A unified method for optimal arbitrary pole placement

We consider the classic problem of pole placement by state feedback. We offer an eigenstructure assignment algorithm to obtain a novel parametric form for the pole-placing feedback matrix that can deliver any set of desired closed-loop eigenvalues, with any desired multiplicities. This parametric formula is then exploited to introduce an unconstrained nonlinear optimisation algorithm to obtain a feedback matrix that delivers the desired pole placement with optimal robustness and minimum gain. Lastly we compare the performance of our method against several others from the recent literature.     

Descent algorithms for optimal periodic output feedback control

Periodic output feedback is investigated in the context of linear-quadratic regulation for finite-dimensional time-invariant linear systems. Discrete output samples are multiplied by a periodic gain function to generate a continuous feedback control. The optimal solution is obtained in two steps by separating the continuous-time from the discrete-time structure. First, the optimal pole placement problem under periodic output feedback is solved explicitly under the assumption that the behavior at the sample times has been specified in terms of a gain matrix G. Then the minimum value, which depends on G, is substituted into the overall objective. This results in a finite-dimensional nonlinear programming problem over all admissible gain matrices G. The solution defines the optimal periodic output feedback control via the formulas of the optimal pole placement problem. A steepest descent and a direct iterative method for solving this problem are formulated and compared. Numerical examples show that the performance using periodic output feedback is almost equivalent to that using optimal continuous-state feedback     

非线性控制系统线性化与极点配置的神经网络方法

提出一种同时实现非线性控制系统反馈线性化和极点配置的神经网络方法,并证明了在 随机逼近意义下该神经网络控制系统的有效性.     

Global exponential stability of recurrent neural networks forsynthesizing linear feedback control systems via pole assignment

Global exponential stability is the most desirable stability property of recurrent neural networks. The paper presents new results for recurrent neural networks applied to online computation of feedback gains of linear time-invariant multivariable systems via pole assignment. The theoretical analysis focuses on the global exponential stability, convergence rates, and selection of design parameters. The theoretical results are further substantiated by simulation results conducted for synthesizing linear feedback control systems with different specifications and design requirements     

针对传感器故障的容错控制问题的数值解法

针对状态反馈闭环系统中的传感器故障容错控制问题，先基于稳定多项式分解导出该容错控制问题状态反馈闭环系统稳定的充分必要条件。在此基础上，基于相容非线性方程组数值解法，提出具有传感器故障容错控制的状态反馈律设计方法。还基于数值优化解方法，提出面向闭环系统极点配置的另一状态反馈容错控制律设计方法。计算机仿真算例表明此方法的有效性。     

Regional pole placement of multivariable systems under controlstructure constraints

Many controller realizations are structurally constrained. Some typical examples are static output feedback, constant gain feedback for multiple operating points of a system, two-controller feedback, and decentralized feedback. A general class of problems of regional pole placement of multivariable systems with such control structure constraints is considered and a unified numerical method is given to solve them. First a problem in this class is converted to a problem of solving a system of equalities and inequalities. This system is then solved by using a modified homotopy method     

广义系统的反馈控制

本文介绍了广义系统的状态（输出）反馈的一种方便计算方法及反馈作用。讨论了系统的能控性结构指数并指出它是广义系统的不变量。最后研究了广义系统的极点结构配置和静态输出反馈下的极点配置。     

A Gauss–Seidel type solver for the fast computation of input-constrained control systems

An important class of nonlinear control systems can be represented as the feedback interconnection of two parts: a linear time-invariant system and a block of decentralized nonlinearities. When the linear time-invariant part has a nontrivial feedthrough term or the nonlinearity has a feedback gain, control computation involves the online implementation of a multivariable algebraic loop which must be resolved at each sampling instant. The requirements for such online computation may result in several implementation issues, especially in real-time and embedded control applications. This paper considers the implementation of such algebraic loops arising from several input-constrained systems. The proposed solution algorithm is globally convergent for a very large class of feedthrough or feedback gains and shows promise for real-time and embedded control applications where a fast but approximate solution is of the essence.     

An algorithm for computing state feedback in multiinput linear systems

A new algorithm is proposed in this note to compute the state feedback for arbitrary pole placement in multiinput linear systems. The algorithm has good numerical behavior. It also allows the user to adjust the eigenstructure of the resulting feedback system and, therefore, improve the condition of the problem and computation.     

广义状态系统的反馈和极点配置

本文讨论线性反馈对连续时间广义状态空间结构的影响和消去脉冲运动的充分必要条件。给出一个利用输出反馈配置极点的方法。     

自平衡控制系统的平衡性仿真

自平衡两轮移动式倒立摆是三个自由度的强耦合和非线性的自然不稳定系统。虽然现代控制论中的状态反馈的极点配置对两轮移动式倒立摆的平衡起到了一定的控制作用,但是在现实多干扰的外界环境中,控制论中极点配置算法的控制作用就很难满足实际控制需要。针对两轮移动式倒立摆数学模型,给出详细的力学推导并在Simulink环境下,对状态反馈的极点配置法控制的两轮移动式倒立摆处于不同的坡度平衡情况进行了仿真和分析。     

Pole placement adaptive control with persistent jumps in the plant parameters

In this paper we consider the problem of adaptively stabilizing, and providing step tracking for, an uncertain linear time-varying system. We propose an adaptive pole placement controller which solves the problem for a single-input single-output plant whose parameters switch at a moderate rate among the elements of a compact set. The output feedback controller incorporates an integrator, and its action emulates the behaviour of a pole placement state feedback compensator; the controller is periodic and mildly nonlinear, is easy to implement, is noise tolerant, and tolerates a degree of unmodelled dynamics.     

Disturbance decoupling in a class of linear systems

The disturbance decoupling problem by state feedback (DDP) is solved for a class of linear systems where the left-invertible systems are included. Also obtained is a complete characterization of the solution as well as the parameterization of all corresponding state feedback matrices. Such characterization enables one to conclude about the solvability of DDP with stability and/or pole placement. A numerical method is proposed for computation of the controllers in an analytical form, suitable to the treatment of complementary design specifications. The solution of the disturbance decoupling problem by measurement feedback (DDPM) is also discussed and, based on the results obtained for DDP, a procedure for design of a reduced-order compensator is proposed     

Output feedback pole placement for linear time-varying systems with application to the control of nonlinear systems

The output feedback pole placement problem is solved in an input-output algebraic formalism for linear time-varying (LTV) systems. The recent extensions of the notions of transfer matrices and poles of the system to the case of LTV systems are exploited here to provide constructive solutions based, as in the linear time-invariant (LTI) case, on the solutions of diophantine equations. Also, differences with the results known in the LTI case are pointed out, especially concerning the possibilities to assign specific dynamics to the closed-loop system and the conditions for tracking and disturbance rejection. This approach is applied to the control of nonlinear systems by linearization around a given trajectory. Several examples are treated in detail to show the computation and implementation issues.     

Feedback norm minimisation with regional pole placement

This article proposes a convex algorithm for minimising an upper bound of the state feedback gain matrix norm with regional pole placement for linear time-invariant multi-input systems. The inherent non-convexity in this optimisation is resolved by a combination of two separate approaches: (1) an inner convex approximation of the polynomial matrix stability region due to Henrion and (2) a novel convex parameterisation of column reduced matrix fraction system representations. Using a sequence of approximations enabled by the above two methods, it is shown that the constraints on closed-loop poles (both pre-specified exact locations and regional placement) define linear matrix inequalities. Finally, the effectiveness of the proposed algorithm is compared with similar pole placement algorithms through numerical examples.     

Mixed‐Objective Robust Dynamic Output Feedback Controller Synthesis for Continuous‐Time Polytopic Lpv Systems

A robust dynamic output feedback controller synthesis algorithm considering H_∞/H₂ performance and regional pole placement is addressed for a nonlinear system with parameter uncertainties and external disturbance. First, the formulation of a gain‐scheduled mixed‐objective robust dynamic output feedback controller for continuous‐time polytopic linear parameter varying (LPV) systems is presented. To reduce conservativeness, some auxiliary slack variables and parameter‐dependent Lyapunov functions are employed in addition to well‐established performance conditions. Then, sufficient conditions for the desired gain‐scheduled mixed‐objective robust dynamic output feedback controllers are cast into an efficiently tractable finite‐dimensional convex optimization problem in terms of linear matrix inequalities (LMIs). Finally, numerical simulation shows the validity of the proposed controller, which has good stability, strong robustness, satisfied disturbance attenuation ability, and smooth dynamic properties.     

Online linearization-based neural predictive control of air–fuel ratio in SI engines with PID feedback correction scheme

Model predictive control (MPC) frequently uses online identification to overcome model mismatch. However, repeated online identification does not suit the real-time controller, due to its heavy computational burden. This work presents a computationally efficient constrained MPC scheme using nonlinear prediction and online linearization based on neural models for controlling air–fuel ratio of spark ignition engine to its stoichiometric value. The neural model for AFR identification has been trained offline. The model mismatch is taken care of by incorporating a PID feedback correction scheme. Quadratic programming using active set method has been applied for nonlinear optimization. The control scheme has been tested on mean value engine model simulations. It has been shown that neural predictive control with online linearization using PID feedback correction gives satisfactory performance and also adapts to the change in engine systems very quickly.     

State feedback with fractional integral control design based on the Bode’s ideal transfer function

State feedback technique through a gain matrix has been a well-known method for pole assignment of a linear system. The technique could encounter a difficulty in eliminating the steady-state errors in some states. Introducing an integral element can effectively eliminate these errors. State feedback with fractional integral control is proposed, in this work, for pole placement of a linear time invariant system. The proposed method yields simple gain formulae. The paper presents the derivation of the design formulae. The method is applied to stabilise an inherently unstable inverted pendulum-cart system. Simulation and experimental results show the effectiveness of the proposed method for set-point tracking, disturbance rejection and stabilising the inverted pendulum. Comparison with the results obtained from applying Achermann’s formula is also presented.     

基于DSC后推法的非线性系统的鲁棒自适应NN控制

针对一类具有不确定系统函数和方向未知的不确定增益函数的非线性系统, 提出了一种鲁棒自适应神经网络控制算法. 本算法采用RBF神经网络(Radial based function neural network, RBF NN)逼近模型不确定性, 外界干扰和建模误差采用非线性阻尼项进行补偿, 将动态面控制(Dynamic surface control, DSC)与后推方法结合, 消除了反推法的计算膨胀问题, 降低了控制器的复杂性; 尤其是采用Nussbaum函数处理系统中方向未知的不确定虚拟控制增益函数, 不仅可以避免可能存在的控制器奇异值问题, 而且还能使得整个系统的在线学习参数显著减少, 与DSC方法优点结合, 使得控制算法的计算量大为减少, 便于计算机实现. 稳定性分析证明了所得闭环系统是半全局一致最终有界(Semi-global uniformly ultimately bounded, SGUUB)的, 并且跟踪误差可以收敛到原点的一个较小邻域. 最后, 计算机仿真结果表明了本文所提出控制器的有效性.     

Multi-objective control for stochastic model reference systems based on LMI approach and sliding mode control concept

In this article, a multi-objective control u(t) is designed for stochastic model reference systems to achieve the following three objectives simultaneously: the pole placement constraint, H _∞-norm constraint and individual error state variance constraint. Using the invariance property of sliding mode control, the reference model input and the plant error term will disappear on the sliding mode of the error system. By combining the upper bound covariance control theory, pole placement skill and H _∞-norm control theory, a controller, in which the control feedback gain matrix is synthesised utilising linear matrix inequality (LMI) approach, is derived to achieve the above multiple objectives. Furthermore, a practical example for the problem of ship yaw-motion systems is adopted to illustrate the proposed method.