Robust stochastic maximum principle for multi-model worst case optimization

This paper develops a version of the robust maximum principle applied to the minimax Mayer problem formulated for stochastic differential equations with a control-dependent diffusion term. The parametric families of first and second order adjoint stochastic processes are introduced to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic Hamiltonians corresponding to each possible value of the parameter. The cost function is defined on a finite horizon and contains the mathematical expectation of a terminal term. A terminal condition, given by a vector function, is also considered. The optimal control strategies, adapted for available information, for the wide class of multi-model systems given by a stochastic differential equation with parameters from a given finite set are constructed. This problem belongs to the class of minimax stochastic optimization problems. The proof is based on the recent results obtained for deterministic minimax Mayer problem by Boltyanski and Poznyak as well as on the results of Zhou and of Yong and Zhou, obtained for stochastic maximum principle for non-linear stochastic systems with a single-valued parameter. Two illustrative examples, dealing with production planning and reinsurance-dividend management, conclude this study.     

Robust optimal control for minimax stochastic linear quadratic problem

The robust maximum principle applied to the minimax linear quadratic problem is derived for stochastic differential equations containing a control-dependent diffusion term. The parametric families of the first and second order adjoint stochastic processes are obtained to construct the corresponding Hamiltonian formalism. The Hamiltonian function used for the construction of the robust optimal control is shown to be equal to the sum of the standard stochastic Hamiltonians corresponding to each value of the uncertain parameter from a given finite set. The cost function is considered on a finite horizon (contains the mathematical expectation of both an integral and a terminal term) and on an infinite one (a time-averaged losses function). These problems belong to the class of minimax stochastic optimization problems. It is shown that the construction of the minimax optimal controller can be reduced to an optimization problem on a finitedimensional simplex and consists in the analysis of the dependence of Riccati equation solution on the weight parameters to be found.     

Mini-max integral sliding-mode control for multimodel linear uncertain systems

An original linear time-varying system with matched and unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external unmatched bounded disturbances. Such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system containing only matched uncertainties leads to a linear multi-model system with known unmatched bounded disturbances and unknown matched disturbances as well. Each model from a given finite set is characterized by a quadratic performance index. The developed minimax integral sliding mode control strategy gives an optimal minimax linear quadratic (LQ)-control with additional integral sliding mode term. The design of this controller is reduced to a solution of an equivalent mini-max LQ problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex. The additional integral sliding mode controller part completely dismisses the influence of matched uncertainties from the initial time instant. Two numerical examples illustrate this study.     

Decentralized state-feedback stabilization and robust control of uncertain large-scale systems with integrally constrained interconnections

This paper is concerned with a problem of stabilization and robust control design for interconnected uncertain systems. A new class of uncertain large-scale systems is considered in which interconnections between subsystems as well as uncertainties in each subsystem are described by integral quadratic constraints. The problem is to design a set of local (decentralized) controllers which stabilize the overall system and guarantee robust disturbance attenuation in the presence of the uncertainty in interconnections between subsystems as well as in each subsystem. The paper presents necessary and sufficient conditions for the existence of such a controller. The proposed design is based on recent absolute stabilization and minimax optimal control results and employs solutions of a set of game-type Riccati algebraic equations arising in H_∞ control.     

Min-max sliding-mode control for multimodel linear time varying systems

An original linear time-varying system with unmatched disturbances and uncertainties is replaced by a finite set of dynamic models such that each one describes a particular uncertain case including exact realizations of possible dynamic equations as well as external bounded disturbances. Such a tradeoff between an original uncertain linear time varying dynamic system and a corresponding higher order multimodel system with a complete knowledge leads to a linear multi-model system with known bounded disturbances. Each model from a given finite set is characterized by a quadratic performance index. The developed min-max sliding-mode control strategy gives an optimal robust sliding-surface design algorithm, which is reduced to a solution of an equivalent linear quadratic problem that corresponds to the weighted performance indices with weights from a finite dimensional simplex. An illustrative numerical example is presented.     

Robust tracking problem for continuous time stochastic uncertain systems

We propose a finite‐horizon robust minimax tracking controller design method for time‐varying continuous time stochastic uncertain systems. The uncertainty in the system is characterized by a set of probability measures under which stochastic noises, driving the system, are defined. A minimax optimal tracking controller is derived from the solution of a risk‐sensitive linear quadratic Gaussian control problem. Also a numerical example is presented to illustrate the characteristics of proposed tracking controller. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Convex and finite-dimensional conditions for controller synthesiswith dynamic integral constraints

In this paper, the problem of synthesizing controllers when the allowable disturbances and the cost criterion are defined via a finite number of matrix valued dynamic integral constraints, is solved. This allows one to design optimal control systems for plants subject to fixed-input disturbances (such as steps, sinusoids, and periodic disturbances), disturbances of fixed and known spectrum, disturbances defined as the set of signals in the kernel of a given operator, and combinations thereof, to penalize particular frequency components of the error variables in the control design process, to penalize the maximum amplitude of an error variable, and to consider other general cost criteria in the optimization, and to solve a large class of robust control synthesis problems. Necessary and sufficient conditions for the general problem to have a solution are in terms of a computationally attractive, finite dimensional linear matrix inequality     

Minimax estimation by probabilistic criterion

A minimax estimation problem in multidimensional linear regression model containing uncertain parameters and random quantities is considered. Simultaneous distribution of random quantities that are a part of the observation model is not prescribed exactly; however, it has a fixed mean and a covariance matrix from the given set. For estimation algorithm optimization, we applied a minimax approach with the risk measure in the form of the exceedance probability of the estimate of a prescribed level by an error. It was shown that a linear estimation problem is equivalent to the minimax problem with the mean-square criterion. In addition, the corresponding linear estimate will be the best (in the minimax sense) by the probabilistic criterion at the class of all unbiased estimates. The least favorable distribution of random model parameters is also constructed. Several partial cases and a numerical example are considered.     

Minimax linear observers and regulators for stochastic systems with uncertain second-order statistics

The problem of minimax design of linear observers and regulators for linear time-varying multivariable stochastic systems with uncertain models of their second-order statistics is treated in this paper. General classes of allowable covariance matrices and means of the process and observation noises and of the random initial condition are considered. A game formulation of the problem is adopted and it is shown that the optimal filter for the least favorable set of covariances is minimax robust for each of the filtering situations analyzed. Conditions satisfied by the saddle-point solutions are given, and their utility for finding the worst case covariances is illustrated by way of several examples of uncertainty classes of practical interest.     

Design of robust stable controls for nonlinear objects

We consider a problem of discrete control for a class of nonlinear time-varying objects. Only set estimations for object parameters are available. The aim is to design controls that ensure robust stability of closed-loop systems in a given domain of state space. Since the considered class of objects is large enough not to have a stabilizing control, the proposed design method has to verify at the last step if the obtained conditions of robust stability are satisfied for a nonlinear system “in a given domain.”     

Design of optimal control under uncertain initial conditions: A minimax approach

The need for a game-theoretical formulation of the problem of linear-quadratic control under nonmeasurable plant state where the functional depends on the uncertain initial state was substantiated. The minimax output control law in terms of linear matrix inequalities that may be regarded as the optimal robust control law for the given set of the plant initial states was established assuming that the measurable components of the initial state were known and the nonmeasurable components take on values within the given ellipsoid. The results obtained were generalized to the plants with uncertain parameters.     

An optimal control approach to the design of vibrating elastic-viscoelastic sandwich beams

This work features the application of an optimal control algorithm to a new class of continuous one-dimensional structural design problems. A sandwich beam of rectangular cross-section is considered. It has a variable-thickness core and two equal variable-thickness cover layers and is subjected to harmonic forced vibrations. The objective is to distribute both the core and layer mass so as to minimize a measure of dynamic compliance for forced steady-state vibration and fixed material volumes. Either or both materials may be viscoelastic. Any constitutive relation may be used provided it is linear and time-invariant.The design problem is formulated as an optimal control problem. The resulting problem contains ten state variables, two control functions, four control parameters, and six terminal state constraints. Simple transformations are used to treat the minimum-gage constraints. A conjugate gradient/gradient projection optimal control algorithm is then used to obtain numerical solutions. Several optimal beam designs are presented and compared for a variety of problem parameter values.     

Robust control of a linear object under parametric uncertainty

A robust control problem for a linear nonstationary system under condition of incomplete information about its parameters is considered. Problems of robust and d-robust stabilization are formulated, and conditions of existence of their solutions are found. To construct robust control, minimax technique is used. Conditions of equivalence of solutions to the classical control problem and to that in the minimax formulation are determined.     

不确定噪声下确保控制性能的鲁棒LQG调节器

本文讨论了一类具有不确定噪声的离散时间随机线性系统的鲁棒ＬＱＧ问题，文章给出了确保控制性能的不确定噪声协方差矩阵的扰动上界，以及极小极大鲁棒ＬＱＧ调节器的设计方法，采用这种调节器不仅能极小化不确定下的最坏性能，而且也能确保控制性能指标达到给定的自由度内。     

Towards the optimal control of Markov chains with constraints

An optimal control problem with constraints is considered on a finite interval for a non-stationary Markov chain with a finite state space. The constraints are given as a set of inequalities. The optimal solution existence is proved under a natural assumption that the set of admissible controls is non-empty. The stochastic control problem is reduced to a deterministic one and it is shown that the optimal solution satisfies the maximum principle, moreover it can be chosen within a class of Markov controls. On the basis of this result an approach to the numerical solution is proposed and its implementation is illustrated by examples.     

Optimization over state feedback policies for robust control with constraints

This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.     

Robust Control System Design for an Uncertain Nonlinear System Using Minimax LQG Design Method

A systematic approach to design a nonlinear controller using minimax linear quadratic Gaussian regulator (LQG) control is proposed for a class of multi‐input multi‐output nonlinear uncertain systems. In this approach, a robust feedback linearization method and a notion of uncertain diffeomorphism are used to obtain an uncertain linearized model for the corresponding uncertain nonlinear system. A robust minimax LQG controller is then proposed for reference command tracking and stabilization of the nonlinear system in the presence of uncertain parameters. The uncertainties are assumed to satisfy a certain integral quadratic constraint condition. In this method, conventional feedback linearization is used to cancel nominal nonlinear terms and the uncertain nonlinear terms are linearized in a robust way. To demonstrate the effectiveness of the proposed approach, a minimax LQG‐based robust controller is designed for a nonlinear uncertain model of an air‐breathing hypersonic flight vehicle (AHFV) with flexibility and input coupling. Here, the problem of constructing a guaranteed cost controller which minimizes a guaranteed cost bound has been considered and the tracking of velocity and altitude is achieved under inertial and aerodynamic uncertainties.     

Reducing the computation required to solve a standard minimax problem

Minimax optimization problems have a long and rich history in the area of control. We show how the computation required to find the solution of a popular and widely applicable minimax problem can be significantly reduced. This reduction in computation results from an observation concerning the inner level (finite) maximization. In particular, we show that the number of parameter combinations that must be considered may be significantly reduced. We next indicate how this optimization problem can be used to synthesize a robust state feedback control for a system with parameter uncertainty. We conclude with an example robust control design problem that has 15 independent uncertain parameters and would not be practical were it not for the reduced computational requirement.     

Worst-case estimation for econometric models with unobservable components

A worst-case estimator for econometric models containing unobservable components, based on minimax principles for optimal selection of parameters, is proposed. Worst-case estimators are robust against the averse effects of unobservables. Computing worst-case estimators involves solving a minimax continuous problem, which is quite a challenging task. Large sample theory is considered, and a Monte Carlo study of finite-sample properties is conducted. A financial application is considered.     

Uncertainty modeling and robust minimax LQR control of multivariable nonlinear systems with application to hypersonic flight

For a class of multi‐input and multi‐output nonlinear uncertainty systems, a novel approach to design a nonlinear controller using minimax linear quadratic regulator (LQR) control is proposed. The proposed method combines a feedback linearization method with the robust minimax LQR approach in the presence of time‐varying uncertain parameters. The uncertainties, which are assumed to satisfy a certain integral quadratic constraint condition, do not necessarily satisfy a generalized matching condition. The procedure consists of feedback linearization of the nominal model and linearization of the remaining nonlinear uncertain terms with respect to each individual uncertainty at a local operating point. This two‐stage linearization process, followed by a robust minimax LQR control design, provides a robustly stable closed loop system. To demonstrate the effectiveness of the proposed approach, an application study is provided for a flight control problem of an air‐breathing hypersonic flight vehicle (AHFV), where the outputs to be controlled are the longitudinal velocity and altitude, and the control variables are the throttle setting and elevator deflection. The proposed method is used to derive a linearized uncertainty model for the longitudinal motion dynamics of the AHFV first, and then a robust minimax LQR controller is designed, which is based on this uncertainty model. The controller is synthesized considering seven uncertain aerodynamic and inertial parameters. The stability and performance of the synthesized controller is evaluated numerically via single scenario simulations for particular cruise conditions as well as a Monte‐Carlo type simulation based on numerous cases. It is observed that the control scheme proposed in this paper performs better, especially from the aspect of robustness to large ranges of uncertainties, than some controller design schemes previously published in the literature. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society