Nonlinear Control Synthesis Research Articles

Nonlinear control of spatially inhomogenous aerosol processes

This article proposes a nonlinear feedback control methodology for spatially inhomogeneous aerosol processes for which the manipulated inputs, the control objectives and the measurements are distributed in space. Initially, a general nonlinear partial integro-differential equation model which describes the spatio-temporal evolution of the aerosol size distribution, as well as the evolution of the concentrations of species and temperature of the continuous phase is presented. The model accounts for simultaneous chemical reaction, nucleation, condensation, coagulation and convective transport. Then, under the assumption of lognormal aerosol size distribution, the method of moments is employed to reduce the original model into a set of first-order hyperbolic partial differential equations (PDEs) which accurately describes the spatio-temporal evolution of the three leading moments needed to precisely characterize the aerosol size distribution. This hyperbolic PDE system is then used as the basis for the synthesis of nonlinear distributed output feedback controllers that enforce closed-loop stability and achieve an aerosol size distribution with desired characteristics. The proposed nonlinear control method is successfully applied through simulations to a typical aerosol process and is shown to outperform a conventional proportional integral control scheme and deal effectively with disturbances in the feed to the process.

Nonlinear control of particulate processes

AbstractA general methodology is proposed for the synthesis of practically‐implementable nonlinear output feedback controllers for spatially‐homogeneous particulate processes modeled by population balance equations. Initially, a nonlinear model reduction procedure based on a combination of the method of weighted residuals and the concept of approximate inertial manifold is presented for the construction of low‐order ordinary differential equation (ODE) systems that accurately reproduce the dominant dynamics of the particulate process. These ODE systems are then used for the synthesis of nonlinear low‐order output feedback controllers that enforce exponential stability in the closed‐loop system and achieve particle‐size distributions with desired characteristics. Precise closed‐loop stability conditions are given and controller implementation issues are discussed. The proposed nonlinear control method is successfully applied to a continuous crystallizer, and is shown to outperform a proportional‐integral controller and cope effectively with model uncertainty and measurement delays.

Robust output feedback control of quasi-linear parabolic PDE systems

This paper proposes a methodology for the synthesis of nonlinear robust output feedback controllers for systems of quasi-linear parabolic partial differential equations with time-varying uncertain variables. The method is successfully applied to a typical diffusion–reaction process with uncertainty.

Robust control of parabolic PDE systems

This article proposes a methodology for the synthesis of nonlinear robust feedback controllers for diffusion–convection–reaction processes with time-varying uncertain variables described by systems of quasi-linear parabolic partial differential equations (PDEs), for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional (possibly unstable) slow one and an infinite-dimensional stable fast complement. Combination of Galerkin’s method with approximate inertial manifolds is used to derive ordinary differential equation (ODE) systems of dimension equal to the number of slow modes that accurately describe the dominant dynamics of the PDE system. These ODE systems are used for the synthesis of robust controllers that guarantee boundedness of the state and output tracking with arbitrary degree of asymptotic attenuation of the effect of the uncertain variables on the output of the closed-loop system. The robust controllers are synthesized via Lyapunov’s direct method and utilize bounding functions on the magnitude of the uncertain terms. The developed methodology is successfully applied to a catalytic packed-bed reactor with unknown heat of reaction.

Multivariable nonlinear controller synthesis in discrete-time

For a nonlinear process with a generically singular characteristic matrix, input–output linearization is usually not achievable with a static state feedback, and one has to use a dynamic state feedback. A systematic and general approach to input–output linearizing feedback control of nonlinear processes with a egenerically singular or nonsingular characteristic matrix is presented. Mixed error- and state-feedback, error-feedback, and mixed error- and output-feedback control laws are derived. The derived feedback control laws reject step disturbances asymptotically and induce linear, input–output, closed-loop behavior to the process under consideration. The theoretical connections between the derived control laws and (a) model predictive control and (b) several already-available linear deadtime compensation methods are established. The application and performance of the derived control laws are illustrated and compared with those of model predictive control, by numerical simulation of a chemical reactor.

Discrete‐Time nonlinear control of processes with actuator saturation

This note presents discrete-time nonlinear model-based control laws for multivariable processes with actuator saturation. This work is a continuation of the nonlinear controller synthesis results for unconstrained processes (Soroush and Kravaris, 1996). The control laws presented in this note are input-output linearizing in the absence of input constraints, can provide significant improvement in control quality in the presence of active input constraints, and are model-predictive in the sense that they are exact solutions to a moving-horizon optimization problem. The connections between the derived control laws and (a) model state feedback control (Coulibaly et al., 1995) and (b) modified internal model control (IMC) (Zheng et al., 1994) are explained briefly. This note begins with a description of the scope of the work. Dynamic control laws are derived first for processes with full state measurements and then for processes with incomplete state measurements. The application and performance of one of the nonlinear control laws are demonstrated by a chemical reactor example.

A homotopy approach for nonlinear control synthesis

In this paper, the authors propose a control method for nonlinear systems which relies on the homotopy method for solving nonlinear equations and tangent linearized control systems. They present a mathematically rigorous treatment of the theoretical foundations, including a constructive proof of the exponential error decay for the controlled system which contains an explicit control design instruction. For illustration, this design instruction is applied to an example system. The behavior of the controlled system is examined by numerical simulations. To demonstrate the advantages and drawbacks of the proposed method they include a short comparison with two usual nonlinear control techniques by means of the considered example system.

Finite-Dimensional Control of Parabolic PDE Systems Using Approximate Inertial Manifolds

This paper introduces a methodology for the synthesis of nonlinear finite-dimensional output feedback controllers for systems of quasi-linear parabolic partial differential equations (PDEs), for which the eigenspectrum of the spatial differential operator can be partitioned into a finite-dimensional slow one and an infinite-dimensional stable fast complement. Combination of Galerkin's method with a novel procedure for the construction of approximate inertial manifolds for the PDE system is employed for the derivation of ordinary differential equation (ODE) systems (whose dimension is equal to the number of slow modes) that yield solutions which are close, up to a desired accuracy, to the ones of the PDE system, for almost all times. These ODE systems are used as the basis for the synthesis of nonlinear output feedback controllers that guarantee stability and enforce the output of the closed-loop system to follow up to a desired accuracy, a prespecified response for almost all times.

Nonlinear Control Synthesis

Abstract The aggregation method in nonlinear control synthesis for the standard nonlinear dynamic subsystems is considered. The proposed approach is relatively simple and effective but nevertheless it does not demand a deep knowledge of the optimal control theory. The merit of the described nonlinear control synthesis is the design of nonadaptive, adaptive or combined controls which ensure feedback linearization of closed-loop control systems. An application is shown in an example.

Nonlinear control of mechanical systems: A Lagrangian perspective

Recent advances in geometric mechanics, motivated in large part by applications in control theory, have introduced new tools for understanding and utilizing the structure present in mechanical systems. In particular, the use of geometric methods for analyzing Lagrangian systems with both symmetries and non-integrable (or nonholonomic) constraints has led to a unified formulation of the dynamics that has important implications for a wide class of mechanical control systems. This paper presents a survey of recent results in this area, focusing on the relationships between geometric phases, controllability, and curvature, and the role of trajectory generation in nonlinear controller synthesis. Examples are drawn from robotics and flight control systems, with an emphasis on motion control problems.

拡張2次形式リアプノフ関数による非線形状態フィードバック制御の構成法

拡張2次形式リアプノフ関数による非線形状態フィードバック制御の構成法

Model-based synthesis of nonlinear PI and PID controllers

PI and PID controllers continue to be popular in industrial applications. It is well known that linear PI and PID controllers result from the application of model-based controller design methods to linear first and second-order systems. This work shows that nonlinear PI and PID controllers result from application of nonlinear controller design methods to nonlinear lirst and second-order systems. Examples illustrating nonlinear PI and PID controllers are given for chemical process models.

Adaptive neuro-control for spacecraft attitude control

Spacecraft attitude control is approached as a nonlinear adaptive control problem and neuro-control, which combines concepts from artificial neural networks and adaptive control, is investigated as an alternative to linear control approaches. Three capabilities of neuro-controllers are demonstrated using a nonlinear model of the Space Station Freedom. These capabilities are: (a) synthesis of robust nonlinear controllers using neural networks; (b) copying an existing control law using neural networks; and (c) adaptively modifying neuro-controller characteristics for varying inertia characteristics. The main components of the adaptive neuro-controllers are an identification network and a controller network. Both these networks are trained using the back-propagation of error learning paradigm. To ensure robustness of the neuro-controller, optimally connected neural networks are synthesized for the identification and the controller networks. For the on-line adaptive control problem, a backpropagation of error technique using a linear adaptive critic is introduced in place of the backpropagation through time technique. Performances of the nonlinear neuro-controllers for the three cases listed above are verified using a nonlinear simulation of the Space Station. Results presented substantiate the feasibility of using neural networks in robust nonlinear adaptive control of spacecraft.

Nonlinear Control of Mechanical Systems: A Lagrangian Perspective

Recent advances in geometric mechanics, motivated in large part by applications in control theory, have introduced new tools for understanding and utilizing the structure present in mechanical systems. In particular, the use of geometric methods for analyzing Lagrangian systems with both symmetries and non-integrable (or nonholonomic) constraints has led to a unified formulation of the dynamics that has important implications for a wide class of mechanical control systems. This paper presents a survey of recent results in this area, focusing on the relationships between geometric phases, controllability, and curvature, and the role of trajectory generation in nonlinear controller synthesis. Examples are drawn from robotics and flight control systems, with an emphasis on motion control problems.

Nonlinear model-based control using second-order Volterra models

A nonlinear controller synthesis scheme is presented that retains the original spirit and characteristics of conventional (linear) model predictive control (MPC) while extending its capabilities to nonlinear systems. The scheme employs a Volterra model—a simple and convenient nonlinear extension of the linear convolution model employed by conventional MPC—and gives rise to a controller composed of a conventional linear controller augmented by an auxiliary loop of nonlinear ‘corrections’. Simulation case studies involving two different examples representative of the typical spectrum of nonlinear behavior in real chemical processes—an industrial polymerization reactor and an isothermal reactor exhibiting inverse response—are used to demonstrate the practical utility of the control scheme and to evaluate its performance.

Industrial applications of fuzzy logic at General Electric

Fuzzy logic control (FLC) technology has drastically reduced the development time and deployment cost for the synthesis of nonlinear controllers for dynamic systems. As a result we have experienced an increased number of FLC applications. We illustrate some of our efforts in FLC technology transfer, covering projects in turboshaft aircraft engine control, steam turbine startup, steam turbine cycling optimization, resonant converter power supply control, and data-induced modeling of the nonlinear relationship between process variables in a rolling mill stand. We compare these applications in a cost/complexity framework, and examine the driving factors that led to the use of FLCs in each application. We emphasize the role of fuzzy logic in developing supervisory controllers and in maintaining explicit tradeoff criteria used to manage multiple control strategies. Finally, we describe some of our FLC technology research efforts in automatic rule base tuning and generation, leading to a suite of programs for reinforcement learning, supervised learning, genetic algorithms, steepest descent algorithms, and rule clustering. >

Dynamically equivalent outputs and their use in nonlinear controller synthesis

A new notion, dynamically equivalent outputs, is developed to aid in controller synthesis for nonlinear systems. A dynamically equivalent output simplifies the controller design problem and, when controlled to its set point, guarantees that the system's primary output is also controlled to its set point. The use of this notion is demonstrated for three classes of systems. For systems with linear state equations but a nonlinear state/output map, conditions are derived for the existence of a linear dynamically equivalent output. Analogous conditions are derived for nonlinear systems for which the state equations can be made linear by means of a coordinate transformation and state feedback. Design of a controller in terms of the new output is straightforward, leading to a response which is nonlinear in the primary output. Simulations for a chemical reactor system using a coordinate transformation, state feedback, and a dynamically equivalent linear output are given. Finally, for general nonlinear systems, the use of input/output linearizing feedback in terms of a dynamically equivalent output is explored. Another chemical reactor system is simulated to demonstrate this approach.

Synthesis of a helicopter nonlinear flight controller using approximate model inversion

This paper considers synthesis of a helicopter full authority flight controller using an approximate inversion of the nonlinear model of the vehicle. Based on the natural time scale separation between position and attitude dynamics of the vehicle, the vehicle attitudes are treated as pseudo-command variables. In order to simplify the controller, approximations to the body axes forces are used in the controller calculations. The first approximation involves neglecting the cyclic and pedal control force terms, and the second approximation involves neglecting the body x- and y-axis force components in the controller calculations. The impact of these approximations on the performance of the closed-loop system is evaluated through nonlinear simulation of the Apache helicopter using typical command maneuvers.

A Design Method for Nonlinear Control Systems

The suboptimal synthesis of nonlinear control by dinamic systems is proposed. The method under consideration is redused to the suboptimal solution of Bellman's equation with previously given degree of precision.

Terminal slider control of robot systems

Many robotic systems would, in the future, be required to operate in environments that are highly unstructured (with varying dynamical properties) and active (possessing means of self-actuation). Although a significant volume of results exist in model-based, robust and adaptive control literature, many issues pertinent to the stabilization of contact interactions with unpredictable environments remain unresolved, especially in dealing with large magnitude and high frequency parametric uncertainties. The primary intent of this paper is nonlinear control synthesis for robotic operations in unstructured environments. We introduce the notion oftime constrained terminal convergence for controlled systems, and propose an approach to nonlinear control synthesis based upon a new class of sliding modes, denotedterminal sliders. Terminal controllers that enforce finite convergence to equilibrium are synthesized for an example nonlinear system (with and without parametric uncertainties). Improved performance is demonstrated through the elimination of high frequency control switching, employed previously for robustness to parametric uncertainties [2]. The dependence of terminal slider stability upon the rate of change of uncertainties over the sliding surface, rather than the magnitude of the uncertainty itself, results in improved control robustness. Improved reliability is demonstrated through the elimination ofinterpolation regions [2]. Finally, improved (guaranteed) precision is argued for through an analysis of steady state behavior.