High-gain filters for non-linear stochastic systems

The use of high-gains in the filtering problem for non-linear stochastic systems is studied. It is shown that a high-gain filter has the same structure as a Luenberger observer. The use of high-gains is studied in the cancellation of non-linearities in order to simplify the filter design for non-linear systems. Singular perturbation techniques are used to show how the error dynamics reaches stable equilibrium in a very fast transient time, ensuring that the slow dynamics of the filter are just those of the non-linear stochastic system.     

Local separation principle for non-linear systems

In this paper, we consider the problem of locally stabilizing non-linear systems by output feedback control. To be precise, the separation principle for non-linear systems are studied, which allows the design of the control law and the observer to be carried out independently. While the previous results assumed the existence of Lyapunov function for the observation error, we assume the existence of a state observer which asymptotically estimates the true state. As a matter of fact, it turns out that the latter condition is necessary for the former condition, but the converse is not true. The separation principle with the reduced order observer is also presented. Under the assumption that a non-linear system has asymptotically stable zero dynamics and is locally detectable, it is shown that the asymptotic output tracking by output feedback control can be achieved.     

Output-feedback adaptive stabilization control design for non-holonomic systems with strong non-linear drifts

This paper investigates the problem of output-feedback adaptive stabilization control design for non-holonomic chained systems with strong non-linear drifts, including modelled non-linear dynamics, unmodelled dynamics, and those modelled but with unknown parameters. An observer and an estimator are introduced for state and parameter estimates, respectively. By using the integrator backstepping approach and based on the observer and parameter estimator, a constructive design procedure for output-feedback adaptive stabilization control is given. It is shown that, under some conditions, the control design ensures the closed-loop system is globally asymptotically stable when there is no non-linear drift in the first subsystem, and semiglobally asymptotically stable, otherwise. An example is given to show the effectiveness of the theory.     

Robust adaptive control of uncertain non-linear systems using neural networks

This paper presents a robust adaptive output feedback control design method for uncertain non-affine non-linear systems, which does not rely on state estimation. The approach is applicable to systems with unknown but bounded dimensions and with known relative degree. A neural network is employed to approximate the unknown modelling error. In fact, a neural network is considered to approximate and adaptively make ineffective unknown plant non-linearities. An adaptive law for the weights in the hidden layer and the output layer of the neural network are also established so that the entire closed-loop system is stable in the sense of Lyapunov. Moreover, the robustness of the system against the approximation error of neural network is achieved with the aid of an additional adaptive robustifying control term. In addition, the tracking error is guaranteed to be uniformly and asymptotically stable, rather than uniformly ultimately bounded, by using this additional control term. The proposed control algorithm is relatively straightforward and no restrictive conditions on the design parameters for achieving the systems stability are required. The effectiveness of the proposed scheme is shown through simulations of a non-affine non-linear system with unmodelled dynamics, and is compared with a second-sliding mode controller.     

Synthesis of upper-triangular non-linear systems with marginally unstable free dynamics using state-dependent saturation

In this paper we present sufficient conditions under which a fairly large class of single-input non-linear systems including feedforward systems and the well-known ball-and-beam model, are globally asymptotically and locally exponentially stabilizable by smooth state feedback. A nested saturation controller with state-dependent saturation levels is constructed explicitly, using a novel design approach which combines the nested saturation strategy for marginally unstable linear systems subject to input saturation, with the small feedback design technique, developed for global asymptotic stabilization of general non-affine systems with marginally stable free dynamics. The power of the state-dependent saturation design method is demonstrated by solving a number of non-linear control problems, particularly, the global stabilization problem of a class of two-dimensional non-linear systems and the ball-and-beam system.     

Robust output feedback stabilization of a class of time-varying non-linear systems

This paper deals with the problem of robust output feedback stabilization of a class of time-varying non-linear systems. This class of systems involves two kinds of time-varying uncertainties: those norm-bounded and those bounded by a smooth non-linear function of the output. Under the assumption that the zero dynamics of the system are uniformly asymptotically stable and some additional mild conditions, we show via a Lyapunov function approach that the uncertain system can be robustly stabilized by a time-varying non-linear output feedback controller. The order of this controller turns out to be one less than the relative degree of the uncertain system. A systematic design procedure is given for constructing the controller. Illustrative examples are given. Note that the results generalize several previous results on robust output feedback stabilization.     

Generalized normal form and stabilization of non-linear systems

This paper considers the normal form of non-linear control systems. First we propose a generalized relative degree (relative degree vector) for non-linear single (respectively, multiple) input control system, which is called the point relative degree (respectively, point relative degree vector). For the systems without output, the concepts of essential relative degree (respectively, essential relative degree vector) and the essential point relative degree (respectively, essential point relative degree vector) are defined. Unlike the classical definition which requires regularity, the point relative degree (vector) is always well defined. Using these new concepts the generalized normal form is obtained. Its relationship with the Jacobian linearization is investigated. Using it, a straightforward computation algorithm is provided to achieve the generalized normal form. Based on the generalized normal form we prove that with an additional condition, if the zero-dynamics is stable the overall system is stabilizable by using pseudo-linear state feedback control. For the systems under generalized normal form with unstable zero dynamics, the centre manifold approach is applied. It is shown that the stabilization technique via a designed centre manifold is still applicable to this kind of general non-linear control system.     

Reduction and non-linear controllability of symmetric distributed systems

This paper considers the ‘reduction’ problem for distributed control systems. In particular, we consider controllability of systems containing multiple instances of diffeomorphic components where the overall system dynamics is invariant with respect to a discrete group action. A subclass of such systems are systems with a set of identical components where the overall system dynamics are invariant with respect to physically interchanging these components. The main result is a proposition which shows that for an equivalence class of symmetric systems of this type, controllability of the entire class of systems can be determined by analysing the member of the equivalence class with the smallest state space. The reduction methods developed are illustrated by considering the controllability of a team of mobile robots and a platoon of underwater vehicles.     

On designing observers for time-delay systems with non-linear disturbances

In this paper, the observer design problem is studied for a class of time-delay non-linear systems. The system under consideration is subject to delayed state and non-linear disturbances. The time-delay is allowed to be time-varying, and the non-linearities are assumed to satisfy global Lipschitz conditions. The problem addressed is the design of state observers such that, for the admissible time-delay as well as non-linear disturbances, the dynamics of the observation error is globally exponentially stable. An effective algebraic matrix inequality approach is developed to solve the non-linear observer design problem. Specifically, some conditions for the existence of the desired observers are derived, and an explicit expression of desired observers is given in terms of some free parameters. A simulation example is included to illustrate the practical applicability of the proposed theory.     

Robust near-optimal output feedback control of non-linear systems

This work proposes a robust near-optimal non-linear output feedback controller design for a broad class of non-linear systems with time-varying bounded uncertain variables. Both vanishing and non-vanishing uncertainties are considered. Under the assumptions of input-to-state stable (ISS) inverse dynamics and vanishing uncertainty, a robust dynamic output feedback controller is constructed through combination of a high-gain observer with a robust optimal state feedback controller synthesized via Lyapunov's direct method and the inverse optimal approach. The controller enforces exponential stability and robust asymptotic output tracking with arbitrary degree of attenuation of the effect of the uncertain variables on the output of the closed-loop system, for initial conditions and uncertainty in arbitrarily large compact sets, provided that the observer gain is sufficiently large. Utilizing the inverse optimal control approach and singular perturbation techniques, the controller is shown to be near-optimal in the sense that its performance can be made arbitrarily close to the optimal performance of the robust optimal state feedback controller on the infinite time-interval by selecting the observer gain to be sufficiently large. For systems with non-vanishing uncertainties, the same controller is shown to ensure boundedness of the states, uncertainty attenuation and near-optimality on a finite time-interval. The developed controller is successfully applied to a chemical reactor example.     

Adaptive non-linear compensation control based on neural networks for non-linear systems with time delay

This article investigates a new adaptive non-linear compensation controller for a class of time-delay non-linear systems with partly known dynamics. First, a non-linear neural-network(NN)-based identification model that includes a prior knowledge about the plant dynamics is discussed by using the approximation capabilities of NNs. Then, the adaptive non-linear compensation controller is developed to produce the desired tracking performance. The proposed controller based on the NN can reduce the effect of modelling uncertainties and provide the time-delay compensation, while stability of the closed-loop system is guaranteed. The effectiveness of the proposed scheme is demonstrated through the application to the control of a continuous stirred tank reactor.     

Indirect adaptive control of non-linear systems using multiple identification models and switching

Transient performance improvement of adaptive control of a class of single-input single-output (SISO) non-linear systems is considered in this study. The system under study is assumed to be minimum-phase and input-output linearisable. The non-linear dynamics is also assumed to be linearly parameterised in terms of the unknown parameters. The improvement in the transient performance under large parametric uncertainties is obtained with the use of multiple identification models and switching, and the closed loop system is shown to be stable with the switching mechanism. A new methodology is proposed for the quantitative evaluation of the transient performance. The study is verified by simulation of a non-linear physical system.     

Contraction analysis of non-linear distributed systems

Contraction theory is a comparatively recent dynamic analysis and non-linear control system design tool based on an exact differential analysis of convergence. This paper extends contraction theory to local and global stability analysis of important classes of non-linear distributed dynamics, such as convection-diffusion-reaction processes, Lagrangian and Hamilton–Jacobi dynamics, and optimal controllers and observers. By contrast with stability proofs based on energy dissipation, stability and convergence can be determined for energy-based systems excited by time-varying inputs.

The Hamilton–Jacobi–Bellman controller and a similar optimal non-linear observer design are studied based on explicitly computable conditions on the convexity of the cost function. These stability conditions extend the well-known conditions on controllability and observability Grammians for linear time-varying systems, without requiring the unknown transition matrix of the underlying differential dynamics.     

Design of observers for non-linear time-varying systems

Novel stable observers are proposed for discrete, time-varying systems having time-varying, sector-bounded non-linearities. First, the form of the non-linearity is assumed to be known for all time and a stable non-linear filter is presented. Then assuming that the form of the non-linearity is not known but only a sector bound is given, we propose a linear filter and an upper bound on the sector constant to obtain a stable observer. The convergence conditions for the proposed observers are given directly in terms of the design parameters and are, therefore, very easy to check.     

Set-point changing for non-linear systems

In many control systems, we sometimes face a situation such that we have to change the set-point of the system in the course of operation. Theoretically ‘set-point changing’ means letting the system be at another equilibrium point. If the system is linear, this operation presents no problem because the properties of linear systems are always global. However, in non-linear systems, this type of global operation has many unsolved problems. We only know experimentally that, if the system is locally stable at every equilibrium point, then we can slowly change the set-point without exciting unstable motion. This paper proposes a theoretically guaranteed method of changing the set-point of non-linear systems and showing a sufficient condition needed for this operation.     

Output tracking of a non-linear non-minimum phase PVTOL aircraft based on non-linear state feedback control

In this paper, a special class of square multi-input multi-output (MIMO) non-linear non-minimum phase systems is considered whereby the internal dynamics does not depend explicitly on the inputs. For such systems, a new output tracking control aproach is proposed and applied to a planar vertical takeoff and landing (PVTOL) aircraft. This control approach first generates input-output linearization of the original non-linear system. Then the internal dynamics is rewritten by separating its linear part from its non-linear part. Finally, a non-linear auxiliary input is introduced to stabilize the overall closed-loop system by a Lyapunov-based technique and a minimum-norm strategy. The effectiveness and excellent performance of the resulting non-linear state feedback controller are demonstrated by using the simulation results.     

Stability of non-linear transportation systems

Lyapunov' direct method is applied to the stability analysis of non-linear distributed parameter transportation systems with lumped parameter dynamic feedback control. Frequency criteria for global asymptotic closed-loop stability are obtained, as well as a method for constructing regions of attraction of a non-globally stable set point.     

Trajectory control of a non-linear one-link flexible arm

The trajectory-tracking control problem is considered for a one-link flexible arm described by a non-linear model. Two meaningful system outputs are chosen; namely, the joint angle and the angular position of a suitable point along the link. The common goal is to stiffen the behaviour of the flexible link with respect to the chosen output. Based on the input-output inversion algorithm, a state-feedback control law is designed that enables exact tracking of any smooth trajectory specified for the output. In the closed loop an unobservable dynamics naturally arises, related to the variables describing the arm's distributed flexibility. Joint-based design is shown to be always stable, whereas in the link-point design the closed-loop dynamics may become unstable depending on the location of the output along the link. Open- versus closed-loop strategies are developed and compared. Extensive simulation results are included.     

Error-feedback servomechanism in non-linear systems

We consider the controller design of the error-feedback servomechanism in exponentially stable non-linear systems. Using the concepts of integral manifold and singular perturbation, a non-linear controller is constructed that ensures a desired steady state tracking rate for unknown constant set-points. The servo rate in the error-feedback servomechanism is shown to be governed by not only the plant characteristics but also the initial conditions of the plant and controller. A self-tuning scheme is proposed to speed up the servo process.