Lie algebraic methods in optimal control of stochastic systems with exponential-of-integral cost

The purpose of this paper is to formulate and study the optimal control of partially observed stochastic systems with exponential-of-integral-sample cost, known as risk-sensitive problems, using Lie algebraic tools. This leads to the introduction of the sufficient statistic algebra, , through which one can determine á priori the maximum order of the controller. When , the construction of the control laws is addressed through extensions of the Wei–Norman method, as in nonlinear filtering problems. Aside from specific known finite-dimensional examples which are studied in order to delineate the application of the Lie algebraic tools, new classes of finite-dimensional controllers are identified as well. In addition, relations with minimax dynamic games are explored to best assess the importance and generality of the finite-dimensional control systems.     

Step response of a class of second-order non-minimum phase transfer functions†

This paper investigates the problem of controlling a discrete-time linear system with jump parameters. A review of the literature is presented as well as a development of the application of dynamic programming to this class of control problems. Dynamic programming has been applied by many researchers and it was observed that no closed-form analytical solution could be constructed because of the ‘dual’ aspects of the controller. The main contribution of the present work is an algorithm, suitable for computer implementation, for the optimal dual control. The construction of the algorithm is based on transforming the dynamic programming relations into a space of sufficient statistics and using a finite-dimensional optimization procedure to obtain the optimal control as a function of the statistics. This is achieved by first developing a suitable recursive realization of a ‘filter’ which generates the sufficient statistics for the problem and then embedding this filter into the dynamic programming equations. Several examples are presented to illustrate the use of the algorithm for constructing optimal and suboptimal controllers.     

Global stabilization of the Kuramoto–Sivashinsky equation via distributed output feedback control

This work addresses the problem of global exponential stabilization of the Kuramoto–Sivashinsky equation (KSE) subject to periodic boundary conditions via distributed static output feedback control. Under the assumption that the number of measurements is equal to the total number of unstable and critically stable eigenvalues of the KSE and a necessary and sufficient stability condition is satisfied, linear static output feedback controllers are designed that globally exponentially stabilize the zero solution of the KSE. The controllers are designed on the basis of finite-dimensional approximations of the KSE which are obtained through Galerkin's method. The theoretical results are confirmed by computer simulations of the closed-loop system.     

Finite-dimensional constrained fuzzy control for a class of nonlinear distributed process systems.

This correspondence studies the problem of finite-dimensional constrained fuzzy control for a class of systems described by nonlinear parabolic partial differential equations (PDEs). Initially, Galerkin's method is applied to the PDE system to derive a nonlinear ordinary differential equation (ODE) system that accurately describes the dynamics of the dominant (slow) modes of the PDE system. Subsequently, a systematic modeling procedure is given to construct exactly a Takagi-Sugeno (T-S) fuzzy model for the finite-dimensional ODE system under state constraints. Then, based on the T-S fuzzy model, a sufficient condition for the existence of a stabilizing fuzzy controller is derived, which guarantees that the state constraints are satisfied and provides an upper bound on the quadratic performance function for the finite-dimensional slow system. The resulting fuzzy controllers can also guarantee the exponential stability of the closed-loop PDE system. Moreover, a local optimization algorithm based on the linear matrix inequalities is proposed to compute the feedback gain matrices of a suboptimal fuzzy controller in the sense of minimizing the quadratic performance bound. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod.     

On an assumed convergence result in the LQG/LTR technique

The finite-dimensional linear quadratic Gaussian/loop transfer recovery (LQG/LTR) technique has proven to be a useful design tool for enhancing the stability robustness properties of systems controlled with LQG controllers. However, there exists an assumed convergence result in the derivation of the technique that has not, to the authors' knowledge, been proven. An alternate formulation that is sufficient to ensure the validity of the LQG/LTR technique is presented     

Stationary LQG control of singular systems

The stationary linear-quadratic-Gaussian control problem is formulated and solved for single-input single-output singular systems. The control system is required to be internally proper and stable in order to avoid both impulsive and unstable exponential behavior. The set of all controllers resulting in such a control system is specified in parametric form. All controllers that yield finite cost are identified, once again in parametric form, within this set. Necessary and sufficient conditions are then established for an optimal controller to exist. All optimal controllers are shown to possess the same transfer function. The problem is analyzed in the complex domain. The transfer functions are expressed as quotients of proper, strict-Hurwitz rational functions. By means of this maneuver, the powerful tools of algebra are made available. The synthesis of the optimal controller is reduced to the solution of two linear Diophantine equations whose coefficients are obtained by spectral factorization.     

On ℋ∞ control for dead-time systems

A mixed sensitivity ℋ_∞ problem is solved for dead-time systems. It is shown that for a given bound on the ℋ_∞-norm causal stabilizing controllers exist that achieve this bound if and only if a related finite-dimensional Riccati equation has a solution with a certain nonsingularity property. In the case of zero time delay, the Riccati equation is a standard Riccati equation and the nonsingularity condition is that the solution be nonnegative definite. For nonzero time delay, the nonsingularity condition is more involved but still allows us to obtain controllers. All suboptimal controllers are parameterized, and the central controller is shown to be a feedback interconnection of a finite-dimensional system and a finite memory system, both of which can be implemented. Some ℋ_∞ problems are rewritten as pure rational ℋ _∞, problems using a Smith predictor parameterization of the controller     

Algorithmic Minimal Sufficient Statistics: a New Approach

We introduce the notion of a strong sufficient statistic for a given data string. We show that strong sufficient statistics have better properties than just sufficient statistics. We prove that there are “strange” data strings, whose minimal strong sufficient statistic have much larger complexity than the minimal sufficient statistic.     

Finite settling time stabilisation for multivariable discrete-time systems: a polynomial equation approach

The multivariable case of Finite Settling Time Stabilisation (FSTS) of linear discrete-time systems is considered in this paper. An algebraic approach is adopted which leads to the solution of a polynomial matrix Diophantine equation. This gives rise to the parametrisation of all FSTS controllers in a Ku?era–Youla–Bonjiorno sense and the FSTS problem is further reduced to a linear algebra problem over the real numbers. Subsequently, the family of all causal FSTS controllers is parametrised, and necessary and sufficient conditions for strong FSTS (stable controllers) are derived. The minimal McMillan degree solution and minimal complexity controllers are examined and new bounds are given. The analysis provides the means for the parametrisation of families of FSTS controllers with certain complexity. Finally the problems of tracking, disturbance rejection and partially assigned dynamics in FST sense are considered and conditions for their solvability are given.     

The graph topology for linear plants with applications to nonlinearrobust stabilization

The graph topology is defined for the set of finite-dimensional linear time-varying plants that are internally stabilizable by output feedback. This follows some earlier results of M. Vidyasagar et al. (1982) on time-invariant systems. Feedback connections of linear time-varying plants and general nonlinear time-varying controllers are considered, and it is shown that the graph topology is the weakest topology in which feedback stabilization is a robust property. Using this result, necessary and sufficient conditions for robust stabilizability of families of linear time-varying plants with parametric uncertainty are derived     

Decentralized blocking zeros and the decentralized strongstabilization problem

This paper is concerned with a new system theoretic concept, decentralized blocking zeros, and its applications in the design of decentralized controllers for linear time-invariant finite-dimensional systems. The concept of decentralized blocking zeros is a generalization of its centralized counterpart to multichannel systems under decentralized control. Decentralized blocking zeros are defined as the common blocking zeros of the main diagonal transfer matrices and various complementary transfer matrices of a given plant. As an application of this concept, we consider the decentralized strong stabilization problem (DSSP) where the objective is to stabilize a plant using a stable decentralized controller. It is shown that a parity interlacing property should be satisfied among the real unstable poles and real unstable decentralized blocking zeros of the plant for the DSSP to be solvable. That parity interlacing property is also sufficient for the solution of the DSSP for a large class of plants satisfying a certain connectivity condition. The DSSP is exploited in the solution of a special decentralized simultaneous stabilization problem, called the decentralized concurrent stabilization problem (DCSP). Various applications of the DCSP in the design of controllers for large-scale systems are also discussed     

On Solvability of a Decentralized Supervisory Control Problem With Communication

We study a decentralized control problem of discrete event systems with communication. We first give a general automata-theoretic formalism that enables us to handle various types of communication, including delay. The decentralized control problem is shown to be undecidable under this formalism. Next we propose a semi-decision procedure for computing finite-state controllers to achieve a given specification in the sense of bisimilarity between the controlled system and a given specification. Using this procedure, we prove decidability of the problem for two special cases, one is the case in which the communication behavior is given as $k$ -bounded-delay communication, and the other is the case in which any cycle in the state transition diagram of the system contains an event observable by all controllers. We also show a method for optimizing controllers based on a graph problem.      

Designing control systems using exact and symbolic manipulations of formulae

The potential of symbolic algebra for the design of control systems is illustrated through several examples. It has been shown in particular, that (a) the exact calculations provided by symbolic algebra can be used to utilize some of the direct (but numerically error prone) methods efficiently in control system design, and (b) symbolic manipulations by a computer can help control engineers at several stages of the design. The block diagram reduction, calculation of all stabilizing controllers, dominant pole assignment and robust pole assignment are taken as case studies.     

Synthesis of MIMO controllers for extending the stability range of periodic solutions in forced nonlinear systems

This paper deals with a central issue in bifurcations and chaos control applications, i.e., the stabilization of periodic motions in sinusoidally forced nonlinear systems. Specifically, the problem of designing multi-input-multi-output (MIMO) finite-dimensional linear time-invariant controllers maximizing the amplitude of the sinusoidal input for which the corresponding periodic solutions are guaranteed to be stable, is considered. By exploiting linearization techniques and the multi-variable circle criterion, a synthesis algorithm is developed to determine the controller which maximizes a suitable lower bound of the amplitude of the input. The algorithm requires the solution of a sequence of linear matrix inequalities (LMIs) of increasing size. The Brusselator oscillator is employed as a case study to show that the synthesized controllers, though optimizing a lower bound, provide quite satisfactory control performance.     

Boundary control of hyperbolic conservation laws using a frequency domain approach

The paper uses a frequency domain method for the boundary control of hyperbolic conservation laws. We show that the transfer function of the hyperbolic system belongs to the Callier-Desoer algebra, which opens the way of sound results, and in particular to the existence of a necessary and sufficient condition for the closed loop stability and the use of Nyquist type test. We examine the link between input-output stability and exponential stability of the state. Specific results are then derived for the case of proportional diagonal boundary controllers. The results are illustrated in the case of diagonal boundary control of open channel flow.     

H∞ control for linear time-varying systems:controller parameterization

In this correspondence, the H^∞ control problem is studied for finite-dimensional linear time-varying (FDLTV) systems. State-space formulas are derived for all FDLTV controllers solving the problem. These controllers are parameterized by the interconnection of the “central controller” with an exponentially stable, free system having H^∞ norm strictly less than γ. Our approach is drawn from some changes of variables, a time-varying version of a strict bounded real lemma, and Youla parameterization; thus, the proofs given are simple and clear     

An operator-theoretic approach to the mixed-sensitivity minimization problem

We consider the mixed-sensitivity minimization problem (scalar case). It gives rise to the so-called two-block problem on the algebra H^∞; we analyze this problem from an operator point of view, using Krein space theory. We obtain a necessary and sufficient condition for the uniqueness of the solution and a parameterization of all solutions in the non-uniqueness case. Moreover, an interpolation interpretation is given for the finite-dimensional case.     

A novel signal detection approach for fading channels with parameter statistics characteristics unknown

A novel signal detection approach is proposed in this paper. The approach exhibits the UMVUE of the probability density distribution function with respect to the received signal detection threshold value in wireless fading channels with unknown channel statistics characteristics. Multiscale particle filter is employed to extract the channel state information (CSI) from noisy observations available and to obtain a sufficient complete statistics of parameters. Based on the sufficient complete statistics, a conditional posterior probability density distribution for detection threshold value is derived via the Bayes statistic inference theorem in the presence of the additive white Gaussian noise (AWGN). Simulation results are given to illustrate the proposed novel scheme advantages.     

不确定广义系统的保性能控制

针对一类具有范数有界不确定性的连续广义系统和一种具有时间乘积形式的二次性能指标,研究其状态反馈保性能控制器的设计问题.这种性能指标能够使系统状态响应随着时间的增加更快地趋于零,从而缩短系统达到稳定所需要的时间.文中给出了最优保性能控制器存在的充分必要条件并基于双线性矩阵不等式(BMI)、二次矩阵不等式(QMI)和LMI给出其设计方法,最后利用SDP软件和BMI软件举例说明所得结论的可行性.     

A switched system approach to stabilization of networked control systems

A switched system approach is proposed to model networked control systems (NCSs) with communication constraints. This enables us to apply the rich theory of switched systems to analyzing such NCSs. Sufficient conditions are presented on the stabilization of NCSs. Stabilizing state/output feedback controllers can be constructed by using the feasible solutions of some linear matrix inequalities (LMIs). The merit of our proposed approach is that the behavior of the NCSs can be studied by considering switched system without augmenting the system. A simulation example is worked out to illustrate the effectiveness of the proposed approach.