistent with the measurements obtained be minimized. Two different measurement régimes are considered. In the first, measurements are obtained throughout the history of the process and, in the second, only an initial state measurement is obtained. Under certain circumstances, it is shown that the optimization problems for the two régimes are equivalent. The general solution of the problem for the second régime is given for the case when the dynamics are linear, and the cost function quadratic.The research activity of this author was supported in part by ONR.     

Linear programming approach to the control of discrete-time periodic systems with uncertain inputs

The problem is considered of finding a control strategy for a linear discrete-time periodic system with state and control bounds in the presence of unknown disturbances that are only known to belong to a given compact set. This kind of problem arises in practice in resource distribution systems where the demand has typically a periodic behavior, but cannot be estimated a priori without an uncertainty margin. An infinite-horizon keeping problem is formulated, which consists in confining the state within its constraint set using the allowable control, whatever the allowed disturbances may be. To face this problem, the concepts of periodically invariant set and sequence are introduced. They are used to formulate a solution strategy that solves the keeping problem. For the case of polyhedral state, control, and disturbance constraints, a computationally feasible procedure is proposed. In particular, it is shown that periodically invariant sequences may be computed off-line, and then they may be used to synthesize on-line a control strategy. Finally, an optimization criterion for the control law is discussed.