Stochastic Control of Linear and Nonlinear Econometric Models: Some Computational Aspects

This paper considers the optimal control of small econometric models applying the OPTCON algorithm. OPTCON determines approximate numerical solutions to optimum control problems for nonlinear stochastic systems. These optimum control problems consist in minimizing a quadratic objective function for linear and nonlinear econometric models with additive and multiplicative (parameter) uncertainties. The algorithm was programmed in C# and in MATLAB and allows for stochastic control with open-loop and passive learning (open-loop feedback) information patterns. Here we compare the results of applying the OPTCON2 version of the algorithm to two macroeconomic models for Slovenia, the nonlinear model SLOVNL and the linear model SLOVL. The results for both models are similar, with open-loop feedback controls giving better results on average and less outliers than open-loop controls. The number of outliers is higher in the nonlinear case and especially under high parameter uncertainty.     

On robustness in control and LTI identification: Near-linearity and non-conic uncertainty

Robustness issues are studied in the context of linear models in linear controller design and in system identification when the true system is nonlinear. The notion of nearly linear system is generalized to include time-varying and open-loop unstable systems. This class of systems, although it presents the simplest possible (global) generalization of linear systems, is in many ways nontrivial for the purpose of linear controller design and linear model identification from input-output data. This is mainly due to the presence of non-conic uncertainty, which is not included in standard treatments of robust control theory and stochastic system identification theory. Signal distribution theory, a realistic non-stochastic signal analysis tool, is used to study the limiting least squares estimates of linear finite impulse response (FIR) and autoregressive with external input (ARX) model parametrizations for two classes of nonlinear systems. Some connections to worst-case analysis of linear model identification are also discussed.     

基于非线性传递函数的非线性闭环控制系统的稳定性研究

基于Ｖｏｌｔｅｒｒａ级数的非线性传递函数表示,针对一类离散非线性系统,提出了直接利用开环稳定性判断系统闭环稳定性的新方法,并用实例仿真验证了此判据的有效性。     

Infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems

This paper studies exponential convergence index assignment of stochastic control systems from the viewpoint of backward stochastic differential equation. Like deterministic control systems, it is shown that the exact controllability of an open-loop stochastic system is equivalent to the possibility of assigning an arbitrary exponential convergence index to the solution of the closed-loop stochastic system, formed by means of suitable linear feedback of the states. As an application, a sufficient and necessary condition for the existence and uniqueness of the solution of a class of infinite horizon forward-backward stochastic differential equations is provided.     

Proof of quasi-adaptivity for the m-measurement feedback class of stochastic control policies

It is shown that them-measurement feedback (mM) policy for nonlinear stochastic control, performs as well or better than the open-loop optimal policy and hence becomes one of only a few suboptimal policies presently known to be quasi-adaptive in the sense of Witsenhausen.     

一种新的随机多变量自适应极点配置控制器

本文提出的随机多变量自适应极点配置控制器,由于采用了一种新的极点配置结构,它不仅可以避免求解伪交换矩阵方程而实现任意极点配置,跟踪时变参考输入,而且可以抑制随机噪声干扰.该控制器不仅可以控制开环不稳定或非最小相位系统,而且还可以控制具有任意、未知或变化的延时结构和具有任意输入输出个数的随机多变量系统。     

A modeling and control approach to magnetic levitation system based on state-dependent ARX model

Magnetic levitation (Maglev) systems are usually strongly nonlinear, open-loop unstable and fast responding. In order to control the position of the steel ball in a Maglev system, a data-driven modeling approach and control strategy is presented in this paper. A state-dependent AutoRegressive with eXogenous input (SD-ARX) model is built to represent the dynamic behavior between the current of electromagnetic coil and the position of the ball. State-dependent functional coefficients of the SD-ARX model are approximated by Gaussian radial basis function (RBF) neural networks. The model parameters are identified offline by applying the structured nonlinear parameter optimization method (SNPOM). Based on the model, a predictive controller is designed to stabilize the magnetic levitation ball to a given position or to make it track a desired trajectory. The real-time control results of the proposed approach and the comparisons with other two approaches are given, which demonstrate that the modeling and control method presented in this paper are very effective and superior in controlling the fast-responding, strongly nonlinear and open-loop unstable system. This paper gives the real experimental evidence that the RBF-ARX model is capable of not only globally, but also locally capturing and quantifying a nonlinear and fast-response system's behavior, and the model-based predictive control strategy is able to work quite well in a wide working-range of the nonlinear system.     

Feedback stabilization of discrete-time quantum systems subject to non-demolition measurements with imperfections and delays

We consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. In open-loop, the measurements are assumed to be quantum non-demolition (QND). The eigenstates of the measured observable are thus the open-loop stationary states: they are used to construct a closed-loop supermartingale playing the role of a strict control Lyapunov function. The parameters of this supermartingale are calculated by inverting a Metzler matrix that characterizes the impact of the control input on the Kraus operators defining the Markov process. The resulting state feedback scheme, taking into account a known constant delay, provides the almost sure convergence to the target state. This convergence is ensured even in the case where the filter equation results from imperfect measurements corrupted by random errors with conditional probabilities given as a left stochastic matrix. Closed-loop simulations corroborated by experimental data illustrate the interest of such nonlinear feedback scheme for the photon box, a cavity quantum electrodynamics system.     

随机部分可积拟哈密顿系统的概率密度追踪控制

目前非线性随机系统的控制方法存在设计复杂,计算成本高,以及缺乏稳定性或收敛性证明等缺点,针对这些问题,本文在作者前期研究的基础上发展了一种全新的针对部分可积的非线性随机系统的反馈控制,使得受控系统输出的稳态概率密度逼近事先给定的目标概率密度,并利用Lyapunov函数法证明受控系统的收敛性.数学仿真结果证明了这种方法的可行性和正确性.     

非线性模型经P控制后的奇异性分析*

对于一类开环具有三个定态的非线性系统,应用三阶台劳级数近似表达式足以揭示经P控制后的闭环系统的奇异特性。特别当设定点是拐点时,闭环系统的定态解分支呈音叉型分枝特性。此方法能减少计算工作量,亦能揭示设定的误差所引起闭环系统定态解分枝的分裂现象。这样的简化模型有利于反应器的设计和调节。     

Sensitivity limitations in nonlinear feedback control

In this paper we define nonlinear sensitivity and complementary sensitivity operators of a feedback control loop and show that they satisfy a complemntarity constraint. We then consider the case of general nonlinear open-loop operators that give rise to nonlinear sensitivities that are Lipschitz operators on some Banach space. Under these conditions, we obtain lower bounds on the Lipschitz constants of both operators for open-loop nonminimum phase and unstable nonlinear systems. These results parallel those known in linear control theory on the H_∞ norms of S and T. We finally point to the relevance of the defined nonlinear sensitivities in robustness issues.     

Stochastic observability in network state estimation and control

A hidden Markov model for the traffic congestion control problem in transmission control protocol (TCP) networks is developed, and the question of observability of this system is posed. Of specific interest are the dependence of observability on the congestion control law and the interaction between observability ideas and the effectiveness of feedback control. Analysis proceeds with a survey of observability concepts and an extension of some available definitions for linear and nonlinear stochastic systems. The key idea is to link the improvement of state estimator performance to the conditioning on the output data sequence. The observability development proceeds from linear deterministic systems to linear Gaussian systems, nonlinear systems, etc., with backwards compatibility to deterministic ideas. The principal concepts relate to the entropy decrease of scalar functions of the state, which in the linear case are describable in terms of covariance matrices. A feature of nonlinear systems is that the estimator properties may affect the closed-loop control performance. Results are derived linking stochastic reconstructibility to strict improvement of the optimal closed-loop control performance over open-loop control for the hidden Markov model. The entropy provides a means to quantify and thus order simulation results for a simplified TCP network. Motivated by the link between feedback control and reconstructibility, the entropy formulation is also explored as a means to discriminate between different control strategies for improving estimator performance. This approach has connections to dual-adaptive control ideas, where the control has the simultaneous and opposing goals of regulating the system and of exciting the system to prevent estimator divergence.     

离散非线性系统开闭环P型迭代学习控制律及其收敛性

本文在讨论了一般开环与闭环迭代学习控制的不足后，针对一类离散非线性系统，提出了新的开闭环ＰＧ型迭代学习控制律，给出了它的收敛性证明，仿真结果表明：开闭环Ｐ型迭代律优于单纯的开环或产才环Ｐ型迭代 律。     

Sensitivity analysis of linear control systems with random plan parameters

Abstract-The effects of random variations of the plant parameters in multivariable linear control systems are analyzed in terms of stochastic sensitivity operators. Both tie-varying and time-invariant control systems are analyzed. Conditions are derived for the stochastic sensitivity operator which are sufficient to guarantee that the closed-loop realization of the control system is less sensitive to the random parameter variations than the nominally equivalent open-loop system.     

一类双悬臂含间隙振动系统的混沌控制策略研究

由于一类双悬臂含间隙振动系统具有典型非光滑特性和有明显的非线性,这直接导致了系统发生分叉与混沌现象的可能性.为此针对该系统的混沌现象,利用基于能量的开环控制策略,构造有界控制器对混沌行为进行控制,混沌运动可被引导到稳定的目标周期轨道,并对控制的收敛速度进行分析,数值模拟结果表明了该控制策略的有效性与可行性,可为碰振系统的优化设计,振动控制和安全运行提供了理论参考.     

非线性随机系统的概率密度追踪控制

针对目前非线性随机系统控制方法的设计复杂、 计算成本高以及缺乏稳定性或收敛性证明等缺点, 提出了一种全新的基于等效非线性系统法求近似稳态解的思想设计的非线性随机系统的反馈控制, 使受控系统输出的稳态概率密度函数逼近事先给定的目标概率密度函数. 利用 Lyapunov 函数法证明了受控系统的收敛性. 数学仿真结果证明了这种方法的可行性和正确性.     

Exponential stabilization of a class of unstable bilinear systems

Considers the control design of bilinear systems with multiplicative control inputs. Previous control designs for such systems normally assume that the open-loop bilinear system is (neutrally) stable. In this paper, a nonlinear control design is proposed for open-loop unstable bilinear systems. The new control stabilizes the bilinear system globally and exponentially if a sufficient stability condition, which can be checked by off-line computer simulations in advance of the control, is satisfied     

Discrete-time optimal control for stochastic nonlinear polynomial systems

This paper presents a solution to the discrete-time optimal control problem for stochastic nonlinear polynomial systems over linear observations and a quadratic criterion. The solution is obtained in two steps: the optimal control algorithm is developed for nonlinear polynomial systems by considering complete information when generating a control law. Then, the state estimate equations for discrete-time stochastic nonlinear polynomial system over linear observations are employed. The closed-form solution is finally obtained substituting the state estimates into the obtained control law. The designed optimal control algorithm can be applied to both distributed and lumped systems. To show effectiveness of the proposed controller, an illustrative example is presented for a second degree polynomial system. The obtained results are compared to the optimal control for the linearized system.     

On the open-loop solution of linear stochastic optimal control problems

We consider open-loop solutions of linear stochastic optimal control problems with constraints on control variables and probabilistic constraints on state variables. It is shown that this problem reduces to an equivalent linear deterministic optimal control problem with similar constraints and with a new criterion to minimize. Concavity or convexity is preserved. Hence, the machinery available for solving deterministic optimal control problems can be used to get an open-loop solution of the stochastic problem. The convex case is investigated and a bound on the difference between closed-loop and open-loop optimal costs is given.     

Approximate controllability of fractional stochastic evolution equations

A class of dynamic control systems described by nonlinear fractional stochastic differential equations in Hilbert spaces is considered. Using fixed point technique, fractional calculations, stochastic analysis technique and methods adopted directly from deterministic control problems, a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations is formulated and proved. In particular, we discuss the approximate controllability of nonlinear fractional stochastic control system under the assumptions that the corresponding linear system is approximately controllable. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result. Finally as a remark, the compactness of semigroup is not assumed and subsequently the conditions are obtained for exact controllability result.