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PERIODICA POLYTECHNICA SER. TRANSP. ENG. VOL. 28, NO. 1–2, PP. 131–142 (2000) LQG/LTR CONTROLLER DESIGN FOR AN AIRCRAFT MODEL Balázs KULCSÁR Department of Control and Transport Automation Budapest University of Technology and Economics H–1111, Budapest, Bertalan L. u. 2., Hungary Fax: +36-1-463-3087 e-mail: [email protected] Phone: +36-1-463-3089 Received: Nov. 10, 2000 Abstract The automation of the operation of aircraft equipment lightens and diminishes the manipulation of the pilot during the flight. This article gives some introduction about the fundamental aspects of the LQG/LTR control theory, and it shows an application option through an example. The movement of an aircraft can be modeled with a linear time invariant dynamic system, which must be controlled by a flight controller. This article contains the synthesis and analysis of the stability and other qualitative control parameters of the flight control. Keywords: LQG/LTR controller, aircaft. 1. Introduction In this paper we shall give a short view of the so-called Linear Quadratic Gaussian theory, which can be consulted for more details in [1] and [2]. KWAKERNAAK and SILVAN ,ANDERSON and MOORE ,DAVIS and VINTER ,ASTRÖM and WIT- TENMARK ,FRANKLIN and POWEL and many others worked on this theory. Then we revise the main stages of a Linear Quadratic Gaussian/Loop Transfer Recovery method, which was elaborated by Doyle and Stein [16]. This article dicusses an example of an aircraft flight controller design, using first LQR and LQG, after- wards the LQG/LTR methods. We can refer to [3], [4] and [5], [7] where we can find some basic applications for Linear Quadratic controller design for simplified aircraft models. First of all we will examine the traditional optimal controller design of the aircraft flight controller system using LQR (Linear Quadratic Regulator) and LQG (Linear Quadratic Gaussian) method. With LQG/LTR method we recover the sta- bility margin of the Kalman filter at the plant output. In the LQG case we can use the separation principle, which means that we are able to design the LQG controller in two steps. First, the design of the LQR (Linear Quadratic Regulator), and then we have to find a state estimator, an LQE (Linear Quadratic Estimator) applying a modified cost function. where x is the state vector, r is the referential signal, y is the output vector, w( t ) is the state noise, v( t ) is the sensor noise, ˆ x is the estimated state vector, e is the error
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132 B. KULCSÁR B A C r L A C o y o x o r x v y w G -K -I e o r x o x Controller Plant Fig. 1 . The LQG controlled plant signal, ˆ y is the estimated output vector, A is the state matrix, B is the input matrix, C is the output matrix, G is the disturbance input matrix, K is the static feedback gain matrix, L is the stationary Kalman filter static gain matrix, the observer gain, I is the identity matrix.
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