State feedback linearization of nonlinear control systems on homogeneous time scales

The paper addresses the state feedback linearization problem for nonlinear systems, defined on homogeneous time scale. Necessary and sufficient solvability conditions are given within the algebraic framework of differential one-forms. The conditions concerning the exact dynamic state feedback linearization are equivalent to the property of differential flatness of the system. An output function which defines a right invertible system without zero-dynamics is shown to exist if and only if the basis of some space of one-forms can be transformed, via polynomial matrix operator over the field of meromorphic functions, into a system of exact one-forms. The results extend the corresponding results for the continuous-time case.     

Realistic feedback control of turbogenerators

In this paper, the optimal control of a turboalternator connected to an infinite bus is considered. The alternator is controlled through a linear feedback of the state variables. The feedback parameters are obtained by solving a two-point nonlinear boundary-value problem. The values obtained for these parameters depend on the strength and duration of the disturbance, since the model is nonlinear, contrary to the usual feedback control of a linear model. In contrast to the model used in Ref. 1, the model used here include the transfer functions of the governor, the turbine, and the voltage regulator.This work was supported in part by the National Research Council of Canada, Grant No. A-4146.     

Optimal feedback control policies for stochastic,distributed-parameter systems

This paper considers optimal feedback control policies for a class of discrete stochastic distributed-parameter systems. The class under consideration has the property that the random variable in the dynamic systems depends only on the time and possesses the Markovian property with stationary transition probabilities. A necessary condition for optimality of a feedback control policy, which has form similar to the Hamiltonian form in the deterministic case, is derived via a dynamic programming approach.     

Optimal nonlinear feedback control of quasi-Hamiltonian systems

An innovative strategy for optimal nonlinear feedback control of linear or nonlinear stochastic dynamic systems is proposed based on the stochastic averaging method for quasi-Hamiltonian systems and stochastic dynamic programming principle. Feedback control forces of a system are divided into conservative parts and dissipative parts. The conservative parts are so selected that the energy distribution in the controlled system is as requested as possible. Then the response of the system with known conservative control forces is reduced to a controlled diffusion process by using the stochastic averaging method. The dissipative parts of control forces are obtained from solving the stochastic dynamic programming equation. Project supported by the National Natural Science Foundation of China (Grant No. 19672054) and Cao Guangbiao High Science and Technology Development Foundation of Zhejiang University.     

Interface feedback control stabilization of a nonlinear fluid-structure interaction

We consider a model of fluid-structure interaction in a bounded domain Ω∈Rⁿ, n=2, where Ω is comprised of two open adjacent sub-domains occupied, respectively, by the solid and the fluid. This leads to a study of the Navier-Stokes equation coupled on the boundary with the dynamic system of elasticity. We shall consider models where the elastic body exhibits small but rapid oscillations. These are established models arising in engineering applications when the structure is immersed in a viscous flow of liquid. Questions related to the stability of finite energy solutions are of paramount interest.It was shown in Lasiecka and Lu (2011) [14] that all data of finite energy produce solutions whose energy converges strongly to zero. The cited result holds under “partial flatness” geometric condition whose role is to control the effects of the pressure in the NS equation. Related conditions has been used in Avalos and Triggiani (2008) [23] for the analysis of the linear model. The goal of the present work is to study uniform stability of all finite energy solutions corresponding to nonlinear interaction. This particular question, of interest in its own rights, is also a necessary preliminary step for the analysis of optimal control strategies arising in infinite-horizon control problems associated with the structure. It is shown in this paper that a stress type feedback control applied on the interface of the structure produces solutions whose energy is exponentially stable.     

Multiple positive periodic solutions of nonlinear functional differential system with feedback control

In this paper, we employ Avery-Henderson fixed point theorem to study the existence of positive periodic solutions to the following nonlinear nonautonomous functional differential system with feedback control:

     

Output feedback control for MIMO nonlinear systems using factorization

A new output feedback adaptive control scheme for multi-input and multi-output (MIMO) nonlinear systems is presented based on the high frequency gain matrix factorization and the backstepping approach with vector form. The only required prior knowledge about the high frequency gain matrix of the linear part of the system is the signs of its leading principal minors. The proposed controller is a dynamic one that only needs the measurement of the system output, and the observer and the filters are introduced in order to construct a virtual estimate of the unmeasured system states. The global stability of the closed-loop systems is guaranteed through this control scheme, and the tracking error converges to zero. Finally, the numerical simulation results illustrate the effectiveness of the proposed scheme.     

Lyapunov type functions for classes of autonomous parabolic feedback control problems and applications

In this note we provide sufficient conditions for the existence of a Lyapunov function for a class of parabolic feedback control problems. The results are applied to the long-time behavior of state functions for the following problems: (i) a model of combustion in porous media; (ii) a model of conduction of electrical impulses in nerve axons; and (iii) a climate energy balance model.     

Robust static output feedback variable structure control for nonlinear systems

In this paper, a robust stabilization problem for nonlinearsystems is investigated using static output feedback. Matchedand mismatched uncertainties are considered and their boundingfunctions take general forms. Based on a Lyapunov analysis approachand sliding mode techniques, some conclusions are presented.Then, a robust nonlinear variable structure control scheme issynthesized to stabilize the system globally, and it is notnecessary for the nominal system to be linearizable globally.The conservatism is reduced by using uncertainty bounds in thecontrol design. Finally, a mass–spring system is employedto illustrate the effectiveness of the results.     

Power-series approximations for the optimal feedback control of noisy nonlinear systems

Power-series methods are developed for designing approximately optimal state regulators for a nonlinear system subject to white Gaussian random disturbances. The performance index of the control is an ensemble average of a quadratic form. A perfect observation of the system state is assumed. When the system nonlinearity is small and it is characterized by a polynomial function of the state, a definite method is presented to construct a suboptimal feedback control of a power-series form in a small nonlinearity parameter. If the variance of noise is small, an alternative method is also applicable which yields a suboptimal control in a power series with respect to a variance parameter. A simple one-dimensional problem is examined to make comparison between controls of the two different forms.     

Viable set computation for hybrid systems

In this paper, we revisit the problem of computing viability sets for hybrid systems with nonlinear continuous dynamics and competing inputs. As usual in the literature, an iterative algorithm, based on the alternating application of a continuous and a discrete operator, is employed. Different cases, depending on whether the continuous evolution and the number of discrete transitions are finite or infinite, are considered. A complete characterization of the reach-avoid computation (involved in the continuous time calculation) is provided based on dynamic programming. Moreover, for a certain class of automata, we show convergence of the iterative process by using a constructive version of Tarski’s fixed point theorem, to determine the maximal fixed point of a monotone operator on a complete lattice of closed sets. The viability algorithm is applied to a benchmark example and to the problem of voltage stability for a single machine-load system in case of a line fault.     

Geometry of cascade feedback linearizable control systems

Cascade feedback linearization provides geometric insights on explicit integrability of nonlinear control systems with symmetry. A central piece of the theory requires that the partial contact curve reduction of the contact sub-connection be static feedback linearizable. This work establishes new necessary conditions on the equations of Lie type - in the abelian case - that arise in a contact sub-connection with the desired static feedback linearizability property via families of codimension one partial contact curves. Furthermore, an explicit class of contact sub-connections admitting static feedback linearizable contact curve reductions is presented, hinting at a possible classification of all such contact sub-connections. Key tools in proving, and stating, the main results of this paper are truncated versions of the total derivative and Euler operators. Additionally, the Battilotti-Califano system with three inputs is used as a clarifying example of both cascade feedback linearization and the new necessary conditions.     

Asymmetric games for convolution systems with applications to feedback control of constrained parabolic equations

The paper is devoted to the study of some classes of feedback control problems for linear parabolic equations subject to hard/pointwise constraints on both Dirichlet boundary controls and state dynamic/output functions in the presence of uncertain perturbations within given regions. The underlying problem under consideration, originally motivated by automatic control of the groundwater regime in irrigation networks, is formalized as a minimax problem of optimal control, where the control strategy is sought as a feedback law. Problems of this type are among the most important in control theory and applications — while most challenging and difficult. Based on the Maximum Principle for parabolic equations and on the time convolution structure, we reformulate the problems under consideration as certain asymmetric games, which become the main object of our study in this paper. We establish some simple conditions for the existence of winning and losing strategies for the game players, which then allow us to clarify controllability issues in the feedback control problem for such constrained parabolic systems.     

Relaxed fuzzy observer‐based output feedback control synthesis of discrete‐time nonlinear control systems

This work develops the development of observer‐based output feedback control design of discrete‐time nonlinear systems in the form of Takagi–Sugeno fuzzy model. Lately, previous results have been improved in virtue of a two‐step method. From a technical point of view, it is not flawless and related problems have not been completely resolved. In this study, more advanced two‐steps approach is further developed while the relative sizes among different normalized fuzzy weighting functions are utilized by introducing some additional matrix variables. As a result of the above work, those main defects of the existing method can be redressed and a desired solution in aspect of not only reducing the conservatism but also alleviating the computation complexity is provided for some special cases. Moreover, the effectiveness of the proposed result is shown at length by means of an illustrative example. © 2016 Wiley Periodicals, Inc. Complexity 21: 593–601, 2016     

Chaos synchronization between two different chaotic systems via nonlinear feedback control

This work presents chaos synchronization between two different chaotic systems via nonlinear feedback control. On the basis of a converse Lyapunov theorem and balanced gain scheme, control gains of controller are derived to achieve chaos synchronization for the unified chaotic systems. Numerical simulations are shown to verify the results.     

Output feedback regulation control for a class of cascade nonlinear systems and its application to fan speed control

The output feedback regulation problem is considered for a class of nonlinear systems with integral input-to-state stable (iISS) inverse dynamics and unknown control direction. The system output together with the complete unmeasured state components appears in the system uncertainties. A systematic output feedback control scheme is presented with the help of a dynamic observer, whose gain comes from an off-line time-varying Riccati matrix differential equation. The proposed scheme can be applied to the analysis of the speed tracking control of a fan. The simulation results demonstrate the validity of the presented algorithm.