The problem is to determine an output feedback law that is optimal in the sense of minimizing the expected value of a quadratic cost criterion. Finally, the results are reformulated for the general, nonstandard case. Linear quadratic gaussian lqg when we use the combination of an optimal estimator not discussed in this course and an optimal regulator to design the controller, the compensator is called linear quadratic gaussian lqg special case of the controllers. The discrete time linearquadraticgaussian stochastic. Control theory for linear systems university of groningen. Psfrag replacements yt 2 rm zt 2 r ut 2 rk controller process figure1. Makila institute of automation and control tampere university of technology fin33101 tampere, finland email. By means of a counterexample, it is shown that a commonly mentioned adapative control scheme can lead to severe problems.
We consider the problem of stochastic finite and infinitehorizon linear quadratic control under power constraints. Linearquadraticgaussian lqg control is a modern statespace technique for designing optimal dynamic regulators and servo controllers with integral action also known as setpoint trackers. It is shown that modified versions of certainty equivalence and the separation theorem still hold, that. Ileg iterative, linear, exponentialquadratic optimal control under gaussian process noise is an iterative optimization method for solving the optimal control problem for a general nonlinear system with an exponential cost function which is affected by gaussian process noise. Kxt which requires the availability of the process measurement when the state variables are not accessible, one can use ut. Finite horizon linear quadratic gaussian density regulator. We consider the problem of adaptively controlling an unknown lineargaussian system with a standard quadratic cost criterion, including a control cost. Linear quadratic gaussian lqg when we use the combination of an optimal estimator not discussed in this course and an optimal regulator to design the controller, the compensator is called linear quadratic gaussian lqg special case of the controllers that can be designed using the sep aration principle. Pasikduncan,senior member, ieee abstract the adaptive linear quadratic gaussian control problem, where the linear transformation of the state a and the linear transformation of. Linear quadratic gaussian control of a quartercar suspension article pdf available in vehicle system dynamics 326 december 1999 with 436 reads how we measure reads. Adaptive continuoustime linear quadratic gaussian control t. The problem of false data injection through compromised cyber links to a physical control system modeled by linear quadratic gaussian dynamics is studied in this paper. This technique allows you to trade off regulationtracker performance and control effort, and to take into account process disturbances and measurement.
Lqg control is the combination of linear quadratic regulator lqr state feedback and kalman. Linearquadraticgaussian synthesis with reduced parameter sensitivity international journal of control, vol. Linear quadratic gaussian synthesis with reduced parameter sensitivity international journal of control, vol. This lqg technique widely used in medical process controllers, in nuclear power plants and motor control systems. The acknowledgment refers to acknowledgment signals, which are sent by the actuator to notify the controller whether the control packets are successfully received or not. Linear quadratic gaussian control with noisy measurement 18. These developments were based on dynamic programming, developed by richard bellman in the 1950s, and also on kalman. The basic problem in engineering control system design almost invariably involves the online i. Request pdf linear quadratic gaussian control spectral factorizationcontroller designsimplified lqg control designintroduction to performance analysis of controllersmatlab codeproblems find.
The conventional linearquadraticgaussian assumptions are modified in that minimisation of the expectation of cost g defined by 2 is replaced by minimisation of the criterion function 5. In control theory, the linear quadratic gaussian lqg control problem is one of the most fundamental optimal control problems. Asymptotic stability of the lqg regulator 19 system equations with lqg control. Optimal linear quadratic regulator theory, currently referred to as the linear quadratic gaussian lqg problem ref. The typical implementation of the lqr solution requires that the plant states be estimated, which can be posed as an lqg problem. The control input stream is compromised by an attacker who modifies the cyber control signals transmitted with the objective of increasing the quadratic cost incurred by the physical controller whilst maintaining. Stealthy control signal attacks in linear quadratic gaussian control systems. Model predictive and linear quadratic gaussian control of a. Linearquadraticgaussian lqg controllers computer science. Steadystate lqg control and the compensator assume timeinvariant plant dynamics. Linear quadratic gaussian control for two interacting.
The mathematical model of the flexible missile is subject to uncertainties that may arise. It concerns linear systems driven by additive white gaussian noise. As with the standard lqg problem, extension to linear timevarying systems requires that each node. The paper deals with the optimal design and analysis of quarter car vehicle suspension system based on the theory of linear optimal control because linear quadratic gaussian lqg offers the possibility to emphasize quantifiable issues like ride comfort or road holding very easily by altering the weighting factor of a quadratic criterion. Ileg designs locallyoptimal feedback control for nonlinear. Linear quadratic gaussian control jitkomut songsiri. Prediction control waylinear quadratic gaussian control a classical prediction equation of lqg control can be described as follow. Finite horizon linear quadratic gaussian density regulator with wasserstein terminal cost abhishek haldery and eric d.
Wendelz abstractwe formulate and solve an optimal control problem where a. Ece5530, linear quadratic gaussian 62 this is called the separation principleusing estimatex. Optimal linear quadratic control problems with incomplete state information were also solved in the 1960s. First, we explain how the original stochastic linear quadratic gaussian problem can be reformulated as the determinis.
Generation and load balance using linear quadratic gaussian. Linear quadratic gaussian lqg control is a modern statespace technique for designing optimal dynamic regulators and servo controllers with integral action also known as setpoint trackers. In, roughly, the intermediate period, from 1970 to 1985, much attention was at. Stealthy control signal attacks in linear quadratic gaussian. Extended decentralized linearquadraticgaussian control. In linear quadratic regulatorlqr model, the resulting control law are linear with respect to state variable. We investigated the applicability of linear quadratic gaussian lqg methodology to the subcutaneous blood glucose regulation problem. Abstract algebraic necessary conditions for the minimization of a quadratic cost function are derived for nonlinear stochastic regulator systems, on the assumption that the statistics of the state and output random variables can be represented satisfactorily by a gaussian distribution.
The linear quadratic gaussian lqg control for onedimensional 1d systems has been known to be one of the fundamental and significant methods in linear system theory. T in this paper, the general formulation of the linear quadratic e w tw. There are numerous books dedicated to control system design. Model predictive and linear quadratic gaussian controllers are designed for a 5mw variable. Linear quadratic gaussianbased closedloop control of type 1 diabetes patek j diabetes sci technol vol 1, issue 6, november 2007. Linearquadraticgaussian mixed games with continuum. Linearquadraticgaussian lqg controllers and kalman filters. Introduction to linear quadratic regulation robert platt computer science and engineering suny at buffalo february, 20 1 linear systems a linear system has dynamics that can be represented as a linear equation. The lqg controller is optimal in the sense that the lqr state feedback control law requires minimal control signal deviations and the kalman. This nonlinear quadratic gaussian nqg technique is related to the use of statistical linearization and. Gaussian control lqg along with the design procedures and 0, if t 6. The singular linear quadratic gaussian control problem. Pasikduncan,senior member, ieee abstract the adaptive linear quadratic gaussian control problem, where the linear transformation of the state a and the linear transformation of the control b are unknown, is. Optimisation and control of vehicle suspension using.
This technique allows you to trade off regulationtracker performance and control effort, and to take into account process disturbances and measurement noise. This paper investigates the solution of the optimal linear quadratic gaussian control lqg problem for systems with packet losses over nonacknowledgment networks. Note the negative feedback and the absence of a reference signal. Linearquadraticgaussian control linearquadraticgaussian control this term should make use of endashes and not hyphens. The linear quadratic gaussian lqg control problem is an optimal control problem where a quadratic cost function is minimized when the plant has random initial conditions, white noise disturbance input, and white measurement noise. Let u t2rmdenote the action also called the control taken by the system at. This is exactly the same separation operation that we performed before when we designed statespace controllers via poleplacement techniques. We consider the problem of adaptively controlling an unknown linear gaussian system with a standard quadratic cost criterion, including a control cost. Abstract algebraic necessary conditions for the minimization of a quadratic cost function are derived for non linear stochastic regulator systems, on the assumption that the statistics of the state and output random variables can be represented satisfactorily by a gaussian distribution. Linear quadratic gaussian an overview sciencedirect topics. Within this class of optimal control problems, the infiniteduration timeinvariant version, which. Approximate solution to optimal linear quadratic gaussian. Linearquadraticgaussian lqg controllers emo todorov. Linearquadraticgaussian lqg control is a statespace technique that allows you to trade off regulationtracker performance and control effort, and to take into account process disturbances and measurement noise.
Linear quadratic gaussianbased closedloop control of type 1. Kalman filtering and linear quadratic gaussian control. If that is the case, this behavior should be changed so that wikipedia doesnt. Let x t2rndenote the state 1 of the system at time t. Model predictive and linear quadratic gaussian control of. Linearquadratic control with noisefree measurement linearquadraticgaussian control with noisy measurement 18. Stealthy control signal attacks in linear quadratic.
The calculations of the optimal control law can be done offline as in the classical linear quadratic gaussian control theory using dynamic programming, which turns out to be a special case of the new theory developed in this technical note. The linear quadratic gaussian lqg control problem is an optimal control problem where a quadratic cost function is minimized when the plant has random. As literally formulated in 1, the approach is valid for linear timeinvariant systems only. Linear quadratic gaussian control of 2dimensional systems. Kalman filtering and linear quadratic gaussian control p. The above texts on lqg design include many such aspects. If w k, v k are also gaussian, it will also turn out this is the best. Encoding targets as quadratic costs the matrices a,b,q,r can be timevarying, which is useful for specifying reference trajectories x k, and for approximating nonlqg problems. Apr 17, 2014 linear quadratic gaussian control for linear timedelay systems.
For some reason though, i dont seem to be able to make the move myself. Updated estimate is a linear combination of previous estimate and most recent measurement. C is a linear equation of the form xoc kd l 0 xo cl ky k. Generation and load balance using linear quadratic. Modern treatments of the design in a general setting include doyle, j. Linear quadratic gaussian control for linear timedelay systems. Stoorvogel november, 1990 abstract inthispaperwe discuss thestandardlqg controlproblemfor linear, finitedimensional timeinvariantsystems without any assumptions on the system parameters. Linear quadratic gaussian lqg control of wind turbines.
In control theory, the linearquadraticgaussian lqg control problem is one of the most fundamental optimal control problems. Spectral factorization controller design simplified lqg control design introduction to performance analysis of controllers matlab code problems linear quadratic gaussian control digital control wiley online library. Quadratic gaussian design is a optimal control theory which has many application in control engineering problem. The engineer, usuallylikes to work with, and benefits from. Optimisation and control of vehicle suspension using linear. Linear quadratic gaussianbased closedloop control of. Pdf linear quadratic gaussian control of a quartercar. Adaptive continuoustime linear quadratic gaussian control. Optimal adaptive control of linearquadraticgaussian systems.
Optimal adaptive control of linearquadraticgaussian. These problems are chosen because of their simplicity, ubiquitous application, wellde. Thus the optimal control law u x,t is the same for stochastic and deterministic systems the latter is called lqr. Experimental study on modified linear quadratic gaussian. Frequencyshaped cost functionals extension of linear. The solution of the linear quadratic gaussian lqg control problem resulted in the famous separation. Optimal linear quadratic regulator theory, currently referred to as the linearquadraticgaussian lqg problem ref. We give an explicit formulafor the infimumover all internallystabilizing strictly proper. The typical implementation of the lqr solution requires that the plant states be estimated, which can be posed as. Most of the research efforts on the linear quadratic regulator problem and the linear quadratic gaussian problem took place in the period up to 1975, whereas in particular h.
1561 919 1100 478 577 588 1544 1079 204 654 615 576 921 439 592 1340 524 267 1344 218 1221 713 460 106 108 765 1100 737 461