 LinearquadraticGaussian control

In control theory, the linearquadraticGaussian (LQG) control problem is one of the most fundamental optimal control problems. It concerns uncertain linear systems disturbed by additive white Gaussian noise, having incomplete state information (i.e. not all the state variables are measured and available for feedback) and undergoing control subject to quadratic costs. Moreover the solution is unique and constitutes a linear dynamic feedback control law that is easily computed and implemented. Finally the LQG controller is also fundamental to the optimal perturbation control of nonlinear systems^{[1]}.
The LQG controller is simply the combination of a Kalman filter i.e. a linearquadratic estimator (LQE) with a linearquadratic regulator (LQR). The separation principle guarantees that these can be designed and computed independently. LQG control applies to both linear timeinvariant systems as well as linear timevarying systems. The application to linear timeinvariant systems is well known. The application to linear timevarying systems enables the design of linear feedback controllers for nonlinear uncertain systems.
The LQG controller itself is a dynamic system like the system it controls. Both systems have the same state dimension. Therefore implementing the LQG controller may be problematic if the dimension of the system state is large. The reducedorder LQG problem (fixedorder LQG problem) overcomes this by fixing apriori the number of states of the LQG controller. This problem is more difficult to solve because it is no longer separable. Also the solution is no longer unique. Despite these facts numerical algorithms are available^{[2]}^{[3]}^{[4]}^{[5]} to solve the associated optimal projection equations^{[6]}^{[7]} which constitute necessary and sufficient conditions for a locally optimal reducedorder LQG controller^{[2]}.
Finally, a word of caution. LQG optimality does not automatically ensure good robustness properties.^{[8]} The robust stability of the closed loop system must be checked separately after the LQG controller has been designed. To promote robustness some of the system parameters may be assumed stochastic instead of deterministic. The associated more difficult control problem leads to a similar optimal controller of which only the controller parameters are different^{[3]}.
Contents
Mathematical description of the problem and solution
Continuous time
Consider the linear dynamic system,
where represents the vector of state variables of the system, the vector of control inputs and the vector of measured outputs available for feedback. Both additive white Gaussian system noise and additive white Gaussian measurement noise affect the system. Given this system the objective is to find the control input history which at every time may depend only on the past measurements such that the following cost function is minimized,
where denotes the expected value. The final time (horizon) may be either finite or infinite. If the horizon tends to infinity the first term of the cost function becomes negligible and irrelevant to the problem. Also to keep the costs finite the cost function has to be taken to be .
The LQG controller that solves the LQG control problem is specified by the following equations,
The matrix is called the Kalman gain of the associated Kalman filter represented by the first equation. At each time this filter generates estimates of the state using the past measurements and inputs. The Kalman gain is computed from the matrices , the two intensity matrices , associated to the white Gaussian noises and and finally . These five matrices determine the Kalman gain through the following associated matrix Riccati differential equation,
Given the solution the Kalman gain equals,
The matrix is called the feedback gain matrix. This matrix is determined by the matrices and through the following associated matrix Riccati differential equation,
Given the solution the feedback gain equals,
Observe the similarity of the two matrix Riccati differential equations, the first one running forward in time, the second one running backward in time. This similarity is called duality. The first matrix Riccati differential equation solves the linearquadratic estimation problem (LQE). The second matrix Riccati differential equation solves the linearquadratic regulator problem (LQR). These problems are dual and together they solve the linearquadraticGaussian control problem (LQG). So the LQG problem separates into the LQE and LQR problem that can be solved independently. Therefore the LQG problem is called separable.
When and the noise intensity matrices , do not depend on and when tends to infinity the LQG controller becomes a timeinvariant dynamic system. In that case both matrix Riccati differential equations may be replaced by the two associated algebraic Riccati equations.
Discrete time
Since the discretetime LQG control problem is similar to the one in continuoustime the description below focuses on the mathematical equations.
Discretetime linear system equations:
Here represents the discrete time index and represent discretetime Gaussian white noise processes with covariance matrices respectively.
The quadratic cost function to be minimized:
The discretetime LQG controller:
 ,
The Kalman gain equals,
where is determined by the following matrix Riccati difference equation that runs forward in time,
The feedback gain matrix equals,
where is determined by the following matrix Riccati difference equation that runs backward in time,
If all the matrices in the problem formulation are timeinvariant and if the horizon tends to infinity the discretetime LQG controller becomes timeinvariant. In that case the matrix Riccati difference equations may be replaced by their associated discretetime algebraic Riccati equations. These determine the timeinvarant linearquadratic estimator and the timeinvariant linearquadratic regulator in discretetime. To keep the costs finite instead of one has to consider in this case.
See also
 Stochastic control
 Witsenhausen's counterexample
References
 ^ Athans M. (1971). "The role and use of the stochastic LinearQuadraticGaussian problem in control system design". IEEE Transaction on Automatic Control AC16 (6): 529–552. doi:10.1109/TAC.1971.1099818.
 ^ ^{a} ^{b} Van Willigenburg L.G., De Koning W.L. (2000). "Numerical algorithms and issues concerning the discretetime optimal projection equations". European Journal of Control 6 (1): 93–100. Associated software download from Matlab Central.
 ^ ^{a} ^{b} Van Willigenburg L.G., De Koning W.L. (1999). "Optimal reducedorder compensators for timevarying discretetime systems with deterministic and white parameters". Automatica 35: 129–138. doi:10.1016/S00051098(98)001381. Associated software download from Matlab Central.
 ^ Zigic D., Watson L.T., Collins E.G., Haddad W.M., Ying S. (1996). "Homotopy methods for solving the optimal projection equations for the H2 reduced order model problem". International Journal of Control 56 (1): 173–191. doi:10.1080/00207179208934308.
 ^ Collins Jr. E.G, Haddad W.M., Ying S. (1996). "A homotopy algorithm for reducedorder dynamic compensation using the HylandBernstein optimal projection equations". Journal of Guidance Control & Dynamics 19 (2): 407–417. doi:10.2514/3.21633.
 ^ Hyland D.C, Bernstein D.S. (1984). "The optimal projection equations for fixed order dynamic compensation". IEEE Transaction on Automatic Control AC29 (11): 1034–1037. doi:10.1109/TAC.1984.1103418.
 ^ Bernstein D.S., Davis L.D., Hyland D.C. (1986). "The optimal projection equations for reducedorder discretetime modeling estimation and control". Journal of Guidance Control and Dynamics 9 (3): 288–293. doi:10.2514/3.20105.
 ^ Green, Limebeer: Linear Robust Control, p. 27
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