S. Nikitin
The paper addresses the stabilization of nonlinear systems with semi-quadratic cost: quadratic with respect to controls and nonlinear for state variables. Paper presents the effective new feedback synthesis procedure. The novel feedback design procedure is based on the ideas borrowed from nonlinear optics and the theory of semi-classical asymptotics.
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This paper is concerned with uniform stabilization and social optimality for general mean field linear quadratic control systems, where subsystems are coupled via individual dynamics and costs, and the state weight is not assumed with the definiteness condition. For the finite-horizon problem, we first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback-type control by decoupling the FBSDEs. For the infinite-horizon problem, by using solutions to two Riccati equations, we design a set of decentralized control laws, whi…
Boundary Stabilization of a Bending and Twisting Wing by Linear Quadratic Gaussian Theory
本文提出无限维 LQR 边界控制与 Kalman 滤波器相结合的 LQG 方法，用于镇定弯扭悬臂梁，并基于 Wagner 函数构建机翼气动模型。
Quantitative rapid stabilization for parabolic equations via the linear quadratic theory
针对非自伴抛物方程，利用线性二次型理论和定量可观测性不等式，实现定量快速镇定，并揭示二者等价性；推广至Navier-Stokes方程及自伴情形下的有限维反馈律。
This paper mainly investigates the optimal control and stabilization problems for linear discrete-time Markov jump systems. The general case for the finite-horizon optimal controller is considered, where the input weighting matrix in the performance index is just required to be positive semi-definite. The necessary and sufficient condition for the existence of the optimal controller in finite-horizon is given explicitly from a set of coupled difference Riccati equations (CDRE). One of the key techniques is to solve the forward and backward stochastic difference equation (FDSDE) which is obtain…
We consider the problem of voltage stability and reactive power balancing in islanded small-scale electrical networks outfitted with DC/AC inverters ("microgrids"). A droop-like voltage feedback controller is proposed which is quadratic in the local voltage magnitude, allowing for the application of circuit-theoretic analysis techniques to the closed-loop system. The operating points of the closed-loop microgrid are in exact correspondence with the solutions of a reduced power flow equation, and we provide explicit solutions and small-signal stability analyses under several static and dynamic …
This paper studies switching stabilization problems for general switched nonlinear systems. A piecewise smooth control-Lyapunov function (PSCLF) approach is proposed and a constructive way to design a stabilizing switching law is developed. The switching law is constructed via the directional derivatives of the PSCLF with a careful discussion on various technical issues that may occur on the nonsmooth surfaces. Sufficient conditions are derived to ensure stability of the closed-loop Filippov solutions including possible sliding motions. The proposed PSCLF approach contains many existing result…
Stabilization of Strictly Pre-Dissipative Receding Horizon Linear Quadratic Control by Terminal Costs
本文研究严格预耗散滚动时域控制中终端约束的替代问题，证明在线性二次框架下可用适当终端代价保证渐近稳定性，并获得比一般非线性情形更强的结论。
It is shown that an oblique projection based feedback control is able to stabilize the state of the Kuramoto-Sivashinsky equation, evolving in rectangular domains, to a given time-dependent trajectory. The number of actuators is finite and consists of a finite number of indicator functions supported in small subdomains. Simulations are presented, in the one-dimensional case, showing the stabilizing performance of the feedback control.
This paper studies an infinite horizon optimal control problem for discrete-time linear systems and quadratic criteria, both with random parameters which are independent and identically distributed with respect to time. A classical approach is to solve an algebraic Riccati equation that involves mathematical expectations and requires certain statistical information of the parameters. In this paper, we propose an online iterative algorithm in the spirit of Q-learning for the situation where only one random sample of parameters emerges at each time step. The first theorem proves the equivalence …
A learning approach for optimal feedback gains for nonlinear continuous time control systems is proposed and analysed. The goal is to establish a rigorous framework for computing approximating optimal feedback gains using neural networks. The approach rests on two main ingredients. First, an optimal control formulation involving an ensemble of trajectories with 'control' variables given by the feedback gain functions. Second, an approximation to the feedback functions via realizations of neural networks. Based on universal approximation properties we prove the existence and convergence of opti…
Local Stability and Stabilization of Quadratic-Bilinear Systems using Petersen's Lemma
本文提出一种基于Petersen引理的可扩展LMI方法，用于二次-双线性系统的局部稳定性分析与镇定控制，可处理数百维状态变量，并在三个基准问题上验证了有效性。
Observer design and boundary output feedback stabilization for semilinear parabolic system over general multidimensional domain
本文提出了一种基于观测器和有限维反馈控制器的边界输出反馈控制策略，用于解决一般多维域上半线性抛物型系统的稳定化问题。
We study global finite-dimensional observer-based stabilization of a semilinear 1D heat equation with globally Lipschitz semilinearity in the state variable. We consider Neumann actuation and point measurement. Using dynamic extension and modal decomposition we derive nonlinear ODEs for the modes of the state. We propose a controller that is based on a nonlinear finite-dimensional Luenberger observer. Our Lypunov $H^1$-stability analysis leads to LMIs, which are shown to be feasible for a large enough observer dimension and small enough Lipschitz constant. Next, we consider the case of a const…
In this paper, we prove the stabilizability of abstract Parabolic Integro-Differential Equations (PIDE) in a Hilbert space with decay rate $e^{-γt} $ for certain $γ> 0,$ by means of a finite dimensional controller in the feedback form. We determine a linear feedback law which is obtained by solving an algebraic Riccati equation. To prove the existence of the Riccati operator, we consider a linear quadratic optimal control problem with unbounded observation operator. The abstract theory of stabilization developed here is applied to specific problems related to viscoelastic fluids, e.g. Oldroyd …
This paper studies uniform stabilization and social optimality for linear quadratic (LQ) mean field control problems with multiplicative noise, where agents are coupled via dynamics and individual costs. The state and control weights in cost functionals are not limited to be positive semi-definite. This leads to an indefinite LQ mean field control problem, which may still be well-posed due to deep nature of multiplicative noise. We first obtain a set of forward-backward stochastic differential equations (FBSDEs) from variational analysis, and construct a feedback control by decoupling the FBSD…
For nonlinear affine in the control systems, a Lie algebraic sufficient condition for sampled-data feedback semi-global stabilization is established. We use this result, in order to derive sufficient conditions for sampled-data feedback stabilization for a couple of three-dimensional systems.
It is shown that an explicit oblique projection nonlinear feedback controller is able to stabilize semilinear parabolic equations, with time-dependent dynamics and with a polynomial nonlinearity. The actuators are typically modeled by a finite number of indicator functions of small subdomains. No constraint is imposed on the sign of the polynomial nonlinearity. The norm of the initial condition can be arbitrarily large, and the total volume covered by the actuators can be arbitrarily small. The number of actuators depend on the operator norm of the oblique projection, on the polynomial degree …
Data Informativity for Quadratic Stabilization under Data Perturbation
本文提出数据扰动概念，统一处理系统噪声和测量噪声，通过扩展矩阵S过程的关键引理，解决了数据信息度评估中的核心问题，无需假设噪声特性。
The feedback exponential stabilization to trajectories for semilinear parabolic equations in a given bounded domain is addressed. The controls take values in a finite-dimensional space and are supported in a small region. Both internal and boundary controls are considered.
In standard linear quadratic (LQ) control, the first step in investigating infinite-horizon optimal control is to derive the stabilization condition with the optimal LQ controller. This paper focuses on the stabilization of an Ito stochastic system with indefinite control and state weighting matrices in the cost functional. A generalized algebraic Riccati equation (GARE) is obtained via the convergence of the generalized differential Riccati equation (GDRE) in the finite-horizon case. More importantly, the necessary and sufficient stabilization conditions for indefinite stochastic control are …
This paper addresses the topic of global output feedback stabilization of semilinear reaction-diffusion PDEs. The semilinearity is assumed to be confined into a sector condition. We consider two different types of actuation configurations, namely: bounded control operator and right Robin boundary control. The measurement is selected as a left Dirichlet trace. The control strategy is finite dimensional and is designed based on a linear version of the plant. We derive a set of sufficient conditions ensuring the global exponential stabilization of the semilinear reaction-diffusion PDE. These cond…
A switched equilibrium of a switched system of two subsystems is a such a point where the vector fields of the two subsystems point strictly towards one another. Using the concept of stable convex combination that was developed by Wicks-Peleties-DeCarlo (1998) for linear systems, Bolzern-Spinelli (2004) offered a design of a state feedback switching rule that is capable to stabilize an affine switched system to any switched equilibrium. The state feedback switching rule of Bolzern-Spinelli gives a nonlinear (quadratic) switching threshold passing through the switched equilibrium. In this paper…
In this paper, we investigate the problem of semi-global minimal time robust stabilization of analytic control systems with controls entering linearly, by means of a hybrid state feedback law. It is shown that, in the absence of minimal time singular trajectories, the solutions of the closed-loop system converge to the origin in quasi minimal time (for a given bound on the controller) with a robustness property with respect to small measurement noise, external disturbances and actuator noise.
We prove the well-posedness of a linear closed-loop system with an explicit (already known) feedback leading to arbitrarily large decay rates. We define a mild solution of the closed-loop problem using a dual equation and we prove that the original operator perturbed by the feedback is (up to the use of an extension) the infinitesimal generator of a strongly continuous group. We also give a justification to the exponential decay of the solutions. Our method is direct and avoids the use of optimal control theory.
The aim of this work is to design an explicit finite dimensional boundary feedback controller of sampled-data form for locally exponentially stabilizing the equilibrium solutions to semilinear parabolic equations. The feedback controller is expressed in terms of the eigenfunctions corresponding to unstable eigenvalues of the linearized equation. This stabilizing procedure is applicable for any sampling rate, not necessary to be small enough, and it tends to the continuous-times version when the sampling period tends to zero.
In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) -- in the sense that large motions (deflections, rotations) are accounted for in addition to shearing -- and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional …
This paper is concerned with the decay estimate of solutions to the semilinear wave equation subject to two localized dampings in a bounded domain. The first one is of the nonlinear Kelvin-Voigt type and is distributed around a neighborhood of the boundary according to the Geometric Control Condition. While the second one is a frictional damping and we consider it hurting the geometric condition of control. We show uniform decay rate results of the corresponding energy for all initial data taken in bounded sets of finite energy phase-space. The proof is based on obtaining an observability ineq…
The goal of this paper is to provide a tutorial on the so-called informativity framework for direct data-driven analysis and control. This framework achieves certified data-based analysis and control by assessing system properties and determining controllers for sets of systems unfalsified by the data. We will first introduce the informativity approach at an abstract level. Thereafter, we will report case studies where we highlight the strength of the framework in the context of various problems involving both noiseless and noisy data. In particular, we will treat controllability and stabiliza…
Universal Formula Families for Safe Stabilization of Single-Input Nonlinear Systems
本文提出一种无需优化的框架，利用给定的控制李雅普诺夫函数和控制屏障函数实现单输入非线性系统的安全稳定化，通过通用稳定器公式构造闭环反馈律，实现渐近稳定和安全集内不变性。
A Less Conservative Sufficient Condition for PID Stabilization of Scalar Second-Order Nonlinear Uncertain Systems
针对标量二阶非线性不确定系统，提出一种端点平衡的二次加积分Lyapunov函数，以降低PID镇定条件的保守性，保证全局渐近调节。
Superlinear and quadratic convergence of a stabilized sequential quadratic semidefinite programming method for nonlinear semidefinite programming problems
本文提出一种稳定化SQSDP方法求解非线性半定规划，在严格Robinson约束规范和二阶充分条件下证明了其超线性与二次局部收敛性。
Semi-global Output Feedback Stabilization of Non-Minimum Phase Nonlinear Systems
本文提出一种方法，利用全局稳定反馈控制方案实现非最小相位非线性系统的半全局稳定化，并通过高增益观测器和扩展卡尔曼滤波器恢复辅助系统的性能。
We show that quadratic growth of a semi-algebraic function is equivalent to strong metric subregularity of the subdifferential --- a kind of stability of generalized critical points. In contrast, this equivalence can easily fail outside of the semi-algebraic setting. As a consequence, we derive necessary conditions and sufficient conditions for optimality in subdifferential terms.
We consider the task of remote state estimation and stabilization of disturbed linear plants via noisy communication channels. In 2007 Matveev and Savkin established a surprising link between this problem and Shannon's theory of zero-error communication. By applying very recent results of computability of the channel reliability function and computability of the zero-error capacity of noisy channels by Boche and Deppe, we analyze if, on the set of linear time-invariant systems paired with a noisy communication channel, it is uniformly decidable by means of a Turing machine whether remote state…
It is well known from the seminal Brockett's theorem that the openness property of the mapping on the right-hand side of a given nonlinear ODE control system is a necessary condition for the existence of locally asymptotically stabilizing continuous stationary feedback laws. However, this condition fails to be sufficient for such a feedback stabilization. In this paper we develop an approach of variational analysis to continuous feedback stabilization of nonlinear control systems with replacing openness by the linear openness property, which has been well understood and characterized in variat…
Constraint Identification and Algorithm Stabilization for Degenerate Nonlinear Programs
针对退化非线性规划，提出区分强弱积极约束的技术，并据此改进序列二次规划算法，在更弱假设下实现超线性收敛。
In this paper, we study the output feedback stabilization for a scalar conservation law with a nonlocal velocity, that models a highly re-entrant manufacturing system as encountered in semi-conductor production. By spectral analysis, we obtain a complete result on the exponential stabilization for the linearized control system. Moreover, by using a Lyapunov function approach, we also prove the exponential stabilization results for the nonlinear control system in certain cases.
Global controllability and stabilization of the wave maps equation from a circle to a sphere
本文研究$(1+1)$维波映射方程，证明目标为$\mathbb{S}^k$（$k>1$）时的全局精确能控性，并确立$2\pi$能量界是连续时变反馈一致渐近镇定的严格阈值。
We prove the semi-global controllability and stabilization of the $(1+1)$-dimensional wave maps equation with spatial domain $\mathbb{S}^1$ and target $\mathbb{S}^k$. First we show that damping stabilizes the system when the energy is strictly below the threshold $2π$, where harmonic maps appear as obstruction for global stabilization. Then, we adapt an iterative control procedure to get low-energy exact controllability of the wave maps equation. This result is optimal in the case $k=1$.
Dynamical State Feedback Control for Linear Input Delay Systems, Part I: Dissipative Stabilization via Semidefinite Programming
本文提出了一种基于半正定规划的框架，用于解决输入延迟系统的稳定问题，通过参数化线性动态状态反馈方法实现耗散稳定。
Small-Gain-Based Boundary Feedback Design for Global Exponential Stabilization of 1-D Semilinear Parabolic PDEs
本文提出了一种用于1-D半线性抛物型PDEs边界反馈稳定器设计的新方法，基于小增益论证，适用于满足线性增长条件的非线性项，包括非局部项，并展示了反馈设计在抛物型情况下的根本限制。
Global stabilization of viscous Burgers' equation around constant steady state solution has been discussed in the literature. The main objective of this paper is to show global stabilization results for the 2D forced viscous Burgers' equation around a nonconstant steady state solution using nonlinear Neumann boundary feedback control law, under some smallness condition on that steady state solution. On discretizing in space using $C^0$ piecewise linear elements keeping time variable continuous, a semidiscrete scheme is obtained. Moreover, global stabilization results for the semidiscrete solut…
Results on the problem of stabilizing a nonlinear continuous-time system by a finite number of control or measurement values are presented. The basic tool is a discontinuous version of the so-called semi-global backstepping lemma. We derive robust practical stabilizability results by quantized and ternary controllers and apply them to some significant control problems.
Stabilization and control of the nonlinear plate equation
本文基于非线性板方程的线性可观察性，提出了一种新的唯一延续性质，实现了半全局稳定性和精确控制，通过分析非线性能量衰减和控制属性。
Semi-Global Approximate stabilization of an infinite dimensional quantum stochastic system
针对无限维量子随机系统，通过离散时间马尔可夫链建模，利用严格Lyapunov函数和Prohorov定理，实现目标状态的半全局近似状态反馈镇定。
Stabilization to trajectories of nonisothermal Cahn-Hilliard equations
本文证明非等温Cahn-Hilliard方程对时间依赖轨迹的半全局指数镇定，提出基于斜投影的显式反馈控制，并验证等温情形及数值性能。
This paper studies the $H^0$ norm and $H^1$ seminorm of quadratic functions. The (semi)norms are expressed explicitly in terms of the coefficients of the quadratic function under consideration when the underlying domain is an $l_p$-ball (1 \leq p \leq \infty) in $R^n$ .
These notes are issued from a short course given by the author in a summer school in Chamb{é}ry in June 2015. We consider general semilinear PDE's and we address the following two questions: 1) How to design an efficient feedback control locally stabilizing the equation asymptotically to 0? 2) How to construct such a stabilizing feedback from approximation schemes? To address these issues, we distinguish between parabolic and hyperbolic semilinear PDE's. By parabolic, we mean that the linear operator underlying the system generates an analytic semi-group. By hyperbolic, we mean that this opera…
Formulas for Data-driven Control: Stabilization, Optimality and Robustness
本文基于持续激励数据，提出了一种线性反馈系统的参数化方法，仅使用数据依赖的线性矩阵不等式解决控制问题，无需显式识别系统矩阵，解决了状态和输出反馈稳定性、线性二次调节问题以及非线性系统不稳定平衡点的鲁棒性问题。
Inverse Optimal Feedback Stabilization of 3D Nonholonomic Vehicles in Spherical Coordinates
本文基于球坐标下三维非完整车辆的严格CLFs，提出一种通用逆最优镇定控制器，采用阻尼$L_gV$反馈，通过两种罚函数选择展示灵活性，并验证了性能与约束影响。
We consider linear control systems of the form $\dot{y}(t)=Ay(t)-μB C y(t)$ where $μ$ is a positive real parameter, $A$ is the state operator and generates a linear $C_0-$semigroup of contractions $S(t) $ on a Banach space $X$, $B$ and $C$ are respectively the operators of control and observability, which are defined in appropriate spaces in which they are unbounded in some sense. We aim to show the exponential stability of the above system under sufficient conditions which are expressed in term of admissibility and observability properties. The uniform exponential stabilization using bilinear…
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