Time-scale separation and robust controller design for uncertain nonlinear singularly perturbed systems under perfect state measurements

We study time-scale separation and robust controller design for a class of singularly perturbed nonlinear systems under perfect state measurements. The system dynamics are taken to be jointly linear in the fast state variables, control and disturbance inputs, but nonlinear in the slow state variables. Since global timescale separation may not always be possible for nonlinear singularly perturbed systems, we restrict our attention here to some closed subset of the state space, on which a timescale separation holds for sufficiently small values of the singular perturbation parameter. We construct a slow controller and a composite controller based on the solutions of particular slow and fast games obtained using time-scale separation. For the class of systems for which the slow controller can be selected to be robust with respect to small regular structural perturbations on the slow subsystem, we show under some growth conditions that the composite controller can achieve any desired level of performance that is larger than the maximum of the performance levels for the slow and fast subsystems,. A slow controller, however, is not generally as robust as the composite controller; but, still under some conditions which are delineated in the paper, the fast dynamics can be totally ignored. The paper also presents a numerical example to illustrate the theoretical results.     

Decomposition of a linear-quadratic optimal control problem with fast and slow variables

A linear-quadratic optimal control problem for a discrete different time-scale system is studied. The decomposition of the boundary value problem for the maximum principle is based on the geometric approach using the properties of invariant manifolds of slow and fast motions. This approach aids in constructing a transformation for reducing the initial problem to a boundary-value problem for slow variables and two initial-value problems for fast variables. The transformation is expressed as an asymptotic siries in powers of a small parameter.     

Multi-time-scale analysis of a power system

A time-scale separation procedure is outlined and applied to a three machine interconnected power system modeled with flux linkage and voltage regulator dynamics. Partial models such as the electromechanical model and single machine-infinite bus model are used to identify the slow and fast states of the systems. Linear simulation results in two- and four-time-scales demonstrate the potential applicability of the singular perturbation approach to long-term dynamic studies of power systems.     

Time-scale separation redesigns for stabilization and performance recovery of uncertain nonlinear systems

In this paper we propose two different time-scale separation based robust redesign techniques which recover the trajectories of a nominal control design in the presence of uncertain nonlinearities. We first consider additive input uncertainties and design a high-gain filter to estimate the uncertainty. We then employ the fast variables arising from this filter in the feedback control law to cancel the effect of the uncertainties in the plant. We next extend this design to systems with uncertain input nonlinearities in which case we design two sets of high gain filters—the first to estimate the input uncertainty over a fast time-scale, and the second to force this estimate to converge to the nominal input on an intermediate time-scale. Using singular perturbation theory we prove that the trajectories of the respective two-time-scale and three-time scale redesigned systems approach those of the nominal system when the filter gains are increased. We illustrate the redesigns by applying them to various physically motivated examples.     

On applications of singular perturbation techniques in nonlinear optimal control

The technique of singular perturbations (SPT) has been applied with considerable success in several nonlinear optimal control problems. In many cases the zero-order approximation of the optimal control function has been expressed in a feedback form. This paper deals with topics involved in such closed-loop application, which seem to merit further discussion. It is formally demonstrated that a ‘forced’ singular perturbation model (obtained by artificial insertion of the perturbation parameter) results in the same zero-order composite feedback control solution as a classical singularly perturbed model (where a small parameter of physical significance appears as a consequence of a scaling transformation). The accuracy of the zero-order feedback approximation depends in both cases on the actual time scale separation of the variables. Two inherent limitations of the feedback solution are also pointed out: (1) first and higher-order correction terms of the zero-order approximation have to be computed by a predictive or off-line integration; (2) on-line implementation of SPT control strategy in a terminal boundary layer requires iterative computations. A simple pursuit problem serves as an illustrative example.     

Analysis for a class of singularly perturbed hybrid systems via averaging

A class of singularly perturbed hybrid dynamical systems is analyzed. The fast states are restricted to a compact set a priori. The continuous-time boundary layer dynamics produce solutions that are assumed to generate a well-defined average vector field for the slow dynamics. This average, the projection of the jump map in the direction of the slow states, and flow and jump sets from the original dynamics define the reduced, or average, hybrid dynamical system. Assumptions about the average system lead to conclusions about the original, higher-dimensional system. For example, forward pre-completeness for the average system leads to a result on closeness of solutions between the original and average system on compact time domains. In addition, global asymptotic stability for the average system implies semiglobal, practical asymptotic stability for the original system. We give examples to illustrate the averaging concept and to relate it to classical singular perturbation results as well as to other singular perturbation results that have appeared recently for hybrid systems. We also use an example to show that our results can be used as an analysis tool to design hybrid feedbacks for continuous-time plants implemented by fast but continuous actuators.     

分形微积分算子的定义及其应用

基于隐式微积分建模方法,提出分形维空间基本解的概念,从而定义分形维上的微积分算子,用以描述分形材料的各种力学行为.分形微积分算子极大地推广经典的连续介质力学微积分建模方法的使用范围,是分形导数概念的进一步发展.运用奇异边界法成功地数值模拟分形维拉普拉斯算子方程唯象描述的分形材料势问题.     

Power system excitation and governor control design via time-scale decomposition

Power system excitation and governor control design using time-scale decomposition and optimal control theoretic concepts are presented. A synchronous machine with an excitation system and a prime-mover is decomposed into slow and fast subsystems using the iterative separation of time scales. The slow subsystem model is employed as the design model. The effectiveness of the proposed controller is studied through digital simulation. The simulation results are compared with those obtained via the classical quasi-steady-state (qss) technique. The approach presented does not require an explicit knowledge of the perturbation parameter.     

Area aggregation and time-scale modeling for sparse nonlinear networks

Model reduction and aggregation are of key importance for simulation and analysis of large-scale systems, such as molecular dynamics, large swarms of robotic vehicles, and animal aggregations. We study a nonlinear network which exhibits areas of internally dense and externally sparse interconnections. The densely connected nodes in these areas synchronize in the fast time-scale, and behave as aggregate nodes that dominate the slow dynamics of the network. We first derive a singular perturbation model which makes this time-scale separation explicit and, next, prove the validity of the reduced-model approximation on the infinite time interval.     

Model reduction and control of reactor–heat exchanger networks

This paper focuses on the dynamics and control of process networks consisting of a reactor connected with an external heat exchanger through a large material recycle stream that acts as an energy carrier. Using singular perturbation arguments, we show that such networks exhibit a dynamic behavior featuring two time scales: a fast one, in which the energy balance variables evolve, and a slow time scale that captures the evolution of the terms in the material balance equations. We present a procedure for deriving reduced-order, non-stiff models for the fast and slow dynamics, and a framework for rational control system design that accounts for the time scale separation exhibited by the system dynamics. The theoretical developments are illustrated with an example and numerical simulation results.     

Dynamics and control of vapor recompression distillation

Vapor recompression distillation is an energy integrated distillation configuration which works on the principle of a heat pump. The tight material and energy integration in such columns shows a potential for intricate dynamics. In this paper, a systematic modeling framework is developed which explicitly captures the discrepancies between different material and energy flows present in such columns. This feature is documented to lead to stiff dynamic equations and multiple time-scale dynamics. Through a nested application of singular perturbations, reduced order non-stiff models are derived which capture the dynamics in each time-scale. A hierarchical control strategy is then proposed exploiting this time-scale multiplicity. A case of propane–propylene separation is considered to illustrate these results and demonstrate the effectiveness of the proposed control strategy via a simulation case study.     

Order-reducing approximation of two-time-scale discrete linear time-varying systems

In this paper we study a two-time-scale discrete-time linear time-varying system. We heuristically find a reduced-order approximation to its asymptotic behaviour as the time-scale separation tends to infinity. This approximation results in a white noise representation of the fast state vector, and a corresponding approximation error in the slow state vector. After introducing the approximate system we define concepts of continuity and rate of variation which are needed in the discrete-time analysis. We then prove that as the time-scale separation increases, the state of the reduced-order system asymptotically coincides with the slow state of the original system in the mean-square sense on compact time intervals. We also find the order of the slow state approximation error covariance.     

Stability Analysis of a Vision-Based UAV Controller

The stability analysis of a vision-based control strategy for a quad rotorcraft UAV is addressed. In the present application, the imaging sensing system provides the required states for performing autonomous navigation missions, however, it introduces latencies and time-delays from the time of capture to the time when measurements are available. To overcome this issue, a hierarchical controller is designed considering a time-scale separation between fast and slow dynamics. The dynamics of the fast-time system are stabilized using classical proportional derivative controllers. Additionally, delay frequency and time domain techniques are explored to design a controller for the slow-time system. Simulations and experimental results consisting on a vision-based road following task are presented.     

First-order corrections in optimal feedback control of singularly perturbed nonlinear systems

In this paper first-order correction terms, developed by using the method of matched asymptotic expansions, are incorporated in the feedback solution of a class of singularly perturbed nonlinear optimal control problems frequently encountered in aerospace applications. This improvement is based on an explicit solution of the integrals arising from the first-order matching conditions and leads to correct the initial values of the slow costate variables in the boundary layer. Consequently, a uniformly valid feedback control law, corrected to the first-order, can be synthesized. The new method is applied to an example of a constant speed minimum-time interception problem. Comparison of the zeroth- and first-order feedback control laws to the exact optimal solution demonstrates that first-order corrections greatly extend the domain of validity of the approximation obtained by singular perturbation methods.     

Robust Stabilization and Performance Recovery of Nonlinear Systems With Unmodeled Dynamics

In this technical note we present a time-scale separation redesign for stabilization and performance recovery of nonlinear systems with unmodeled dynamics. The unmodeled dynamics studied do not change the relative degree of the plant and are minimum phase. We design two sets of high gain filters—the first to estimate the uncertain input to the plant over a fast time-scale, and the second to force this estimate to converge to the nominal input on an intermediate time-scale. The control input then acts over the slow time-scale and guarantees that the closed-loop trajectories approach those of the nominal system.      

Singular perturbation in piecewise-linear systems

A singular perturbation technique is developed that allows for a decoupling of a continuous piecewise-linear system into slow and fast subsystems. Under the assumption of asymptotic stability, the fast variable is found to decay in the boundary layer to its quasi-steady-state solution, which is given by a continuous implicit function of the slow variable. The solution is found using a finite step algorithm. Sufficient conditions for the approximation to be accurate to an order of O(μ), where μ is a parameter of the system, are given. The technique is illustrated by a numerical example     

Enhanced measurement and estimation methodology for flexible link arm control

The two objectives of this article are (1) to study the effects of truncation of the infinite series in the assumed mode shapes approach for flexible-link robot arms, and (2) to design an improved Kalman filter for estimation of the rigid and flexible modes for controls purposes. A singular perturbation approach is used to derive an improved reduced-order model that does not correspond simply to truncating the finite series, but includes some additional information about the neglected residual modes. Then, a second time-scale separation is used to design a slow/fast Kalman filter with improved performance for flexible-link arms. © 1994 John Wiley & Sons, Inc.     

Time-scale decomposition of the reachable set of constrained linear systems

We consider a linear control system with a multiparameter singular perturbation representing multiple time scales and with constraints for the control and the slow state. The Hausdorff limit of the reachable set when the small parameters tend to zero is found. The result provides a basis for a time-scale approximation of the reachable set.on leave from the Institute of Mathematics, Bulgarian Academy of Sciences, Sofia, Bulgaria.     

Robustness analysis of uncertain linear singular systems withoutput feedback control

The robustness analysis for a linear singular system with uncertain parameters and static output feedback control is considered. The problem is transformed into a robust nonsingularity problem. Based on the linear fractional transformation (LFT) approach, the robustness bounds to preserve regularity, impulse immunity, and stability are found in terms of the structured singular value μ with respect to parametric uncertainties. The LFT approach provides a unified framework for robustness analysis of both uncertain linear continuous/discrete-time singular systems     

Singular perturbation and iterative separation of time scales

This tutorial paper presents an iterative method for the separation of slow and fast modes, which removes the inconsistencies of the classical quasi-steady-state approach and systematically improves the accuracy of the lower order models. It also serves as a self-contained introduction to singular perturbations. State variable reformulation and time scale identification are discussed and illustrated with power system examples. A correction procedure for nonlinear systems is also presented.