Mean-field stochastic linear–quadratic optimal control with Markov jump parameters

This paper considers a class of mean-field stochastic linear–quadratic optimal control problems with Markov jump parameters. The new feature of these problems is that means of state and control are incorporated into the systems and the cost functional. Based on the modes of Markov chain, the corresponding decomposition technique of augmented state and control is introduced. It is shown that, under some appropriate conditions, there exists a unique optimal control, which can be explicitly given via solutions of two generalized difference Riccati equations. A numerical example sheds light on the theoretical results established.     

Time optimal control of linear systems with delay†

Necessary conditions, analogous to those of Pontryagin's maximum principle are developed for the problem of reducing the state, of linear systems with delay, to zero in minimum time. An example is presented that demonstrates the technique of finding the time optimal control.     

A new approach on designing l1 optimal regulator with minimum order for SISO linear systems

For a SISO linear discrete-time system with a specified input signal, a novel method to realize optimal l1 regulation control is presented. Utilizing the technique of converting a polynomial equation to its corresponding matrix equation, a linear programming problem to get an optimal l1 norm of the system output error map is developed which includes the first term and the last term of the map sequence in the objective function and the right vector of its constraint matrix equation, respectively. The adjustability for the width of the constraint matrix makes the trade-off between the order of the optimal regulator and the value of the minimum objective norm become possible, especially for achieving the optimal regulator with minimum order. By norm scaling rules for the constraint matrix equation, the optimal solution can be scaled directly or be obtained by solving a linear programming problem with l\ norm objective.     

A simple iterative method for sub-optimal control of linear time delay systems with quadratic cost†

The usual approach for obtaining the optimal control For a linear time-delay system with a quadratic cost consists of first deriving a set of necessary conditions for optimality and then using conventional iterative numerical methods to find a control satisfying those conditions. The burden of computation in this approach is enormous. The iterative scheme proposed in this paper does not proceed along these lines. Instead, the delay term is treated liko an extra perturbing input. A linear non-delay system is optimized at each stage, and the resulting sequence of control functions converges rapidly to the sub-optimal control for the original problem, as illustrated by two numerical examples.     

H∞ control of continuous-time Markov jump linear systems with detector-based mode information†

This paper features new results on H_∞ analysis and control of linear systems with Markov jump disturbances, in a scenario of partial observations of the jump process. We consider the situations in which the jump process can only be measured through a suitable detector. A distinctive feature of the approach here is that it is general enough to encompass particular scenarios such as that of perfect information, no information (mode independent) and cluster observations of the Markov jump process. The main results, comprising a new bounded real lemma and a condition for state feedback synthesis, are expressed via linear matrix inequality-based optimisation problems. The method devised for the design of H_∞ controllers is applied to the control of an unmanned aerial vehicle model.     

Stability analysis of linear systems with an interval time-varying delay – A delay-range-partition approach

In this paper, the stability analysis problem of linear systems with an interval time-varying delay is investigated. Firstly, an augmented Lyapunov-Krasovskii functional is constructed, which includes more information of the delay’s range and the delay’s derivative. Secondly, based on two improved integral inequalities which are less conservative than Jensen’s integral inequalities, a delay-range-partition (DRP) approach is proposed to estimate the upper bound of the derivative of the augmented Lyapunov-Krasovskii functional. Then, less conservative stability criteria in the form of linear matrix inequality (LMI) are established no matter whether the lower bound of delay is zero or not. Finally, to illustrate the effectiveness of the stability criteria proposed in this paper, two numerical examples are given, and their results are compared with the existing results.     

Decentralized linear–quadratic–Gaussian control of networked control systems with asymmetric information

The decentralized linear–quadratic–Gaussian (LQG) control problem for networked control systems (NCSs) with asymmetric information is investigated, where controller 1 shares its historical information with controller 2, and not vice versa. The asymmetry of the information structure leads to the coupling between controller 2 and estimator 1, and hence the classical separation principle fails. Through the assumption of linear control strategy, the coupling between controller 2 and estimator 1 (CCE) is decoupled, but the estimation gain is still coupled with the control gain. It is noted that the control gain conforms to the backward Riccati equation while estimation gain abides by the forward equation, which is computationally challenging. Applying the stochastic maximum principle, the solvability of the decentralized LQG control problem is reduced to that of corresponding forward and backward stochastic difference equations (FBSDEs). Further, necessary and sufficient conditions for the solvability of optimal control problem are presented by two Riccati equations, one of which is nonsymmetric. Moreover, a novel iterative forward method is proposed to calculate the coupled backward control gain and forward estimation gain.     

Inversion-based optimal output tracking–transition switching with preview for nonminimum-phase linear systems

In this article, the problem of nonperiodic tracking–transition switching with preview is considered. Such a control problem exists in applications including nanoscale material property mapping, robot manipulation, and probe-based nanofabrication, where the output needs to track the desired trajectory during the tracking sections, and rapidly transit to another point during the transition sections with no post-transition oscillations. Due to the coupling between the control of the tracking sections and that of the transition ones, and the potential mismatch of the boundary system state at the tracking–transition switching instants, these control objectives become challenging for nonminimum-phase systems. In the proposed approach, the optimal desired output trajectory for the transition sections is designed through a direct minimization of the output energy, and the needed control input that maintains the smoothness of both the output and the system state across all tracking–transition switching is obtained through a preview-based stable-inversion approach. The needed preview time is quantified by the characteristics of the system dynamics, and can be minimized via the recently developed optimal preview-based inversion technique. The proposed approach is illustrated through a nanomanipulation example in simulation.     

A computational method for convex optimal control problems involving linear hereditary systems†

We consider a class of convex optimal control problems involving a linear hereditary system. The main aim of the paper is to devise a computational algorithm for generating a minimizing sequence of controls such that the sequence converges to the optimal control in both the weak? topology of L∞ and the almost-everywhere topology.     

Integral action in the optimal control of linear systems with some inaccessible state variables†

Optimal feedback control for linear systems with the usual quadratic performance index which penalizes the state and the magnitude of the control is often difficult to apply because of the requirement of complete state measurement and lack of integral mode. A technique is presented, which introduces integral mode into the optimal control law and does not require the measurement of all of the state variables during the dynamic period. The control produced is equivalent to the original optimal Kalman feedback when external disturbance is absent, and during the period of constant disturbance the method yields a trajectory which can be made arbitrarily close to the optimal trajectory without disturbance.     

Robust ℓ 1 performance analysis for linear systems with parametric uncertainties

In this contribution, a computational approach for analysing the robust ?_∞-gain (or the robust ?₁ performance) of uncertain linear systems is developed. In particular, the system's state-space matrices may have a rational dependence on structured parametric time-invariant or time-varying uncertainties. The computation is based on robust semi-definite programming and provides a trade-off between accuracy and computational effort. A novel matrix inequality condition to determine the star-norm of discrete-time systems is derived as an auxiliary result.     

ℓ1-control using linear programming for systems with asymmetric bounds

In this paper, a novel way to handle the analysis and design of control systems with bounded asymmetrical signals is presented. Instead of the standard peak-to-peak norm, normally used in ?₁-optimal control, this paper proposes to use an asymmetric objective functional, which simplifies the consideration of asymmetrically bounded signals. Necessary and sufficient conditions are derived to check if some performance specifications, expressed using this objective functional, are fulfilled. The analysis and the synthesis problems are also formulated with respect to this objective functional: as in the standard ?₁ control, this approach leads to a linear programming problem which can be solved by available optimization tools. A numerical example is also studied to clarify this proposed approach.     

A pascal program for the solution of sparse linear systems – Part I

A Pascal program that implements the Gaussian elimination strategy for the solution of (very) sparse linear systems is presented.     

Delay-fractioning-dependent robust H ∞ control for linear systems with delay in state or input and its applications

A new delay fraction technique is proposed to investigate the H _∞ control of uncertain systems with time delay in state or input. First, the variation interval of the time-varying delay is divided into several subintervals with equal length. By checking the variation of derivative of the Lyapunov functional in every subinterval, some new criteria on H _∞ performance analysis of the systems are derived in terms of the convexity properties of a matrix inequality and some other new analysis techniques. Then, criteria for the H _∞ control design are presented, which are concluded from the H _∞ performance analysis results. As applications of the derived results, H _∞ performance analysis and H _∞ control design are carried out for some systems, including two numerical systems, linearised truck trailer system and traffic network system. Discussions show that, the proposed method is effective for these systems and much less conservativeness appears than those given in the existing references.     

Remarks on the paper “Optimal control of a class of linear multivariable systems with integral quadratic energy constraint ”

An optimal periodic control problem for a system described by differential equations is considered. Control units are assumed to generate control actions with the square-integrable derivative. The above problem is approximated by a sequence of discretized problems containing trigonometric polynomials, which approximate the state and control variables, and the functions in the criterion and differential equations. The conditions for a sequence of optimal solutions to discretized problems, which are to be a generalized minimizing sequence for the basic problem, are given. Extensions to more general problem formulations are presented. The possibility of application is illustrated by the example of an optimal periodic control problem for a chemical reactor.     

Comment on “Time optimal control of linear systems with delay” and “Optimal control for systems which include pure delays”

A basic numerical matrix of infinite dimension is highlighted which is of significance to the development of computer programmes for the design of sampled data systems. A few elegant properties of this matrix are briefly outlined