Dynamics and control of mechanical systems

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1 Dynamics and control of mechanical systems Date Day 1 (03/05) - 05/05 Day 2 (07/05) Day 3 (09/05) Day 4 (11/05) Day 5 (14/05) Day 6 (16/05) Content Review of the basics of mechanics. Kinematics of rigid bodies - coordinate transformation, angular velocity vector, description of velocity and acceleration in relatively moving frames. Euler angles, Review of methods of momentum and angular momentum of system of particles, inertia tensor of rigid body. Dynamics of rigid bodies - Euler's equation, application to motion of symmetric tops and gyroscopes and problems of system of bodies. Kinetic energy of a rigid body, virtual displacement and classification of constraints. D Alembert s principle. Introduction to generalized coordinates, derivation of Lagrange's equation from D Alembert s principle. Small oscillations, matrix formulation, Eigen value problem and numerical solutions. Modelling mechanical systems, Introduction to MATLAB, computer generation and solution of equations of motion. Introduction to complex analytic functions, Laplace and Fourier transform. PID controllers, Phase lag and Phase lead compensation. Analysis of Control systems in state space, pole placement, computer simulation through MATLAB. 1

2 Contents Focus on Introduction to control systems Basic elements of a control system Types of control system Control system connections Feedback connections Reduction of feedback systems Introduction to PID controllers Example 2

3 Introduction What is Control of Mechanical System? - It refers to the process of modifying the dynamic behavior of a system in order to achieve some desired outputs. - It provides the desired response by controlling the output, i.e. output is controlled by varying input Primary objectives of control system analysis - determination of the degree of the system stability, the steady-state performance, and characteristics of transient response. Procedures to analyze a control system : - the derivation of the equation of motion (transfer function (TF)) for every component; - representation of the system, such as using a block diagram; and - determination of system characteristics. - Block diagrams contain transfer functions of each component or element of the system 3 INPUT CONTROL SYSTEM OUTPUT

4 Basic elements of a control system Blocks: The two blocks G(s) and H(s) represent TFs - a block has single input and single output Summing point: - has two or more inputs and single output. - produces the algebraic sum of the inputs. - performs the summation or subtraction or combination of summation and subtraction of the inputs based on the polarity/sign of the inputs. 4

5 Basic elements of a control system Take-off point: a point from which the same input signal can be passed through more than one branch. take-off point can be used to apply the same input to one or more blocks, summing points. 5

6 Basic elements of a control system Transfer function (TF) - a mathematical operator that gives the ratio of the Laplace transformed output to Laplace transformed input. GG ss = YY(ss) XX(ss) TF for open-loop control OOOO 6 GG = YY XX

7 Types of control systems Diverse classifications of control systems Process Control or Regulator (e.g. flyball governor) - maintains constant output despite disturbances Compensator (e.g. elevator) - drive system from an initial to a final state according to specifications on the transient response Tracking (e.g. space robot) - match output to a non-stationary input despite disturbances Optimal control - drive system from an initial to a final state while optimizing a merit function (e.g. minimum time to target or minimum energy consumption) Combinations of the above 7

8 Types of control systems Diverse classifications of control systems Based on type of signal: Continuous time and Discretetime Control Systems Based on number of inputs and outputs: Single Input and Single Output (SISO) and Multiple Input and Multiple Output (MIMO) Control Systems Based on existence of feedback path: Open-loop and closed-loop control systems 8

9 Types of control systems Open-loop and closed-loop control systems - In open-loop control systems, output is not fedback to the input. So, the control action is independent of the desired output, example: a traffic light control system - Open-loop is a special case of closed loop control system - Open-control is simpler 9

10 Types of control systems Closed-loop control systems In closed-loop control systems, output is fed back to the input. So, the control action is dependent on the desired output. The error detector produces an error signal- the difference between the input and the feedback signal. This feedback signal is obtained from the block by considering the output of the overall system as an input to this block. 10 Example: a traffic control system with a sensor at its input

11 Types of control systems Open-loop and closed-loop control systems Open-loop control system Control action is independent of the desired output. There exists no feedback path in the block diagram Easy to design Closed-loop control system Control action is dependent of the desired output. There exists feedback path Difficult to design Inaccurate control not useful in industrial plant control Economical Uncontrolled system Better accurate control Expensive Controlled system 11

12 Control System Connections Block diagrams in series connection GG = YY XX 11 = XX 22 XX 11 XX 33 XX 22 YY XX 33 = GG 11 GG 22 GG 33 Block diagrams in parallell connection GG = YY XX G = 11 nn GG ii where 12 Y = YY 11 +YY 22 +YY 33 = GG 11 XX +GG 22 XX + GG 33 XX = (GG 11 +GG 22 +GG 33 )XX G = 1 nn GG ii

13 Two forms Feedback connections - (1) Positive feedback and (2) Negative feedback Negative feedback Y = E.G where E = X H.Y Y = (X H.Y)G = X.G H.Y.G Y(1+H.G) = X.G Negative feedback TF G = GG 11+GG.HH 13 YY XX = GG 11 + GG. HH Exercise: Derive the TF for a positive feedback

14 Block diagram reduction Block diagram reduction is applied to simplify the procedure of getting closed-loop TF of complex control system (1) Shifting take-off point after TF X Y X Y Equivalent B = X B = YY GG Y = X.G where X = YY GG B = YY GG 14 B = XX.GG GG The above block diagrams are equivalent = X

15 Block diagram reduction (2) Shifting summation point after TF Y 1 X Y X Y Equivalent Y = X.G = (X + B)G = X.G + B.G The above block diagrams are equivalent 15 Y 1 = X.G B 1 = B.G and Y = Y 1 + B 1 = X.G + B.G

16 Block diagram reduction - Example X Y 16

17 Block diagram reduction Example... after first reduction X Y 17

18 Block diagram reduction Example final feedback control system Y X 18

Introduction to PID Controller PID: P Proportional I Integral D Derivative - PID controllers are typical closed-loop controllers - PID controllers

19 Introduction to PID Controller PID: P Proportional I Integral D Derivative - PID controllers are typical closed-loop controllers - PID controllers represent about 95% of industrial control in the form of microcontrollers - PID controllers are simple, provide good stability and rapid response 19

20 Introduction to PID Controller Applications of PID controller: Temperature control: Let us take an example of AC (air-conditioner) of any plant/process. - Setpoint is temperature (20 0 C) and current measured temperature by sensor is 28 0 C. Our aim is to run AC at desired temperature (20 0 C). - Now, controller of AC, generate signal according to error (8 0 C) and this signal is given to the AC. - According to this signal, output of AC is changed and temperature decrease to 25 0 C. - The same process will repeat until temperature sensor measures desired temperature. - When error is zero, controller will give stop command to AC and again temperature will increase up to certain value and again error will generate and same process repeated continuously. 20

21 Summary This lecture has focused on Some fundamentals of controls systems - Types of control systems, - Basic components - Block diagram and TF Reduction of complex feedback systems Brief introduction to PID control system THAT IS THE END!? 21

Fundamental principles

Fundamental principles Dynamics and control of mechanical systems Date Day 1 (03/05) - 05/05 Day (07/05) Day 3 (09/05) Day 4 (11/05) Day 5 (14/05) Day 6 (16/05) Content Review of the basics of mechanics. Kinematics of rigid