Suboptimal mean controllers for bounded and dynamic stochastic distributions

Based on the recently developed algorithms for the modelling and control of bounded dynamic stochastic systems (H. Wang, J. Zhang, Bounded stochastic distributions control for pseudo ARMAX stochastic systems, IEEE Transactions on Automatic control, 486–490), this paper presents the design of a subotpimal nonlinear mean controller for bounded dynamic stochastic systems with guaranteed stability. The B-spline functional expansion based square root model is used to represent the output probability density function of the system. This is then followed by the design of a mean controller of the output distribution of the system using nonlinear output tracking concept. A nonlinear quadratic optimization is performed using the well known Hamilton–Jacobi–Bellman equation. This leads to a controller which consists of a static unit, a state feedback part and an equivalent output feedback loop. In order to achieve high precision for the output tracking, the output feedback gain is determined by a learning process, where the Lyapunov stability analysis is performed to show the asymptotic stability of the closed loop system under some conditions. A simulation example is included to demonstrate the use of the algorithm and encouraging results have been obtained.     

Robust H-infinity fuzzy control for uncertainMarkovian jump nonlinear singular systems withwiener process

This paper deals with the problems of robust stochastic stabilization and H-infinity control for Markovian jump nonlinear singular systems with Wiener process via a fuzzy-control approach. The Takagi-Sugeno (T-S) fuzzy model is employed to represent a nonlinear singular system. The purpose of the robust stochastic stabilization problem is to design a state feedback fuzzy controller such that the closed-loop fuzzy system is robustly stochastically stable for all admissible uncertainties. In the robust H-infinity control problem, in addition to the stochastic stability requirement, a prescribed performance is required to be achieved. Linear matrix inequality (LMI) sufficient conditions are developed to solve these problems, respectively. The expressions of desired state feedback fuzzy controllers are given. Finally, a numerical simulation is given to illustrate the effectiveness of the proposed method.     

Nonlinear–nonquadratic optimal and inverse optimal control for stochastic dynamical systems

In this paper, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control for nonlinear stochastic dynamical systems. Specifically, we provide a simplified and tutorial framework for stochastic optimal control and focus on connections between stochastic Lyapunov theory and stochastic Hamilton–Jacobi–Bellman theory. In particular, we show that asymptotic stability in probability of the closed‐loop nonlinear system is guaranteed by means of a Lyapunov function that can clearly be seen to be the solution to the steady‐state form of the stochastic Hamilton–Jacobi–Bellman equation and, hence, guaranteeing both stochastic stability and optimality. In addition, we develop optimal feedback controllers for affine nonlinear systems using an inverse optimality framework tailored to the stochastic stabilization problem. These results are then used to provide extensions of the nonlinear feedback controllers obtained in the literature that minimize general polynomial and multilinear performance criteria. Copyright © 2017 John Wiley & Sons, Ltd.     

Comparative evaluation of nonlinear model predictive and flatness-based two-degree-of-freedom control design in view of industrial application

Two advanced nonlinear model-based control design methods – nonlinear model predictive control (NMPC) and a two-degree-of-freedom control-scheme with flatness-based feedforward control design and decentralised PI-controllers (FB-2DOF) – are compared in view of industrial application. The comparative evaluation is carried out on a setpoint-transition of the Klatt–Engell reactor model. Based on an analysis of simulation scenarios, the controllers are compared with respect to controller performance, robustness criteria, and implementation issues. Thereby, the choice of the control task and the comparison methodology are oriented on industrial practice.In the considered comparative evaluation, NMPC exhibits performance advantages when it comes to time-efficient setpoint-transitions in the nominal case, in which FB-2DOF control design is restricted by the existing input constraints. In return, robustness of stability of the FB-2DOF controller is determined only by the feedback controll part; it is therefore independent from the setpoint-transition performance – determined by the feedforward controll part – whereas the NMPC suffers from degradation of robustness properties if it is tuned for time-efficiency only. NMPC allows direct incorporation of process models and constraints, but, as it employs computationally expensive online optimisation, has to be connected to the digital control system (DCS) via some standard interface. The FB-2DOF controller in contrast can be directly implemented in a DCS, whereby the feedforward-part can be realised as an extension of an already existing feedback-part.     

Finite-time output feedback control for a class of stochastic low-order nonlinear systems

This paper discusses the problem of finite-time stabilisation for a class of stochastic low-order nonlinear systems via output feedback. By generalising the adding a power integrator technique, constructing an implementable reduced-order observer and using the stochastic finite-time stability criterion, a finite-time output feedback controller is presented to guarantee that the closed-loop system is finite-time stable in probability. A simulation example is provided to verify the effectiveness of the proposed design method.     

Finite-dimensional optimal controllers for nonlinear plants

Optimal risk sensitive feedback controllers are now available for very general stochastic nonlinear plants and performance indices. They consist of nonlinear static feedback of so called information states from an information state filter. In general, these filters are linear, but infinite dimensional, and the information state feedback gains are derived from (doubly) infinite dimensional dynamic programming. The challenge is to achieve optimal finite dimensional controllers using finite dimensional calculations for practical implementation.This paper derives risk sensitive optimality results for finite-dimensional controllers. The controllers can be conveniently derived for ‘linearized’ (approximate) models (applied to nonlinear stochastic systems). Performance indices for which the controllers are optimal for the nonlinear plants are revealed. That is, inverse risk-sensitive optimal control results for nonlinear stochastic systems with finite dimensional linear controllers are generated. It is instructive to see from these results that as the nonlinear plants approach linearity, the risk sensitive finite dimensional controllers designed using linearized plant models and risk sensitive indices with quadratic cost kernels, are optimal for a risk sensitive cost index which approaches one with a quadratic cost kernel. Also even far from plant linearity, as the linearized model noise variance becomes suitably large, the index optimized is dominated by terms which can have an interesting and practical interpretation.Limiting versions of the results as the noise variances approach zero apply in a purely deterministic nonlinear H_∞ setting. Risk neutral and continuous-time results are summarized.More general indices than risk sensitive indices are introduced with the view to giving useful inverse optimal control results in non-Gaussian noise environments.     

Stochastic differential games and inverse optimal control and stopper policies

In this paper, we consider a two-player stochastic differential game problem over an infinite time horizon where the players invoke controller and stopper strategies on a nonlinear stochastic differential game problem driven by Brownian motion. The optimal strategies for the two players are given explicitly by exploiting connections between stochastic Lyapunov stability theory and stochastic Hamilton–Jacobi–Isaacs theory. In particular, we show that asymptotic stability in probability of the differential game problem is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Isaacs equation, and hence, guaranteeing both stochastic stability and optimality of the closed-loop control and stopper policies. In addition, we develop optimal feedback controller and stopper policies for affine nonlinear systems using an inverse optimality framework tailored to the stochastic differential game problem. These results are then used to provide extensions of the linear feedback controller and stopper policies obtained in the literature to nonlinear feedback controllers and stoppers that minimise and maximise general polynomial and multilinear performance criteria.     

Reduced-order compensators via balancing and central control design for a structural control problem

Real-time control of a physical system necessitates controllers that are low order. In this article, we compare two balanced truncation methods as a means of designing low-order controllers for a nonlinear cable-mass system. The first is the standard technique of balanced truncation. The second, linear quadratic Gaussian (LQG) balanced truncation, can be thought of as balancing based on the controller, and states that are important from the perspective of control and filter design are retained. The control design applied to each reduced-order model is the central controller. We provide an overview of the central controller and devote attention to the design of this controller in the presence of balancing. Also described in this article is a method for reducing computational time in solving algebraic Riccati equations for the design of low-order LQG balanced controllers.     

output feedback control for uncertain stochastic systems with time-varying delays

This paper deals with the problem of H_∞ output feedback control for uncertain stochastic systems with time-varying delays. The parameter uncertainties are assumed to be time-varying norm-bounded. The aim is the design of a full-order dynamic output feedback controller ensuring robust exponential mean-square stability and a prescribed H_∞ performance level for the resulting closed-loop system, irrespective of the uncertainties. A sufficient condition for the existence of such an output feedback controller is obtained and the expression of desired controllers is given.     

Multi-objective constraint optimizing IOL control of distillation column with nonlinear observer

Performance of input–output linearizing (IOL) controllers suffers due to constraints on input and output variables. This problem is successfully tackled by augmenting IOL controllers with quadratic dynamic matrix controller (QDMC). However, this has created a constraint-mapping problem for coupled MIMO systems like distillation column. A multi-objective optimization problem needs to be solved to map the constraints on inputs. A suitable transformation technique is proposed to convert this multi-objective optimization problem to a single objective one. This makes the controller less computationally intensive and easy to implement. This controller (IOL-QDMC) along with nonlinear observer is implemented on a binary distillation column for dual composition control. Its performance is evaluated against a quadratic dynamic matrix controller (QDMC) and input–output linearization with PI controller (IOL-PI).     

The stability of an oceanic structure with T–S fuzzy models

In this study we construct and derive analytical solutions for a mathematical model of an oceanic environment in which wave-induced flow fields cause structural surge motion after which a fuzzy control technique is developed to alleviate structural vibration. Specifically the Takagi–Sugeno (T–S) fuzzy model is employed to approximate the oceanic structure and a parallel-distributed-compensation (PDC) scheme is utilized in a control procedure designed to reduce the structural response. All local state feedback controllers are integrated to construct a global fuzzy logic controller. The Lyapunov method is used to achieve structural stability. The interaction between the wave motion and the structural response is explained using the separation of variables method. The surge motion is related to the characteristics of the wave and the structure. A parametric approach is utilized to show these effects. Other parameters remain constant. In an oceanic structural system, platform migration is often caused by the wave force. The stability of an oceanic structure can be proven theoretically based on stability analysis. The decay of the displacement and velocity due to the use of the proposed fuzzy controllers is demonstrated by a numerical simulation.     

Reduced-Order Adaptive Controllers for Fluid Flows Using POD

This article presents a reduced-order adaptive controller design for fluid flows. Frequently, reduced-order models are derived from low-order bases computed by applying proper orthogonal decomposition (POD) on an a priori ensemble of data of the Navier–Stokes model. This reduced-order model is then used to derive a reduced-order controller. The approach discussed here differ from these approaches. It uses an adaptive procedure that improves the reduced-order model by successively updating the ensemble of data. The idea is to begin with an ensemble to form a reduced-order control problem. The resulting control is then applied back to the Navier–Stokes model to generate a new ensemble. This new ensemble then replaces the previous ensemble to derive a new reduced-order model. This iteration is repeated until convergence is achieved. The adaptive reduced-order controllers effectiveness in flow control applications is shown on a recirculation control problem in channel flow using blowing (actuation) on the boundary. Optimal placement for actuators is explored. Numerical implementations and results are provided illustrating the various issues discussed.     

Service process control: conceptualising a service as a feedback control system

This paper shows how a service process can be recast into a feedback control structure and how chemical process control principles can be applied. The outputs of the ‘service plant’ are process characteristics that meet the requirements of the plant's various stakeholders; the manipulated and disturbance variables are those variables that are within and outside a service manager's control respectively; the model is a process flowchart; and the controller is the service manager. Good stability, performance and robustness are also required because customers value reliability, responsiveness, and consistency of service in the face of variations caused by involvement of people in the service delivery. To illustrate these concepts, a patient treatment process within an emergency ward was recast as a control system, and an off-line model-based optimal controller was applied to this stochastic process. The benefits of recasting services as feedback control systems flow both ways: service design and operation will be improved by the application of engineering principles; and chemical process control will benefit from techniques specially developed to handle market fluctuations.     

Finite-time stabilisation of a class of stochastic nonlinear systems with dead-zone input

This paper addresses the problem of finite-time stabilisation by state feedback for a class of stochastic nonlinear systems in strict-feedback form. Remarkably, the system under consideration possesses high-order and low-order nonlinear growth rates and the dead-zone input. By delicately combining sign function with adding a power integrator technique, a state feedback controller is designed such that the closed-loop system is finite-time stable in probability. Simulation results of the parallel active suspension system are provided to show the effectiveness of the proposed method.     

A novel approach to output feedback control of fuzzy stochastic systems

This paper investigates the problem of Hankel-norm output feedback controller design for a class of T–S fuzzy stochastic systems. The full-order output feedback controller design technique with the Hankel-norm performance is proposed by the fuzzy-basis-dependent Lyapunov function approach and the conversion on the Hankel-norm controller parameters. Sufficient conditions are established to design the controllers such that the resulting closed-loop system is stochastically stable and satisfies a prescribed performance. The desired output feedback controller can be obtained by solving a convex optimization problem, which can be efficiently solved by standard numerical algorithms. Finally, a Henon map system is used to illustrate the effectiveness of the proposed techniques.     

Implications of dissipativity,inverse optimal control,and stability margins for nonlinear stochastic feedback regulators

In this paper, we derive stability margins for optimal and inverse optimal stochastic feedback regulators. Specifically, gain, sector, and disk margin guarantees are obtained for nonlinear stochastic dynamical systems controlled by nonlinear optimal and inverse optimal Hamilton‐Jacobi‐Bellman controllers that minimize a nonlinear‐nonquadratic performance criterion with cross‐weighting terms. Furthermore, using the newly developed notion of stochastic dissipativity, we derive a return difference inequality to provide connections between stochastic dissipativity and optimality of nonlinear controllers for stochastic dynamical systems. In particular, using extended Kalman‐Yakubovich‐Popov conditions characterizing stochastic dissipativity, we show that our optimal feedback control law satisfies a return difference inequality predicated on the infinitesimal generator of a controlled Markov diffusion process if and only if the controller is stochastically dissipative with respect to a specific quadratic supply rate.     

Continuous finite‐time state feedback stabilizers for some nonlinear stochastic systems

This paper is concerned with the problem of finite‐time stabilization for some nonlinear stochastic systems. Based on the stochastic Lyapunov theorem on finite‐time stability that has been established by the authors in the paper, it is proven that Euler‐type stochastic nonlinear systems can be finite‐time stabilized via a family of continuous feedback controllers. Using the technique of adding a power integrator, a continuous, global state feedback controller is constructed to stabilize in finite time a large class of two‐dimensional lower‐triangular stochastic nonlinear systems. Also, for a class of three‐dimensional lower‐triangular stochastic nonlinear systems, a recursive design scheme of finite‐time stabilization is given by developing the technique of adding a power integrator and constructing a continuous feedback controller. Finally, a simulation example is given to illustrate the theoretical results. Copyright © 2014 John Wiley & Sons, Ltd.     

-like control for nonlinear stochastic systems

In this paper we develop a H_∞-type theory, from the dissipation point of view, for a large class of time-continuous stochastic nonlinear systems. In particular, we introduce the notion of stochastic dissipative systems analogously to the familiar notion of dissipation associated with deterministic systems and utilize it as a basis for the development of our theory. Having discussed certain properties of stochastic dissipative systems, we consider time-varying nonlinear systems for which we establish a connection between what is called the L₂-gain property and the solution to a certain Hamilton–Jacobi inequality (HJI), that may be viewed as a bounded real lemma for stochastic nonlinear systems. The time-invariant case with infinite horizon is also considered, where for this case we synthesize a worst case-based stabilizing controller. Stability in this case is taken to be in the mean-square sense. In the stationary case, the problem of robust state feedback control is considered in the case of norm-bounded uncertainties. A solution is then derived in terms of linear matrix inequalities.     

深海自持式智能浮标双闭环模糊PID定深控制

针对深海自持式智能浮标运动模型非线性、强耦合性的特点,提出了一种基于双闭环反馈回路的模糊比例-积分-微分（proportion-integral-derivative,PID）定深控制器.根据浮标的浮力调节机构,分析了浮标的运动过程,建立了非线性运动方程.针对外环深度反馈回路,设计了模糊控制器.基于内环速度反馈回路与模糊控制器,设计了联级模糊PID定深控制器.传统PID定深控制器超调量5.6%,最终在目标深度±30 m范围内震荡,而双闭环模糊PID定深控制器在相同的上升时间内,超调量2.0%,深度误差控制在1.0%以内.存在外界扰动的情况下,通过双闭环模糊PID定深控制器的调节,浮标仍可以稳定在目标深度内.仿真结果表明,所建立的双闭环模糊PID定深控制系统具有良好的控制效果和稳定性.