Nonlinear Control Systems II
Abstract
From the Publisher:
"This book incorporates recent advances in the design of feedback laws to the purpose of globally stabilizing nonlinear systems via state or output feedback. It is a continuation of the first volume by Alberto Isidori on Nonlinear Control Systems. Specifically this second volume will cover: Stability analysis of interconnected nonlinear systems; the notion of input-to-state stability and its role in analysing stability of cascade-connected or feedback-connected systems; the notion of dissipativity and its consequences (passivity and "gain"); robust stabilization in the case of parametric uncertainties; the case of state feedback (global or semi-global stabilization); the case of output feedback (semi-global stabilization); robust stabilization in the case of unstructured perturbations; feedback design via the small-gain approach; robust semi-global stabilization via output feedback; methods for asymptotic tracking, disturbance rejection and model following; global and semi-global analysis; normal forms for multi-input multi-output nonlinear systems form a global point of view; and their role in feedback design."--BOOK JACKET.
Chapters (5)
For convenience of the reader, this section provides a quick review of the notion of comparison functions and their role in the well-known criterion of Lyapunov for determining stability and asymptotic stability.
The purpose of this Chapter is to describe some important tools for the design of feedback laws which globally asymptotically stabilize a nonlinear system in the presence of parameter uncertainties. We consider the case in which the mathematical model of the system to be controlled depends on a vector μ ∈ ℝp
of parameters, which are assumed to be constant, but whose actual values are unknown to the designer. The vector µ of unknown parameters could be any vector in some a priori given set \(
\mathcal{P} \), and the goal of the design is to find a feedback law (obviously independent of μ) which globally asymptotically stabilizes the system for each value of \(
\mu \in \mathcal{P} \). A problem of this type is usually referred to as a problem of robust stabilization.
In section 9.3 we have introduced the concept of semiglobal stabilizability, and we have shown (Theorem 9.3.1) how, using a linear feedback, it is possible to stabilize in a semiglobal sense (i.e. imposing that the domain of attraction of the equilibrium contains a prescribed compact set) a system of the form (9.23), under the hypothesis that the equilibrium z = 0 of its zero dynamics is globally asymptotically stable. In this section, in preparation to the subsequent study of the problem of robust semiglobal stabilization using output feedback, we extend the result of Theorem 9.3.1 to the case of a system modeled by equations of the form
$$\begin{array}{*{20}{l}}
{\dot z}& = &{{f_0}(z,\xi )} \\
{{{\dot \xi }_1}}& = &{{\xi _2}} \\
{{{\dot \xi }_2}}& = &{{\xi _3}} \\
{}&{}& \cdots \\
{{{\dot \xi }_r}}& = &{q(z,{\xi _1}, \ldots ,{\xi _r},\mu ) + b(z,{\xi _1}, \ldots ,{\xi _r},\mu )u,}
\end{array}$$ (1)
in which z ∈ ℝn
, ξi
∈ ℝ for i=1,…,r, u ∈ ℝ and \(\mu \in \mathcal{P} \subset {\mathbb{R}^p}\)
is a vector of unknown parameters, ranging over a compact set \(\mathcal{P}\).
In this Chapter we will study problems of global stabilization of systems that can be modeled as feedback interconnection of two subsystems, one of which is accurately known while the other one is uncertain but has a finite L
2 gain, for which an upper bound is available. More precisely, we consider systems modeled by equations of the form $$\begin{array}{*{20}{l}}
{{{\dot x}_1}}& = &{{f_1}({x_1},{h_2}({x_2}),u)} \\
{{{\dot x}_2}}& = &{{f_2}({x_2},{h_1}({x_1})),}
\end{array}$$ (13.1) which describe the feedback interconnection of a system
$$\begin{array}{*{20}{l}}
{{{\dot x}_1}}& = &{{f_1}({x_1},w,u)} \\
y& = &{{h_1}({x_1})}
\end{array}$$ (13.2) in which \({x_1} \in {\mathbb{R}^{{n_1}}}\), w ∈ ℝ, u ∈ ℝ, y ∈ ℝ and f
1(0,0,0)=0, h
1(0)=0, a system $$\begin{array}{*{20}{l}}
{{{\dot x}_2}}& = &{{f_2}({x_2},y)} \\
w& = &{{h_2}({x_2})}
\end{array}$$ (13.3) in which \({x_2} \in {\mathbb{R}^{{n_2}}}\) and f
2(0,0)=0,h
2(0)=0.
In this Chapter, we describe methods for global (robust) stabilization of nonlinear systems, by means of memoryless feedback, in cases in which the amplitude of the control input cannot exceed a fixed bound. Of course, if such a hard constraint is imposed on the amplitude of the control input, one cannot expect — in general — that global asymptotic stability is possible, unless the uncontrolled system already possesses this property to a certain extent. The simplest case in which so happens is when there exists a positive definite and proper function, whose derivative along the trajectories of the uncontrolled system is negative semi-definite but, possibly, not negative definite. In this case, in fact, under mild hypotheses, it is possible to find a smooth feedback law, whose amplitude does not exceed any (arbitrarily small) a priori fixed number, yielding global asymptotic stability. We discuss this special case first, as a point of departure for the analysis of more general structures that will be presented in the subsequent sections of the Chapter.
... where the known boundsd 0 , d n1 and d n2 can be retrieved for instance from the physical characteristics of the plant. Taking into account the foregoing problem formulation, the control objective is to design the control laws u 0 and u 1 appearing in (1) such that [x 0 , x ⊤ ] ⊤ , as t → ∞, converges to a small vicinity of the equilibrium point, which will be formally defined in the sequel of the paper relying on the concept of Input-to-State Stability [35], and all the other signals in the closed-loop system are bounded. Note that the triangular structure of system (1) allows us to design the control inputs u 0 and u 1 in two separate steps. ...
... with k j > 0, substituting in (35), exploiting (31) and posinĝ ∆ j =ŵ ⊤ j b j , one haṡ ...
... Then, by relying on the concept of Input-to-State-Stability (ISS) [35], the following result can be proved. ...
The problem of stabilizing a class of nonholonomic systems in chained form affected by both matched and unmatched uncertainties is addressed in this paper. The proposed design methodology is based on a discontinuous transformation of the perturbed nonholonomic system to which an adaptive multiple-surface sliding mode technique is applied. The generation of a sliding mode allows to eliminate the effect of matched uncertainties, while a suitable function approximation technique enables to deal with the residual uncertainties, which are unmatched. The control problem is solved by choosing a particular sliding manifold upon which a second order sliding mode is enforced via a continuous control with discontinuous derivative. A positive feature of the present proposal, apart from the fact of being capable of dealing with the presence of both matched and unmatched uncertainties, is that no knowledge of the bounds of the unmatched uncertainty terms is required. Moreover, the fact of producing a continuous control makes the proposed approach particularly appropriate in nonholonomic applications, such as those of mechanical nature.
... Los campos vectoriales f (t, z) y g (t, z) en R n son suaves e inciertos, y la dimensión n puede ser también incierta. Un problema básico de control consiste en la estabilización robusta del origen z = 0 de (1), a pesar de las incertidumbres y/o perturbaciones presentes en f y g (Khalil, 2002;Isidori, 1995Isidori, , 1999. Nótese que problemas de seguimiento robusto de referencias variantes en el tiempo pueden ser reformulados como problemas de estabilización robusta. ...
... Los campos vectoriales f (t, z) y g (t, z) en R n son suaves e inciertos, y la dimensión n puede ser también incierta. Un problema básico de control consiste en la estabilización robusta del origen z = 0 de (1), a pesar de las incertidumbres y/o perturbaciones presentes en f y g (Khalil, 2002;Isidori, 1995Isidori, , 1999. Nótese que problemas de seguimiento robusto de referencias variantes en el tiempo pueden ser reformulados como problemas de estabilización robusta. ...
... dónde σ ∈ R y h : R × R n → R es una función suave. σ debe ser tal que: a) σ tenga grado relativo ρ ≥ 1 bien definido (Isidori, 1995) con respecto a u, es decir, ...
En este trabajo se presenta una panorámica del desarrollo de los métodos básicos de análisis y diseño de controladores y observadores por modos deslizantes de orden superior. Inicialmente se describen los controladores por retroalimentación de estados con una ley de control discontinua, que generan un modo deslizante de cualquier orden. Posteriormente se presenta una nueva clase de algoritmos por modos deslizantes de orden superior, que consisten en una retroalimentación de estados continua y una acción de control integral discontinua. Se describen también observadores por modos deslizantes, que estiman los estados del sistema en tiempo finito, y que permiten obtener un controlador por retroalimentación de la salida. Todos los diseños presentados se basan en el uso de funciones de Lyapunov (explícitas), que constituyen una contribución importante del grupo de trabajo de los autores en la Universidad Nacional Autónoma de México. La presentación es tutorial y solo se dan los resultados, dejando a un lado la formalización rigurosa y las pruebas matemáticas. Para ello se refiere al lector a la literatura pertinente. Se ilustran los resultados mediante simulaciones y la validación experimental en un sistema de levitación magnética.
... In the linear case this inequality reduces to a more tractable matrix algebraic Riccati inequality. Furthermore, the powerful concept of Input-to-State Stability (ISS) is strongly related to L ∞ −stability, and constitutes an important generalization [2], [6], [8]- [10]. ...
... Apparently, it is important to understand the concepts of L p −stability and L p −gain for homogeneous systems. Since, in general, homogeneous systems may be non-smooth and their linearizations (when meaningful) are trivial, many of the standard results for non-linear systems [2], [6]- [8] cannot be used. ...
... An inequality of the type of (22) with p = q = 2 is classically used to charaterize the L 2 −gain of a non-linear system [2], [6]- [8]. In the linear case this reduces to the well-known Riccati inequality, when choosing a quadratic V (x). ...
The motivation of this paper comes from the fact that
$\mathcal {L}_{p}-$
stability and
$\mathcal {L}_{p}-$
gain, using the classical signal norms, is not well-defined for arbitrary continuous weighted homogeneous systems. However, using homogeneous signal norms it is possible to show that every internally stable homogeneous system has a globally defined finite homogeneous
$\mathcal {L}_{p}-$
gain, for
$p$
sufficiently large. If the system has a homogeneous approximation, the homogeneous
$\mathcal {L}_{p}-$
gain is inherited locally. Homogeneous
$\mathcal {L}_{p}-$
stability can be characterized by a homogeneous dissipation inequality, which in the input affine case can be transformed to a homogeneous Hamilton-Jacobi inequality. An estimation of an upper bound for the homogeneous
$\mathcal {L}_{p}-$
gain can be derived from these inequalities. Homogeneous
$\mathcal {L}_{\infty }-$
stability is also considered and its strong relationship to Input-to-State stability is studied. These results are extensions to arbitrary homogeneous systems of the well-known situation for linear time-invariant systems, where the Hamilton-Jacobi inequality reduces to an algebraic Riccati inequality. A natural application of finite-gain homogeneous
$\mathcal {L}_{p}-$
stability is in the study of stability for interconnected systems. An extension of the small-gain theorem for negative feedback systems and results for systems in cascade are derived for different homogeneous norms. Previous results in the literature use classical signal norms, hence, they can only be applied to a restricted class of homogeneous systems. The results are illustrated by several examples.
... Hence, nonlinear systems are typical research models in various control fields and can be used to describe the dynamics of the system states. In order to handle nonlinear control problems, many classical control methods were proposed based on the nonlinear control theory [1][2][3], such as the baskstepping control method, feedback linearization technique, and nonlinear disturbance observer-based control scheme. Based on these nonlinear control theories, a number of advanced control techniques were proposed for various control systems. ...
... A. The Calculation of d((k 5 + H 25 + (H 24 /4‖u 1 ‖ (4/3)) + (3ε 25 /4‖u 1 ...
In this paper, an antidisturbance controller is presented for helicopter stochastic systems under disturbances. To enhance the antidisturbance abilities, the nonlinear disturbance observer method is applied to reject the time-varying disturbances. Then, the antidisturbance nonlinear controller is designed by combining the backstepping control scheme. And the stochastic theory is used to guarantee that the closed-loop system is asymptotically bounded in mean square while the proposed control method is shown via some traditional nonlinear control techniques, which still show some common issues such as “dimension explosion” or others. The result of this paper can be regarded as a typical case of the nonlinear control method to help and promote the generation of advanced methods.
... However, even though ML researchers have provided some theoretical guarantees (Sutton and Barto, 1998), this rests in contrast to the CT paradigm, where provable guarantees are a design specification (e.g., see feedback linearization (Isidori, 1999) for convergence and prescribed performance control (Bechlioulis and Rovithakis, 2008) for output constraints). For example, in the context of the motion planning problem, there exist a variety of conventional algorithms that provide provable guarantees, whereas ML approachesmostly aim at achieving a desired control performance, without imposing the latter by design, hence, a statistical analysis may be applied to asses the effectiveness of the method. ...
In control theory, reactive methods have been widely celebrated owing to their success in providing robust, provably convergent solutions to control problems. Even though such methods have long been formulated for motion planning, optimality has largely been left untreated through reactive means, with the community focusing on discrete/graph-based solutions. Although the latter exhibit certain advantages (completeness, complicated state-spaces), the recent rise in Reinforcement Learning (RL), provides novel ways to address the limitations of reactive methods. The goal of this paper is to treat the reactive optimal motion planning problem through an RL framework. A policy iteration RL scheme is formulated in a consistent manner with the control-theoretic results, thus utilizing the advantages of each approach in a complementary way; RL is employed to construct the optimal input without necessitating the solution of a hard, non-linear partial differential equation. Conversely, safety, convergence and policy improvement are guaranteed through control theoretic arguments. The proposed method is validated in simulated synthetic workspaces, and compared against reactive methods as well as a PRM and an RRT⋆ approach. The proposed method outperforms or closely matches the latter methods, indicating the near global optimality of the former, while providing a solution for planning from anywhere within the workspace to the goal position.
... with x ∈ R nx , u ∈ R and f, g to be sufficiently smooth of suitable dimension. Let q : R nx → R be a C 1 function and let r ∈ N. We say that the function y = q(x) has a relative degree 1 r with respect to system (8) and input u in the sense of [9,Section 4] ...
... Combining the Equations (26), (28), and (29), the extended system state space model of ADRC and controlled system is established. ...
Wind has been admitted as one of the most promising renewable energy resources in multinational regionalization policies. However, the energy conversion and utilization are challenging due to the technique reliability and cost issues. Hydraulic wind turbine (HWT) may solve the above problems. HWT is taken as a research object, and the maximum power point tracking (MPPT) control strategy is proposed collaborating with active disturbance rejection control (ADRC) and linear quadratic regulator (LQR) control methods, to solve multiplicative nonlinearity problems in the plant models and the influence of external disturbance on control performance in the MPPT control process. A nonlinear simulation model is built to explain the main findings from the experiments and obtain a better understanding of the effect of time‐varying system parameters and random fluctuation in wind speed. The collaborative control algorithm is experimentally verified on a 24‐kW HWT semi‐physical test platform that results in a promising energy conversion rate, plus the hydraulic parameters can satisfy the demand, accordingly. Ultimately, the potential challenges of implementing this technique in a smart wind energy conversion system are discussed to give a further design guidance, either theoretically or practically.
... Fortunately, for the 3D overhead crane with 6-Dof, all dependent states are weakly stable when the control-directly states are stable at the equibrilium points. This property suggests a method to design a feedback controller to make the system to be stable and it is also satisfying Alberto Isidori's sufficient condition for the stability of the underactuated system studied in [9]. ...
This paper proposes a novel scheme for stabilization of the 6-DoF Overhead Crane to control the movements of this system without the state measurements through the separation principle. The movements of the system studied in this paper are described more closely than the existing crane model when considering six behaviors simultaneously: the movement of the trolley and the bridge, two swing angles along the x-axis and y-axis of the payload, the hoisting drum rotation, and axial payload oscillation. The mission of the proposal is to make all the above behaviors of the 6-DOF crane system to be stable accurately enough at the desired positions and minimize both the horizontal swings of the payload and the axial oscillation of the cable by utilizing information about the position of the trolley, the bridge, and the length of cable. To guarantee these objectives, a cooperation regime controller comprising a new asymptotic state observer which is elaborately constructed via a neural network, and the output feedback backstepping hierarchical sliding mode vehicle is integrated into the controller for improving the adaption of the closed-loop system. This cooperation controller is also evolved to prop up the fault injection data onslaught by eliminating all the information about the input system into the procedure. Furthermore, to enhance the flexibility of the closed scheme, an updating law for the observer’s parameter is developed based on the data-driven principle. All of them are developed and designed not only to handle these above tasks but also to avoid the undesired finite-escape-time (FET) phenomenon. All the results are proven systematically by three theorems and validated their performance through two scenarios and simulated by Matlab/Simulink platform.
... For this purpose, consider a system which, in addition to the control input u, is also influenced by a disturbance w. In the case with affinity to the input, the system can be described as followṡ (56) by the vector fields f , g 1 , g 2 : R n → R n . Definition 1 can be generalized as follows [7,66,67]: ...
This paper deals with systematic approaches for the analysis of stability properties and controller design for nonlinear dynamical systems. Numerical methods based on sum-of-squares decomposition or algebraic methods based on quantifier elimination are used. Starting from Lyapunov’s direct method, these methods can be used to derive conditions for the automatic verification of Lyapunov functions as well as for the structural determination of control laws. This contribution describes methods for the automatic verification of (control) Lyapunov functions as well as for the constructive determination of control laws.
... Nonlinear control systems are widely used in aerospace, robotics, power systems, and mechatronics fields. [12,13]. ...
... Upon (13), one observes that in the conventional ADRC system the tracking error dynamics and a terminal upper bound of error (4) are forced by the aggregated estimation deviation ∆(t). The residual deviation lim sup t→∞ |∆(t)|, and as a consequence of (13) -by the Input-to-State-Stability result, [10] -also lim sup →∞ |e(t)|, can be made sufficiently small (at least in the perfect measurement conditions) by increasing the bandwidth of LESO, that is, by increasing the parameter ω o , see [15]. This well known high-gain observation strategy has substantial practical limitation due to the possible presence of a high-frequency measurement noise z corrupting the output signal y. ...
... . The stability analysis of cascaded systems has been thoroughly explored, see for example Isidori (1999); Sepulchre et al. (1997), and two main methods exist to infer the asymptotic stability of the cascaded system. These methods rely on satisfying either: ...
In this paper, we analyse the behaviour of a buck converter network that contains arbitrary, up to mild regularity assumptions, loads. Our analysis of the network begins with the study of the current dynamics; we propose a novel Lyapunov function for the current in closed-loop with a bounded integrator. We leverage on these results to analyse the interaction properties between voltages and bounded currents as well as between node voltages and to propose a two-layer optimal controller that keeps network voltages within a compact neighbourhood of the nominal operational voltage. We analyse the stability of the closed loop system in two ways: one considering the interconnection properties which yields a weaker ISS type property and a second that contemplates the network in closed loop with a distributed optimal controller. For the latter, we propose a novel distributed way of controlling a Laplacian network using neighbouring information which results in asymptotic stability. We demonstrate our results in a meshed topology network containing 6 power converters, each converter feeding an individual constant power load with values chaning arbitrarily within a pre-specified range.
Towards magnetically levitated slice motor (MLSM) which is a seriously coupled and nonlinear multi-input–multi-output system with fast dynamics, this paper proposes a novel nonlinear disturbance observer-based model predictive control scheme that decouples the outputs and enhances the system response speed and anti-disturbance capability while guaranteeing robustness and static performances. Based on the nominal mathematical model of MLSM, the closed-form MPC is employed to derive the optimal control inputs offline. With the derived inputs applied, the outputs are decoupled and meanwhile, a fast dynamic response is ensured. Nevertheless, the outputs suffer from various disturbances, including matching and mismatching ones. Aiming to eliminate disturbances from the outputs, a disturbance compensation matrix is deduced while a nonlinear disturbance observer is utilized to give the disturbance estimation. Finally, experiments together with simulations demonstrate that the proposed controller presents superiorities when compared to the dual-loop PID-PI controller. In the experiments, the settling times of speed step and displacement step are reduced by 54.5% and 95%, respectively and overshoots are avoided. Displacement deviation induced by speed step is 0 mm while it is an average of 0.27635 mm under PID-PI controller. Moreover, when disturbed, recovery time and deviation value of the system are reduced simultaneously by over 50%.
In this article, a novel observer‐based output feedback control approach is proposed to address the distributed optimal coordination problem of uncertain nonlinear multi‐agent systems in the normal form over unbalanced directed graphs. The main challenges of the concerned problem lie in unbalanced directed graphs and nonlinearities of multi‐agent systems with their agent states not available for feedback control. Based on a two‐layer controller structure, a distributed optimal coordinator is first designed to convert the considered problem into a reference‐tracking problem. Then a decentralized output feedback controller is developed to stabilize the resulting augmented system. A high‐gain observer is exploited in controller design to estimate the agent states in the presence of uncertainties and disturbances so that the proposed controller relies only on agent outputs. The semi‐global convergence of the agent outputs toward the optimal solution that minimizes the sum of all local cost functions is proved under standard assumptions. A key feature of the obtained results is that the nonlinear agents under consideration are only required to be locally Lipschitz and possess globally asymptotically stable and locally exponentially stable zero dynamics.
A port-Hamiltonian model is suitable to represent the dynamics of many physical processes, which has an essential feature of underscoring a significance of an energy function, an interconnection pattern, and a dissipation for physical systems. Hence, adaptive path-tracking control with a passivity-based observer is designed by the port-Hamiltonian model for autonomous vehicles in this paper. Firstly, a passivity-based observer of lateral velocity is established by designing a gain matrix to assign the interconnection and damping parts, which makes an observation error system transform into a desired port-Hamiltonian system and renders the observer passive. At the same time, it's convergence and stability analysis is given. Then, a port-Hamiltonian model with perturbations and actuator saturations is constructed to describe lateral vehicle dynamics in terms of tracking errors for path-tracking, and an L2-gain disturbance attenuation strategy is developed to overcome the adverse influence caused of external disturbances. Furthermore, the robust stability of the controller is proved in the presence of observation errors and external disturbances. Finally, simulations and experiments are performed to demonstrate the effectiveness of the proposed control strategy.
The purpose of this chapter is to review some notions of fundamental importance on the analysis and design of feedback laws for nonlinear control systems. The first part of the chapter begins by reviewing the notion of dynamical system and continues with a discussion of the concepts of stability and asymptotic stability of an equilibrium, with emphasis on the method of Lyapunov. Then, it continues by reviewing the notion of input-to-state stability and discussing its role in the analysis of interconnections of systems. Then, the first part is concluded by the analysis of the asymptotic behavior in the presence of persistent inputs. The second part of the chapter is devoted to the presentation of systematic methods for stabilization of relevant classes of nonlinear systems, namely, those possessing a globally defined normal form. Methods for the design of full-state feedback and, also, observer-based dynamic output feedback are presented. A nonlinear separation principle, based on the use of a high-gain observer, is discussed. A special role, in this context, is played by the methods based on feedback linearization, of which a robust version, based on the use of extended high-gain observer, is also presented.KeywordsDynamical systemsStabilityMethod of LyapunovInput-to-state stabilitySteady-state behaviorNormal formsFeedback linearizationBacksteppingHigh-gain observersSeparation principleExtended observers
This work aims to obtain reduced-order models for fluid flows in porous media that can be used for optimal well-control design and are they are equipped with input-output tracking capabilities. Meeting the net-zero emission paradigm will require a realignment of hydrocarbon production strategies with other forms of energy production, such as hydrogen and geothermal. Profiting from all these energy sources is only possible if accurate and timely predictions of the injection-production behavior of fluids, including geomechanics issues in the subsurface, can be attained. High-fidelity reservoir simulation provides accurate characterizations of complex flow dynamics in the subsurface. Still, it is unsuitable for production or uncertainty quantification due to its prohibitive computational complexity.
Balanced truncation (BT) is a well-known model reduction technique for linear systems. It is input-output invariant and does not require a training phase once the system can be written in a linear state-space form, unlike other methods (Proper Orthogonal Decomposition - POD, Deep learning, among others). However, reduced-order models are unsuitable for long-term simulations as these simulations exhibit highly nonlinear behavior. This paper builds upon the bilinear formulation of dynamical systems to construct a suitable reduced-order model. A combination of data-driven model reduction strategies and machine learning (deep-neural networks ANN) will be used to simultaneously predict state and the best correlated input-output matching. We remove the states that are hard to control and observe in the bilinear space by introducing a loss function to the Artificial Neural Network (ANN) training process based on the variational interpretation of the controllability and observability gramians. Both these matrices are related respectively to the energy demanded to control a state (i.e., how hard is it to change a gridblock pressure by controlling the injector wellťs bottom-hole pressure) and to the energy produced by a state (i.e., if we can infer the pressure in a gridblock by measuring the rate of a producing well).
We applied this new framework to a two-dimensional two-phase (oil and water) reservoir under waterflooding with three wells (one injector and two producers). The proposed method is a non-intrusive data-driven method as it does not need access to the reservoir simulation's internal structure; thus, it can be easily applied with any commercial reservoir simulator and is extensible to other studies. Although state information is well preserved during truncation, the output, e.g., cumulative production, presents a slightly worse response than simply applying POD. This is because we also identify the output matrices (C and D) and enforce the orthogonalization of the projection matrices through a loss function and not by construction.
As far as we know, it is the first attempt to apply balanced truncation of bilinear reservoir models to solve well-control problems. It has the potential to lead the trend of generating robust reduced-order (proxy) models. In this paper, we propose a novel data-driven framework to construct the proxy model while using as much physical information as possible to guide the neural network to best correlates the input well-control and output well-response, making it ideal for well-control optimization.
Motion control methods of manipulators with flexible-joint or flexible-link are different from rigid manipulators due to different composition and characteristics Yan et al. (IEEE Trans. Syst., Man, Cybern.: Syst. 51(3):1671–1678, 2021), Reis and da Costa (J. Sound Vib. 331(14):3255–3270, 2012).
This paper presents a Lyapunov formulation of the small-gain theorem for the finite-time input-to-state stability (FTISS) of an interconnected nonlinear system composed of two or more FTISS subsystems. In addition, an FTISS-Lyapunov function for the interconnected system is constructed from the FTISS-Lyapunov functions of the subsystems. With respect to the previously developed nonlinear, Lyapunov-based small-gain theorem restricted to input-to-state stability, a new power-function-based scaling technique is proposed to deal with the challenge that a nonlinearly scaled FTISS-Lyapunov function may not retain a decreasing rate as a power function with a positive power less than one.
This article proposes a new control scheme with a nonlinear quadratic regulator (NQR) associated with a hybrid switching stage for permanent magnet synchronous generator-based wind turbines (WTs). The proposed method shows a new approach combining a nonlinear optimal control method with a hybrid switching stage for WT generations (WTGs). As the WTG is a nonlinear system, NQR control design is directly employed to deal with the nonlinearity while a hybrid switching scheme can deal with constrained control inputs as discrete nonlinear characteristics of switch-mode power converters. A polynomial-based dynamic model is derived for the new NQR design. Unlike the previous state-dependent Riccatti equation (SDRE) technique where the solution of the nonlinear optimal controller is approximated, this technique finds the exact online solution of SDRE with nonlinear feedback gains flexibly depending on the tracking error. Then, the optimal control laws are implemented via a hybrid switching stage ensuring a smooth current regulation under a reduced switching frequency while avoiding any complicated modulator or finite state search. The proposed control design is investigated via comparative studies with linear quadratic regulator (LQR), direct switching, and space-vector pulse-width-modulation stages (SV-PWMs) implemented in both platforms of MATLAB/Simulink and real-time OPAL-RT. Comparative results exhibit flexible optimal performances via online hyper-parameter tuning to reduce the speed tracking error and low current ripples under a reduced switching frequency.
An objective of a control theory is to influence the dynamics of a system to guarantee its desired behavior. Challenges arising in this way are manifold and include nonlinearity of a system, a need to ensure robustness (or reliability) of designed controllers in spite of actuator and observation errors, hidden (unmodeled) dynamics of a system, and external disturbances acting on the system. Furthermore, networks to be controlled may be huge with a nontrivial topological structure.
This paper investigates the sliding mode extremum seeking control problem for nonlinear systems with disturbances to improve the disturbance rejection capability, delivering promising searching accuracy and efficiency under disturbances. The existing results in sliding mode extremum seeking control suffer from the drawback of unsatisfactory transient performance and slow convergence speed when disturbances, especially unmatched disturbances, are injected into the system. To this end, a disturbance rejection sliding mode extremum seeking control approach is proposed in this paper. A nonlinear disturbance observer is developed for disturbance estimation, and then a disturbance rejection sliding mode extremum seeking control algorithm is proposed by exploiting the disturbance estimation. Under the proposed approach, the disturbances can be online estimated and compensated. As a result, the cost function is less disturbed by unknown disturbances, and faster convergence speed as well as shorter recovery time can be obtained. The proposed method is applied for directional antenna automatic tracking system. Simulation results show that the newly proposed searching control approach outperforms the baseline extremum searching approaches in terms of much more reliable and high efficiency searching performance under disturbances.
The prescribed‐time stabilization problem for a general class of uncertain nonlinear systems with unknown input gain and appended dynamics (with unmeasured state) is addressed. Unlike the asymptotic stabilization problem, the prescribed‐time stabilization objective requires convergence of the state vector to the origin in a finite time interval that can be arbitrarily picked (i.e., “prescribed”) by the control system designer irrespective of the system's initial condition. The class of systems considered is allowed to have general nonlinear uncertain terms throughout the system dynamics as well as uncertain appended dynamics (that effectively generate a time‐varying non‐vanishing disturbance signal input into the nominal system). The control design is based on a nonlinear transformation of the time scale, dynamic high‐gain scaling, adaptation dynamics with temporal forcing terms, and a composite control law that includes two components. The first component in the composite control law is analogous to prior dynamic high‐gain scaling‐based control designs, but with a time‐dependent function in place of the unknown input gain, while the second component has a non‐smooth form with time‐dependent terms that ensure prescribed‐time convergence in spite of the unknown input gain and the disturbances. The efficacy of the proposed control design is illustrated through numerical simulation studies on two example systems (a “synthetic” fifth‐order system and a “real‐world” electromechanical system).
Torque ripple of permanent magnet synchronous motor (PMSM) and load disturbances are the main influencing factors that restrict the application of PMSM in low-speed tracking systems. To improve the speed tracking precision of the gimbal servo system in control moment gyro (CMG) with multisource disturbances, this article proposes a composite control method based on parameter-optimized extended state observers (ESOs) with sliding mode control in the speed loop and PI control with a feed-forward compensation in the current loop. Three ESOs are employed to estimate the disturbances on the
$d$
-axis current, the
$q$
-axis current, and the load torque so that the controller can have the corresponding parts to compensate for the disturbances. In particular, aiming at the contradiction between the estimation quality of the observer and the sensitivity to the measurement noise, an appropriate cost function is designed to take a compromise between the lumped disturbances and measurement noise and optimize the parameters of the ESOs. Finally, simulation and experimental results on the gimbal servo system of magnetically suspended control moment gyroscope (MSCMG) show the effectiveness of the proposed methods.
A family of homogeneous controllers is proposed, which utilize noisy state measurements for stabilization of integrator chains subject to system faults and disturbances. In the absence of sampling noises, the exact system stabilization is ensured in spite of some system faults. The same controllers also practically stabilize generally disturbed integrator chains. Controllers demonstrate strong noise‐reduction capabilities, and system faults are robustly detected. The stabilization accuracy is estimated in the presence of noises and faults. Simulation confirms the theoretical results.
This article investigates the optimal resource allocation of nonlinear high‐order multi‐agent systems with coupling uncertainties. The objective is to design an output‐feedback distributed algorithm for each agent to optimize the global cost function under network constraints. The global cost function is the sum of local cost functions. The available information to design the algorithm for each agent is only its own output and cost function, the outputs of its neighbors. To address the key technical challenge arising from the uncertainties and nonlinearities of network systems, a novel distributed resource allocation algorithm is proposed using active disturbance rejection strategy. The simulation results in the economic dispatch of a smart grid validate the effectiveness of the proposed method.
In this article, a novel distributed secondary control scheme is proposed to solve the voltage regulation and power sharing problems of an islanded direct current (DC) microgrid. It not only eliminates the voltage deviation caused by droop control but also minimizes the total generation cost of the DC microgrid. The proposed secondary control of distributed generators (DGs) does not require global information, but only current information of neighbors and the DC bus voltage. Different from the existing secondary frequency control, the corresponding closed-loop system of the DC microgrid under the proposed scheme is a nonlinear networked system. With the help of the properties of cascade systems and the input-to-state stability theory, it is found that the DC microgrid under the proposed scheme can achieve voltage regulation and minimization of the total generation cost if the corresponding communication graph of DGs is connected. A simulation model is established to verify the proposed scheme. Finally, a DC microgrid platform with two DGs in parallel is used for the experimental verification.
In this article, the distributed nonlinear placement problem for a class of multicluster Euler–Lagrange systems is considered. The problem is first converted into a time-varying noncooperative game. A distributed Nash equilibrium seeking algorithm composed of an auxiliary double-integrator system and a coordinated-tracking observer is designed to solve the problem. The convergence results are established by an iterative approach and the small gain theorem. The effectiveness of the algorithm is demonstrated via simulations.
This paper designs an event-triggering based communication strategy for the global attitude synchronization of a network of rigid bodies. To overcome the topological constraint on the manifold
$SO(3)$
, the quaternion-based hybrid control strategy is designed using a binary logic variable, relying on the relative measurements of adjacent rigid bodies, to determine the torque orientation. The Zeno-free distributed event-triggering strategies (ETSs) are designed combining with the reset of the binary logic variable to generate discrete communication instants, where only the corresponding parts of the control inputs are updated at those discrete instants. By assuming perfect knowledge of the rigid bodies’ dynamics and considering uncertainties and/or exogenous disturbances simultaneously, nominal and robust cases are analyzed to ensure the global attitude synchronization, respectively. The effectiveness of the main results is demonstrated by considering the attitude synchronization of six miniature quadrotor prototypes.
The problem of synchronization in networks of linear systems with nonlinear diffusive coupling and a connected undirected graph is studied. By means of a coordinate transformation, the system is reduced to the form of mean-field dynamics and a synchronization-error system. The network synchronization conditions are established based on the stability conditions of the synchronization-error system obtained using the circle criterion, and the results are used to derive the condition for synchronization in a network of neural-mass-model populations with a connected undirected graph. Simulation examples are presented to illustrate the obtained results.
Control input acting on the system in the form of nonlinear implicit tends to exert great impacts on the dynamic behavior of the multiple-input-multiple-output (MIMO) nonaffine nonlinear system with actuator saturation, posing a great challenge to its tracking control and dynamic regulation. Performance optimization and design simplicity are hardly taken simultaneously into account by existing controllers. For this sake, the multidimensional Taylor network (MTN) controller, which involves both transient improvement and design simplicity, is proposed here. The MTN controller, as a novel one with a fixed structure, is capable of making the proposed system asymptotically track the reference with no need of system information except for the control direction. A performance index related to the transient improvement by dynamic regulation is established. Parameters of the MTN controller are adapted for the optimization problem in line with this performance index of the arbitrary approximation by the polynomial function. The existence and the realizability of the solution to the optimization problem are demonstrated. Specific steps of the controller design are given. The effectiveness of the proposed controller is confirmed by its theoretical analysis as well as its application to a two-link manipulator system.
In this paper, we investigate the attitude trajectory tracking problem of a rigid spacecraft subject to external disturbances and full-state constraint under the event-triggered scheme. By incorporating a barrier Lyapunov function (BLF) into the dynamic surface technique, we develop a coordinate-free integral sliding mode controller which can not only deal with the state constraint but also enforce the tracking error converge into a small region containing the origin in the presence of the external disturbance. To save the computation and communication resources, a modified integral sliding mode based robust event-triggered controller is proposed, in which the control torque generation and the information transmission of attitude and control signal only need to be implemented at some discrete time when the predefined measurement error exceeds a constant threshold. Furthermore, a lower bound estimation on the inter-execution intervals is also given to show that Zeno behavior can be avoided. Finally, numerical simulations are presented to demonstrate the effectiveness and robustness of the proposed controller.
This paper proposes a controller design method to stabilize a class of nonlinear, non-autonomous second-order systems in finite time. This method is developed based on exact-linearization and Pontryagin's minimum time principle. It is shown that the system can be stabilized in a finite time of which the upper bound can be chosen according to the initial states of the system. Simulation results are given to validate the theoretical analysis.
In this study, we address limitations of the existing methods of disturbance attenuation for nonlinear control systems. We focus on scenarios with trajectory tracking in differentially flat systems. In this context, we propose a framework of a time-varying input-to-state stability tracking control Lyapunov function (time-varying ISS-TCLF) and prove that a Sontag-type time-varying ISS-CLF based controller compensates for a predefined ISS-gain performance. As an example, we design a time-varying ISS-TCLF for trajectory tracking control of a vessel. The effectiveness of our proposed method is confirmed by computer simulation.
This paper proposes a distributed controller for a network of agents of second-order dynamics to fence a moving target of unknown velocity within their convex hull. Moreover, the agents form a regular polygon formation and avoid collision during the entire evolution. In the control scheme, the agents are not necessarily labelled and the nearest angle rules are applied. Each controller is composed of the functionalities of estimation of the target's velocity, regulation of distance between the agents and the target, and angle repulsion among agents resulting in an equal distribution. The latter two also guarantee collision avoidance. The asymptotical behavior of the closed-loop system is rigorously analyzed especially subject to the velocity estimation error. The effectiveness of the controller is also demonstrated by numerical simulation.
In this paper, we propose a notion of high-order (zeroing) barrier functions that generalizes the concept of zeroing barrier functions and guarantees set forward invariance by checking their higher order derivatives. The proposed formulation guarantees asymptotic stability of the forward invariant set, which is highly favorable for robustness with respect to model perturbations. No forward completeness assumption is needed in our setting in contrast to existing high order barrier function methods. For the case of controlled dynamical systems, we relax the requirement of uniform relative degree and propose a singularity-free control scheme that yields a locally Lipschitz control signal and guarantees safety. Furthermore, the proposed formulation accounts for "performance-critical" control: it guarantees that a subset of the forward invariant set will admit any existing, bounded control law, while still ensuring forward invariance of the set. Finally, a non-trivial case study with rigid-body attitude dynamics and interconnected cell regions as the safe region is investigated.
This note studies the robust output feedback stabilization problem of multi-input multi-output invertible nonlinear systems with output-dependent multipliers. An "ideal" state feedback is first designed under certain mild assumptions. Then, a set of extended low-power high-gain observers is systematically designed, providing a complete estimation of the "ideal" feedback law. This yields a robust output feedback stabilizer such that the origin of the closed-loop system is semiglobally asymptotically stable, while improving the numerical implementation with the power of high-gain parameters up to 2.
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