2. Самарский государственный аэрокосмический университет имени академика С. П. Королева 1 I. E. DAVYDOV Modern Methods of Rockets Stability and Controllability Estimation RUSSIAN MINISTRY OF EDUCATION AND SCIENCE SAMARA STATE AEROSPACE UNIVERSITY I. E. DAVYDOV Modern Methods of Rockets Stability and Controllability Estimation Samara 2015

3. Самарский государственный аэрокосмический университет имени академика С. П. Королева 2 Module 1. Dynamic Characteristics of Carrier Rocket Module 2. General Ideas of Automatic Control Theory Module 3. Stability Criteria of Linear Automatic Control Systems Module 4. Quality of Control Process Module 5. Non-linear Automatic Control Systems TRAINING PLAN: Module 1. Dynamic Characteristics of Carrier Rocket Module 2. General Ideas of Automatic Control Theory Module 3. Stability Criteria of Linear Automatic Control Systems Module 4. Quality of Control Process Module 5. Non-linear Automatic Control Systems

4. Самарский государственный аэрокосмический университет имени академика С. П. Королева 3 Module 1. Dynamic Characteristics of Carrier Rocket Picture 1.1., 1.2 – Speed and principal coordinate systems The system of equations:

5. Самарский государственный аэрокосмический университет имени академика С. П. Королева 4 Equations of disturbed motion are written in matrix form: Module 1. Dynamic Characteristics of Carrier Rocket X – column vector (n  1) X – column vector (n  1) of system status with components:  coefficient’s matrices of equations of disturbed motion sized (n  n) (n=3+m+h); F – column vector of disturbances sized (n  1);

6. Самарский государственный аэрокосмический университет имени академика С. П. Королева 5 Module 2. General Ideas of Automatic Control Theory Picture 2.1 — Simplified analysis plan of CR disturbed motion in heel channel Where: γ - heel angle; γпр – programmed value of heel angle; Оxyz – undisturbed principal coordinate system; - disturbed principal coordinate system; 1 – pitch gyro (AT – angle transmitter) ; 2 – servodrive amplifier; 3 – actuator; 4 – transfer function of controlled object (CO) which describe CR movement in heel channel; 5 – angular speed sensor (ASS). Carrier Rocket (CR) ACS in Heel Channel

7. Самарский государственный аэрокосмический университет имени академика С. П. Королева 6 Module 2. General Ideas of Automatic Control Theory ACS Classification: Example of disturbed and undisturbed movement of SpCr :

8. Самарский государственный аэрокосмический университет имени академика С. П. Королева 7 Module 3. Stability Criteria of Linear Automatic Control Systems - Vyshnegradsky stability criterion. - Algebraic stability criteria (Gurwits, Gauss). - Frequency stability criteria (Michailov, Naichlist). Mikhailov Curve: The change of amplitude of vector jω - pk because of frequency growth ω from -∞ to +∞

9. Самарский государственный аэрокосмический университет имени академика С. П. Королева 8 Module 3. Stability Criteria of Linear Automatic Control Systems Modal Forming Method of Dynamic System Properties where Ds- location area on plane of complex variable S spectrums of systems assemblage; pj – elements k - project (forming) system parameters vectors; Pf – multitude of possible project parameters; P – multitude of system project parameters. In general linear stationary object ‘CR-SD’ is described by equations in vector-matrix form:

10. Самарский государственный аэрокосмический университет имени академика С. П. Королева 9 Module 3. Stability Criteria of Linear Automatic Control Systems Modal Forming Method of Dynamic System Properties Operator L acting on any matrix A s, chosen from, transfers its spectrum to points multitude contained in where 0 – zero matrix.

11. Самарский государственный аэрокосмический университет имени академика С. П. Королева 10 Module 3. Stability Criteria of Linear Automatic Control Systems Modal Forming Method of Dynamic System Properties

12. Самарский государственный аэрокосмический университет имени академика С. П. Королева 11 Module 4. Quality of Control Process Structural plan of the system Contents of calculating-graphic work: 1. Work topic. 2. Purpose of the work. 3. Structural plan of automatic system with values. 4. Original system transformation to single circuit and defining transfer function of an open and close system. 5. Defining characteristic polynomial of a close system. 6. Defining system stability by: - algebraic Gurwits stability criterion; - particular Mikhailov stability criterion. 7. Conclusions for fulfilled work.

13. Самарский государственный аэрокосмический университет имени академика С. П. Королева 12 Module 4. Quality of Control Process Structural plan of the system Dynamic links parameters: Variant number Coefficients k1k1 T2T2 k4k4 T4T4 k5k5 T5T5 C5C5 110.03250.10.50.3160.3 21.50.2373.50.081.120.50.3 31.20.1691.80.151.580.3160.5 420.2630.11.2850.3460.6 51.80.261.60.0851.550.310.4 61.50.832.50.21.260.220.3 720.19630.121.2850.270.35 81.30.452.80.150.850.3160.25 91.40.091.50.12.250.450.5 101.20.022.20.052.50.780.25 111.60.151.80.072.30.60.35 121.80.3520.081.740.660.2 131.20.52.50.22.250.45 141.50.830.251.250.35 1520.541.50.181.250.330.4 161.80.5620.121.50.3 1710.3230.12.250.40.25 1830.5510.0820.560.25 192.50.061.60.052.80.840.35 201.50.182.50.151.50.350.6

14. Самарский государственный аэрокосмический университет имени академика С. П. Королева 13 Module 4. Quality of Control Process Structural plan of the system Mikhailov curve

15. Самарский государственный аэрокосмический университет имени академика С. П. Королева 14 Module 5. Non-linear Automatic Control Systems Structural diagram of a nonlinear system 1 - the servo drive amplifier described by equation of aperiodic link; 2 - nonlinearity of the "dead zone" type with current saturation due to power limitation; 3 - speed characteristic of the servo unit; 4 - non-linear element of the saturation zone type in respect of deviation of the servo unites due to design features. K  angular transfer function ratio; Ky  overall gain; а 4  saturation of speed characteristic; Wрн (р)  SR angular transfer function; Kw  attitude rate transfer function ratio.

16. Самарский государственный аэрокосмический университет имени академика С. П. Королева 15 Module 5. Non-linear Automatic Control Systems Structural diagram of a nonlinear system Problem 1. In accordance with diagram: - Get transfer function of the control actuator. - Get AS pitch angle equation. Assume that the nonlinear link 3 has a slope of К рм.

17. Самарский государственный аэрокосмический университет имени академика С. П. Королева 16 Module 5. Non-linear Automatic Control Systems Structural diagram of a nonlinear system Problem 2. Diagram describes link 3 by the function:  nonlinear link., where А and   are respectively the amplitude and frequency of self-oscillation at the inlet of the nonlinear link. Than, for the given link we have at the output where q(A)  harmonic linearization of the nonlinear link. Take the equation of attitude (angular) motion by the pitch plane as: Get:- - the transfer function of the linear part and nonlinear link. - Determine the amplitude and frequency of self-oscillations.

18. Самарский государственный аэрокосмический университет имени академика С. П. Королева 17 Module 5. Non-linear Automatic Control Systems Structural diagram of a nonlinear system Problem 3. High-frequency interference with the frequency  (t) and amplitude В (f = B Sin  t) enters to the input of the measuring device (gyro). Equation of measuring device to be represented in the form (blocks 1, 5): The amplitude of the high-frequency forced oscillations at the input of servo unit is determined by the formula: Assess vibration noise immunity applying the methodology of calculation of the stability areas by the modal method of forming design parameters of the dynamic system (in this case, the AC parameters).

19. Самарский государственный аэрокосмический университет имени академика С. П. Королева 18 Final attestation Modern Methods of Rockets Stability and Controllability Estimation