Controller Design for Ascent Phase of Reusable Launch Vehicle Using Backstepping

A nonlinear feedback control law that achieves global asymptotic stabilization of a 2D thermal convection loop is presented. The loop consists of a viscous Newtonian fluid contained in between two concentric cylinders standing in a vertical plane. The lower half of the loop is heated while the upper half is cooled. Stability analysis of the thermal convection loop shows that the no-motion steady state for the uncontrolled case is unstable for the values of the non-dimensional Rayleigh number R/sub /spl alpha//>1. The objective is to stabilize the unstable no-motion steady state using boundary control of velocity and temperature on the outer cylinder. In our controller design we start by discretizing the original PDE model in space using a finite difference method which gives a high order system of coupled nonlinear ODEs in 2D. Then, using backstepping design, we obtain a discretized coordinate transformation that transforms the original coupled system into two uncoupled systems that are asymptotically stable in 12-norm with homogeneous Dirichlet boundary conditions. Using the property that the discretized coordinate transformation is smoothly invertible for an arbitrary grid choice, we conclude that the discretized version of the original system is globally asymptotically, stable and obtain nonlinear feedback boundary control laws for velocity and temperature in the original set of coordinates. The control design is accompanied by an extensive simulation study. Numerical results show that the feedback control law designed on a very coarse grid can successfully stabilize the system for a very wide range of the Rayleigh number. This means that an excellent closed loop performance is achieved using just a few measurements of the flow and temperature fields implying that the proposed backstepping design has a potential to be successfully applied in a real experimental setting.

International audience

Recently, model predictive controller design for the FO model of dynamical systems has attracted many researchers. In [1], authors discuss how to design generalized predictive control for a fuel cell using fractional calculus. In [2], authors discuss the designing of switched state MPC for fractional-order discrete-time systems. In paper [4], discrete-time FO systems have been considered for MPC controller design. Articles [3] and [5] uses the same methodology to design control law to control the response of the system efficiently. More recently, paper [6] discuss developing a fractional MPC which can be used to control the temperature efficiently in industrial processes. In [3], an idea of how to design MPC for FO systems is presented which is built upon in this chapter to obtain the controller design for a different system. In this chapter, a FO model of a system is chosen using the method discussed in Chap. 3 and then the MPC controller is obtained for efficient control of the considered system. The fractional MPC (FMPC) control design method has tremendous potential to improve system performance. However, some properties of FMPC controller design is at present a subject of in-depth analysis and experimentation.

Theproblem ofapproximatedecoupling of nonlinear systemsby outputfeedback is considered. The structure of the controller is high-gain proportional plus integral. Linearization of the nonlinear closed-loop equations results in the standard singular perturbation form. Based on the singular perturbation concept, a control law is found that accomplishes approximate decoupling with stability for nonlinear systems. Application oftheproposed design schemeto thefullsix-degree-of-freedomnonlinearaircraftdynamicsmodelispresented,demonstrating itsvalidity. I. Introduction A SYSTEM is said to be decoupled if each component of the output depends only on the corresponding component of the input. The problem of decoupling a multivariable system has been extensively treated in the literature in the past 30 years. The necessary and sufe cient conditions were developed for linear 1;2 and nonlinear 3i6 systems. However, no systematic procedures for the controller design were considered. Nonlinear system controller design is typically approached by linearizing the system about a nominal operating point and applying linear system design methods. This will provide a linear control law that yields approximate noninteracting control and satisfy design objectives in the neighborhood of the constant operating point. For the purpose of this work, the design objectives for this highly coupled system of e ight dynamics are to closely track individual desired outputs, with zero steady-state response from others. However, the problem of computing the decoupling feedback control laws for a nonlinear system is quite dife cult. Approximate decoupling control for nonlinear systems is based onsingularperturbationconcepts.Thestructureofcontrollerishighgain output feedback with proportional plus integral (PI) control.It will be shown that the linearization of nonlinear closed-loop system about the nominal operating point yields the standard singular perturbation equation form when a scalar gain weighting g ! 1. Then,thesingularperturbationconcept,i.e.,necessaryandsufe cient conditions, is utilized to e nd a decoupling control law. This approach is a new, conceptually simple, but practically useful design method for approximate noninteracting control law design with stability for nonlinear systems. The results developed are easy to follow and are easily applicable to aircraft control decoupling problems. This paper introduces the application of this design method to approximate noninteracting e ight control system design.

Adaptive fuzzy tracking controller design is addressed for a class of uncertain nonlinear system in strict-feedback form. The system dynamics are unknown, and external disturbances satisfy triangular bounds. We utilize Tagaki-Sugeno (T-S) fuzzy logic system to approximate the unknown system functions. The proposed scheme can guarantee the ultimate uniform boundedness of all the signals in the closed-loop system. Compensated tracking errors are employed to construct Lyapunov candidate function. The proposed design circumvents the repeated differential of virtual control law, which renders the controller structures simple and easy to implement. Simulation researches are carried out. The simulation results demonstrate the effectiveness and usage of the proposed design.

The backstepping control design procedure has been used to develop stabilizing controllers for time invariant plants that are linear or belong to some class of nonlinear systems. The use of such a procedure to design stabilizing controllers for plants with time varying parameters has been an open problem. In this paper we consider the backstepping design procedure for linear time varying (LTV) plants with known and unknown parameters. We first show that a backstepping controller can be designed for an LTV plant by following the same steps as in the linear time-invariant (LTI) case and treating the plant parameters as constants at each time t. Its stability properties however cannot be established by using the same Lyapunov function and techniques as in the LTI case. We then introduce a new parametrization and filter structure that takes into account the plant parameter variations leading to a new backstepping controller. The new control design guarantees exponential convergence of the tracking error to zero if the plant parameters are exactly known. If the parameters are not precisely known but the time variations of the parameters associated with the system zeros are known, the appropriate choice of certain design parameters, without using any adaptive law, leads to closed-loop stability and perfect regulation. This new control design is modified and supplemented with an update law to be applicable to LTV plants with unknown parameters. In the adaptive control design, the notion of structured parameter variations is used in order to include possible a priori information about the plant parameter variations. With this formulation, only the unstructured plant parameters are estimated and are required to be slowly time varying, and the structured plant parameters are allowed to have any finite speed of variation. The adaptive controller is shown to be robust with respect to the unknown but slow parameter variations in the global sense. We derive performance bounds which can be used to select certain design parameters for performance improvement. The properties of the proposed control scheme are demonstrated using simulation results.

Design of ‘softer’ robust nonlinear control laws

On local [formula omitted] switched controller design for uncertain T–S fuzzy systems subject to actuator saturation with unknown membership functions

In recent years, research on speed cruise control for high-speed trains has primarily focused on control design based on multi-body models, leading to significant achievements in various aspects. However, these control methods adopt a single-system approach for the entire train, referred to as the typical method in this paper. Yet, a malfunction in one car can impact the control laws and controller design of other cars. To address this limitation, this paper proposes a large-scale system control, which decouples the control laws and controller design for each car. The control law of an individual car is based solely on its own position and velocity, with separate processes for solving the controllers for each car. Additionally, the speed cruise control for high-speed trains considers both <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {H}}_{2}$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal {H}}_{\infty } $ </tex-math></inline-formula> performance in the presence of uncertain parameters. This study validates the feasibility of such decoupled methods in this domain through simulation-based comparative analysis. This study aims to verify the effectiveness of large-scale system control methods in normal train operations and to lay the foundation for more complex fault-tolerant and model-switching challenges. In these areas, large-scale system control has greater potential for application and offers significant advantages.

This paper investigates adaptive fuzzy output feedback fault-tolerant optimal control problem for a class of single-input and single-output nonlinear systems in strict feedback form. The considered nonlinear systems contain unknown nonaffine nonlinear faults and unmeasured states. Fuzzy logic systems are used to approximate cost function and unknown nonlinear functions, respectively. It is assumed that the states of the systems to be controlled are unmeasurable, thus an adaptive state observer is developed. To solve the nonaffine nonlinear fault control design problem, filtered signals are introduced into the adaptive backstepping control design procedures, and in the framework of adaptive critic technique and fault-tolerant control technique, a novel adaptive fuzzy fault-tolerant optimal control scheme is developed. The stability of the closed-loop system is proved by using Lyapunov stability theory. The simulation results verify the effectiveness of the proposed control strategy.

Asynchronous motor system has the characteristics of high order, strong coupling, and nonlinearity. From the dynamical model, it is the underactuated mechanical system, which means that the dimension of its input space is fewer than the degree of freedom. Following this perspective, the energy based nonlinear control technology-CL (controlled Lagrangians) method is used to solve the control problem in this paper. Based on the expected controlled energy and its derivative with respect to time, controlled Lagrangians and generalized force are constructed, and they produce the controlled equations. In order to ensure the complete matching between the controlled equation and the original equation, the gyroscopic forces containing the first-order term of velocity are innovatively introduced into the generalized force, and the matching conditions are obtained. By solving the matching conditions composed of some partial differential equations, the nonlinear smooth feedback control law can realize the global asymptotic stabilization of not only velocity but also position. Finally, the controlled energy is selected as the Lyapunov function, and the stability is proved according to the LaSalle invariant theorem. The effectiveness of the designed control law is demonstrated in the results of the simulation.

In this paper, a design of a signal synthesis adaptive controller for a class of linear time-varying uncertain plants using model reference adaptive control techniques is presented. Following a discussion of the general concept, the design of an adaptive controller for unconstrained and constrained control conditions is given. The small ultimate boundness of the state error is considered as the adaptation criterion, which is shown to be satisfied by a Liapunov type stability theorem. To handle the uncertainties that are associated with the plant dynamics and its environment, a min-max concept is employed in the design of the controller. The signal synthesis adaptive approach presented in this paper does not require as many structural assumptions as do most other adaptive approachers, and this is its principal advantage. The state vectors of the plant and model are assumed to be both accessible. Some simulation results are presented which illustrate the effectiveness of the design given here.

This paper is concerned with control design for a generalized discrete-time Takagi-Sugeno fuzzy system. The Takagi-Sugeno fuzzy system generally describes nonlinear systems by employing local linear system representations. It is a great tool to analyze a nonlinear system and to design a stabilizing controller for it. However, there is a case where a controller design for a nonlinear system based on the Takagi-Sugeno fuzzy system becomes difficult due to its representation. A generalized fuzzy system to be considered in this paper describes even a wider class of nonlinear systems by representing locally nonlinear systems, and gives a different representation from the standard Takagi-Sugeno fuzzy system, which leads to a rather easy controller design. For such a generalized system, a stabilizing controller design method is proposed by introducing a new class of nonlinear controllers. Then, the resulting closed-loop system becomes a fuzzy system with linear system structures locally, which makes the control design easier. Stabilizing controller design conditions are given in terms of a set of linear matrix inequalities(LMIs), which are easily numerically solvable. A relaxation method is used to reduce the conservatism of design conditions. Finally, numerical examples are given to illustrate our nonlinear control design and to show the effectiveness over other existing results.

Concerns nonlinear model predictive control, and particularly the nonlinear optimization problem. Usually the control sequence can be determined by using some effective numerical iteration approaches, especially for multistep predictive control. This work focuses on the multistep adaptive NMPC controller design using neural-net. The main ideas are (A) initialisation of the multistep control laws by using one-step ahead predictive control law; (B) linearization of the neural-net predictor at every operating point; and (C) tuning of the neural-net predictor through online learning using teacher signals generated by closed-loop system input-output data. As an illustrative example of our approach, an explicit control laws are derived for the control horizon Nu=2 case.

This paper deals with the design of adaptive linearizing controllers of process components at the output of fixed bed reactors. The control design is based on the distributed parameter model of the process (late lumping). It takes advantage of the well-known part of the (nonlinear) structure (basically the hydrodynamics) and does not require the knowledge of the kinetics, which are estimated on-line. Two examples are considered: the control of the concentration of one process component and of the gas production rate.