Optimal Linear Quadratic Gaussian Research Articles

Robust stabilization of LTI negative imaginary systems using the nearest negative imaginary controller

AbstractThis paper considers the problem of robust stabilization of linear time‐invariant systems with respect to unmodelled dynamics and structure uncertainties. To that end, a methodology to find the nearest negative imaginary system for a given non‐negative imaginary system is presented first. Then, this result is employed to construct a near optimal linear quadratic Gaussian controller achieving desired performance measures. The problem is formulated using port‐Hamiltonian method and the required conditions are defined in terms of linear matrix inequalities. The technique is presented using the fast gradient method to solve the problem systematically. The designed controller satisfies a negative imaginary property and guarantees a robust feedback loop. The effectiveness of the approach is demonstrated by a simulation on a numerical example.

Quickest physical watermarking-based detection of measurement replacement attacks in networked control systems

In this paper, we propose and analyze an attack detection scheme for securing the physical layer of a networked control system (NCS) with a wireless sensor network against attacks where the adversary replaces the true observations with stationary false data. An independent and identically distributed watermarking signal is added to the optimal linear quadratic Gaussian (LQG) control inputs, and a cumulative sum (CUSUM) test is carried out using the joint distribution of the innovation signal and the watermarking signal for quickest attack detection. We derive the expressions of the supremum of the average detection delay (SADD) for a multi-input and multi-output (MIMO) system under the optimal and sub-optimal CUSUM tests. The SADD is asymptotically inversely proportional to the expected Kullback–Leibler divergence (KLD) under certain conditions. The expressions for the MIMO case are simplified for multi-input and single-output systems and explored further to distil design insights. We provide insights into the design of an optimal watermarking signal to maximize KLD for a given fixed increase in LQG control cost when there is no attack. Furthermore, we investigate how the attacker and the control system designer can accomplish their respective objectives by changing the relative power of the attack signal and the watermarking signal. Simulations and numerical studies are carried out to validate the theoretical results.

LQG Reference Tracking With Safety and Reachability Guarantees Under Unknown False Data Injection Attacks

We investigate a linear quadratic Gaussian (LQG) tracking problem with safety and reachability constraints in the presence of an adversary who mounts a false data injection attack on an unknown set of sensors. For each possible set of compromised sensors, we maintain a state estimator disregarding the sensors in that set, and calculate the optimal LQG control input at each time based on this estimate. We propose a control policy which constrains the control input to lie within a fixed distance of the optimal control input corresponding to each state estimate. The control input is obtained at each time step by solving a quadratically constrained quadratic program. We prove that our policy can achieve a desired probability of safety and reachability using the barrier certificate method. Our control policy is evaluated via a numerical case study.

Fast hierarchical coordinated controller for distributed battery energy storage systems to mitigate voltage and frequency deviations

This paper proposes a novel hierarchical optimal control framework to support frequency and voltage in multi-area transmission systems, integrating battery energy storage systems (BESSs). The design is based on the coordinated active and reactive power injection from the BESSs over conventional synchronous generator-based control for fast and timely mitigation of voltage and frequency deviations. The principle of this new idea is to use two hierarchical schemes, one physical and one logical. The objective of the first scheme prioritises the power injection from the BESSs installed in the area where a contingency occurs, consequently reducing the disturbance of the dynamics in the neighbouring areas. In the second scheme, operational rules for aggregated BESSs in each are incorporated, increasing the safety of the asset. The proposed approach exploits the advantages of time-synchronised measurements, the eigensystem realisation algorithm (ERA) identification technique, the optimal linear quadratic Gaussian (LQG) controllers and a new aggregating agent that coordinates the power injection of BESSs in a hierarchical and scalable scheme to precisely regulate frequency and voltage of modern transmission grids, increasing their reliability and stability. The feasibility and robustness of the proposal is demonstrated using simulated scenarios with significant load changes and three-phase, three-cycle faults on a modified Kundur-system with four interconnected areas, mitigating frequency and voltage contingencies in less than 450 ms.

Privacy-Preserving Design of Scalar LQG Control.

This paper studies the agent identity privacy problem in the scalar linear quadratic Gaussian (LQG) control system. The agent identity is a binary hypothesis: Agent A or Agent B. An eavesdropper is assumed to make a hypothesis testing the agent identity based on the intercepted environment state sequence. The privacy risk is measured by the Kullback–Leibler divergence between the probability distributions of state sequences under two hypotheses. By taking into account both the accumulative control reward and privacy risk, an optimization problem of the policy of Agent B is formulated. This paper shows that the optimal deterministic privacy-preserving LQG policy of Agent B is a linear mapping. A sufficient condition is given to guarantee that the optimal deterministic privacy-preserving policy is time-invariant in the asymptotic regime. It is also shown that adding an independent Gaussian random process noise to the linear mapping of the optimal deterministic privacy-preserving policy cannot improve the performance of Agent B. The numerical experiments justify the theoretic results and illustrate the reward–privacy trade-off.

An optimized LQG servo controller design using LQI tracker for VSP-based AC/DC interconnected systems

This paper proposes a novel application for the optimal Linear Quadratic Gaussian (LQG) servo controller to enable a proper coordination of the AC/HVDC interconnected system with Virtual Synchronous Power (VSP) based inertia emulation. Particularly, the proposed control design takes the process disturbances and measurement noise of the studied VSP-HVDC system into account, while few studies have focused on this perspective. The proposed LQG controller with modifications is designed by means of a combination of Kalman Filter (state estimator) and an added Linear Quadratic Integrator (LQI) to observe the system model’s states and track the reference commands while rejecting the effects of system noise. Besides, we utilize a swarm-based optimization algorithm to operate as the search process for the tuning of the elements in the weighting matrices involved in the controller design. The role of the proposed optimal LQG controller is to stabilize such AC/DC interconnected system with VSP-based inertia emulator while minimizing the associated performance index. According to the obtained simulation results, in addition to the advancement from the VSP-based approach for damping frequency oscillations excited by faults, application of the proposed LQG servo controller can achieve the targets on both estimating the state variables and tracking the reference signals with satisfactory performance, comparing with the conventional LQG regulator.

Robust-optimal integrated control design technique for a pressurized water-type nuclear power plant

A control design scheme is formulated for a pressurized water type nuclear power plant by integrating the optimal linear quadratic Gaussian (LQG) control with the robust integral sliding mode (ISM) technique. A novel robust-optimal hybrid control scheme is further proposed by integrating the LQG-ISM technique with the loop transfer recovery approach to enhance the effectiveness and robustness capability. The control architecture offers robust performance with minimum control efforts and tracks the reference set-point effectively in the presence of disturbances and parametric uncertainties. The multi-input-multi-output nuclear power plant model adopted in this work is characterized by 38 state variables. The nonlinear plant model is linearized around steady-state operating conditions to obtain a linear model for the controller design. The efficacy of the proposed controllers is demonstrated by nuclear power plant subsystem simulations. The control performance of the proposed technique is also compared with other classical control design schemes. Numerical measures are employed to quantitatively analyse and compare the performance of the different controllers that are studied in the work.

An LQG Controller Based on Real System Identification for an Active Hydraulically Interconnected Suspension

Rollover prevention is always one of the research hotspots in vehicle design. Active hydraulically interconnected suspension (HIS) is a promising technology to reduce vehicle body roll angle caused by different driving inputs and road conditions. This paper proposes a novel actuator of the active HIS system. The actuator consists of two cylinders, a ball screw, and only one motor. The actuator proposed can reduce the number of motors needed in the system. Meanwhile, forced vibration identification (FVI) is used to identify the transfer function of a half-car physical model and a Kalman state observer is applied to eliminate the influence of sensor noise. The FVI method can eliminate most model uncertainties and hidden variables. Aggressive and moderate optimal linear quadratic Gaussian (LQG) methods are implemented to control the motion of the vehicle body based on the identified transfer function of the physical model. The performance of an active HIS system with an aggressive and moderate LQG controller is compared with that of a passive HIS system. The effectiveness of the LQG controller is validated by simulation and experimental results. Also, the obtained results show that the stabilization speed of the active HIS system is 20% faster than that of the passive HIS system and the roll angle can be reduced up to 55% than that of the passive HIS system.

Proximity Operations and Three Degree-of-Freedom Maneuvers Using the Smartphone Video Guidance Sensor

This paper presents the first demonstration of NASA’s Smartphone Video Guidance Sensor (SVGS) as real-time position and attitude estimator for proximity and formation maneuvers. An optimal linear quadratic Gaussian controller was used, combining a linear quadratic regulator and a Kalman filter. The system was demonstrated controlling the 3-degree of freedom planar motion of the RINGS ground units (Resonant Inductive Near-field Generation Systems). A state-space model of the system’s 3-DOF motion dynamics was derived, and model parameters extracted using a system identification technique. The system’s motion control performance is experimentally demonstrated in both tracking and formation maneuvers. The results highlight the capabilities and performance of the Smartphone Video Guidance Sensor (SVGS) as a vision-based real-time position and attitude sensor for motion control, formation flight and proximity operations. A leader-follower formation maneuver approach is demonstrated, as well as position hold and path following.

On-sky results for adaptive optics control with data-driven models on low-order modes

ABSTRACT Dedicated tip–tilt loops are commonly implemented on adaptive optics (AO) systems. In addition, a number of recent high-performance systems feature tip–tilt controllers that are more efficient than the integral action controller. In this context, linear–quadratic–Gaussian (LQG) tip–tilt regulators based on stochastic models identified from AO telemetry have demonstrated their capacity to effectively compensate for the cumulated effects of atmospheric disturbance, windshake and vibrations. These tip–tilt LQG regulators can also be periodically retuned during AO operations, thus allowing to track changes in the disturbances’ temporal dynamics. This paper investigates the potential benefit of extending the number of low-order modes to be controlled using models identified from AO telemetry. The global stochastic dynamical model of a chosen number of turbulent low-order modes is identified through data-driven modelling from wavefront sensor measurements. The remaining higher modes are modelled using priors with autoregressive models of order 2. The loop is then globally controlled using the optimal LQG regulator build from all these models. Our control strategy allows for combining a dedicated tip–tilt loop with a deformable mirror that corrects for the remaining low-order modes and for the higher orders altogether, without resorting to mode decoupling. Performance results are obtained through evaluation of the Strehl ratio computed on H-band images from the scientific camera, or in replay mode using on-sky AO telemetry recorded in 2019 July on the CANARY instrument.

Optimal linear quadratic Gaussian control for discrete time‐varying system with simultaneous input delay and state/control‐dependent noises

SummaryThis article focuses on the problem of linear quadratic Gaussian (LQG) control for discrete time‐varying system with input delay and state/control‐dependent noises. When the state variables can be exactly observed, first, we obtain the maximum principle by applying the method of variation. Second, a nonhomogeneous relationship between the state and the costate is developed in virtue of the obtained maximum principle and the mathematical induction. It is noted that the nonhomogeneous relationship is the solution to the forward and backward stochastic difference equations (FBSDEs). Finally, based on the solution to the FBSDEs, a necessary and sufficient condition for the optimal LQG control is derived in terms of coupled Riccati equations. Moreover, the analytical expression of the optimal controller is presented. The derived results are applied in networked control systems with packet dropout. Numerical examples are shown to illustrate the effectiveness of the proposed algorithm.

LQG control on mixed H2/H∞ problem: the discrete-time case

ABSTRACTIn this paper, we are concerned with linear quadratic Gaussian (LQG) control on the mixed problem. The mixed problem can be formulated as a kind of constraint optimisation problems where the control input is to minimise the -norm subject to the -constraint dealt with by the disturbance. The main contributions are twofold. First, when the state variables can be obtained exactly, necessary and sufficient conditions are given to guarantee the existence of the strategies. Second, the optimal LQG controller in terms of three decoupled equations (two Riccati equations and one Lyapunov equation) is obtained. The key technique is to apply the Stackelberg game approach by treating the disturbance as the follower and the control input as the leader respectively. A practical example is given to verify the efficiency of the proposed approach.

Finite Element Model of Vibration Control for an Exponential Functionally Graded Timoshenko Beam with Distributed Piezoelectric Sensor/Actuator

This paper presents a dynamic study of sandwich functionally graded beam with piezoelectric layers that are used as sensors and actuators. This study is exploited later in the formulation of the active control laws, while using the optimal control Linear Quadratic Gaussian (LQG), accompanied by the Kalman filter. The mathematical formulation is based on Timoshenko’s assumptions and the finite element method, which is applied to a flexible beam divided into a finite number of elements. By applying the Hamilton principle, the equations of motion are obtained. The vibration frequencies are found by solving the eigenvalue problem. The structure is analytically then numerically modeled and the results of the simulations are presented in order to visualize the states of their dynamics without and with active control.

Nonlinear SPKF-Based Time-Varying LQG for Inverted Pendulum System

This paper deals with the holing issue of nonlinear inverted pendulum (IP) system. Close to the equilibrium point, IP is considered as a linear system without disruption and linear control is sufficient. However, when the IP swings over a wide range, its nonlinear dynamics becomes significant and the stabilization of IP becomes challenging task due to the inconsistency between its nonlinear dynamics and the controllers designed based on linearized models. Hence, the need for sophisticated control becomes highly demanding. This paper proposes an optimal time-varying linear quadratic Gaussian controller (TV-LQG) that is able to overcome this inconsistency problem. The proposed TV-LQG utilizes a sigma-point Kalman filter (SPKF) and a linear quadratic regulator with a prescribed degree of stability. SPKF is a highly accurate nonlinear state estimator since it does not use any linearization for calculating the state prediction covariance and Kalman gains. This leads, instantaneously, to a more exact nonlinear state estimation. The estimated states are fed to the Jacobian method, which updates, at once, the system dynamics accordingly. Thus, the parameters of the proposed TV-LQG are optimized based on the updated system dynamics. The proposed controller is intensively tested using simulation experiments and compared to the traditional LQG, LQR self-adjusting control and LQR-fuzzy control. The results proved the robustness and competitiveness of the proposed scheme to enhance the performance of the nonlinear IP system.

The particle swarm modified quasi bang-bang controller for seismic vibration control

The frequent occurrence of seismic events results in the damage to the structures. Structural control is one of the popular methods to prevent these damages. In structural control, the main objective is to impart an adequate counterforce. This required counter force is determined by a suitable control algorithm. In this work, an optimized and adaptive control algorithm is proposed based on modified quasi bang-bang control algorithm to generate the adequate counterforce. The constant output weights used in the modified quasi bang-bang controller are optimized for best performance in dynamic loading such as earthquake using the particle swarm optimization (PSO). The proposed optimized controller is then applied to a three storey prototype structure fixed with an MR damper. This structure has been subjected to various seismic events for performance evaluation. The results thus obtained are compared with best performing clipped optimal linear quadratic Gaussian (LQG) controller, quasi bang-bang and modified quasi bang-bang controller. The results establish that the PSO modified quasi bang-bang controller is superior to the other controller in reducing the structural responses i.e. relative displacement, interstorey drift and the absolute accelerations. Further, the voltage comparison shows that the power consumption is less for the proposed controller in attaining better performance.

Design of Optimal Linear Quadratic Gaussian (LQG) Controller for Load Frequency Control (LFC) using Genetic Algorithm (G.A) in Power System

Design of Optimal Linear Quadratic Gaussian (LQG) Controller for Load Frequency Control (LFC) using Genetic Algorithm (G.A) in Power System

Discrete-Time LQG Control With Input Delay and Multiplicative Noise

This paper is concerned with linear quadratic Gaussian (LQG) control problem for discrete-time systems with both multiplicative noises and input delay. The main contributions are two-fold. First, when the state variables can be obtained exactly, the optimal LQG controller is derived which consists of the feedback form of the conditional expectation of the state and one additive deterministic term based on the coupled Riccati equations. Through the completing square approach, the optimal controller is obtained based on the complete solution to the forward and backward stochastic difference equations. Second, when the state variables are partially observed, we derive a suboptimal linear state estimate feedback controller for LQG models involving multiplicative noises and input delay. Since the separation principle does not hold and the optimal estimator cannot be obtained, the suboptimal linear state estimate feedback controller is derived by linearizing the linear optimal estimator and using the obtained results of the optimal LQG control.

Optimal Estimation and Control for Lossy Network: Stability, Convergence, and Performance

In this paper, we study the problems of optimal estimation and control, i.e., the linear quadratic Gaussian (LQG) control, for systems with packet losses but without acknowledgment. Such acknowledgment is a signal sent by the actuator to inform the estimator of the incidence of control packet losses. For such system, which is usually called as a user datagram protocol (UDP)-like system, the optimal estimation is nonlinear and its calculation is time-consuming, making its corresponding optimal LQG problem complicated. We first propose two conditions: 1) the sensor has some computation abilities; and 2) the control command, exerted to the plant, is known to the sensor. For a UDP-like system satisfying these two conditions, we derive the optimal estimation. By constructing the finite and infinite product probability measure spaces for the estimation error covariances (EEC), we give the stability condition for the expected EEC, and show the existence of a measurable function to which the EEC converges in distribution, and propose some practical methods to evaluate the estimation performance. Finally, the LQG controllers are derived, and the conditions for the mean square stability of the closed-loop system are established.

Linear Quadratic Gaussian Control for Discrete-time Systems with Delay and Multiplicative Noise

This paper considers the linear quadratic Gaussian (LQG) control problem for discrete-time systems subject to multiplicative noises and input delay. When the state variables can be obtained exactly, the necessary and sufficient condition for the existence of a unique solution to the LQG control has been presented and an explicit analytical expression is given for the optimal LQG controller. More specially, the optimal LQG controller is derived which consists of the feedback form of the conditional expectation of the state and one additive deterministic term based on the coupled Riccati equations. Through the completing square approach, the optimal controller is obtained based on the solution to the forward and backward stochastic difference equations (FBSDEs).

Optimal energy allocation for linear control with packet loss under energy harvesting constraints

This paper studies a closed loop linear control system over a lossy communication link. A sensor computes a state estimate of the observed discrete-time system and sends it (in the form of packetized transmission) to the controller in the receiver block over a randomly time-varying (fading) packet dropping link. The receiver sends an ACK/NACK packet to the transmitter over an acknowledgement channel which might also be prone to packet loss. It is assumed that the energy used in packet transmission depletes a battery of limited capacity at the sensor, but is also replenished by an energy harvester which has access to a source of everlasting but random harvested energy. Under an assumption of finite-state Markov chain models of the energy harvesting and the fading channel gain processes, the objective is to design an optimal energy allocation policy at the transmitter and an optimal control policy at the receiver so that an average infinite horizon linear quadratic Gaussian (LQG) control cost is minimized. It is shown that in the case of perfect channel feedback a separation principle holds, the optimal LQG controller is linear and the optimal energy allocation policy at the transmitter can be obtained via solving the Bellman dynamic programming equation. A Q-learning algorithm is used to approximate the optimal energy allocation policy in case the system parameters are unknown. Numerical simulation examples show that the dynamic programming based policies outperform various simple heuristic policies, especially at higher battery capacities.