Control of linear singularly perturbed systems with colored noise disturbance

This paper considers the stochastic control of linear-quadratic problems for singularly perturbed systems when the input noise is colored. A near optimal linear output feedback control is obtained by optimizing a slow subsystem only.     

Instability analysis and improvement of robustness of adaptive control

The effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed. Five possible types of instability mechanisms—parameter drift, ‘linear’ instability, ‘fast adaptation’ instability, ‘high frequency’ instability, and ‘throughput’ instability—are analyzed using simple examples. A procedure is used to construct Lyapunov-like functions for a modified adaptive controller applied to a dominant plant of relative degree one, in the presence of parasitics and disturbances, and obtain sufficient conditions under which none of the five types of instability can occur. The modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set. The size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast, the region of attraction becomes the whole space.     

Nonlinear singularly perturbed optimal control problems with singular arcs

A third-order, nonlinear, singularly perturbed optimal control problem is considered under assumptions which assure that the full problem is singular and the reduced problem is nonsingular. The separation between the singular arc of the full problem and the optimal control law of the reduced one, both of which are hypersurfaces in state space, is of the same order as the small parameter of the problem. Boundary-layer solutions are constructed which are stable and reach the outer solution in a finite time. A uniformly valid composite solution is then formed from the reduced and boundary-layer solutions. The value of the approximate solution is that it is relatively easy to obtain and does not involve singular arcs. To illustrate the utility of the results, the technique is used to obtain an approximate solution of a simplified version of the aircraft minimum time-to-climb problem. A numerical example is included.     

Asymptotic methods in the optimal control of distributed systems

A short review of the various problems which arise in connection with the use of asymptotic methods in the optimal control of distributed systems is presented. We consider the cases when the asymptotic analysis comes from the state equation, or from the cost function, or both and also when the state equation is defined in perturbed domains.     

Altitude transitions in energy climbs

The aircraft energy-climb trajectory for configurations with a sharp transonic drag rise is well known to possess two branches in the altitude/Mach-number plane. Transition in altitude between the two branches occurs instantaneously, a ‘corner’ in the minimum-time solution obtained with the energy-state model. If the initial and final values of altitude do not lie on the energy-climb trajectory, then additional jumps (crude approximations to dives and zooms) are required at the initial and terminal points. With a singular-perturbation approach, a ‘boundary-layer’ correction is obtained for each altitude jump, the transonic jump being a so-called ‘internal’ boundary layer, different in character from the initial and terminal layers. The determination of this internal boundary layer is examined and some computational results for an example presented.     

Sliding mode control of singularly perturbed systems and its application in quad-rotors

This paper presents a sliding mode control scheme for tracking control of nonlinear singularly perturbed systems in the presence of model errors and external disturbances. A dual-loop feedback control is developed to provide accurate tracking capability and sufficient robustness to system uncertainties. A sliding mode controller is proposed in the outer-loop feedback design such that the plant states are stabilised for given reference trajectories, while an additional robust controller is designed in the inner loop to increase the adaptability to uncertainties, and reduce the effect of unmodelled high-frequency dynamics on plant dynamics. An appealing feature of the control scheme is the attenuation of chattering. The effectiveness and merits of the new control scheme developed are shown via a verification example of velocity control of a quad-rotor.     

Closed-loop balancing for a class of non-linear singularly perturbed systems

The aim of this article is to investigate the closed-loop balancing reduction method for a class of non-linear singularly perturbed systems. We show that the well-known two-stage strategy involved commonly within the singular perturbation theory can be used to derive an approximate closed-loop balancing. The proposed two-stage method avoids the difficult task of solving high dimensional and ill conditioned non-linear Hamilton–Jacobi equation due to the presence of the small perturbation parameter.     

Two-time scale sliding-mode control for a class of nonlinear systems

The paper deals with the robust asymptotic stabilization of a class of nonlinear singularly perturbed systems using sliding-mode control techniques. The approach consists of decomposing the original system into two reduced order systems, for which stabilizing sliding-mode controllers are applied and combined in a two-time scale sliding-mode control for the full order system. A stability analysis allows us to provide sufficient conditions for the asymptotic stability of the full order closed-loop system. An example illustrates the design procedure and performance of the proposed control scheme. © 1997 by John Wiley & Sons, Ltd.     

Subsystems, time scales and multimodeling

Through a couple of naive examples the control theorists are invited to re-examine the role of modeling in the study of large scale dynamic systems. Instead of assuming the existence of ‘N diagonally dominant blocks’, it is possible to justify one strongly coupled slow core and N weakly coupled fast subsystems. This structure is exhibited with a physically meaningful choice of state variables. The controls are introduced following the recent concept of multimodeling.     

基于PI 控制器的线性系统的鲁棒耗散控制

针对参数具有确定性及不确定性的连续系统,给出两种严格耗散PI控制器的设计方法.首先,系统参数确定时,采用线性矩阵不等式方法,导出了类状态反馈和静态输出反馈严格耗散PI控制器存在的充要条件,并由线性矩阵不等式的可行解构造出严格耗散PI控制器增益的显式表达式;然后,考虑系数矩阵均具有范数有界不确定性时的鲁棒严格耗散控制问题,得到相似的结果;最后,通过数值算例表明了所给方法的有效性.     

Asymptotic stability of nonlinear singularly perturbed systems using higher order corrections

This paper examines a methodology for investigating the asymptotic stability properties of nonlinear singularly perturbed systems. This is achieved by constructing the so-called zeroth order model (uncorrected model). Based on this model a weighted scalar sum of the Lyapunov functions for two lower order subsystems is obtained. The estimates of the region of attraction and the upper bound on the perturbed parameter are established. It is shown that by extending this methodology to higher order corrected models, less conservative results are obtained. These considerable improvements are facilitated by introducing a new fast variable in both uncorrected and corrected models.     

广义Delta算子系统可稳定分析

分析广义Delta算子系统的能稳定性问题,在系统可控的前提下,利用状态反馈的方式,对系统进行稳定性分析,得到了反馈控制的可稳定性结论。     

Stability radii of infinite-dimensional systems subjected to unbounded stochastic perturbations

In this paper, we use the framework of stability radii to study the robust stability of linear deterministic systems on real Hilbert spaces which are subjected to unbounded stochastic perturbations. First, we establish an existence and uniqueness theorem of the solution of the abstract equation describing the system. Then we characterize the stability radius in terms of a Lyapunov equation or equivalently in terms of the norm of an input-output operator.     

二维直线电机的并联型无模型自适应复合控制

针对二维直线电机平台系统的XY轴协同控制问题,将PI与并联型无模型自适应相结合的复合控制方法应用于控制系统.该控制方法通过在每个子系统加入PI控制方法保证其稳定性,再通过并联型的无模型自适应控制方法来提高整个系统的跟踪性能,以减小系统位置误差.基于数据驱动的无模型自适应控制方法设计二维直线电机系统控制器,其优势在于无需...     

Sliding Mode Control and State Estimation for a Class of Nonlinear Singularly Perturbed Systems

This paper is concerned with the design of a controller-observer scheme for the exponential stabilization of a class of singularly perturbed nonlinear systems. The controller design uses a sliding mode technique and is divided in two phases: slow feedback control and fast feedback control so that a final composite control is obtained. Assuming that only the fast state is available and the system's output is a function of the slow state, an observer design is presented. A stability analysis is also made to provide sufficient conditions for the ultimate boundedness of the full order closed-loop system when the slow state is estimated by means of the observer. An application to the model of a permanent magnet stepper motor is given to show the controller-observer methodology and stability analysis.