Constructive dynamic output feedback stabilization of a class of nonlinear systems

We address the problem of giving a new canonical form for a class of (time varying) nonlinear noncontrollable systems. We define and solve the partial linearization problem (PLP) by static state feedback. Our notion of linearization is weaker than the one found in the literature, since we want to transform the given system to a controllable linear system that is affected by a (perhaps nonlinear) autonomous system, which plays the role of its noncontrollable part. For the class of nonlinear systems for which the PLP is solvable, we show that the influence of the noncontrollable part in the linear part can be always removed by a convenient static state feedback. In particular, a new canonical form for this class of noncontrollable nonlinear systems is obtained.

Dynamic Output Feedback Stabilization of Non-uniformly Observable Dissipative Systems

This technical note deals with asymptotic stabilization of general nonlinear discrete-time systems by dynamic output feedback control. First, a sufficient condition for the dynamic output feedback stabilization with the closed-loop domain of attraction (DOA) estimation is given. For a given Lyapunov function of the plant state and the controller state, the negative-definite domain (NDD) in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space is proposed, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z_+,u,z$</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">$y$</tex-math></inline-formula> denote the next controller state, the current control input, the current controller state and the current plant output, respectively. All points <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+;u;z;y)$</tex-math></inline-formula> in the NDD can make the time difference of the Lyapunov function negative-definite for the controller state <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> and all plant states in a set of plant states, which is determined by the controller state <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> and the plant output <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y$</tex-math></inline-formula>. The NDD in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space could be viewed as an unstructured dynamic output feedback stabilization controller set. And any invariant subset of the projection of the NDD in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space onto the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x)$</tex-math></inline-formula>-space is an estimate of the DOA for the closed-loop, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> denotes the current plant state. Second, based on the NDD, the controller design problem is converted to a set approximation problem in Euclidean space. Using numerical set approximation methods, dealing with strong nonlinearities is no longer the bottleneck. An estimation method for the unstructured dynamic output feedback stabilization controller set is proposed based on interval analysis, which is a kind of guaranteed numerical method for approximating sets.

The paper presents a novel robust control of one class of nonlinear single-input single-output (SISO) systems with distributed delay on state used the auxiliary loop method. The problem is solved by introducing an auxiliary loop and using two Khalil observers to compensate for a generalized disturbance signal containing information not only about the parametric uncertainty of the installation model, but also information about distributed delay and disturbances. The designed control law is investigated in MATLAB. To illustrate the obtained results, an example of a nonlinear system with a distributed time delay is given. It is shown that the proposed control law under conditions of parametric uncertainty ensures the achievement of the goal with a given accuracy.

The adaptive output-tracking problem of a class of nonlinear systems with uncontrollable unstable linearization was proposed. Using the modified adaptive adding a power integrator technique as a basic tool, a new smooth adaptive state feedback controller was designed. This controller can guarantee the global boundedness of all variables of the closed-loop systems and the output tracking error arbitrary small.

This work proposes a decentralized super twisting control scheme for group formation and trajectory tracking of multi-agent system networks with a fixed topology. Each agent dynamics is represented by a single-input single-output nonlinear system in its normal form and the information required by each controller is the output of the others agents. It is proved that output consensus and trajectory tracking can be achieved with the proposed super twisting control. Moreover, the state space representation of the agents dynamics is not restricted to be of the same dimension, as usually assumed in the literature. The stability of the controlled system is proved using a Lyapunov function and a simulation example is provided in order to illustrate the application of the proposed control.

This paper investigates the dynamic output feedback control for a class of nonlinear time-delay system with the triangular structure-type uncertain nonlinearity satisfying the linear growth condition. A dynamic observer-based output feedback robust stabilizing controller is presented on the basis of Lyapunov-Razumikhin function design strategy. A numerical example is given to illustrate the effectiveness of this method.

In this paper we address the problem of nonin teracting control with stability for the class of nonlinear square systems for which noninteraction can be achieved (without stability) by means of invertible static state-feedback. The use of both static state-feedback and dynamic state-feedback is investigated. We prove that in both cases the asymptotic stabilizability of certain subsystems is necessary to achieve noninteraction and stability. We use this and some recent results to state a complete set of necessary and sufficient conditions in order to solve the problem.

In this paper, the problem of designing a tracking controller for uncertain nonlinear second-order SISO (single input single output) systems which can suppress the effects of both the model uncertainty and noisy measurements is considered. Based on the framework of linear systems with nonlinear uncertainties whose norm-bounds are known we construct not only a sliding mode controller but a sliding observer for uncertain nonlinear second-order SISO systems, and we show the robust stability of the closed-loop control system. >

The problem of output feedback stabilization of a class of under-actuated benchmark nonlinear systems is discussed. The proposed method utilizes an extended high gain observer (EHGO)-based sliding mode control (SMC) technique to control a class of nonlinear systems which may have unstable zero dynamics. Starting with Lagrangian model of the system and using a suitable coordinate transformation, a generalized normal form representation is derived which decouples the system into an internal and external dynamics. The internal dynamics is utilized to derive an auxiliary system and the full-order EHGO thus obtained is used for estimation of derivative(s) of the system output that are further used in design of an output feedback control law. It is shown that the proposed output feedback controller stabilizes the system and convergence of estimated states is demonstrated with suitable selection of observer parameters. The proposed control scheme is applied to a benchmark nonlinear system, namely inertia wheel pendulum (IWP), in order to demonstrate the efficacy of the technique by simulation.

This paper deals with output feedback stabilisation of a class of nonlinear systems with nonlinearity of the unmeasured state variables. The nonlinearity of the unmeasured state complicates the zero dynamics of the nonlinear system and makes the control design more difficult. In this paper, the zero dynamics analysed for the class of nonlinear systems with respect to the original output and then with respect to a virtual output. It is demonstrated that nonlinear systems with unstable zero dynamics may have stable zero dynamics with the virtual output. Based on the analysis, conditions are then identified for nonlinear systems which can be stabilised by control design with estimated state, including the estimated virtual output, and this removes a restriction of a recent result that requires the transformed system has no zero dynamics. Examples are included to demonstrate that the proposed control design can be used to stabilise nonlinear systems with unstable nonlinear zero dynamics.

Using the Lyapunov function method, this study investigates both state and output feedback stabilisation control design problems for fractional order non‐linear systems in the lower triangular form, and presents a number of new results. First, some new properties for Caputo fractional derivative are presented. Second, by introducing appropriate transformations of coordinates, the feedback stabiliser design problem is converted into the determination of finding some parameters, which can be obtained by solving the Lyapunov equation and relevant matrix inequalities. Finally, based on the Lyapunov function method, both state and output feedback stabilisers are explicitly designed to make the closed‐loop system asymptotically stable. The study of an illustrative example shows that the obtained results are effective in designing feedback stabilisers for fractional order non‐linear systems in the lower triangular form.

This paper studies the characteristic modeling problem for a class of single input single output (SISO) nonlinear systems with a relative degree of two. Firstly, to deal with the complexities of high-order nonlinear systems, we provide a class of low order and slowly time-varying linear systems to represent the original systems. Secondly, based on the characteristic model, we propose a linear controller by using coefficients obtained from system online identification. Through the above two steps, we solve the stabilization problem of closed-loop system composed of the characteristic model and the linear controller. Furthermore, the stability analysis of the closed-loop system composed of an exact discrete-time model and a linear controller is given. Finally, two simulation examples are given to show the effectiveness of the proposed design.

On observer design for a class of Persidskii systems based on steady-state estimation

This paper presents a method of continuous-time simple adaptive control (SAC) using multiple neural networks for a single-input single-output (SISO) nonlinear systems with unknown parameters and dynamics, bounded-input bounded- output, and bounded nonlinearities. The control input is given by the sum of the output of the simple adaptive controller and the sum of the outputs of the parallel small-scale neural networks. The parallel small-scale neural networks are used to compensate the nonlinearity of plant dynamics that is not taken into consideration in the usual SAC. The role of the parallel small- scale neural networks is to construct a linearized model by minimizing the output error caused by nonlinearities in the control systems. Finally, the stability analysis of the proposed method is carried out, and the effectiveness of this method is confirmed through computer simulations.