LQG Optimal Control Research Articles

Structural properties, LQG control and scheduling of a networked control system with bandwidth limitations and transmission delays

The problem of simultaneous LQG control and scheduling of a Networked Control System (NCS) with constant network induced delays at input and output and bandwidth limitations is investigated. Delays are considered at plant as well as controller side. Sufficient conditions for controllability, stabilizability, reconstructibility and detectability of the underlying networked control system are drawn. The proposed conditions extend previous works on structural properties of NCS by capturing both plant and controller side delays together with bandwidth limitations. A framework for computing the optimal LQG controller for the NCS with a fixed scheduling is provided. The proposed modeling approach facilitates use of LQG as well as other control methods for NCSs with delays and bandwidth limitations. In order to optimize performance, a semi-online scheduling procedure is proposed based on an offline look up table. The look up table assigns an optimal schedule with associated optimal LQG controller to initial conditions. The proposed scheme improves previous results by online deployment of schedule and LQG control with stability guarantees and very low computational overhead. A simulation example with communication delays, packet losses and bandwidth limitations in both sensor and actuator sides is included. Static optimal periodic communication sequence, Optimal Pointer Placement (OPP) approach proposed in previous works, a random access scheduling method representing contention based access policies and the proposed method are simulated and compared.

Active tilting-pad journal bearings supporting flexible rotors: Part II–The model-based feedback-controlled lubrication

This is part II of a twofold paper series dealing with the design and implementation of model-based controllers meant for assisting the hybrid and developing the feedback-controlled lubrication regimes in active tilting pad journal bearings (active TPJBs). In both papers theoretical and experimental analyses are presented with focus on the reduction of rotor lateral vibration. This part is devoted to synthesising model-based LQG optimal controllers (LQR regulator + Kalman Filter) for the feedback-controlled lubrication and is based upon the mathematical model of the rotor-bearing system derived in part I. Results show further suppression of resonant vibrations when using the feedback-controlled or active lubrication, overweighting the reduction already achieved with hybrid lubrication, thus improving the whole machine dynamic performance.

Optimal control over multiple erasure channels using a data dropout compensation scheme

In this paper we focus on networked control systems subject to data dropouts. We consider that the controller communicates with the plant through multiple independent erasure channels, and that each actuator sends acknowledgement packages to the controller. We restrict our attention to the optimal design of a class of controllers that embeds a data dropout compensator and where, besides this compensation mechanism, all processing is an affine function of the available data. Our first result shows that the optimal design consists in solving, separately, an estimation problem and a control problem. The optimal controller is a linear function of the estimated state, which is obtained exploiting the information given by the acknowledgement packages and the data dropout compensator. Since we focus on a specific class of controllers, our proposal is suboptimal, however, unlike the optimal LQG controller found in the literature, our proposal is easier to implement, and its steady state behaviour can be simply determined. We also show that, under standard conditions, the affine part of the optimal controller converges to an LTI filter, which can be used as a simple alternative to the time varying controller.

Design of coherent quantum observers for linear quantum systems

Quantum versions of control problems are often more difficult than their classical counterparts because of the additional constraints imposed by quantum dynamics. For example, the quantum LQG and quantum optimal control problems remain open. To make further progress, new, systematic and tractable methods need to be developed. This paper gives three algorithms for designing coherent quantum observers, i.e., quantum systems that are connected to a quantum plant and their outputs provide information about the internal state of the plant. Importantly, coherent quantum observers avoid measurements of the plant outputs. We compare our coherent quantum observers with a classical (measurement-based) observer by way of an example involving an optical cavity with thermal and vacuum noises as inputs.

Jointly Optimal LQG Quantization and Control Policies for Multi-Dimensional Systems

For controlled R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> -valued linear systems driven by Gaussian noise under quadratic cost criteria, we investigate the existence and the structure of optimal quantization and control policies. For fully observed and partially observed systems, we establish the global optimality of a class of predictive encoders and show that an optimal quantization policy exists, provided that the quantizers allowed are ones which have convex codecells. Furthermore, optimal control policies are linear in the conditional estimate of the state, and a form of separation of estimation and control holds.

Wind turbine mechanical stresses reduction and contribution to frequency regulation

The aim of the present research work has been to design an optimal MIMO LQG controller to reduce the drive-train, blades and tower mechanical stresses of a wind turbine (WT), and at the same time, to involve the WT in the grid primary frequency regulation when it is operating in full load (FL) zone. To verify the effectiveness of the proposed controller, the achieved results are compared to those obtained by a base-line controller based on a PI regulator.Simulation results show that thanks to these controllers, WT can effectively contribute to the grid frequency regulation, tracking tightly the generator power reference which depends on that frequency. Compared with the base-line controller, the LQG controller significantly reduces the mechanical stresses of the WT׳s most costly components.

Regeneratively-constrained LQG control of passive networks

In this paper we consider the synthesis of optimal feedback controllers for a stochastically-excited passive electromechanical network, subject to the constraint that in stationarity, the feedback law must be realizable with a regenerative actuation system. Regenerative systems are similar to passive systems but their dynamic constraints are more relaxed, in the sense that they only need to conserve energy in the stationary average sense, rather than at every time instant. In this paper, we examine the design of optimal LQG controllers for passive networks controlled with regenerative actuation. We show that this problem may be posed as a multi-objective LMI problem. We also characterize how close a regeneratively-constrained optimal LQG feedback law is to a passive transfer function. The concepts are demonstrated on a simple example related to vibration suppression.

Optimal signal tracking algorithm for GNSS signal using moving set-point LQG system

This paper proposes moving set-point state feedback for a LQG control system. In contrast to the independent code and carrier tracking loop of a conventional receiver algorithm, the code and carrier tracking variables are correlated in the LQG system. Not only are the multiple states correlated with each other, but also the control inputs are formulated from a combination of the states and the optimal LQG controller gain. In addition, this paper develops a moving set point to estimate the GNSS signal more accurately. To analyze the advantage of the proposed method, a signal tracking simulation using hardware GNSS simulator is performed and the simulation results show that the tracking performance of the moving set-point LQG system is better than that of a conventional loop filter, especially in terms of the code tracking performance.

Limits of Performance in Systems with Periodic Irregular Sampling and Actuation Rates

This paper examines the limits of performance in systems with periodic irregular sampling rate when the actuation is not necessarily synchronized with the sampling. For such a system, three sampling and actuation schemes are considered: when the sampling and control rate are both regular, when they are both irregular, and when the sampling rate is irregular while the control rate is regular. To ascertain the limits of performance of this type of systems under each sampling and actuation scheme, the system is modeled as a linear periodically time-varying (LPTV) system; optimal LQG control design with a variance constraint is applied to find the smallest achievable mean variance of the performance signal subject to a constraint on the mean control effort variance. In addition, to deal with the computational delay of the controller, an innovative discretization method is proposed which does not introduce any extra states into the state space model. The proposed method is exploited to determine the performance of a hard disk drive (HDD) in track-following mode. A simulation study demonstrates that in the presence of 30% irregularity in sampling time, using the irregular sampling and regular control action scheme for the HDD achieves an RMS 3s value of the position error signal (PES) that is 40% smaller than the corresponding value achieved by using a controller provided by our industry partner, in which both the sampling and control rates are irregular.

Optimal State Feedback Control in Operator Domain for Multi-Rate Networked Control Systems with Long Time Delay

By multi-rate networked control systems (NCS), we mean the sampling periods of the sensor, the controller and the actuator in networked control systems are not the same, that is to say there are more than one sampling rate in networked control systems. For the long time delay multi-rate NCS with event-driven controller and actuator, a stochastic discrete model is established under operator. The state feedback control laws for the multi-rate NCS in operator domain are designed by using a dynamic programming approach. The derived optimal LQG controller can be used as a delay-compensator for multi-rate NCS with long time delays. An example is given to verify the theory results of this paper.

Distributed Output-Feedback LQG Control with Delayed Information Sharing

This paper develops a controller synthesis method for distributed LQG control problems under output-feedback. We consider a system consisting of three interconnected linear subsystems with a delayed information sharing structure. While the state-feedback case has previously been solved, the extension to output-feedback is nontrivial as the classical separation principle fails. To find the optimal solution, the controller is decomposed into two independent components: a centralized LQG-optimal controller under delayed state observations, and a sum of correction terms based on additional local information available to decision makers. Explicit discrete-time equations are derived whose solutions are the gains of the optimal controller.

A Simple Unified Formula for Discrete and Continuous-time H∞ “All-solutions” Controllers

In this paper we present a simple, unified formula for discrete and continuous-time H∞ “all-solutions” controllers. By observing a “cost” equivalence between the standard H∞ control problem and a certain LQ optimal regulator problem, an elegant controller structure reminiscent of an LQG optimal controller is developed. Our choice of notation also simplifies the derivation and existence conditions considerably, whereby all unnecessary assumptions on plant state-space matrices and “loop-shifting” transformations are eliminated. Additionally, with our focus entirely on input-output weighting “cost functions” this derivation offers a “behavioral” theory interpretation for all solutions of a standard H∞ control problem.

Central suboptimal H ∞ controller design for linear time-varying systems with unknown parameters

This article presents the central finite-dimensional H ∞ controller for linear time-varying systems with unknown parameters, that is suboptimal for a given threshold γ with respect to a modified Bolza–Meyer quadratic criterion including the attenuation control term with the opposite sign. In contrast to the previously obtained results, this article reduces the original H ∞ controller problem to the corresponding H 2 controller problem, using the technique proposed in Doyle et al. [Doyle, J.C., Glover, K., Khargonekar, P.P., and Francis, B.A. (1989), ‘State-space Solutions to Standard H 2 and H Infinity Control Problems’, IEEE Transactions Automatic Control, 34, 831–847]. This article yields the central suboptimal H ∞ controller for linear systems with unknown parameters in a closed finite-dimensional form, based on the corresponding H 2 controller obtained in Basin and Calderon-Alvarez [Basin, M.V., and Calderon-Alvarez, D. (2008), ‘Optimal LQG Controller for Linear Systems with Unknown Parameters’, Journal of The Franklin Institute, 345, 293–302]. Numerical simulations are conducted to verify performance of the designed central suboptimal controller for uncertain linear systems with unknown parameters against the conventional central suboptimal H ∞ controller for linear systems with exactly known parameter values.

Optimal LQG control over continuous fading channels

AbstractThis paper studies the optimal control of linear systems over continuous valued fading channels. The sensor measurements are sent over a fading wireless channel to a remote controller using the analog amplify and forward technique. The controller then computes a control signal, which is transmitted over another fading channel to the actuator. Under the assumption of full channel state information (CSI) for both wireless links, we derive the optimal LQG control law. In the case where full channel state information is not available but channel statistics are available, we present the optimal linear static estimator and controller. Numerical comparisons are made between the full CSI and statistical CSI solutions.

Optimal Squeezing using Multivariable Integral LQG Control⋆

AbstractIn quantum optics, squeezed states of light can be generated in a nonlinear process by using an optical parametric oscillator (OPO). It is necessary however that the resonator cavity of the OPO maintain a frequency-lock, while at the same time sufficiently suppress classical noises in a chosen optical quadrature. In this paper, we model such an optical system and design an LQG optimal multivariable controller (which includes integral action) that satisfies the practical constraints imposed by this problem.

Performance assessment of advanced supervisory–regulatory control systems with subspace LQG benchmark

In this paper, a subspace method for LQG design and performance assessment is proposed for control systems in which supervisory and regulatory controllers are employed in a cascade structure. Usually, this is the case when an advanced controller is used in a supervisory control layer on the top of the regulatory control system, resulting in a cascade control structure. The objective of this study is to provide the optimal LQG control design in the cascade control structure and also to propose a method for calculation of the LQG ‘trade-off’ curves for performance assessment. The trade-off curve provides the optimal performance limit in terms of the best achievable input and output variances. Three possible scenarios for LQG control design in this supervisory–regulatory structure are discussed in the paper. The problem formulations are presented in the subspace framework to directly derive the control law and LQG trade-off curve without need of the conventional parametric models. A simulation example is provided to demonstrate the proposed method.

Optimal inverse LQG control for certain ode and PDE stationary processes

Following work carried out earlier on linear-quadratic optimal control for linear finite dimensional stationary systems we report, in this article, on extension of some of those results to certain infinite dimensional systems; in particular a class of PDE systems of elliptic type. These systems are studied in the now familiar framework developed by J. L. Lions and E. Magenes, specialized to a subclass of such systems important in a variety of applications. As an extended example this paper studies an optimal redistribution problem in a groundwater flow system governed by Darcy’s equation, presenting both analytic and computational work related to such problems.

Coherent quantum LQG control

Based on a recently developed notion of physical realizability for quantum linear stochastic systems, we formulate a quantum LQG optimal control problem for quantum linear stochastic systems where the controller itself may also be a quantum system and the plant output signal can be fully quantum. Such a control scheme is often referred to in the quantum control literature as “coherent feedback control”. It distinguishes the present work from previous works on the quantum LQG problem where measurement is performed on the plant and the measurement signals are used as the input to a fully classical controller with no quantum degrees of freedom. The difference in our formulation is the presence of additional non-linear and linear constraints on the coefficients of the sought after controller, rendering the problem as a type of constrained controller design problem. Due to the presence of these constraints, our problem is inherently computationally hard and this also distinguishes it in an important way from the standard LQG problem. We propose a numerical procedure for solving this problem based on an alternating projections algorithm and, as an initial demonstration of the feasibility of this approach, we provide fully quantum controller design examples in which numerical solutions to the problem were successfully obtained. For comparison, we also consider the case of classical linear controllers that use direct or indirect measurements, and show that there exists a fully quantum linear controller which offers an improvement in performance over the classical ones.

Performance and robust stability trade-off in minimax LQG control of vibrations in flexible structures

An optimal robust Minimax LQG control of vibration of a flexible beam is studied in this paper. The first six modes of the beam in the frequency range of 0–800 Hz are selected for control purposes. Among these modes, three modes in the frequency range of 100–400 Hz are used for control, while the other three modes are left as the uncertainty of modeling. Both the model and the uncertainty are measured based on experimental data. The nominal model is identified from frequency response data and the uncertainty is presented by a frequency weighted multiplicative modeling method. For the augmented plant consisting of the nominal model and its accompanied uncertainty, a Minimax LQG controller is designed. A trade-off between robust stability and robust performance is shown by selecting two different choices of uncertainty modeling. Simulation results show that the proposed robust controller increases the damping of the system in its resonance frequencies.

Prudent monetary policy and prediction of the output gap

Risk-adjusted LQG optimal control with perfect and imperfect observation of the economy is used to obtain prudent Taylor rules for monetary policies and cautious Kalman filters. A prudent central bank adjusts the nominal interest rate more aggressively to changes in the inflation gap, especially if the volatility of cost-push shocks is large. If the interest rate impacts the output gap after a lag, the interest also responds to the output gap, especially with strong persistence in aggregate demand. Prudence pushes up this reaction coefficient as well. If data are poor and appear with a lag, a prudent central bank responds less strongly to new measurements of the output gap. However, prudence attenuates this policy reaction and biases the prediction of the output gap upwards, particularly if output targeting is important. Finally, prudence requires an extra upward (downward) bias in its estimate of the output gap before it feeds into the policy rule if inflation is above (below) target. This reinforces nominal interest rate reactions. A general lesson is that prudent predictions are neither efficient nor unbiased.