LMI based design of constrained fuzzy predictive control

Predictive control of nonlinear systems subject to output and input constraints is considered. A fuzzy model is used to predict the future behavior. Two new ideas are proposed here. First, an added constraint on the applied control action is used to ensure the decrease of a quadratic Lyapunov function, and so guarantee Lyapunov exponential stability of the closed-loop system. Second, the feasibility of the finite-horizon optimization problem with the added constraints is ensured based on an off-line solution of a set of LMIs. The novel stability method is compared to the existing methods, such as the techniques based on the end-point constraints (terminal constraint set), and the robust stability techniques based on the small gain theory. The proposed method ensures Lyapunov exponential stability, does not need an auxiliary controller and can be used with any feasible controller parameters. Illustrative examples including the predictive control of a highly nonlinear chemical reactor (CSTR) are discussed.     

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.     

A computationally efficient robust tube based MPC for linear switched systems

This article considers the robust regulation problem for a class of constrained linear switched systems with bounded additive disturbances. The proposed solution extends the existing robust tube based model predictive control (RTBMPC) strategy for non-switched linear systems to switched systems. RTBMPC utilizes nominal model predictions, together with tightened sets constraints, to obtain a control policy that guarantees robust stabilization of the dynamic systems in presence of bounded uncertainties. In this work, similar to RTBMPC for non-switched systems, a disturbance rejection proportional controller is used to ensure that the closed loop trajectories of the switched linear system are bounded in a tube centered on the nominal system trajectories. To account for the uncertainty related to all sub-systems, the gain of this controller is chosen to simultaneously stabilize all switching dynamics. The switched system RTBMPC requires an on-line solution of a Mixed Integer Program (MIP), which is computationally expensive. To reduce the complexity of the MIP, a sub-optimal design with respect to the previous formulation is also proposed that uses the notion of a pre-terminal set in addition to the usual terminal set to ensure stability. The RTBMPC design with the pre-terminal set aids in determining the trade-off between the complexity of the control algorithm with the performance of the closed-loop system while ensuring robust stability. Simulation examples, including a Three-tank benchmark case study, are presented to illustrate features of the proposed MPC.     

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.     

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.     

Suboptimal target control for hybrid automata using model predictive control

In this paper, a computationally efficient controller is proposed for the target control problem when the system is modelled by hybrid automata. The design is carried out in two stages. First, we compute off-line the shortest switching path which has the minimum discrete cost from an initial set to the given target set. Next, a controller is derived which successfully drives the system from any given initial state in the initial set to the target set while minimizing a cost function. The model predictive control (MPC) technique is used when the current state is not within a guard set, otherwise the mixed-integer predictive control (MIPC) technique is employed. An on-line, semi-explicit control algorithm is derived by combining these two techniques. When the system is subject to additive bounded disturbance, it is shown that the proposed on-line control algorithm holds if tighter constraints on the original nominal state and controller are imposed. Finally, as an application of the proposed control procedure, the high-speed and energy-saving control problem of the CPU processing is considered.     

Event-triggered dual-mode predictive control for constrained nonlinear systems with continuous/intermittent detection

This paper investigates the problem of event-triggered model predictive control for constrained nonlinear systems. A dual-mode control strategy combined with two different event-triggered mechanisms are introduced to reduce computational and communication loads. For the event-triggered mechanisms, two cases, continuous detection and intermittent detection, are considered, respectively. In order to avoid the transmission of continuous predicted control input trajectories, the actual control signals are generated under a sample-and-hold manner. A decreasing prediction horizon is introduced to reduce the complexity of optimization problems and a tightened state constraint is designed to achieve robust constraint satisfaction. The sufficient conditions are derived to guarantee the feasibility and stability of the closed-loop system. The performance of the proposed strategy is illustrated by a simulation example.     

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.     

输入受限非线性系统的鲁棒广义逐点最小范数控制

面向具有输入约束的非线性不确定系统,根据输入输出有限增益$L_2$稳定的概念,提出了一种新的鲁棒控制Lyapunov函数.根据此概念,在前期研究的广义逐点最小范数控制的基础上,提出了一种对参数不确定性及外部干扰均具有抑制作用的鲁棒广义逐点最小范数控制器设计方法,并研究了其解析形式的求解方法.通过引入``引导函数",新的算法能够在保证鲁棒稳定性的同时更加灵活的考虑各种控制性能指标.最后,通过将新方法与状态相关Riccati方程非线性控制方法相结合验证该方法可用于提高原有控制器的闭环性能,并通过仿真实验验证了方法的可行性及有效性.     

Observer-based robust adaptive variable universe fuzzy control for chaotic system

A novel observer-base output feedback variable universe adaptive fuzzy controller is investigated in this paper. The contraction and expansion factor of variable universe fuzzy controller is on-line tuned and the accuracy of the system is improved. With the state-observer, a novel type of adaptive output feedback control is realized. A supervisory controller is used to force the states to be within the constraint sets. In order to attenuate the effect of both external disturbance and variable parameters on the tracking error and guarantee the states to be within the constraint sets, a robust controller is appended to the variable universe fuzzy controller. Thus, the robustness of system is improved. By Lyapunov method, the observer-controller system is shown to be stable. The overall adaptive control algorithm can guarantee the global stability of the resulting closed-loop system in the sense that all signals involved are uniformly bounded. In the paper, we apply the proposed control algorithms to control the Duffing chaotic system and Chua’s chaotic circuit. Simulation results confirm that the control algorithm is feasible for practical application.     

Small-gain technique-based adaptive output constrained control design of switched networked nonlinear systems via event-triggered communications

This paper investigates the problem of event-triggered tracking control for switched networked nonlinear systems with asymmetric time-varying output constraints. To handle the output constraints, an output-dependent generic constraint function is constructed to describe relationship between the output and the performance requirement. Meanwhile, an event-triggering rule is designed to reduce communication frequency between the controller and the actuator, thereby reducing the burden of the network communication. Based on the common Lyapunov function method and event-triggered control strategy, an adaptive control method is designed, which can guarantee that the closed-loop signals are bounded and avoid the Zeno behavior. Different from existing results considering constraints, the proposed scheme not only relaxes the restricted condition of constraint boundaries but also both the cases with and without output constraints can be addressed simultaneously. Furthermore, the stability of the system can be guaranteed by the small-gain technique. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed scheme.     

非线性系统双模鲁棒预测控制: 不变集切换方法

提出了一种基于不变集切换的非线性系统鲁棒预测控制算法.采用分段蕴含方法将非线性系统动态用一组线性变参数(LPV)系统动态包裹;计算出非线性系统的平衡面,对于每个LPV蕴含模型,针对相应的平衡点构造多面体不变集,得到覆盖非线性系统平衡面的一组相互重叠的不变集;在线根据系统当前状态所处的不变集和LPV区间切换控制律,最终保证闭环系统的稳定性.与传统的非线性预测控制相比,这种方法在构造不变集和确定控制律的计算都是离线进行,而在线只需根据当前状态切换控制律即可,从而避免了求解复杂的非凸非线性规划,在很大程度上降低了在线计算量.     

Robust tube-based MPC of constrained piecewise affine systems with bounded additive disturbances

In this article, we propose a robust tube-based MPC formulation for a class of hybrid systems, namely autonomously switched PWA systems, with bounded additive disturbances. The term tube-based refers to those control techniques whose objective is to maintain all possible trajectories of the uncertain system inside a tube which is a set around the nominal (or reference) system trajectory, that is free from disturbances. Common methods in tube-based control systems consider an error dynamical system as the difference between the state of the nominal system and the state of the perturbed system. However, this definition of the error dynamical system leads to a complicated switched affine system for PWA systems. Therefore, we use a new notion of the reference system similar to the nominal system except that the switching between the various modes of the PWA system is driven by the state of the real system. Using this reference system instead of the nominal system leads us to an error dynamical system that can be modeled as a switched linear system. We employ a switched linear controller to stabilize this error system under arbitrary switching. This auxiliary controller forces the states of the uncertain system to remain in a tube confined to the invariant set around the state of the reference system. We add new constraints and tighten some other constraints of the nominal hybrid MPC for the reference system, in order to ensure convergence of the uncertain system and to guarantee robust exponential stability of the closed-loop system.     

Predictive control of uncertain nonlinear parabolic PDE systems using a Galerkin/neural-network-based model

In this paper, a model predictive control (MPC) scheme for a class of parabolic partial differential equation (PDE) systems with unknown nonlinearities, arising in the context of transport-reaction processes, is proposed. A spatial operator of a parabolic PDE system is characterized by a spectrum that can be partitioned into a finite slow and an infinite fast complement. In this view, first, Galerkin method is used to derive a set of finite dimensional slow ordinary differential equation (ODE) system that captures the dominant dynamics of the initial PDE system. Then, a Multilayer Neural Network (MNN) is employed to parameterize the unknown nonlinearities in the resulting finite dimensional ODE model. Finally, a Galerkin/neural-network-based ODE model is used to predict future states in the MPC algorithm. The proposed controller is applied to stabilize an unstable steady-state of the temperature profile of a catalytic rod subject to input and state constraints.     

New Robust Model Predictive Control for Uncertain Systems with Input Constraints Using Relaxation Matrices

In this paper, we propose a new robust model predictive control (MPC) method for time-varying uncertain systems with input constraints. We formulate the problem as a minimization of the worst-case finite-horizon cost function subject to a new sufficient condition for cost monotonicity. The proposed MPC technique uses relaxation matrices to derive a less conservative terminal inequality condition. The relaxation matrices improve feasibility and system performance. The optimization problem is solved by semidefinite programming involving linear matrix inequalities (LMIs). A numerical example shows the effectiveness of the proposed method. The authors thank the associate editor and two anonymous referees for careful reading and useful suggestions.     

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.     

Robust optimal control for a batch nonlinear enzyme-catalytic switched time-delayed process with noisy output measurements

In this paper, we consider a nonlinear switched time-delayed (NSTD) system with an unknown time-varying function describing the batch culture. The output measurements are noisy. According to the actual fermentation process, this time-varying function appears in the form of a piecewise-linear function with unknown kinetic parameters and switching times. The quantitative definition of biological robustness is given to overcome the difficulty of accurately measuring intracellular material concentrations. Our main goal is to estimate these unknown quantities by using noisy output measurements and biological robustness. This estimation problem is formulated as a robust optimal control problem (ROCP) governed by the NSTD system subject to continuous state inequality constraints. The ROCP is approximated as a sequence of nonlinear programming subproblems by using some techniques. Due to the highly complex nature of these subproblems, we propose a hybrid parallel algorithm, based on Nelder–Mead method, simulated annealing and the gradients of the constraint functions, for solving these subproblems. The paper concludes with simulation results.     

Robust output feedback stabilization for two-dimensional continuous systems in roesser form

This paper considers the problem of robust stabilization via dynamic output feedbackcontrollers for uncertain two-dimensional continuous systems described by the Roesser's state space model. The parameter uncertainties are assumed to be norm-bounded appearing in all the matrices of the system model. A sufficient condition for the existence of dynamic output feedback controllers guaranteeing the asymptotic stability of the closed-loop system for all admissible uncertainties is proposed. A desired dynamic output feedback controller can be constructed by solving a set of linear matrix inequalities. Finally, an illustrative example is provided to demonstrate the applicability and effectiveness of the proposed method.     

Chaos suppression via observer based active control scheme: Application to Duffing’s oscillator

The aim of this paper is the synthesis of a robust control law for chaos suppression of a class of non-linear oscillator with affine control input. A robust state observer based active controller, which provides robustness against model uncertainties and noisy output measurements is proposed. The closed-loop stability for the underlying closed-loop system is done via the regulation and estimation errors dynamics. The performance of the proposed control law is illustrated with numerical simulations. The method is general and can be applied to various non-linear systems which satisfy the conditions required.     

Adaptive-observer-based output-constrained tracking of a class of arbitrarily switched uncertain non-affine nonlinear systems

This paper addresses an adaptive output-feedback tracking problem of arbitrarily switched pure-feedback nonlinear systems with time-varying output constraints and unknown control directions. In this work, the tracking problem of switched non-affine nonlinear systems with output constraints is transformed into the stabilization problem of switched unconstrained affine systems. The main contribution of this paper is to present a universal formula for constructing an adaptive state-observer-based tracking controller with only two adaptive parameters by using the common Lyapunov function method. These adaptive parameters in the proposed control scheme are derived using the function approximation technique and a priori knowledge of the signs of control gain functions is not required. The theoretical analysis is presented for the Lyapunov stability and the constraint satisfaction of the resulting closed-loop system in the presence of arbitrary switchings.