Robust Control System Design for an Uncertain Nonlinear System Using Minimax LQG Design Method

A systematic approach to design a nonlinear controller using minimax linear quadratic Gaussian regulator (LQG) control is proposed for a class of multi‐input multi‐output nonlinear uncertain systems. In this approach, a robust feedback linearization method and a notion of uncertain diffeomorphism are used to obtain an uncertain linearized model for the corresponding uncertain nonlinear system. A robust minimax LQG controller is then proposed for reference command tracking and stabilization of the nonlinear system in the presence of uncertain parameters. The uncertainties are assumed to satisfy a certain integral quadratic constraint condition. In this method, conventional feedback linearization is used to cancel nominal nonlinear terms and the uncertain nonlinear terms are linearized in a robust way. To demonstrate the effectiveness of the proposed approach, a minimax LQG‐based robust controller is designed for a nonlinear uncertain model of an air‐breathing hypersonic flight vehicle (AHFV) with flexibility and input coupling. Here, the problem of constructing a guaranteed cost controller which minimizes a guaranteed cost bound has been considered and the tracking of velocity and altitude is achieved under inertial and aerodynamic uncertainties.     

Synthesis of minimax optimal controllers for uncertain time-delay systems with structured uncertainty

This paper is concerned with the design of robust state feedback controllers for a class of uncertain time-delay systems. The uncertainty is assumed to satisfy a certain integral quadratic constraint. The controller proposed is a minimax optimal controller in the sense that it minimizes the maximum value of a corresponding linear quadratic cost function over all admissible uncertainties. The controller leads to an absolutely stable closed loop uncertain system and is constructed by solving a finite dimensional parameter-dependent algebraic Riccati equation.     

Nonlinear robust state feedback control system design for nonlinear uncertain systems

This paper presents a systematic approach to the design of a nonlinear robust dynamic state feedback controller for nonlinear uncertain systems using copies of the plant nonlinearities. The technique is based on the use of integral quadratic constraints and minimax linear quadratic regulator control, and uses a structured uncertainty representation. The approach combines a linear state feedback guaranteed cost controller and copies of the plant nonlinearities to form a robust nonlinear controller with a novel control architecture. A nonlinear state feedback controller is designed for a synchronous machine using the proposed method. The design provides improved stability and transient response in the presence of uncertainty and nonlinearity in the system and also provides a guaranteed bound on the cost function. An automatic voltage regulator to track reference terminal voltage is also provided by a state feedback equivalent robust nonlinear proportional integral controller. Copyright © 2016 John Wiley & Sons, Ltd.     

不确定时滞系统的全局鲁棒最优滑模控制

针对一类不确定性时滞系统, 研究线性二次型最优调节器的鲁棒性设计问题. 首先基于级数近似方法, 将原标称时滞系统的最优调节器问题转化为迭代求解一族不含时滞的两点边值问题, 从而获得标称时滞系统最优控制的近似解. 然后将滑模控制理论应用于最优调节器的设计, 使得系统对于不确定性具有全局的鲁棒性, 并且其理想滑动模态与标称系统的最优闭环控制系统相一致, 从而实现了全局鲁棒最优滑模控制. 仿真示例将所提出的方法与相应的二次型最优控制进行比较, 验证了该方法的有效性和优越性.     

Minimax optimal control of uncertain systems with structured uncertainty

This paper considers a minimax control problem for an uncertain system containing structured uncertainties. The uncertainties in this system are assumed to satisfy a certain integral quadratic constraint. For a given initial condition, the minimax optimal controller is constructed by solving a parameter-dependent Riccati equation of the game type. This controller leads to a closed-loop uncertain system which is absolutely stable.     

Robust tracking problem for continuous time stochastic uncertain systems

We propose a finite‐horizon robust minimax tracking controller design method for time‐varying continuous time stochastic uncertain systems. The uncertainty in the system is characterized by a set of probability measures under which stochastic noises, driving the system, are defined. A minimax optimal tracking controller is derived from the solution of a risk‐sensitive linear quadratic Gaussian control problem. Also a numerical example is presented to illustrate the characteristics of proposed tracking controller. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust output tracking of nonlinear systems with mismatched uncertainties

The problem of robust output tracking for a class of uncertain nonlinear systems which do not satisfy the conventional matching condition is considered. The main assumption on the uncertainty is that the triangularity condition is satisfied. Based on backstepping method and input/output linearization approach, we propose a class of non-adaptive state feedback controllers which can guarantee exponential stability of the tracking error for the uncertain nonlinear systems first. Next, adaptive control laws are developed so that no prior knowledge of the bounds on the uncertainties is required. By updating these upper bounds, we design a class of adaptive robust controllers. It is shown that under the proposed adaptive robust control the tracking error of the controlled system converges to zero as time approaches infinity.     

Robust controller design for a class of nonlinear uncertain chemical processes

This article considers the issue of designing robust controllers for single-input/single-output nonlinear chemical processes whose uncertainties satisfy the so-called generalized matching condition. The nominal system (mathematical model) is assumed to be input–output linearizable and the only assumption on uncertainties is that they are bounded. A design methodology of combining the techniques of the differential geometric feedback linearization, the sliding mode control strategy and the adaptive state feedback is presented. Based on the nominal system and the related bounds of uncertainties, a hybrid nonlinear controller, which is more practicable and easily implemented than many other existing ones in the literature, is proposed. A Lyapunov-based approach is utilized to guarantee the robust stability and behavior of the closed-loop system. For demonstrating the effectiveness and applicability of the proposed scheme, we applied it to the control of a continuously stirred tank reactor (CSTR) in the presence of uncertainties including unmodeled side reaction, measuring error, and/or extra unmeasured disturbances. The potential use of a sliding observer along with the proposed scheme is also investigated in this work. Extensive simulation results reveal that the proposed scheme appears to be a practical and promising approach to the robust control of nonlinear uncertain chemical processes.     

Mu-based trajectory tracking control for a quad-rotor UAV

In this paper, the design and application of a robust mu-synthesis-based controller for quad-rotor trajectory tracking are presented. The proposed design approach guarantees robust performance over a weakly nonlinear range of operation of the quad-rotor, which is a practical range that suits various applications. The controller considers different structured and unstructured uncertainties, such as unmodeled dynamics and perturbation in the parameters. The controller also provides robustness against external disturbances such as wind gusts and wind turbulence. The proposed controller is fixed and linear; therefore, it has a very low computational cost. Moreover, the controller meets all design specifications without tuning. To validate this control strategy, the proposed approach is compared to a linear quadratic regulator (LQR) controller using a high- fidelity quad-rotor simulation environment. In addition, the experimental results presented show the validity of the proposed control strategy.     

Continuous‐ and discrete‐time fixed‐gain controller designs for the control of vehicle lateral dynamics

In this paper, fixed‐gain feedback linearization controls are presented to stabilize the vehicle lateral dynamics at bifurcation points for both continuous‐time and discrete‐time cases. Based on the assumption of constant driving speed, a second‐order nonlinear lateral dynamics model is adopted for controller design. Via the feedback linearization scheme and the first‐order Taylor series expansion, a time‐invariant feedback linearization control is proposed as a fixed‐gain linear version of the previously proposed nonlinear one. Furthermore, the conventional linear quadratic regulator (LQR) design is applied to facilitate the choice of the fixed‐gain matrix. Refined controls to compensate the model uncertainty and their local stability analysis are provided. Extension of the continuous‐time design results to discrete‐time cases is also addressed. Numerical simulations for an example model demonstrate the effectiveness of the proposed continuous‐time and discrete‐time design results. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

电液伺服系统的多滑模鲁棒自适应控制

针对一类参数与外负载非匹配不确定的非线性高阶系统,提出了一种基于逐步递推方法的多滑模鲁棒自适应控制策略.应用逐步递推的多滑模控制方法简化了高阶系统的控制问题,同时在自适应控制中加入鲁棒控制的方法,以消除不确定性对控制性能的影响.首先利用逐步递推方法与状态反馈精确线性化理论,得出确定系统的多滑模控制器设计方法;然后基于Lyapunov稳定性分析方法,给出不确定系统的参数自适应律,及鲁棒自适应控制器的设计方法.本文把该控制策略应用到电液伺服系统的位置跟踪控制中,仿真结果显示,该控制方法具有较强的鲁棒性及良好的跟踪效果.     

Robust dissipative control for linear systems with dissipative uncertainty and nonlinear perturbation

This paper focuses on the problem of dissipative control for linear systems which are subjected to dissipative uncertainty and matched nonlinear perturbation. Specifically, quadratic dissipative uncertainty is considered, which contains norm-bounded uncertainty, positive real uncertainty and uncertainty satisfying integral quadratic constraints (IQCs) as special cases. We develop a linear matrix inequality (LMI) approach for designing a robust nonlinear state feedback controller such that the closed-loop system is quadratic dissipative for all admissible uncertainties. Furthermore, under some condition on the dissipative uncertainty, we show that the controller also guarantees the asymptotic stability of the closed-loop system. As special cases, robust H_∞ control and robust passive control problems for systems with nonlinear perturbation and norm-bounded uncertainty (respectively, generalized positive real uncertainty) are solved using the LMI approach.     

Robust‐decentralized tracking control for a class of uncertain MIMO nonlinear systems with time‐varying delays

A new control design method based on signal compensation is proposed for a class of uncertain multi‐input multi‐output (MIMO) nonlinear systems in block‐triangular form with nonlinear uncertainties, unknown virtual control coefficients, strongly coupled interconnections, time‐varying delays, and external disturbances. By this method, the controller design is performed in a backstepping manner. At each step of backstepping procedure, a nominal virtual controller is first designed to get desired output tracking for the nominal disturbance‐free subsystem, and then a robust virtual compensator is designed to restrain the effect of the uncertainties, delays involved in the subsystem, and the couplings among the subsystems. The designed controller is linear and time‐invariant, so the explosion of complexity in the control law is avoid. It is proved that robust stability and robust practical tracking property of the closed‐loop system can be ensured, and the tracking errors can be made as small as desired. Copyright © 2013 John Wiley & Sons, Ltd.     

一类不确定非线性系统的指数鲁棒输出跟踪

研究一类不确定非线性系统的鲁棒输出跟踪控制问题。应用输入／输出反馈线性化法和李亚普诺夫方法,提出一种基于不确定项上界的连续型鲁棒输出跟踪控制器设计方法。该控制器不仅可确保闭环系统的状态一致最终有界,使系统输出按指数规律跟踪期望输出,而且计算简单,更易实现。仿真结果证明了该方法的可行性与有效性。     

Probabilistic robust design with linear quadratic regulators

In this paper, we study robust design of uncertain systems in a probabilistic setting by means of linear quadratic regulators (LQR). We consider systems affected by random bounded nonlinear uncertainty so that classical optimization methods based on linear matrix inequalities cannot be used without conservatism. The approach followed here is a blend of randomization techniques for the uncertainty together with convex optimization for the controller parameters. In particular, we propose an iterative algorithm for designing a controller which is based upon subgradient iterations. At each step of the sequence, we first generate a random sample and then we perform a subgradient step for a convex constraint defined by the LQR problem. The main result of the paper is to prove that this iterative algorithm provides a controller which quadratically stabilizes the uncertain system with probability one in a finite number of steps. In addition, at a fixed step, we compute a lower bound of the probability that a quadratically stabilizing controller is found.     

Decentralized state-feedback stabilization and robust control of uncertain large-scale systems with integrally constrained interconnections

This paper is concerned with a problem of stabilization and robust control design for interconnected uncertain systems. A new class of uncertain large-scale systems is considered in which interconnections between subsystems as well as uncertainties in each subsystem are described by integral quadratic constraints. The problem is to design a set of local (decentralized) controllers which stabilize the overall system and guarantee robust disturbance attenuation in the presence of the uncertainty in interconnections between subsystems as well as in each subsystem. The paper presents necessary and sufficient conditions for the existence of such a controller. The proposed design is based on recent absolute stabilization and minimax optimal control results and employs solutions of a set of game-type Riccati algebraic equations arising in H_∞ control.     

Discrete time robust control via state feedback for single inputsystems

A nonlinear robust state feedback controller for discrete-time uncertain systems is proposed. The uncertainties in the system must satisfy the matching condition, but only their bounds need to be known. The controller is designed via the second method of Lyapunov. The detailed derivation of the controller for single-input systems is given. The system response remains in the neighborhood of the zero state if uncertainties are small. The uncertainty bound for the stability condition is derived     

Multi‐parametric extremum seeking‐based iterative feedback gains tuning for nonlinear control

We study in this paper the problem of iterative feedback gains auto‐tuning for a class of nonlinear systems. For the class of input–output linearizable nonlinear systems with bounded additive uncertainties, we first design a nominal input–output linearization‐based robust controller that ensures global uniform boundedness of the output tracking error dynamics. Then, we complement the robust controller with a model‐free multi‐parametric extremum seeking control to iteratively auto‐tune the feedback gains. We analyze the stability of the whole controller, that is, the robust nonlinear controller combined with the multi‐parametric extremum seeking model‐free learning algorithm. We use numerical tests to demonstrate the performance of this method on a mechatronics example. Copyright © 2016 John Wiley & Sons, Ltd.     

Introduction of Feedback Linearization to Robust LQR and LQI Control – Analysis of Results from an Unmanned Bicycle Robot with Reaction Wheel

In this paper, the Jacobian‐linearization‐ and feedback‐linearization‐based techniques of obtaining linearized model approaches are combined with a family of robust LQR control laws to identify the pairing which results in superior control performance of the bicycle robot, despite uncertainty and constraints, what is the main contribution of the paper. The control performance is analyzed using various indices, related, e.g. to energy consumption of the considered laws, with the experiments conducted on a real bicycle robot. As a result, the easily‐implementable controller is obtained, which requires only to perform a set of off‐line computations with a single additional parameter δ in comparison with a standard linear‐quadratic controller, to obtain a state‐feedback vector, which, when implemented to the control system, ensures proper regulation of the output signal of the plant, despite uncertainty or possible actuator failures, obtaining energy‐efficient control law.     

NETWORK‐INDUCED DELAY‐DEPENDENT H∞ CONTROLLER DESIGN FOR A CLASS OF NETWORKED CONTROL SYSTEMS

This paper is concerned with the problem of robust H_∞ controller design for a class of uncertain networked control systems (NCSs). The network‐induced delay is of an interval‐like time‐varying type integer, which means that both lower and upper bounds for such a kind of delay are available. The parameter uncertainties are assumed to be normbounded and possibly time‐varying. Based on Lyapunov‐Krasovskii functional approach, a robust H_∞ controller for uncertain NCSs is designed by using a sum inequality which is first introduced and plays an important role in deriving the controller. A delay‐dependent condition for the existence of a state feedback controller, which ensures internal asymptotic stability and a prescribed H_∞ performance level of the closed‐loop system for all admissible uncertainties, is proposed in terms of a nonlinear matrix inequality which can be solved by a linearization algorithm, and no parameters need to be adjusted. A numerical example about a balancing problem of an inverted pendulum on a cart is given to show the effectiveness of the proposed design method.