Optimal Decentralized Control Problem Research Articles

Decentralized control for multi-sensors networked systems with different transmission delays and packet dropouts

In this paper, the linear quadratic (LQ) optimal decentralized control and stabilization problems are investigated for multi-sensors networked control systems (MSNCSs) with multiple controllers of different information structure. Specifically, for a MSNCS, in view of the packet dropouts and the transmission delays, each controller may access different information sets. To begin with, the sufficient and necessary solvability conditions for the LQ decentralized control problems are developed. Consequently, for the purpose of deriving the optimal decentralized control strategy, an innovative orthogonal decomposition method is proposed to decouple the forward and backward stochastic difference equations (FBSDEs) from the maximum principle. In the following, we show that the optimal decentralized controller can be calculated according to a set of Riccati-type equations. Finally, a stabilizing controller is derived for the stabilization problem.

On the Connectivity Properties of Feasible Regions of Optimal Decentralized Control Problems

The optimal decentralized control (ODC) is an NP-hard problem with many applications in real-world systems. There is a recent trend of using local search algorithms for solving optimal control problems. However, the effectiveness of these methods depends on the connectivity property of the feasible region of the underlying optimization problem. In this article, for ODC problems with static controllers, we develop a novel criterion for certifying the connectivity of the feasible region in the case where the input and output matrices of the system dynamics are identity matrices. This criterion can be checked via an efficient algorithm, and it is used to prove that the number of communication networks leading to connected feasible regions is greater than a square root of the exponential number of possible communication networks (named patterns). For ODC problems with dynamic controllers, we prove that under certain mild conditions, the closure of the feasible region is always connected after some parameterization, for general communication networks and system dynamics.

Optimal Decentralized Control for Uncertain Systems by Symmetric Gauss–Seidel Semi-Proximal ALM

The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> -guaranteed cost decentralized control problem is investigated in this article. More specifically, on the basis of an appropriate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathcal {H}_2$</tex-math></inline-formula> reformulation that we put in place, the optimal control problem in the presence of parameter uncertainties is then suitably characterized by convex restriction and solved in parameter space. It is shown that a set of stabilizing decentralized controller gains for the uncertain system is parameterized in a convex set through appropriate convex restriction, and then an approximated conic optimization problem is constructed. This facilitates the use of the symmetric Gauss–Seidel (sGS) semi-proximal augmented Lagrangian method (ALM), which attains high computational effectiveness. A comprehensive analysis is given on the application of the approach in solving the optimal decentralized control problem; and subsequently, the preserved decentralized structure, robust stability, and robust performance are all suitably guaranteed with the proposed methodology. Furthermore, an illustrative example is presented to demonstrate the effectiveness of the proposed optimization approach.

Decentralized Control for Networked Control Systems With Asymmetric Information

This article considers the decentralized control for networked control systems (NCSs) with asymmetric information. In this NCSs model, the controller 2 (C2) shares its observations and part of its historical control inputs with the controller 1 (C1), whereas C2 cannot obtain the information of C1 due to network constraints. Under the linear control strategies assumption, we present the optimal estimators for C1 and C2 respectively based on asymmetric observations. Since the information for C1 and C2 are asymmetric, the estimation error covariance (EEC) is coupled with the controller which means that the classical separation principle fails. By applying the Pontryagin’s maximum principle, we obtain a solution to the forward and backward stochastic difference equations. Based on this solution, we derive the optimal controllers to minimize a quadratic cost function. Combining the linear optimal controllers with the EEC, the controller C1 is decoupled from the EEC. It should be emphasized that the control gain is dependent on the estimation gain. What is more, the estimation gain satisfies the forward Riccati equation and the control gain satisfies the backward Riccati equation which makes the problem more challenging. We propose iterative solutions to the Riccati equations and give a suboptimal solution to the optimal decentralized control problem.

On the Nearest Quadratically Invariant Information Constraint

Quadratic invariance is a condition which has been shown to allow for optimal decentralized control problems to be cast as convex optimization problems. The condition relates the constraints that the decentralization imposes on the controller to the structure of the plant. In this technical note, we consider the problem of finding the closest subset and superset of the decentralization constraint which are quadratically invariant when the original problem is not. We show that this can itself be cast as a convex problem for the case where the controller is subject to delay constraints between subsystems, but that this fails when we only consider sparsity constraints on the controller. For that case, we develop an algorithm that finds the closest superset in a fixed number of steps, and discuss methods of finding a close subset.

Decentralized Control via Gröbner Bases and Variable Elimination

We consider the problem of optimal decentralized controller synthesis. There are several classes of such problems for which effective solution methods are known, including the quadratically invariant one. In this technical note, we use Grobner bases and elimination theory to characterize all closed-loop maps achievable by forming a feedback loop with decentralized controllers. We show that this approach allows solution of a wide class of optimal decentralized control problems; it includes not only quadratically invariant problems under a technical condition but also some other problems which are not quadratically invariant.

On the Number of Iterations to the Closest Quadratically Invariant Information Constraint

AbstractQuadratic invariance is a condition which has been shown to allow for optimal decentralized control problems to be cast as convex optimization problems. The condition relates the constraints that the decentralization imposes on the controller to the structure of the plant. Recent work considered the problem of finding the closest subset and superset of the decentralization constraint which are quadratically invariant when the original problem is not. It was shown that this can itself be cast as a convex problem for the case where the controller is subject to delay constraints between subsystems, but that this fails when we only consider sparsity constraints on the controller. For that case, an algorithm was developed that finds the closest superset in a fixed number of steps, and it was shown to converge in n2 iterations, where n is the number of subsystems. This paper studies the algorithm further and shows that it actually converges in log2n iterations.

Newton's method for solving parametric linear quadratic control problems

Newton's method is applied to parametric linear quadratic control problems, including the optimal output feedback problem and the optimal decentralized control problem. Newton's equations are obtained as a system of coupled linear matrix equations. They are solved iteratively using the conjugate gradient method. In order to reduce the amount of work associated with the procedure, an inexact newtonian algorithm is also considered. In this algorithm, an approximate solution of the Newton equations is computed in such a way that the asymptotic convergence rate is quadratic.

A descent anderson-moore algorithm for optimal decentralized control

A convergent version of the Anderson-Moore algorithm for the optimal output feedback problem is applied to a class of optimal decentralized control problems. The algorithm is based on a positive definite approximation of the second-order Taylor-series expansion of the loss function. Numerical results show that the algorithm performs favourably when compared with the method of Geromel and Bernussou (1979) ( Automatica, 15, 489) and with variable metric function minimization techniques.