Further results on linear nonquadratic optimal control

Continues an investigation by the authors of minimizing the transient response of a linear system as measured by nonquadratic penalty functions, in particular, penalty functions which have linear growth. First, this note shows that the optimal state feedback which minimizes the transient response in the case of no exogenous inputs also minimizes the induced l¹ norm in case exogenous inputs are present. Second, it considers the case of constrained systems and derives bounds which establish the stability and performance of receding horizon control laws. Finally, this note illustrates the results for scheduling of reliable manufacturing systems     

Further results on the Bellman equation for optimal control problems with exit times and nonnegative Lagrangians

In a series of papers, we proved theorems characterizing the value function in exit time optimal control as the unique viscosity solution of the corresponding Bellman equation that satisfies appropriate side conditions. The results applied to problems which satisfy a positivity condition on the integral of the Lagrangian. This positive integral condition assigned a positive cost for remaining outside the target on any interval of positive length. In this note, we prove a new theorem which characterizes the exit time value function as the unique bounded-from-below viscosity solution of the Bellman equation that vanishes on the target. The theorem applies to problems satisfying an asymptotic condition on the trajectories, including cases where the positive integral condition is not satisfied. Our results are based on an extended version of “Barb lat's lemma”. We apply the theorem to variants of the Fuller problem and other examples where the Lagrangian is degenerate.     

Optimal rejection of persistent bounded disturbances

In this paper, we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude, but is otherwise arbitrary. The objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance. Mathematically, this corresponds to requiring uniformly good disturbance rejection over all time. Since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used (see [7, sect. 5.6]), the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose L2-norm is small, as is the case with the currently popularH_{infty}theory. The present theory is a natural counterpart to the existing theory of optimal disturbance rejection (the so-calledH_{infty}theory) which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square-integrable and has unit energy. It is shown that the problem studied here has quite different features from its predecessor. Complete solutions to the problem are given in several important cases, including those where the plant is minimum phase or when it has only a single unstable zero. In other cases, procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers.     

Further results on the subspace distance

In previous papers [L. Wang, X. Wang, J. Feng, Subspace distance analysis with application to adaptive Bayesian algorithm for face recognition, Pattern Recognition 39(3) (2006) 456-464, L. Wang, X. Wang, J. Feng, Intrapersonal subspace analysis with application to adaptive Bayesian algorithm for face recognition, Pattern Recognition 38(4) (2005) 617-621], we proposed a subspace distance. However, whether the subspace distance satisfies the triangle inequality was left open. In this note, we give positive answer to the open problem and prove our assertion.     

Further results on stability of

Our aim in this communication is to improve the existing results concerning the stability of . It is shown that the unstable characteristic roots could be more accurately located in a bounded and restrictive region of the complex plane. An iterative scheme is presented for numerical implementation of the results. Numerical computations are presented for illustration and comparison with previous results.     

Further results on the asymptotic agreement problem

The problem of reaching a concensus between two decision-makers provided with different information is considered. The problem in which the decision-makers may have different underlying probability models is studied. Results are developed to characterize the likelihood of an agreement being reached eventually in terms of the nature of the inter-decision-maker communications. The problem in which the decision-makers are aware of the possibility that they may have different models is treated. It is found that in this case a deadlock can be reached where neither decision maker can learn additional information from the concensus process and they cannot reach a concensus decision. This result indicates that incorporating human uncertainty in probability assessment into the asymptotic agreement problem can lead to outcomes not anticipated in the general theory previously developed     

Further results on the preconditioned sor method

Several methods have been developed on the preconditioned iterative methodsi. e iterative methods applied to preconditioned linear systems. Usui, Kohno and Niki [4] have proposed the adaptive Gauss-Seidel (GS) method, and the same authors [5] have presented the pre-conditioned SOR method. They have shown, with the aid of numerical examples, that the two methods have a better rate of convergence in comparison with the classical SOR method. In this paper we will prove theoretically the improvement in the rate of convergence.     

Further results on the observer design problem

In this paper the observer design problem for nonlinear systems is considered. Sufficient Lyapunov-like conditions are presented for the existence of a nonlinear observer. The theory we develop considerably improves and extends the results of our recent work [14].     

Further results on trajectory insensitivity

Further contributions are given to the problem of synthesizing an insensitivity generating control law for a linear system. The results refer to the possibility of achieving trajectory insensitivity for any given subspace of initial conditions without modifying the state controllability subspace of the system.     

Set-valued observers and optimal disturbance rejection

A set-valued observer (also called guaranteed state estimator) produces a set of possible states based on output measurements and models of exogenous signals. We consider the guaranteed state estimation problem for linear time-varying systems with a priori magnitude bounds on exogenous signals. We provide an algorithm to propagate the set of possible states based on output measurements and show that the centers of these sets provide optimal estimates in an l^∞-induced norm sense. We then consider the utility of set-valued observers for disturbance rejection with output feedback and derive the following general separation structure. An optimal controller can consist of a set-valued observer followed by a static nonlinear function on the observed set of possible states. A general construction of this function is provided in the scalar control case. Furthermore, in the special case of full-control, i.e., the number of control inputs equals the number of states, optimal output feedback controllers can take the form of an optimal estimate of the full-state feedback controller     

Further results on controllability and sensitivity

The structural uncontrollability of the sensitivity system is defined relative to a class of linear time-invariant systems. Previously obtained results are reviewed, and the new result stating structural uncontrollability, of the sensitivity, system for normal systems having the number of independent parameters not smaller than twice the number of control variables is proved.     

Further results on nonlinear receding-horizon observers

Further results are proposed that concern the design and the convergence of receding-horizon nonlinear observers. The key feature is the definition of observability radius in relation with a prespecified compact set of initial configurations. This enables a semiglobal convergence result to be derived that turns out to be a global convergence result when appropriate regularity assumptions are made. A simple example is proposed to illustrate the basic features of the note     

Further results on optimization of recognition time

Preceding studies on optimization of computational time in pattern recognizers started from strong hypotheses of separability of the classes by a known recognizer and from consideration of naive algorithms which implement the recognizer. Here, we consider weaker separability hypotheses, which allow for doubtful cases, and slightly more sophisticated algorithms. The expressions of the mean computational length and of its total variation actually valid are presented with their relation to the old ones. We give evidence for the fact that the old criterion for deciding about the optimality of an algorithm, by simple ordering of the class-probabilities, is still applicable in this new setting.     

An adaptive control with optimal disturbances rejection

This paper provides a way to optimize the overall disturbances rejection performance of the adaptive control system in the presence of unknown external disturbances.Especially,the updatable non-empty admissible model set,which is consistent to the a priori knowledge of the plant parameter and the online measurements,is computed.With the overall system performance as the criteria,the nominal model is optimally chosen within the admissible model set.The optimal nominal model is subsequently used to synthesize the optimal closed-loop controller based on the 1 design methodology.Combining the above two aspects,an optimal adaptive control scheme is proposed.Because of the consistency of the identification criteria and control object,the adaptive control scheme proposed in this paper can achieve the overall optimal disturbances rejection performance,and the effect of the interplay between the identification and control of the adaptive system can be handled effectively.In addition,the computable optimal performance is also provided.     

Further improvement on dynamic programming for optimal bit allocation

Dynamic programming algorithms based on Lagrange multiplier method is often used for obtaining an optimal bit allocation strategy to minimize the total distortion given a constrained rate budget in both source and channel coding applictions.Due to possible large quantizer set and improper initialization,the algorithm often suffers from heavy computational complexity.There have been may solutions in recent years to the above question.In this paper,a simple but efficient algorithm is presented to further speed up the convergence of the algorithm.This algorithm can be easily realized and get the final solution much faster.The experimental result shows that our new algorithm can figure out the optimal solution with a speed 5-7 times faster than the original algorithm.     

Further results on variable output feedback controllers

Zak and Hui (1993) proposed a sliding mode controller for linear multiple-input-multiple-output (MIMO) systems using static output feedback. The author's previous paper (1996) provides an improvement of the output feedback controller of Zak-Hui for a class of linear single-input-single-output (SISO) systems that eliminated two important limitations: 1) system uncertainties must be bounded by the system output; and 2) a requirement of a matrix inequality. The controller developed previously can guarantee global closed-loop stability. This paper extends the previous results to linear MIMO systems. It is emphasized that the proposed MIMO controller yields global closed-loop stability whereas the one in Zak-Hui can only guarantee local stability. An application of the proposed MIMO controller to an aircraft model is included to show the effectiveness of the method     

Further results on regular splittings and multisplittings

In this article a detailed study of the regularity of induced splitting from the two-stage and multisplitting iterative methods is made. We present sufficient conditions which guarantee that the induced splitting is still a regular one. Meanwhile, we have established two new comparison theorems for the induced splittings characterized by a monotone matrix. In addition, we have also applied these theorems to some concrete methods.