Nonlinear Feedback Control Design Research Articles

Amplitude Saturated Nonlinear Output Feedback Controller Design for Underactuated 5-DOF Tower Cranes Without Velocity Measurement

Amplitude Saturated Nonlinear Output Feedback Controller Design for Underactuated 5-DOF Tower Cranes Without Velocity Measurement

Improved delay-dependent stability of superlinear hybrid stochastic systems with general time-varying delays

In the recent paper (Fei et al., 2019), the study of delay-dependent stability of hybrid stochastic differential delay equations (SDDEs) was generalized to superlinear ones (namely, do not satisfy the usual linear growth condition). However, the theory developed there could not be applied to hybrid SDDEs with non-differentiable time delays, or whose drift coefficients miss the key decomposition in (Fei et al., 2019) (see Assumption 1 below). This paper therefore is to deal with these two challenging problems so that the delay-dependent stability criteria derived in (Fei et al., 2019) could be improved. The decomposition scheme is modified in order to include more general hybrid SDDEs. The differentiability assumption on time-varying delays is replaced by a relatively weaker one. Also the Lyapunov functional used in this paper is modulated to adapt to these new changes. Finally, two interesting examples, an application to mosquito model, and design of nonlinear delay feedback control, respectively, are given to demonstrate the effectiveness of our new theory.

Robust integral controller design based on composite nonlinear feedback in uncertain systems with constrained inputs and external disturbances

AbstractIn this article, to enhance the transient performances in constrained uncertain systems, a control technique based on linear matrix inequality (LMI) is characterized to the robust integral composite nonlinear feedback (CNF) controller design. To achieve this aim, considering the inputs limitations and the uncertainties, the regulation is transformed into a minimization subjected to several LMIs. Solving an optimization in real‐time, the gains of the controller are calculated. So, the regulation and the disturbance rejection are effectively handled employing the presented control law. Compared with the existing integral methodology, the provided simulations validate the applicability of the robust integral CNF control law in uncertain systems.

Feedback control for fluid mixing via advection

This work is concerned with nonlinear feedback control design for the problem of fluid mixing via advection. The overall dynamics is governed by the transport and Stokes equations in an open bounded and connected domain Ω⊂Rd, with d=2 or d=3. The feedback laws are constructed based on the ideas of instantaneous control as well as a direct approximation of the optimality system derived from an optimal open-loop control problem. It can be shown that under appropriate numerical discretization schemes, two approaches generate the same sub-optimal feedback law. On the other hand, different discretization schemes may result in feedback laws of different regularity, which determine different mixing results. The Sobolev norm of the dual space (H1(Ω))′ of H1(Ω) is used as the mix-norm to quantify mixing based on the known property of weak convergence. The major challenge is encountered in the analysis of the asymptotic behavior of the closed-loop systems due to the absence of diffusion in the transport equation together with its nonlinear coupling with the flow equations. To address these issues, we first establish the decay properties of the velocity, which in turn help obtain the estimates on scalar mixing and its long-time behavior. Finally, mixed continuous Galerkin (CG) and discontinuous Galerkin (DG) methods are employed to discretize the closed-loop system. Numerical experiments are conducted to demonstrate our ideas and compare the effectiveness of different feedback laws.

Robust tracking composite nonlinear feedback controller design for time-delay uncertain systems in the presence of input saturation

In this study, a robust control technique is investigated for the reference tracking of uncertain time-delayed systems in the existence of the actuator saturation. Due to emerging of some control complexities, as well as the input limitations, time-varying delay, uncertainty, and external disturbance, such a tracking goal would be realized through suitable design of the composite nonlinear feedback (CNF) controller. Thus, considering the mentioned limitations, a Lyapunov-based procedure is used to determine the control law. Then, the parameters of the CNF input are derived by using the solution of a linear matrix inequality (LMI) problem. The planned tracking idea is numerically implemented in two uncertain control systems. Some performance characteristics (i.e., the tracking error, boundedness, and transient responses) are compared with similar ones. Accordingly, the simulations illustrate the efficiency of the suggested control procedure over the existing CNF approaches.

Toward Embedded System Resources Relaxation Based on the Event-Triggered Feedback Control Approach

The paper describes an event-triggered nonlinear feedback controller design. Event triggering is a real-time controller implementation technique which reduces embedded system utilization and relaxes task scheduling of the real-time system. In contrast to classic time implementation techniques, the event-triggered execution is validated regarding the introduced triggering policy. The triggering rule is a boundary, where the last task value is preserved until the rule is violated. In the given paper, two different event-triggered strategies are designed for the class of dynamic systems with integral behavior. Both methods are based on sliding mode controller design, where the triggering rule of the first design involves only a partial state vector, which is a direct consequence of the triggering rule derivation throughout the Lyapunov stability analysis. In the second approach, the sliding mode controller is designed upon prior stabilized systems with the additional term, which enables derivation of the triggering rule based on the whole state vector. The second approach offers better closed-loop performance and higher relaxation of the system utilization. The selection of triggering boundary is related closely to the derived minimal inter-event time, which impacts the computational burden of the real-time system and closed-loop performance directly. The derived controllers are compared with the classic sample and hold implementation techniques. The real-time results are presented, and system performances are confirmed regarding embedded system task relaxation, lowering the computational intensity and preserving closed-loop dynamics.

An approach to exact nonlinear feedback control design employing recursive approximations and the null space

A strategy to design exact nonlinear feedback controllers based on a recursive application of approximate linearization methods is examined. The computations are algebraic and computationally simpler than solving the set of coupled nonlinear partial differential equations thereby facilitating practical symbolic computer computations enabling discernment of evolving patterns in the approximate solutions as the order of approximation increases. Utilizing the null space that appears at each step in the computations as part of the computations, a family of analytic solutions can be generated asymptotically. There are possibilities for optimizing the performance by judiciously choice of analytic solution that emerge from the selective use of the null space.

Nonlinear state feedback control design for port-Hamiltonian systems with unstructured component

Abstract This paper considers the problem of state feedback controller design to stabilize generalized Hamiltonian systems with an unstructured component. This class of models enable one to exploit the structure of port-Hamiltonian systems for feedback control design while relaxing the constraint of deriving an exact structured port-Hamiltonian representation. For a given stabilizable nonlinear system, and with some assumptions on the unstructured part of the dynamics, a stabilizing control law is designed and asymptotic stability of a desired equilibrium of the system is demonstrated. A numerical illustration of the proposed approach is presented to demonstrate the design method.

Tensor product model transformation‐based control for fractional‐order biological pest control systems

AbstractAs environmental pollution and safety of human and creatures have been an issue of concern, pest management based on nonchemical use is preferable. Biological pest control can satisfy the regulation of pest population without causing environmental problems. Moreover, representation of dynamical systems in the form of fractional‐order dynamic models can explain several phenomena better than the integer‐order dynamic models can. The aim of this study is to determine the biological pest control policy for the fractional‐order pest control systems, which is a complex nonlinear multiple input and multiple output control problem. Here, the system is represented in the form of a fractional‐order Lotka–Volterra model. In order to avoid complexity in the nonlinear feedback controller design process, the tensor product (TP) model transformation was applied to automatically transform the biological pest control system to the TP polytopic model which allows the designer to synthesize the controller under linear framework. Moreover, the design process can be easily carried out by solving an linear matrix inequality (LMI) problem, which is very well applicable to high‐dimensional multiple input and multiple output control problems. Simulation studies were conducted to demonstrate the ease of the TP‐based design process. The simulation results showed that the designed control policy effectively manipulates the pest populations of the fractional‐order Lotka–Volterra model to the desired level which is economically viable.

Optimal tuning of composite nonlinear feedback control in time-delay nonlinear systems

A composite nonlinear feedback tracking control is mainly designed via the linear matrix inequality approach in the time-delay nonlinear systems. For achieving this purpose, the composite nonlinear feedback control design is firstly formulated in the nonlinear systems with time-delay. Then, a sufficient stability condition will be addressed in terms of the linear matrix inequality feasibility checking. Therefore, a stabilizing composite nonlinear feedback control policy can be obtained with solving of the linear matrix inequality. A quadratic cost function is considered as a performance index for optimal tuning of the composite nonlinear feedback control. A minimization problem subject to some linear matrix inequality constraints would be subsequently suggested to the composite nonlinear feedback control tuning. The proposed control method is numerically simulated in two time-delay dynamical systems. The effectiveness of the suggested control law will be shown in comparing with the existing control techniques.

Systematic construction of high dimensional fractional-order hyperchaotic systems

Abstract The problem of designing fractional-order hyperchaotic systems (FOHSs) by a consistent and systematic procedure is still an open one at large. The tedious hit and trial design procedures adopted in literature to generate hyperchaos pose as a hindrance; especially when the dimension of the system is high. The paper thus presents a solution to the above problem and further discusses the application of FOHSs by putting forward three proposals. First is the design of the simplest possible single state nonlinear feedback controller based on fractional-order stability theorems for creating hyperchaos from a nominal system. The design is organised and even higher dimensional FOHSs can be easily constructed. Secondly, a hybrid synchronisation controller is designed so that the FOHSs obtained using the proposed systematic methodology, are both completely synchronised and antisynchronised simultaneously. Thirdly, a public key cryptographic algorithm is presented using the higher unpredictable nature and complexity of FOHSs to encrypt and decrypt private messages. Two representative examples presented in this work distinctly confirm the effectiveness of the proposed techniques.

Single-step full-state feedback control design for nonlinear hyperbolic PDEs

ABSTRACTThe present work proposes an extension of single-step formulation of full-state feedback control design to the class of distributed parameter system described by nonlinear hyperbolic partial differential equations (PDEs). Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear state feedback law, both feedback control and stabilisation design objectives given as target stable dynamics are accomplished in one step. In particular, the mathematical formulation of the problem is realised via a system of first-order quasi-linear singular PDEs. By using Lyapunov's auxiliary theorem for singular PDEs, the necessary and sufficient conditions for solvability are utilised. The solution to the singular PDEs is locally analytic, which enables development of a PDE series solution. Finally, the theory is successfully applied to an exothermic plug-flow reactor system and a damped second-order hyperbolic PDE system demonstrating ability of in-domain nonlinear control law to achieve stabilisation.

Composite nonlinear feedback control for strict-feedback nonlinear systems with input saturation

ABSTRACTThis paper focuses on composite nonlinear feedback (CNF) controller design for tracking control problem of strict-feedback nonlinear systems with input saturation to address the improvement of transient performance. First, without considering the input saturation, a stabilisation control law is designed by using standard backstepping technique for the nonlinear system, then a feedforward control law is added to the backstepping-based stabilisation control law to construct a tracking control law. The tracking control law is tuned to drive the output of the closed-loop system to track a command input with quick response. Then, an additional nonlinear feedback law is constructed and combined with the tracking control law to obtain a CNF control law. The role of this additional nonlinear feedback law is to smoothly change the damping ratio of the closed-loop system while the system output approaches the command input, and to reduce overshoot caused by the tracking control law. It is shown that the extra-adding nonlinear feedback part does not cause the loss of stability of the closed-loop system in its attractive basin.

Two term composite nonlinear feedback controller design for nonlinear time-delay systems

A two term composite nonlinear feedback (2TCNF) control technique is developed here for nonlinear time delay systems in presence of input saturation. The proposed controller consists of two terms that are conventional composite nonlinear feedback (CNF) control and delay state control. CNF control has advantage of fast transient response and delay state control improves damping characteristics. The proposed controller thus has both these advantages and it tracks the reference smoothly. The closed loop asymptotic stability is guaranteed via Lyapunov–Krasovskii analysis. The efficiency of the proposed controller is tested on exothermic chemical reactor and validated through simulation results. Its performance is compared with the conventional CNF control. The superiority of 2TCNF is established in terms of less overshoot, fast transient response and reduced steady state error.

Stochastic finite-time partial stability, partial-state stabilization, and finite-time optimal feedback control

In many practical applications, stability with respect to part of the system’s states is often necessary with finite-time convergence to the equilibrium state of interest. Finite-time partial stability involves dynamical systems whose part of the trajectory converges to an equilibrium state in finite time. In this paper, we address finite-time partial stability in probability and uniform finite-time partial stability in probability for nonlinear stochastic dynamical systems. Specifically, we provide Lyapunov conditions involving a Lyapunov function that is positive definite and decrescent with respect to part of the system state and satisfies a differential inequality involving fractional powers for guaranteeing finite-time partial stability in probability. In addition, we show that finite-time partial stability in probability leads to uniqueness of solutions in forward time and we establish necessary and sufficient conditions for almost sure continuity of the settling-time operator of the nonlinear stochastic dynamical system. Finally, we develop a unified framework to address the problem of optimal nonlinear analysis and feedback control design for finite-time partial stochastic stability and finite-time, partial-state stochastic stabilization. Finite-time partial stability in probability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function that is positive definite and decrescent with respect to part of the system state and can clearly be seen to be the solution to the steady-state form of the stochastic Hamilton–Jacobi–Bellman equation guaranteeing both finite-time, partial-state stability and optimality. The overall framework provides the foundation for extending stochastic optimal linear–quadratic controller synthesis to nonlinear–nonquadratic optimal finite-time, partial-state stochastic stabilization.

Nonlinear state feedback control design to eliminate subcritical limit cycle oscillations in aeroelastic systems

A strictly nonlinear state feedback control law is designed for an aeroelastic system to eliminate subcritical limit cycle oscillations. Numerical continuation techniques and harmonic balance methods are employed to generate analytical estimates of limit cycle oscillation commencement velocity and its sensitivity with respect to the introduced control parameters. The obtained estimates are used in a multiobjective optimization framework to generate optimal control parameters which maximize the limit cycle oscillation commencement velocity while minimizing the control cost. Numerical simulations are used to show that the assumed nonlinear state feedback law with the optimal control parameters successfully eliminates any existing subcritical limit cycle oscillations by converting it to supercritical limit cycle oscillations, thereby guaranteeing safe operation of the system in its flight envelope.

Sum-of-squares approach to feedback control of laminar wake flows

In this paper a novel nonlinear feedback control design methodology for incompressible fluid flows aiming at the optimisation of long-time averages of flow quantities is presented. It applies to reduced-order finite-dimensional models of fluid flows, expressed as a set of first-order nonlinear ordinary differential equations with the right-hand side being a polynomial function in the state variables and in the controls. The key idea, first discussed in Chernyshenkoet al.(Phil. Trans. R. Soc. Lond. A, vol. 372, 2014, 20130350), is that the difficulties of treating and optimising long-time averages of a cost are relaxed by using the upper/lower bounds of such averages as the objective function. In this setting, control design reduces to finding a feedback controller that optimises the bound, subject to a polynomial inequality constraint involving the cost function, the nonlinear system, the controller itself and a tunable polynomial function. A numerically tractable and efficient approach to the solution of such optimisation problems, based on sum-of-squares techniques and semidefinite programming, is proposed. To showcase the methodology, the mitigation of the fluctuation kinetic energy in the unsteady wake behind a circular cylinder in the laminar regime at$Re=100$, via controlled angular motions of the surface, is numerically investigated. A compact reduced-order model that resolves the long-term behaviour of the fluid flow and the effects of actuation, is first derived using proper orthogonal decomposition and Galerkin projection. In a full-information setting, feedback controllers are then designed to reduce the long-time average of the resolved kinetic energy associated with the limit cycle. These controllers are then implemented in direct numerical simulations of the actuated flow. Control performance, total energy efficiency and the physical control mechanisms identified are analysed in detail. Key elements of the methodology, implications and future work are finally discussed.

Exploiting the dynamic properties of covalent modification cycle for the design of synthetic analog biomolecular circuitry.

BackgroundCycles of covalent modification are ubiquitous motifs in cellular signalling. Although such signalling cycles are implemented via a highly concise set of chemical reactions, they have been shown to be capable of producing multiple distinct input-output mapping behaviours – ultrasensitive, hyperbolic, signal-transducing and threshold-hyperbolic.ResultsIn this paper, we show how the set of chemical reactions underlying covalent modification cycles can be exploited for the design of synthetic analog biomolecular circuitry. We show that biomolecular circuits based on the dynamics of covalent modification cycles allow (a) the computation of nonlinear operators using far fewer chemical reactions than purely abstract designs based on chemical reaction network theory, and (b) the design of nonlinear feedback controllers with strong performance and robustness properties.ConclusionsOur designs provide a more efficient route for translation of complex circuits and systems from chemical reactions to DNA strand displacement-based chemistry, thus facilitating their experimental implementation in future Synthetic Biology applications.

Sparse Approximation–Based Collocation Scheme for Nonlinear Optimal Feedback Control Design

This paper discusses the development of a computationally efficient approach to generate optimal feedback control laws for infinite time problems by solving the corresponding Hamilton–Jacobi–Bellman (HJB) equation. The solution process consists of iteratively solving the linear generalized HJB (GHJB) equation starting with an admissible stable controller. The collocation methods are exploited to solve the GHJB equation in the specified domain of interest. Recently developed nonproduct quadrature method known as Conjugate Unscented Transformation is used to manage the curse of dimensionality associated with the growth of collocation points with increase in the state dimension. Furthermore, recent advances in sparse approximation are leveraged to automatically generate the appropriate polynomial basis function set for the collocation-based approximation from an overcomplete dictionary of basis functions. It is demonstrated that the solution process uses the basis function selection process to automatically identify a form for the feedback control law, which is frequently unknown. Several numerical examples demonstrate the efficacy of the proposed approach in accurately generating feedback control policies for nonlinear systems.

Operator-based experimental studies on nonlinear vibration control for an aircraft vertical tail with considering low-order modes

In this paper, a robust nonlinear control design using an operator-based robust right coprime factorization approach is considered for vibration control on an aircraft vertical tail with piezoelectric elements. First, a model of the aircraft vertical tail is derived to describe vibration response using the operator-based approach, where, to stabilize vibration of the tail, piezoelectric elements are used as actuators and a hysteresis nonlinear property of piezoelectric actuators is considered. Simultaneously, positions of the piezoelectric actuators that are stuck on the plate are arranged by using a finite element method. Then based on the obtained operator-based model, a robust nonlinear feedback control design is given by using robust right coprime factorization for the aircraft vertical tail with considering the effect of hysteresis nonlinearity from piezoelectric actuators. In particular, low-order modes are employed to design the control scheme even though vibration is configured by high-order modes. In other words, robustness is considered, and the desired performance of tracking is discussed. Finally, both simulation and experimental results are shown to verify the effectiveness of the proposed control scheme.