On the optimal control of discrete-time linear systems with random parameters

This work is concerned with the optimal control of a discrete-time linear system with random parameters. It is assumed that the parameters of the system vary randomly during the process, namely, the parameters constitute sequences of random variables. These random variables are not necessarily independent. An important particular case occurs where there are unknown constant parameters in the system. The measurements of the state of the system contain additive noise. A quadratic function of the state and controller, with appropriate weighting, serves as the criterion function. The solutions for the open-loop controller and the open-loop feedback controller are presented. The method of solution is based on the dynamic programming approach which leads to functional recurrence equations.     

Optimal control of linear plants with random parameters

The subject of this paper is the optimal control of linear plants whose coefficients and whose reference signals are random processes. The discrete-time as well as the continuous-time problems are treated and some fairly deep-seated distinctions between the two are pointed out. Special emphasis is placed on control over long periods of time. A set of assumptions is laid down under which the transition to infinitely long periods of operation is tractable. It is shown that a discrete-time plant is controllable, in the sense that the rate of the mean-squared error remains finite, only when certain necessary and sufficient conditions are fulfilled by the statistics of the plant parameters.     

An actively adaptive control for linear systems with random parameters via the dual control approach

A new method is presented for controlling a discrete-time linear system with possibly time-varying random parameters in the presence of input and output noise. The cost is assumed to be quadratic in the state and control. Previous algorithms for the above problem when the system had both zeros and poles unknown were of the open-loop feedback type, i.e., they did not take into account that future observations will be made. Therefore, even though these schemes were adaptive, their learning was "accidental." In contrast to this, the new approach uses an expression of the optimal cost-to-go that exhibits the dual purpose of the control, i.e., learning and control. The effect of the present control on the future estimation ("learning") appears explicitly in the cost used in the stochastic dynamic programming equation. The resulting sequence of controls, which is of the closed-loop type, is shown via simulations to appropriately divide its energy between the learning and the control purposes. Therefore, this control is called actively adaptive because it regulates the speed and amount of learning as required by the performance index. The simulations on a third-order system with six unknown parameters also demonstrate the computational feasibility of the proposed algorithm.     

Stability of linear dynamic systems with random time-dependent parameters

We present conditions of asymptotic stability for the solution of a system of linear differential equations that depend on stochastic processes.Translated from Kibernetika, No. 3, pp. 70–72, 75, May–June 1990.     

On the stability of linear time-varying feedback systems with random parameters

The problem of finding frequency-domain conditions which guarantee asymptotic stability with probability 1 of linear systems with a time-varying feedback controller and having white noise parameter variations is considered. It is shown that the stochastic version of the second method of Liapunov can be employed for deriving sufficient conditions for stability of such systems. Stability criteria are derived after taking into account restrictions on the gain as well as on the rate of change of the gain of the controller.     

The optimal control of linear systems with unknown parameters

The optimal open-loop controller for a linear system with some unknown parameters is determined. It is assumed that the unknown parameters do not vary during the process, and that their probability distribution function is given. The criterion for optimality is a quadratic function of the state and the input whoso expectation with respect to the unknown parameters has to be minimized. An explicit solution for the optimal controller is obtained by using the methods of the calculus of variations. Further, an alternate approach is presented which, for the problem solved, leads to the same optimal controller.     

Sensitivity analysis of the linear nonconservative systems with fractional damping

In this paper, the influence of small structural perturbation on a linear, nonconservative dynamical system exhibiting fractional bifurcation was investigated. In considering design problems for nonconservative systems, the integral structural characteristics as fundamental frequencies, critical loads for instability, and the sensitivity analysis play an important part. In this paper, the influence of small perturbation on a linear, nonconservative dynamical system exhibiting a flutter type bifurcation was investigated. The hereditary damping is described by means of fractional derivatives. To study the dynamical instability for nonconservative governing equations with fractional damping, the method of auxiliary eigenvalue problem is applied. The stability conditions of generalized Lyapunov type for the system with hereditary damping were derived. A new analytical framework for the coupled optimization of aero-structural, fractionally damped systems is presented. The approach to obtain aerodynamic sensitivities is based on adjoint systems.     

Suboptimal control of linear stochastic multivariable systems with unknown parameters

An algorithm is proposed for suboptimal control of linear multivariable systems with unknown parameters and output noise covariances. This algorithm is based on the idea of explicitly separating the functions of identification, estimation and control. The parameters and states of the system are estimated in a bootstrap manner by the stochastic approximation method. A suboptimal controller is then obtained which utilizes a deterministic control gain derived using the estimates obtained from the parameter and state estimators. This suboptimal controller will approach the optimal strategy when the estimated system parameters approach their true values.     

Sensitivity analysis of linear systems using orthogonal functions

Some useful properties of coefficients for determining the sensitivity functions of linear systems are given. These properties simplify the numerical computations of sensitivity functions with respect to the system parameters.     

Optimal open-loop feedback control for linear systems with unknown parameters

The problem of controlling a class of linear systems with unknown parameters is considered. The optimal open-loop strategies are obtained for linear systems with unknown parameters in the system matrix with a quadratic performance index. The method of solution is based upon the use of the Hamilton-Jacobi theory.     

具有时延的网络控制系统控制器设计

针对一类存在随机时延的网络控制系统,传感器采用时间驱动,控制器和执行器采用事件驱动,提出了一种新的具有随机时延的网络控制系统的建模方法-离散模糊T-S模型,在此模型的基础上应用并行分布补偿（PDC）原理设计了模糊控制器。应用Lyapunov定理和线性矩阵不等式（LMI）方法,研究了系统的稳定性问题,给出基于LMI的状态反馈模糊控制器的设计方法。通过仿真实例验证控制方法能够保证系统稳定。     

Optimal control of continuous-time linear systems with a time-varying, random delay

We study a finite horizon optimal control problem for a class of linear systems with a randomly varying time-delay. The systems of this type may arise in embedded control applications and in certain applications in economics. The delay value is treated as an unknown variable but with known statistical properties, modelled by a Markov process with a finite number of states. The “probabilistic delay averaging” approach is employed to determine the optimal control in the form which is independent of the delay value.     

Feedback control of a class of linear systems with jump parameters

A class of linear systems are studied which are subject to sudden changes in parameter values. An algorithm similar in form to Kushner's stochastic maximum principle is derived and the relationship between these algorithms discussed. Systems in which the performance measure is quadratic are investigated in detail and a differential equation is derived which yields the optimal feedback gains.     

Distributed control of multi-agent systems with random parameters and a major agent

Distributed control of the multi-agent systems involving a major agent and a large number of minor agents is investigated in this paper. There exist Markov jump parameters in the dynamic equation and random parameters in the index functions. The major agent has salient impact on others. Each minor agent merely has tiny influence, while the average effect of all the minor agents is not negligible, which plays a significant role in the evolution and performance index of each agent. Besides the state of the major agent, each minor agent can only access to the information of its state and parameters. Based on the mean field (MF) theory, a set of distributed control laws is designed. By the probability limit theory, the uniform stability of the closed-loop system and the upper bound of the corresponding index values are obtained. Via a numerical example, the consistency of the MF estimation and the influence of the initial state values and parameters on the index values are demonstrated.     

Sensitivity analysis of LP-MPC cascade control systems

Model predictive control (MPC) has found wide application in the chemical process industry as well as other industrial sectors. Commercial MPC systems are typically implemented in conjunction with a steady-state linear or quadratic programming optimizer, whose key functions are to track the economic optimum and to provide feasible set-points to the model predictive controller. The two-level system is complementary to real-time optimization which typically utilizes more complex models and is executed less frequently. Despite the widespread adoption of LP-MPC systems, occurrences of poor performance have been reported, where large variations in the computed set-points were observed. In this paper, we analyze the sensitivity of the LP solution to variation in the LP model bias, through which feedback to the LP layer occurs. We consider both multi-input, single-output (MISO) and multi-input, multi-output (MIMO) systems. Principles are illustrated through graphical representation as well as case studies. The performance of the two-level LP-MPC closed-loop system is evaluated and explained using results of the LP sensitivity analysis.     

Recursive linear optimal filter for Markovian jump linear systems with multi-step correlated noises and multiplicative random parameters

This paper presents the state estimation problem for discrete-time Markovian jump linear systems with multi-step correlated additive noises and multiplicative random parameters (termed as MCNMP). A recursive linear optimal filter for the considered MCNMP (which is abbreviated as RLMMF) is derived based on state augmentation between the original state and mode uncertainty, with the help of estimating the multi-step correlated additive noises online simultaneously. A maneuvering target tracking example under one-step and two-step correlated additive noises scenarios with different cases (i.e. Gaussian/Gaussian mixture distribution and no multiplicative noises) is simulated to validate the designed filter.     

Output feedback guaranteed cost control of networked linear systems with random packet losses

This article is concerned with the output feedback guaranteed cost control problem for a class of networked control systems (NCSs) with both packet losses and network-induced delays. The packet-loss processes in the forward channel and the backward channel are modelled as two Markov chains. The dynamic output feedback controllers are considered, and the closed-loop NCS is modelled as a discrete-time Markovian system with two modes and unit time delay. By using a properly constructed Lyapunov function and the state transformation technique, a sufficient condition is derived for the closed-loop NCS to be mean-square exponentially stable and ensure a decay rate that can be tuned according to the packet loss situations in the networks. Moreover, design procedures for the guaranteed cost controllers are also presented based on the obtained stability condition and guaranteed cost performance result. Finally, an illustrative example is given to demonstrate the effectiveness of the proposed results.     

Stability analysis and stabilization of networked linear systems with random packet losses

This paper is concerned with the stability analysis and stabilization of networked discrete-time and sampled-data linear systems with random packet losses. Asymptotic stability, mean-square stability, and stochastic stability are considered. For networked discrete-time linear systems, the packet loss period is assumed to be a finite-state Markov chain. We establish that the mean-square stability of a related discrete-time system which evolves in random time implies the mean-square stability of the system in deterministic time by using the equivalence of stability properties of Markovian jump linear systems in random time. We also establish the equivalence of asymptotic stability for the systems in deterministic discrete time and in random time. For networked sampled-data systems, a binary Markov chain is used to characterize the packet loss phenomenon of the network. In this case, the packet loss period between two transmission instants is driven by an identically independently distributed sequence assuming any positive values. Two approaches, namely the Markov jump linear system approach and randomly sampled system approach, are introduced. Based on the stability results derived, we present methods for stabilization of networked sampled-data systems in terms of matrix inequalities. Numerical examples are given to illustrate the design methods of stabilizing controllers.     

Infinite horizon optimal control of linear discrete time systems with stochastic parameters

The infinite horizon optimal control problem is considered in the general case of linear discrete time systems and quadratic criteria, both with stochastic parameters which are independent with respect to time. A stronger stabilizability property and a weaker observability property than usual for deterministic systems are introduced. It is shown that the infinite horizon problem has a solution if the system has the first property. If in addition the problem has the second property the solution is unique and the control system is stable in the mean square sense. A simple necessary and sufficient condition, explicit in the system matrices, is given for the system to have the stronger stabilizability property. This condition also holds for deterministic systems to be stabilizable in the usual sense. The stronger stabilizability and weaker observability properties coincide with the usual ones if the parameters are deterministic.