Approximate linearization of nonlinear control systems

The paper address the problem of approximate linearization of a nonlinear control system by state feedback or dynamic feedback. The main result proved is that if the fast k d-relative degrees are equal to each other, then the input-output response of a nonlinear control system can be linearized to degree k. Any system having linear d-relative degree can be linearized to any degree by dynamic feedbacks. One example of signal tracking using dynamic feedback linearization method to improve the performance is also given.     

Stochastic optimal open-loop feedback control: Approximate solution of the Hamiltonian system

Considering a dynamic control system with random model parameters and using the stochastic Hamilton approach stochastic open-loop feedback controls can be determined by solving a two-point boundary value problem (BVP) that describes the optimal state and costate trajectory. In general an analytical solution of the BVP cannot be found. This paper presents two approaches for approximate solutions, each consisting of two independent approximation stages. One stage consists of an iteration process with linearized BVPs that will terminate when the optimal trajectories are represented. These linearized BVPs are then solved by either approximation fixed-point equations (first approach) or Taylor-Expansions in the underlying stochastic model parameters (second approach). This approximation results in a deterministic linear BVP, which can be handled by solving a matrix Riccati differential equation.     

Adaptive output feedback tracking control of stochastic nonlinear systems with dynamic uncertainties

In this paper, adaptive output feedback tracking control is developed for a class of stochastic nonlinear systems with dynamic uncertainties and unmeasured states. Neural networks are used to approximate the unknown nonlinear functions. K‐filters are designed to estimate the unmeasured states. An available dynamic signal is introduced to dominate the unmodeled dynamics. By combining dynamic surface control technique with backstepping, the condition in which the approximation error is assumed to be bounded is avoided. Using It ô formula and Chebyshev's inequality, it is shown that all signals in the closed‐loop system are bounded in probability, and the error signals are semi‐globally uniformly ultimately bounded in mean square or the sense of four‐moment. Simulation results are provided to illustrate the effectiveness of the proposed approach. Copyright © 2014 John Wiley & Sons, Ltd.     

Fast algorithms for generalized predictive control

Fast algorithms for generalized predictive control (GPC) are derived by adopting an approach whereby dynamic programming and a polynomial formulation are jointly exploited. They consist of a set of coupled linear polynomial recursions by which the dynamic output feedback GPC law is recursively computed wwith only O(Nn) computations for an n-th order plant and N-steps prediction horizon.     

High-gain feedback in non-linear control systems†

High-gain state and output feedback are investigated for non-linear control systems with a single additive input by using singular perturbation techniques.

Classical approximation results (Tihonov-like theorems) in singular perturbation theory are extended to non-linear control systems by defining a composite additive control strategy, a control-dependent fast equilibrium manifold and non-linear change of coordinates.

Those tools and an appropriate change of coordinates show that high-gain state feedback and variable structure control systems can be equivalently used for approximate non-linearity compensation in feedback-linearizable systems.

Next the effect of high-gain output feedback is shown to be related to the strong invertibility property and the relative order of invertibility. For strongly invertible systems the slow reduced subsystem coincides with the dynamics of the inverse system when zero input is applied and with the unobservable dynamics when a certain input-output feedback-linearizable transformation is applied.     

Recursive second-order approximation method for optimization and control of steady-state systems

A recursive second-order approximation approach for the optimization and control of steady-state systems is proposed. The method constructs the hessian matrix of the real performance by using a modification of the Broyden, Fletcher, Goldfart, and Shanno (BFGS) formula which only uses first-order information of the real process. The iterative direction can be obtained by solving a second-order approximate optimization problem which is determined by a modification of the BFGS formula. To obtain a new iterative point, Newton step and one-dimensional search updating strategies are employed. Global convergence and optimality of the derived algorithms are thoroughly investigated. In particular, the R-superlinear properties of the new approach are verified under local conditions. Numerical results show that the new approach is superior to the sequence model approximation (SMA) method.     

Smooth function approximation using neural networks

An algebraic approach for representing multidimensional nonlinear functions by feedforward neural networks is presented. In this paper, the approach is implemented for the approximation of smooth batch data containing the function's input, output, and possibly, gradient information. The training set is associated to the network adjustable parameters by nonlinear weight equations. The cascade structure of these equations reveals that they can be treated as sets of linear systems. Hence, the training process and the network approximation properties can be investigated via linear algebra. Four algorithms are developed to achieve exact or approximate matching of input-output and/or gradient-based training sets. Their application to the design of forward and feedback neurocontrollers shows that algebraic training is characterized by faster execution speeds and better generalization properties than contemporary optimization techniques.     

Pole assignment for linear time-invariant systems by periodic memoryless output feedback

It is well known that the poles of a linear time-invariant controllable and observable system can be assigned arbitrarily by state feedback. When only the output is available, pole assignment is still possible by means of dynamic output feedback. In this paper the potential of time-varying memoryless output feedback is considered. It is shown that, up to some technical conditions, it is indeed possible to allocate the poles of a linear time-invariant discrete-time system by memoryless output feedback with periodic gains. The period of the gains is (n + 1) with n the order of the system. The power of the design technique is proved to be comparable to what can be achieved by the classical dynamic feedback approach.     

Nonlinear stabilization of a thermal convection loop by state feedback

A nonlinear feedback control law that achieves global asymptotic stabilization of a 2D thermal convection loop (widely known for its “Lorenz system” approximation) is presented. The loop consists of viscous Newtonian fluid contained in between two concentric cylinders standing in a vertical plane. The lower half of the loop is heated while the upper half is cooled, which makes the no-motion steady state for the uncontrolled case unstable for values of the non-dimensional Rayleigh number R_a>1. The objective is to stabilize that steady state using boundary control of velocity and temperature on the outer cylinder. We discretize the original nonlinear PDE model in space using finite difference method and get a high order system of coupled nonlinear ODEs in 2D. Then, using backstepping design, we transform the original coupled system into two uncoupled systems that are asymptotically stable in l²-norm with homogeneous Dirichlet boundary conditions. The resulting boundary controls actuate velocity and temperature in the original coordinates. The control design is accompanied by an extensive simulation study which shows that the feedback control law designed on a very coarse grid (using just a few measurements of the flow and temperature fields) can successfully stabilize the actual system for a very wide range of the Rayleigh number.     

A discrete-time dynamic K-winners-take-all neural circuit

A discrete-time mathematical model of K-winners-take-all (KWTA) neural circuit that can quickly identify the K-winning from N neurons, where 1K<N, whose input signals are larger than that of remaining N−K neurons, is given and analyzed. A functional block scheme of the circuit is presented. For N competitors, such circuit is composed of N feedforward and one feedback hard-limiting neurons that are used to determine the dynamic shift of input signals. The circuit has low computational and hardware implementation complexity, high speed of signal processing, can process signals of any finite range, possesses signal order preserving property and does not require resetting and corresponding supervisory circuit that additionally increases a speed of signal processing.     

Output feedback model predictive control of uncertain norm‐bounded linear systems

A constrained output feedback model predictive control (MPC) scheme for uncertain Norm‐Bounded discrete‐time linear systems is presented. This scheme extends recent results achieved by the authors under full‐state availability to the more interesting case of incomplete and noisy state information. The design procedure consists of an off‐line step where a state feedback and an asymptotic observer (dynamic primal controller) are designed via bilinear matrix inequalities and used to robustly stabilize a suitably augmented state plant. The on‐line moving horizon procedure adds N free control moves to the action of the primal controller which are computed by solving a linear matrix inequality optimization problem whose numerical complexity grows up only linearly with the control horizon N. The effectiveness of the proposed MPC strategy is illustrated by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

Convergence of the standard RLS method and UDUT factorisation of covariance matrix for solving the algebraic Riccati equation of the DLQR via heuristic approximate dynamic programming

The main focus of this article is to present a proposal to solve, via UDU^T factorisation, the convergence and numerical stability problems that are related to the covariance matrix ill-conditioning of the recursive least squares (RLS) approach for online approximations of the algebraic Riccati equation (ARE) solution associated with the discrete linear quadratic regulator (DLQR) problem formulated in the actor–critic reinforcement learning and approximate dynamic programming context. The parameterisations of the Bellman equation, utility function and dynamic system as well as the algebra of Kronecker product assemble a framework for the solution of the DLQR problem. The condition number and the positivity parameter of the covariance matrix are associated with statistical metrics for evaluating the approximation performance of the ARE solution via RLS-based estimators. The performance of RLS approximators is also evaluated in terms of consistence and polarisation when associated with reinforcement learning methods. The used methodology contemplates realisations of online designs for DLQR controllers that is evaluated in a multivariable dynamic system model.     

Dynamic output feedback control of a flexible air-breathing hypersonic vehicle via T–S fuzzy approach

By utilising Takagi–Sugeno (T–S) fuzzy set approach, this paper addresses the robust H_∞ dynamic output feedback control for the non-linear longitudinal model of flexible air-breathing hypersonic vehicles (FAHVs). The flight control of FAHVs is highly challenging due to the unique dynamic characteristics, and the intricate couplings between the engine and fight dynamics and external disturbance. Because of the dynamics’ enormous complexity, currently, only the longitudinal dynamics models of FAHVs have been used for controller design. In this work, T–S fuzzy modelling technique is utilised to approach the non-linear dynamics of FAHVs, then a fuzzy model is developed for the output tracking problem of FAHVs. The fuzzy model contains parameter uncertainties and disturbance, which can approach the non-linear dynamics of FAHVs more exactly. The flexible models of FAHVs are difficult to measure because of the complex dynamics and the strong couplings, thus a full-order dynamic output feedback controller is designed for the fuzzy model. A robust H_∞ controller is designed for the obtained closed-loop system. By utilising the Lyapunov functional approach, sufficient solvability conditions for such controllers are established in terms of linear matrix inequalities. Finally, the effectiveness of the proposed T–S fuzzy dynamic output feedback control method is demonstrated by numerical simulations.     

Achieving diagonal interactor matrix for multivariable linear systems with uncertain parameters

The notion of interactor matrix or equivalently the Hermite normal form, is a generalization of relative degree to multivariable systems, and is crucial in problems such as decoupling, inverse dynamics, and adaptive control. In order for a system to be input-output decoupled using static state feedback, the existence of a diagonal interactor matrix must first be established. For a multivariable linear system which does not have a diagonal interactor matrix, dynamic precompensation or dynamic state feedback is required for achieving a diagonal interactor matrix for the compensated system. Such precompensation often depends on the parameters of system, and is thus difficult to implement with accuracy when the system is subject to parameter uncertainty. In this paper we characterize a class of linear systems which can be precompensated to achieve a diagonal interactor matrix without the exact knowledge of the system parameters. More precisely, we present necessary and sufficient conditions on the transfer matrix of the system under which there exists a diagonal dynamic precompensator such that the compensated system has a diagonal interactor matrix. These conditions are associated with the so-called (non)generic singularity of certain matrix related to the system structure but independent of the system parameters. The result of this paper is expected to be useful in robust and adaptive designs.     

H∞ fault-tolerant control for time-varied actuator fault of nonlinear system

This paper studies H_∞ fault-tolerant control for a class of uncertain nonlinear systems subject to time-varied actuator faults. A radial basis function neural network is utilised to approximate the unknown nonlinear functions; an updating rule is designed to estimate on-line time-varied fault of actuator; and the controller with the states feedback and faults estimation is applied to compensate for the effects of fault and minimise H_∞ performance criteria in order to get a desired H_∞ disturbance rejection constraint. Sufficient conditions are derived, which guarantees that the closed-loop system is robustly stable and satisfies the H_∞ performance in both normal and fault cases. In order to reduce computing cost, a simplified algorithm of matrix Riccati inequality is given. A spacecraft model is presented to demonstrate the effectiveness of the proposed methods.     

Approximations for optimal stopping of a piecewise-deterministic process

This paper deals with approximation techniques for the optimal stopping of a piecewise-deterministic Markov process (P.D.P.). Such processes consist of a mixture of deterministic motion and random jumps. In the first part of the paper (Section 3) we study the optimal stopping problem with lower semianalytic gain function; our main result is the construction of ε-optimal stopping times. In the second part (Section 4) we consider a P.D.P. satisfying some smoothness conditions, and forN integer we construct a discretized P.D.P. which retains the main characteristics of the original process. By iterations of the single jump operator from ℝ ^N to ℝ ^N , each iteration consisting ofN one-dimensional minimizations, we can calculate the payoff function of the discretized process. We demonstrate the convergence of the payoff functions, and for the case when the state space is compact we construct ε-optimal stopping times for the original problem using the payoff function of the discretized problem. A numerical example is presented.     

Adaptive Neural Network Control For Smart Materials Robots Using Singular Perturbation Technique

In this paper, an adaptive neural network controller is presented for smart materials robots using Singular Perturbation techniques by modeling the flexible modes and their derivatives as the fast variables and link variables as slow variables. The neural network (NN) controller is to control the slow dynamics in order to eliminate the need tor the tedious dynamic modeling and the error prone process in obtaining the regressor matrix. In addition, inverse dynamic model evaluation is not required and the time‐consuming training process is avoided except for initializing the NNs based on the approximate function values at the initial posture at time t=0. The smart materials bonded along the links are used to active suppress the residue vibration. Simulation results have shown that the controller can control the system successfully and effectively.     

考虑非线性执行器的补偿逼近模型控制

考虑执行器的非线性, 研究了一种带补偿的逼近模型控制系统. 该控制系统包含逼近模型控制器与补偿器.逼近模型控制器根据对象的输入输出线性化关系直接得出控制律, 并由支持向量机辨识对象模型来实现. 补偿部分采用反馈环节来提高系统的鲁棒稳定性, 并采用在线估计得到的逆模型来抵消执行器的非线性特征. 文章分析了该控制系统的稳定性, 针对励磁系统的仿真实验验证了其有效性能.     

An Alternative approach to H∞ control for fuzzy systems

In this paper the problem of H_∞ dynamic feedback control for fuzzy dynamic systems has been studied. First the problem of H_∞ dynamic feedback controller designs for complex nonlinear systems, which can be represented by Takagi‐Sugeno (T‐S) fuzzy systems, is presented. Second, based on a Lyapunov function, four new dynamic feedback H_∞ fuzzy controllers are developed by adequately considering the interactions among all fuzzy sub‐systems and these dynamic feedback H_∞ controllers can be obtained by solving a set of suitable linear matrix inequalities. Finally, two examples are given to demonstrate the effectiveness of the proposed design methods. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society