Composite control of nonlinear singularly perturbed systems via approximate feedback linearization

One of the most common methods of synthesis of the nonlinear control systems is the method of a feedback linearization (FL). The idea of this method consists in conversion of the original nonlinear system into a linear one by means of a state feedback and coordinate transformation. Then, the methods of control theory for the linear systems are used for the system design. If the original nonlinear system cannot be linearized exactly by the state feedback, the method of the approximate feedback linearization (AFL) is used. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system (not of the entire system). In this paper, the author proposes a method of an approximate feedback linearization control of the nonlinear singularly perturbed (SP) systems. The proposed method is based on a decomposition of the original SP system and construction of AFL control in the form of composite FL controls for the slow and fast subsystems. In general, a nonlinear SP system cannot be easily separated into slow and fast subsystems, because the conditions of Tikhonov theorem are not complied. In order to overcome this, the author proposes to perform the feedback linearization method at first for the system's part, which describes the fast state variables. Thus, a fast control is chosen, so that the conditions of Tikhonov theorem would be met. Then, using a standard singular perturbation technique, we obtain a slow subsystem. Further the problem of FL control for a slow subsystem is solved. The resulting AFL control is obtained in the form of a composite control. Application of the proposed approach is illustrated with two examples.

In this paper, the regulation problem of a class of nonlinear singularly perturbed discrete-time systems is investigated. Using the theory of singular perturbations and time scales, the nonlinear system is decoupled into reduced-order slow and fast (boundary layer) subsystems. Then, a composite controller consisting of two sub-controllers for the slow and fast subsystems is developed using the discrete-time state-dependent Riccati equation (D-SDRE). It is proved that the equilibrium point of the original closed-loop system with a composite controller is locally asymptotically stable. Moreover, the region of attraction of the closed-loop system is estimated by using linear matrix inequality. One example is given to illustrate the effectiveness of the results obtained.

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.

Robust adaptive output tracking for uncertain nonlinear systems based on approximate feedback linearization

In this paper the H/sub /spl infin//-optimal control and risk-sensitive control of linear singularly perturbed, discrete-time systems are considered. It is shown that the Riccati equation associated with the solution of the H/sub /spl infin//-optimal control problem, can be approximated by an outer series solution, and a boundary-layer correction series solution. As the singular perturbation parameter /spl epsiv/ tends to zero, it is shown that the full-order Riccati equation can be approximated by the zeroth-order outer-series solution. It is also recognized that the outer-series solution and the boundary-layer solution, are equivalent to the resulting asymptotic series expansion in powers of /spl epsiv/, of the Riccati equation associated with a certain risk-sensitive control problem. When a two-time-scale decomposition is considered, it is shown that an approximate composite control can be formed, as the sum of the slow and fast subsystems. Unlike the asymptotic expansion approach, it is shown that the original system can be approximated by the maximum of the performance associated with the slow and fast subsystems. Finally, it is shown that the slow and fast subsystems, are equivalent to certain slow and fast subsystems, respectively, resulting from applying a two-time-scale decomposition to a certain risk-sensitive control problem. >

The emerging field of nonlinear control theory has attempted to alleviate the problem associated with applying linear control theory to nonlinear problems. A segment of nonlinear control theory, called exact feedback linearization. has proven useful in a class of problems satisfying certain controllability and integrability constraints. Approximate feedback linearization has enlarged this class by weakening the integrability conditions, but application of both these techniques remains limited to problems in which a series of linear partial differential equations can easily be solved. By use of the idea of normal forms, from dynamical systems theory, an efficient method of obtaining the necessary coordinate transformation and nonlinear feedback rules is given. This method, which involves the solution of a set of linear algebraic equations, is valid for any dimensional system and any order nonlinearity provided it meets the approximate feedback linearization conditions.

New stability conditions for discrete singularly perturbed systems are presented in this study. The corresponding slow and fast subsystems of the original discrete singularly perturbed system are first derived. The observer-based controllers for the slow and the fast subsystems are then separately designed and a composite observer-based controller for the original system is subsequently synthesized from these observer-based controllers. Finally, a frequency domain ε-dependent stability criterion for the original discrete singularly perturbed system under the composite observer-based controller is proposed. If any one condition of this criterion is fulfilled, stability of the original system by establishing that of its corresponding slow and fast subsystems is thus investigated. An illustrative example is given to demonstrate that the upper bound of the singular perturbation parameter ε can be obtained by examining this criterion.

A singularly perturbed model is proposed for a system comprised of a PEM Fuel Cell(PEM-FC) with Natural Gas Hydrogen Reformer (NG-HR). This eighteenth order system is decomposedinto slow and fast lower order subsystems using singular perturbation techniques that provides tools forseparation and order reduction. Then, three different types of controllers, namely an optimal full-order,a near-optimal composite controller based on the slow and the fast subsystems, and a near-optimalreduced-order controller based on the reduced-order model, are designed. The comparison of closedloopresponses of these three controllers shows that there are minimal degradations in the performanceof the composite and the reduced order controllers.

This paper mainly focuses on dynamic modelling and real-time control for a parallel manipulator with flexible link. The Lagrange principle and assumed modes method (AMM) substructure technique is presented to formulate the dynamic modelling of a two-degrees-of-freedom (DOF) parallel manipulator with flexible links. Then, the singular perturbation technique (SPT) is used to decompose the nonlinear dynamic system into slow time-scale and fast time-scale subsystems. Furthermore, the SPT is employed to transform the differential algebraic equations (DAEs) for kinematic constraints into explicit ordinary differential equations (ODEs), which makes real-time control possible. In addition, a novel composite control scheme is presented; the computed torque control is applied for a slow subsystem and the H ∞ technique for the fast subsystem, taking account of the model uncertainty and outside disturbance. The simulation results show the composite control can effectively achieve fast and accurate tracking control.

This paper studies the guaranteed cost control for singularly perturbed uncertain systems. The system is decomposed into slow and fast sub-systems. The uncertainties in the fast subsystem are transferred to the composite control gain and are considered as the controller gain perturbations. The composite control is achieved based on the slow subsystem under controller gain perturbations and the fast subsystem

In this paper, we propose a sliding mode controller (SMC) based on singular perturbation theory for Sawyer motors. Since the dynamics of Sawyer motors can be divided into slow and fast subsystem, the current measurements are removed by using singular perturbation theory and passivity of the dynamics of Sawyer motors. Input voltages including auxiliary inputs for slow subsystem are designed to stabilize the fast dynamics. The quasi-steady states of fast subsystem, which become the inputs of reduced model corresponding to slow subsystem, is a function of the auxiliary inputs. We design the auxiliary inputs with SMC to stabilize the reduced model. Therefore, the output feedback position controller for Sawyer motors can be achieved by the proposed method. The effectiveness of proposed method is demonstrated by simulation results.

Second-order sliding mode controllers for nonlinear singular perturbation systems

For the control of 2-link flexible joint manipulators, a control method based on the singular perturbation theory and the Udwadia-Kalaba (U-K) method was proposed. The control design was implemented by 2 steps. First, the system order was reduced based on the singular perturbation method and the system was divided into fast and slow sub-systems, to simplify the solution process and overcome the system flexibility. Second, the state feedback constraint following control law for the fast and slow sub-systems was designed with the U-K method, to make the constraint following errors of the fast and slow sub-systems converge to zero, even if the system can't initially satisfy the constraints. The proposed method can deal with holonomic and nonholonomic servo constraints at the same time without the auxiliary variables of the Lagrange multiplier and the pseudo generalized velocity. The method was applied to 2-link flexible joint manipulator systems to solve the flexible oscillation and constraint following problems. Through simulations on MATLAB, and was compared with the traditional PID control, to verify the effectiveness and superiority.

An adaptive neural network controller is presented for flexible link robots (FLRs) using the singular perturbation technique. The complex full model of FLRs can be decomposed into two separate time scale subsystems by modelling the elastic forces as the fast variables and the joint variables as the slow variables. A composite control strategy is adopted to control the two-time scale model with the slow subsystem being controlled by a neural network controller. The weights of the NNs are updated online based on direct adaptive techniques. A robust control term is also added for closed-loop stability. The fast subsystem is stabilized by a simple LQR control around the equilibrium trajectory defined by the slow subsystem.

Disturbance observer based reliable H∞ fuzzy attitude tracking control for Mars entry vehicles with actuator failures