Controlo em Espaço de Estados

Transcription

1 1 Controlo em Espaço de Estados 2016/ a) x x 1 João Miranda Lemos Professor Catedrático

2 2 Docents João Miranda Lemos (Theoretical lectures, course coordinator) , Office at INESC-ID Students: Send an or phone to decide an hour and day. José António Gaspar (Lab and Problems) , Office at ISR

3 3 Course organization Lectures: Theory (slides in 4 parts, see Fénix) Problems (see problems at Fénix) Lab (guide at Fénix) Self-study: Read and study the slides Self-study problems (Fénix) Bibliography (see Fénix) o Franklin, Powell, Emami-Naeini. Feedback Control of Dynamic Systems (chaps. 7, 9) Non-exhaustive!

4 4 Evaluation and grading Theory o 2 tests (strongly recomemded) or 1 exam o Not possible to repeat the tests Não há repescagem dos testes o Aproval: Theory grade (average of tests or exam) minimum of 9,3 o There is no minimum grade in each individual test. o If approved by tests, may improve the theory grade in the exam Laboratory o 1 work, report in groups of 3, grading individual by student Final grade (remark the constraint on the lab grade) o 0,7T+0.3min(L,T+6)

5 5 Laboratory Objective: Design and test a state feedback controller to equilibrate an inverted pendulum. Emphasis on the design of computer controlled systems with experimental verification Cyber Physical Systems 1. Design an optimal state feedback controller based on a given model. MATLAB 2. Test in simulation. Review the design. SIMULINK 3. Test on the real plant. Evaluate. Improve. SIMULINK MATLAB is increasingly popular in industry to develop initial versions of new ideas.

6 6 Why have you chosen this course? Mas afinal, esta é, ou não é, a reunião do Sindicato dos Padeiros? After all, is this, or is not, the meeting of the Baker s Union?

7 7 The state model u y Input/output model (differential equation or transfer function): Alternative: Two first-order differential equations (state model) d 2 dt y 2 u dx1 dt x1 ( t) y( t), x2( t) y ( t) dx dt 2 x 2 u

8 8 State of a system: A set of variables such that, you known at an instant, if we we know the set of external forces that act on the system, allow the computation of their values for all future time. The state variables satisfy a set of 1 st order differential equations known as the state model. Example in the linear case: dx dt = Ax + Bu (Model of dynamics and actuators) y = Cx (Sensor model) We will study the non-linear case as well.

9 9 Course objective Study of methods for analysis and design of control systems, based on the state model. Syllabus 1. Sate model analysis (model conversion, time response). 2. Controller design based on state variable feedback and state estimation with observers. 3. Stability and controller design for nonlinear systems. 4. Optimal Control design using Pontryagin s principle.

10 10 What are the main new things to learn in this course? A new model (state model, linear and nonlinear) A new technique to study stability (Lyapunov s direct method) New controller design methods o Pole placement by linear state feedback o Non-linear control o Optimal control o Multivariable control o Adaptive control (learn the model while the controller operates)

11 11 Major new ideas addressed Sate model and state feedback. Asymptotic observers. Kalman filter. Lyapunov s direct method Optimal control. Pontryagin s Principle.

12 12 Where did these ideas came from? Joseph-Louis Lagrange Analytical Mechanics Aleksandr Lyapunov Stability of nonlinear systems Rudolph Kalman Filtro. State model in Control

13 13 Johann Bernouilli ( ) and the brachistochrone What is the curve of fastest descent when sliding between (0,0) and the final point?

14 14 L. Euler Variational Calculus. Euler-Lagrange equation Lev Pontryagin Principle to solve optimal control problems

15 15 Example: Water delivery canal y 1 M1 u 1 u 2 u 3 u 4 M2 M3 M4 y 2 y 3 y 4 Q o Pool 1 G1 Pool 2 G2 Pool 3 G3 Pool 4 G4 Q 1 Q 2 Q 3 Q 4

16 J 3 [mm] J 2 [mm] J 1 [mm] Course presentation 16 Isolated PID s (no coordination) Gate Gate Gate

17 J 4 [mm] J 3 [mm] J 2 [mm] J 1 [mm] Course presentation 17 Canal optimal multivariable state feedback control J. M. Lemos and L. F. Pinto (2012). Distributed Linear-Quadratic Control of Serially Chained Systems -- Application to a Water Delivery Canal. IEEE Control Systems Mag., 32(6): Multivariable Controller u 1 u 2 u m y 1 y 2 y p Plant Gate Gate Gate Gate

18 18 State estimation: Localization using GPS dx f ( x) y h( x) dt Estimate x using the measures of y. Observers

19 19 Parabolic trough solar thermal fields y=temperatura u=caudal Espelho Keep the outlet temperature constant despite changes in solar radiation. An example of a nonlinear system to be tackled in the course.

20 20 Control for anesthesia J. M. Lemos et. al. (2014). Robust Control of Maintenance Phase Anesthesia. IEEE Control Systems, 34 (6): Available in Fénix

21 21

22 22 Optimal control for sea wave energy Source: Luis Gato, 2014 How to open and close the air valve to maximize the power transmitted from the wave train to the turbine? J. Henriques, L. Gato, J. M. Lemos, R. Gomes, A. Falcão (2016). Peak-power control of a grid-integrated oscillating water column wave energy converter. Energy, 109:

23 23 Put knowledge in action This is a course that addresses theoretical basis. However, many of them can be directly applied in a wide variety of fields with economical impact (Medicine, Niotechnology, Agriculture, Aerospace, Mechatronics, Robotics, Energy, Tellecommunications, Management ) There is nothing more practical than a good theory (Boltzman) Makers community (share knowledge to change the world):