National Aviation University

Institute of Aerospace Control Systems

Aircraft Control Systems Department

TERM PAPER

ANALYTICAL DESIGN OF AUTOPILOT FOR GIVEN OPEN-LOOP SYSTEM “ACTUATOR + AIRCRAFT”

on the discipline “Optimal Control Systems”

Submitted by student of IACS 405 Bogutska A.O.

Checked by Tunik A.A.

Kyiv 2013

Contents

Task………………………………………………………………………………...3

1. Theoretical information………………………………………………………….5

2. Results of MatLab code………………………………………………………….9

2.1. Determining of the characteristics of the system……………………….9

2.2. Solving the problem of analytical design of controller (autopilot) for given open-loop system…………………………………………………………...16

2.3. Determining the characteristics of closed-loop system with matrix of optimal gains of controller in feedback…………………………………………...19

Conclusions……………………………………………………………………….26

Appendix 1………………………………………………………………………..27

Task

For given models of aircraft and actuator it is necessary:

I. To describe the aircraft motion (including all states, inputs and outputs) and mode according to variant.

II. To create the program in MatLab and using it:

1. It is necessary to determine:

1. State space matrices of series connection “actuator+aircraft”;

2. Transfer functions of this connection from all inputs to all outputs;

3. Frequency responses (Nyquist and Bode diagrams) of this connection from all inputs to n-th output;

4. Eigenvalues of state matrix of the closed loop system (to close the open-loop system by unit negative feedback);

5. Graphic representation of the eigenvalues of the received closed loop system on the complex plane;

6. H₂-norm for the closed loop system;

7. H_∞-norm for the closed loop system.

2. To solve the problem of analytical design of controller (autopilot) for given open loop system.

3. To determine:

1. Eigenvalues of state matrix of closed loop system and show them on the complex plane;

2. H₂-norm for the closed loop system;

3. H_∞-norm for the closed loop system;

4. Transfer functions of closed loop system from all inputs to all outputs;

5. Frequency responses (Nyquist and Bode diagrams) of closed loop system from all inputs to n-th output.

III. To draw the conclusion of the work.

Variant 2

Model of short-range airplane. Longitudinal channel. Approach mode. State space variables:1)V-velocity, 2)alpha-angle of attack, 3)q-pitch rate,4)theta-pitch angle, 5)h-altitude, 6)n-engine's rpm. Control variables: 1)elevator, 2)thrust control. State space matrices of the aircraft:

Aair=[0.0027 6.1616 0 -9.8007 0 0.0288;

-0.0011 -0.9623 1 -0.0026 0 0;

-0.0004 -5.6916 -1.926 0.0013 0 0.0011;

0 0 1 0 0 0;

0.0349 134.1 0 -134.1 0 0;

0 0 0 0 0 -0.6667]

Bair=[-0.2326 0;

-0.0636 0;

-3.4498 0;

0 0;

0 0;

0 0.6667]

Cair=eye(6,6)

Dair=zeros(6,2)

n=1;

Actuator:

Ta=0.1

Aac=[-1/Ta 0;

0 -1/Ta]

Bac=[1/Ta 0;

0 1/Ta]

Cac=eye(2)

Dac=zeros(2)