The linear quadratic regulator problem for a class of controlled systems modeled by singularly perturbed Ito differential equations

This paper discusses an infinite-horizon linear quadratic (LQ) optimal control problem involving state- and control-dependent noise in singularly perturbed stochastic systems. First, an asymptotic structure along with a stabilizing solution for the stochastic algebraic Riccati equation (ARE) are newly established. It is shown that the dominant part of this solution can be obtained by solving a parameter-independent system of coupled Riccati-type equations. Moreover, sufficient conditions for the existence of the stabilizing solution to the problem are given. A new sequential numerical algorithm for solving the reduced-order AREs is also described. Based on the asymptotic behavior of the ARE, a class of O(√ε) approximate controller that stabilizes the system is obtained. Unlike the existing results in singularly perturbed deterministic systems, it is noteworthy that the resulting controller achieves an O(ε) approximation to the optimal cost of the original LQ optimal control problem. As a result, the proposed control methodology can be applied to practical applications even if the value of the small parameter ε is not precisely known. © 2012 Society for Industrial and Applied Mathematics.Vasile Dragan, Hiroaki Mukaidani and Peng Sh

Improvement of parametric stability margin under pole assignment

In this paper, the improvement of the parametric stability margin of state-space uncertain systems via a maximization formulation under the constraints of pole assignment is investigated. The class of systems considered is where the uncertainty may be modeled as the, possibly nonlinear, variation of a parameter appearing in the entries of the system and input matrices. The continuity and differentiability properties of the stability margin are discussed. A gradient-based approach is presented for the improvement of the stability margin and a compact formula to compute the gradient is provided. Numerical examples are used to demonstrate the effectiveness of the approach.published_or_final_versio

A Singular Perturbation Approach for Time-Domain Assessment of Phase Margin

This paper considers the problem of time-domain assessment of the Phase Margin (PM) of a Single Input Single Output (SISO) Linear Time-Invariant (LTI) system using a singular perturbation approach, where a SISO LTI fast loop system, whose phase lag increases monotonically with frequency, is introduced into the loop as a singular perturbation with a singular perturbation (time-scale separation) parameter Epsilon. First, a bijective relationship between the Singular Perturbation Margin (SPM) max and the PM of the nominal (slow) system is established with an approximation error on the order of Epsilon(exp 2). In proving this result, relationships between the singular perturbation parameter Epsilon, PM of the perturbed system, PM and SPM of the nominal system, and the (monotonically increasing) phase of the fast system are also revealed. These results make it possible to assess the PM of the nominal system in the time-domain for SISO LTI systems using the SPM with a standardized testing system called "PM-gauge," as demonstrated by examples. PM is a widely used stability margin for LTI control system design and certification. Unfortunately, it is not applicable to Linear Time-Varying (LTV) and Nonlinear Time-Varying (NLTV) systems. The approach developed here can be used to establish a theoretical as well as practical metric of stability margin for LTV and NLTV systems using a standardized SPM that is backward compatible with PM

Finite Frequency H

This paper investigates the problem of finite frequency (FF) H∞ filtering for time-delayed singularly perturbed systems. Our attention is focused on designing filters guaranteeing asymptotic stability and FF H∞ disturbance attenuation level of the filtering error system. By the generalized Kalman-Yakubovich-Popov (KYP) lemma, the existence conditions of FF H∞ filters are obtained in terms of solving an optimization problem, which is delay-independent. The main contribution of this paper is that systematic methods are proposed for designing H∞ filters for delayed singularly perturbed systems

Anti‐windup controller design for singularly perturbed systems subject to actuator saturation

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/166157/1/cth2bf00153.pd

Final Scientific Report: Control Strategies for Complex Systems for Use in Aerospace Avionics

Coordinated Science Laboratory was formerly known as Control Systems LaboratoryAir Force Office of Scientific Research (AFOSR), U.S. Air Force / AF-AFOSR 73-257

Two dimensional agonistic control

The conventional method of precise multiple-axis motion control entails use of a multiple axis positioning system with each axis carrying not only the workpiece but also the positioning system of the remaining axes. The resultant structure is heavy, sluggish, and expensive. An alternative positioning technique is being investigated in which the motion of the workpiece is controlled by pulling it with tendons, each of which has its own actuator. Since the actuators can be mounted on the base of the structure instead of being carried by motion system of the other axes, they can be relatively large and powerful without the need for a massive structure such as is found in a conventional motion control system. This method of control is given the appellation agonistic, based on the usages of the word suggesting tension or a contest. Agonistic control system can be used for low cost accurate positioning of workpiece. The control task can be moving the workpiece from one point to another point and kept there or tracking a given trajectory. While the workpiece moves, the tendons should be always kept in tension. In this thesis, the model of two dimensional agonistic control (in the case of tendons of infinite elastic modulus) is established. It leads to a nonlinear multi-variable control problem. Based on this nonlinear model, a full-state feedback control law is synthesized. It is composed of two parts. The first part is a feedforward control to cancel the nonlinear dynamics. The second part is a PD control term which requires velocity information. In the practice, velocity measurement may be contaminated by noise. In order of only using position measurement in the control law, a nonlinear observer is designed to provide the velocity information. Numerical simulation is performed to verify the ability of the proposed control law. In reality, the tendon has some elasticity. This finite elasticity, if not accounted for, can render the closed-loop system unstable. The investigation shows that the effect of elastic tendons can be compensated for by appropriately modifying the control law designed for inelastic tendons. In particular, the control law is synthesized using the singular perturbation method. It consists of a fast control and a slow control. The fast control is used to stablize the oscillations incurred by the finite elasticity of the tendon. The slow control drives the system to track the desired trajectory. Robustness of the controller is enhanced by using sliding mode control. In the chapter 4, the design of observer in the elastic case is addressed. Linear uncertain system theory is used. The observer is globally stable. The use of decentralized control scheme makes very simple the controller design and reduces the computational complexity. It is very useful for real time agonistic control. A design approach is presented for the decentralized control scheme. A simple linear second order model is used instead of complex nonlinear model used in centralized version. In this approach, the tension in each tendon is treated as disturbance, estimated by an observer, to be compensated