Control Laws For Nonlinear Systems Research Articles

Stabilizing a Class of Nonlinear Systems Based on Approximate Feedback Linearization

We present a method of stabilizing a class of nonlinear systems which are not necessarily feedback linearizable. First, we show a new way of constructing a diffeomorphism to transform a class of nonlinear systems to the feedback linearized form with perturbation. Then, we propose a semi-globally stabilizing control law for nonlinear systems that are connected by a chain of integrator perturbed by arbitrary nonlinear terms. In our approach, we have flexibility in choosing a diffeomorphism where the system is not restricted to involutivity and this leads to reduction in computational burden and flexibility in controller design.

Nearly optimal control laws for nonlinear systems with saturating actuators using a neural network HJB approach

The Hamilton–Jacobi–Bellman (HJB) equation corresponding to constrained control is formulated using a suitable nonquadratic functional. It is shown that the constrained optimal control law has the largest region of asymptotic stability (RAS). The value function of this HJB equation is solved for by solving for a sequence of cost functions satisfying a sequence of Lyapunov equations (LE). A neural network is used to approximate the cost function associated with each LE using the method of least-squares on a well-defined region of attraction of an initial stabilizing controller. As the order of the neural network is increased, the least-squares solution of the HJB equation converges uniformly to the exact solution of the inherently nonlinear HJB equation associated with the saturating control inputs. The result is a nearly optimal constrained state feedback controller that has been tuned a priori off-line.

OUTPUT TRACKING WITH CONSTRAINED INPUTS VIA ADAPTIVE RECURRENT NEURAL CONTROL

This paper extends previous results to the output tracking problem of nonlinear systems with unmodelled dynamics and constrained inputs. A recurrent high order neural network is used to identify the unknown system dynamics and a learning law is obtained using the Lyapunov methodology. A stabilizing control law for the output tracking error dynamics is developed using the Lyapunov methodology and the Sontag control law for nonlinear systems with constrained inputs.

Problems and Techniques

Suppose you are a bicycle messenger in the busy business district of a large city. Here is what your day looks like: You are to deliver incoming mail to building i, and you are to pick up outgoing mail destined only for building i + 1. All in all you are responsible for m buildings. Your bike, however, is not of the highest quality: It can accommodate only mail for a single delivery. Riding from building i to building j requires tij minutes. Once you arrive at a building, you need d minutes to get off your bike, run into the building, and get onto your bike again. Moreover, once incoming mail has arrived at building i, it takes pi minutes for the outgoing mail to be ready (if this is too long, you may want to make another delivery and pickup in the meantime). You are to visit each building k times. Question: In which order should you visit the buildings so that you finish your work as fast as possible? The above description is a simplified version of the problem discussed in the paper by Milind W. Dawande, H. Neil Geismar, and Suresh P. Sethi. There the messenger is a robot, the buildings are machines, and the mail represents parts to be processed by the machines. The authors show that there exists a cyclic schedule that maximizes long-term throughput. Cyclic schedules are preferred in industrial environments, because they are easy to implement and control. Since the literature on robotic cell scheduling is full of different models for different kinds of industrial applications, it is important to know that all optimal schedules can be reduced to cyclic schedules. The authors end their paper by describing several challenging open problems. The paper by Miguel Torres-Torriti and Hannah Michalska describes a software package (LTP), implemented in Maple, for the symbolic manipulation of expressions that occur in the context of Lie algebra theory. This theory has found applications in classical and quantum mechanics, analysis of dynamical systems, construction of nonlinear filters, and the design of feedback control laws for nonlinear systems. Since the symbolic computations are often complex and tedious, the development of software for applications of Lie algebra theory is crucial. The LTP software package is targeted at applications such as solution of differential equations evolving on Lie groups, and structure analysis of general dynamical systems.

Scheduling of Local Robust Control Laws for Nonlinear Systems

This paper presents a systematic approach for nonlinear control design by using the gain scheduling technique to insure the transition of a nonlinear dynamical process from an actual operating condition to a desired one. The nonlinear equations of the system are imbedded in the interior of an inclusion polyhedron. A robust control law is built so as to insure asymptotic stability to a given equilibrium within a maximal ellipsoidal region contained in the interior of the polyhedron. State feedback and output feedback for the local robust control synthesis are considered, together with local performance criteria. A numerical experiment is provided to illustrate both the effectiveness of the synthesis and the performance achieved.

Robust inverse optimal control laws for nonlinear systems

AbstractThis work proposes a robust inverse optimal controller design for a class of nonlinear systems with bounded, time‐varying uncertain variables. The basic idea is that of re‐shaping the scalar nonlinear gain of an LgV controller, based on Sontag's formula, so as to guarantee certain uncertainty attenuation properties in the closed‐loop system. The proposed gain re‐shaping is shown to yield a control law that enforces global boundedness of the closed‐loop trajectories, robust asymptotic output tracking with an arbitrary degree of attenuation of the effect of uncertainty on the output, and inverse optimality with respect to a meaningful cost that penalizes the tracking error and the control action. The performance of the control law is illustrated through a simulation example and compared with other controller designs. Copyright © 2003 John Wiley & Sons, Ltd.

Immersion and invariance: a new tool for stabilization and adaptive control of nonlinear systems

A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method relies upon the notions of system immersion and manifold invariance and, in principle, does not require the knowledge of a (control) Lyapunov function. The construction of the stabilizing control laws resembles the procedure used in nonlinear regulator theory to derive the (invariant) output zeroing manifold and its friend. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to higher order dynamics. This is achieved by designing a control law that asymptotically immerses the full system dynamics into the reduced order one. We also show that in adaptive control problems the method yields stabilizing schemes that counter the effect of the uncertain parameters adopting a robustness perspective. Our construction does not invoke certainty equivalence, nor requires a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states. Finally, it is shown that the proposed approach is directly applicable to systems in feedback and feedforward form, yielding new stabilizing control laws. We illustrate the method with several academic and practical examples, including a mechanical system with flexibility modes, an electromechanical system with parasitic actuator dynamics and an adaptive nonlinearly parameterized visual servoing application.

CONTROLLERS FOR NONLINEAR SYSTEMS USING NORMAL FORMS

We describe a method for the design of feedback stabilization control laws for nonlinear systems using the theory of normal forms and the results from optimal control theory. We show that the resulting controllers can provide a larger region of stability than local linear controllers designed to perform the same task.

Synthesis of Control Law for Non-Linear Systems in Critical Case

Stabilization (control system design) is one of fundamental areas investigated in the system theory and many articles have already been dedicated to it in technical literature. The paper deals with a stabilization method for a certain class of non-linear systems. Non-linear systems belonged to the considered class have (after performed approximative linearization) some poles on the imagine axis which are uncontrollable. As the theoretical base of the approach presented in the paper the Center manifold theory (which enables us to solve the stabilization problem) has been chosen.

Immersion and Invariance: A New Tool for Stabilization and Adaptive Control of Nonlinear Systems

A new method to design asymptotically stabilizing and adaptive control laws for nonlinear systems is presented. The method is well suited in situations where we know a stabilizing controller of a nominal reduced order model, which we would like to robustify with respect to high order dynamics. We also show that in this new framework the adaptive control problem can be formulated from a new perspective that, under some suitable structural assumptions, allows to modify the classical certainty equivalent controller and derive parameter update laws such that stabilization is achieved. It is interesting to note that our construction does not require a linear parameterization, furthermore, viewed from a Lyapunov perspective, it provides a procedure to add cross terms between the parameter estimates and the plant states in the Lyapunov function.

System Identification and Control using Probabilistic Incremental Program Evolution Algorithm

An indispensable ability for intelligent control is to comprehend and learn about plants, disturbances, environment, and operating conditions. In this paper, the Probabilistic Incremental Program Evolution (PIPE) algorithm, with its self-organizing and learning ability, is used as a promising tool for such purposes. The previous work on evolutionary control by using tree structure based evolutionary algorithm was inverse control in general. In this case, Genetic Programming (GP) was usually used to evolving a directly control law of nonlinear systems. It is difficult to design a better fitness function that should reflect the characteristics of nonlinear systems, and a prior knowledge about operating conditions is usually needed. In this paper, a new identification and control method is proposed without prior knowledge of the plant. Firstly, the input-output behavior of the discrete-time nonlinear system is approximated by the individual structure of PIPE (PIPE Emulator). Secondly, a model based evolutionary controller (PIPE Emulator-based controller) of nonlinear system is designed. Simulation results for a typical nonlinear discrete-time system show the feasibility and effectiveness of the proposed method.

Application of true linearization in stochastic quasi‐optimal control problems

AbstractThe problem of quasi‐optimal linear feedback control laws for nonlinear systems with quadratic performance criteria under Gaussian white noise excitations is considered. To determine the quasi‐optimal control a modified version of standard iterative procedure is used, where three versions of statistical linearization methods and true linearization method were combined with the optimal control method for linear systems with the mean‐square criterion. The validity of this method is shown by analytical and numerical calculations for examples.

Nonlinear sliding surface design in the presence of uncertainty *

This paper combines the block control, sliding mode and high gain robust control techniques to derive a stabilizing control law for nonlinear systems with matched and unmatched uncertainties.

Suboptimal model predictive control (feasibility implies stability)

Practical difficulties involved in implementing stabilizing model predictive control laws for nonlinear systems are well known. Stabilizing formulations of the method normally rely on the assumption that global and exact solutions of nonconvex, nonlinear optimization problems are possible in limited computational time. In the paper, we first establish conditions under which suboptimal model predictive control (MPC) controllers are stabilizing; the conditions are mild holding out the hope that many existing controllers remain stabilizing even if optimality is lost. Second, we present and analyze two suboptimal MPC schemes that are guaranteed to be stabilizing, provided an initial feasible solution is available and for which the computational requirements are more reasonable.

Improving the performance of stabilizing controls for nonlinear systems

There are a variety of tools for computing stabilizing feedback control laws for nonlinear systems. The difficulty is that these tools usually do not take into account the performance of the control, and therefore systematic improvement of an arbitrary stabilizing control law is extremely difficult and often impossible. The objective of this article is to present a design algorithm that addresses this problem. The algorithm that we present iteratively computes a sequence of control laws with increasingly improved performance. We also consider implementation issues and discuss some of the successes and difficulties that we have encountered. Finally, we present a number of illustrative examples and compare our algorithm with perturbation methods.

Static-state feedback laws for output regulation of non-linear systems

In this paper static-state feedback laws are considered which address the problem of output stabilization of non-linear dynamic control systems without drift. It is shown that as long as a dynamic-state control law exists which input-output decouples a given system, it is possible to construct a static-state control which stabilizes the system's output. The design procedure sheds light on the inherent difficulties in designing internally stable control laws for non-linear systems. To demonstrate the theory, it is applied to the task of designing an output stabilizing feedback for a simple model of a mobile robot.

An Introduction to the Theory of Exact Stochastic Feedback Linearization for Nonlinear Stochastic Systems

In the design of feedback control laws for nonlinear systems, the concept of exact feedback linearization has emerged as a promising technique. In its current form, the concept of feedback linearization is limited to deterministic systems. The purpose of this paper is to extend this concept to a class of nonlinear stochastic systems. The results of this paper open up a new avenue towards designing nonlinear controllers for stochastic systems.

Sliding Mode Block Control of Uncertain Nonlinear Plants

This paper combines the block control, sliding mode and high gain robust control techniques to form a stabilising control law for nonlinear systems with matched and unmatched uncertainties. A generalised observer is presented to cater for plants subjected to an unknown matched disturbance. The approach is applied to form a non-chatiering sliding-mode observer-based control law for robotic manipulators driven by DC motors.

A controller design method for nonlinear systems via a staged-transformation technique

The paper addresses a controller design approach for nonlinear systems based on a staged-transformation technique. A new nonlinear state transformation is proposed to transform a complex single-input nonlinear system into a simple intermediate nonlinear model, a nonlinear state feedback is then introduced to transform the intermediate nonlinear model into a linear controllable canonical form. The optimal nonlinear controller to the primitive nonlinear system is derived via a staged-design method. Furthermore, an optimal tracking control law for nonlinear systems is developed. The simulation result of an inverted pendulum example shows that this approach is quite effective, and the nonlinear controller is robust to parameter perturbations.

Nonlinear inversion flight control for a supermaneuverable aircraft

Nonlinear dynamic inversion affords the control system designer a straightforward means of deriving control laws for nonlinear systems. The control inputs are used to cancel unwanted terms in the equations of motion using negative feedback of these terms. In this paper, we discuss the use of nonlinear dynamic inversion in the design of a flight control system for a Supermaneuvera ble aircraft. First, the dynamics to be controlled are separated into fast and slow variables. The fast variables are the three angular rates and the slow variables are the angle of attack, sideslip angle, and bank angle. A dynamic inversion control law is designed for the fast variables using the aerodynamic control surfaces and thrust vectoring control as inputs. Next, dynamic inversion is applied to the control of the slow states using commands for the fast states as inputs. The dynamic inversion system was compared with a more conventional, gain-scheduled system and was shown to yield better performance in terms of lateral acceleration, sideslip, and control deflections.