1 / 55

Introduction to System Modeling and Control

Introduction to System Modeling and Control. Introduction Basic Definitions Different Model Types System Identification Neural Network Modeling. Mathematical Modeling (MM). A mathematical model represent a physical system in terms of mathematical equations

bao
Download Presentation

Introduction to System Modeling and Control

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Introduction to System Modeling and Control • Introduction • Basic Definitions • Different Model Types • System Identification • Neural Network Modeling

  2. Mathematical Modeling (MM) • A mathematical modelrepresent a physical system in terms of mathematical equations • It is derived based on physical laws (e.g.,Newton’s law, Hooke’s, circuit laws, etc.) in combination with experimental data. • It quantifies the essential features and behavior of a physical system or process. • It may be used for prediction, design modification and control.

  3. Numerical Solution Theory Solution Data Data Engineering System Math. Model Model Reduction Control Design Graphical Visualization/Animation Engineering Modeling Process • Example: Automobile • Engine Design and Control • Heat & Vibration Analysis • Structural Analysis

  4. System Variables Every system is associated with 3 variables: • Input variables (u) originate outside the system and are not affected by what happens in the system • State variables (x) constitute a minimum set of system variables necessary to describe completely the state of the system at any given time. • Output variables (y) are a subset or a functional combination of state variables, which one is interested to monitor or regulate. y u System x

  5. Mathematical Model Types discrete-event distributed Lumped-parameter Most General Input-Output Model Linear-Time invariant (LTI) LTI Input-Output Model Discrete-time model: Transfer Function Model

  6. u x x fs fs M M fd fd Example: Accelerometer (Text 6.6.1) Consider the mass-spring-damper (may be used as accelerometer or seismograph) system shown below: Free-Body-Diagram fs(y): position dependent spring force, y=u-x fd(y): velocity dependent spring force Newton’s 2nd law Linearizaed model:

  7. Acceleromter Transfer Function • Accelerometer Model: • Transfer Function: Y/A=1/(s2+2ns+n2) • n=(k/m)1/2, =b/2n • Natural Frequency n, damping factor  • Model can be used to evaluate the sensitivity of the accelerometer • Impulse Response • Frequency Response

  8. Impulse Response

  9. Frequency Response /n

  10. Mixed Systems • Most systems in mechatronics are of the mixed type, e.g., electromechanical, hydromechanical, etc • Each subsystem within a mixed system can be modeled as single discipline system first • Power transformation among various subsystems are used to integrate them into the entire system • Overall mathematical model may be assembled into a system of equations, or a transfer function

  11. Ra La B ia dc u J  Electro-Mechanical Example Input: voltage u Output: Angular velocity  Elecrical Subsystem (loop method): Mechanical Subsystem

  12. Ra La B ia dc u  Electro-Mechanical Example Power Transformation: Torque-Current: Voltage-Speed: where Kt: torque constant, Kb: velocity constant For an ideal motor Combing previous equations results in the following mathematical model:

  13. System identification Experimental determination of system model. There are two methods of system identification: • Parametric Identification: The input-output model coefficients are estimated to “fit” the input-output data. • Frequency-Domain (non-parametric): The Bode diagram [G(j) vs.  in log-log scale] is estimated directly form the input-output data. The input can either be a sweeping sinusoidal or random signal.

  14. Ra La B ia Kt u 12  10 u 8 t 6 Amplitude 4 T 2 0 0 0.1 0.2 0.3 0.4 0.5 Time (secs) Electro-Mechanical Example Transfer Function, La=0: k=10, T=0.1

  15. Comments on First Order Identification Graphical method is • difficult to optimize with noisy data and multiple data sets • only applicable to low order systems • difficult to automate

  16. Least Squares Estimation • Given a linear system with uniformly sampled input output data, (u(k),y(k)), then • Least squares curve-fitting technique may be used to estimate the coefficients of the above model called ARMA (Auto Regressive Moving Average) model.

  17. Nonlinear System Modeling& Control Neural Network Approach

  18. Introduction • Real world nonlinear systems often difficult to characterize by first principle modeling • First principle models are oftensuitable for control design • Modeling often accomplished with input-output maps of experimental data from the system • Neural networks provide a powerful tool for data-driven modeling of nonlinear systems

  19. Input-Output (NARMA) Model

  20. What is a Neural Network? • Artificial Neural Networks (ANN) are massively parallel computational machines (program or hardware) patterned after biological neural nets. • ANN’s are used in a wide array of applications requiring reasoning/information processing including • pattern recognition/classification • monitoring/diagnostics • system identification & control • forecasting • optimization

  21. Advantages and Disadvantages of ANN’s • Advantages: • Learning from • Parallel architecture • Adaptability • Fault tolerance and redundancy • Disadvantages: • Hard to design • Unpredictable behavior • Slow Training • “Curse” of dimensionality

  22. Biological Neural Nets • A neuron is a building block of biological networks • A single cell neuron consists of the cell body (soma), dendrites, and axon. • The dendrites receive signals from axons of other neurons. • The pathway between neurons is synapse with variable strength

  23. Artificial Neural Networks • They are used to learn a given input-output relationship from input-output data (exemplars). • The neural network type depends primarily on its activation function • Most popular ANNs: • Sigmoidal Multilayer Networks • Radial basis function • NLPN (Sadegh et al 1998,2010)

  24. x1 y x2 weights activation function Multilayer Perceptron • MLP is used to learn, store, and produce input output relationships • The activation function (x) is a suitable nonlinear function: • Sigmidal: (x)=tanh(x) • Gaussian: (x)=e-x2 • Triangualr (to be described later)

  25. Sigmoidal and Gaussian Activation Functions

  26. y x W0 Wp Multilayer Netwoks Wk,ij: Weight from node i in layer k-1 to node j in layer k

  27. Universal Approximation Theorem (UAT) Comments: • The UAT does not say how large the network should be • Optimal design and training may be difficult A single hidden layer perceptron network with a sufficiently large number of neurons can approximate any continuous function arbitrarily close.

  28. Training • Objective: Given a set of training input-output data (x,yt) FIND the network weights that minimize the expected error • Steepest Descent Method: Adjust weights in the direction of steepest descent of L to make dL as negative as possible.

  29. Neural Networks with Local Basis Functions • These networks employ basis (or activation) functions that exist locally, i.e., they are activated only by a certain type of stimuli • Examples: • Cerebellar Model Articulation Controller (CMAC, Albus) • B-Spline CMAC • Radial Basis Functions • Nodal Link Perceptron Network (NLPN, Sadegh)

  30. Biological Underpinnings • Cerebellum: Responsible for complex voluntary movement and balance in umans • Purkinje cells in cerebellar cortex is believed to have CMAC like architecture

  31. Nodal Link Perceptron Network (NLPN) [Sadegh, 95,98] • Piecewise multilinear network (extension of 1-dimensional spline) • Good approximation capability (2nd order) • Convergent training algorithm • Globally optimal training is possible • Has been used in real world control applications

  32. wi y x NLPN Architecture • Input-Output Equation • Basis Function: • Each ij is a 1-dimensional triangular basis function over a finite interval

  33. wi+1 wi ai ai+1 NLPN Approximation: 1-D Functions • Consider a scalar function f(x) • f(x) on interval [ai,ai+1] can be approximated by a line

  34. ai-1 ai ai+1 Basis Function Approximation • Defining the activation/basis functions • Function f can expressed as • This is also similar to fuzzy-logic approximation with “triangular” membership functions. (1st order B-spline CMAC)

  35. Neural Network Approximation of NARMA Model y u[k-1] y[k-m] Question: Is an arbitrary neural network model consistent with a physical system (i.e., one that has an internal realization)?

  36. State-Space Model u y system States: x1,…,xn

  37. A Class of Observable State Space Realizable Models • Consider the input-output model: • When does the input-output model have a state-space realization?

  38. Comments on State Realization of Input-Output Model • A Generic input-Output Model does not necessarily have a state-space realization (Sadegh 2001, IEEE Trans. On Auto. Control) • There are necessary and sufficient conditions for realizability • Once these conditions are satisfied the state-space model may be symbolically or computationally constructed • A general class of input-Output Models may be constructed that is guaranteed to admit a state-space realization

  39. The Model Form • The following Input-Output Model always admits a minimal state realization:

  40. State Space Realization • The state-model of the input-output model is as follows with y=x1:

  41. Neural Networks • Reduced coupling results in sub-networks: • Can’t use prepackaged software, but standard training methods are the same

  42. Nodal Link Perceptron Networks • Local basis functions, similar to CMAC networks. Reduced Coupling also results in sub-networks:

  43. Simulation Example • Nonlinear mass spring damper • Data sampled at 0.01s, output is the velocity of the 2nd mass

  44. Simulation Results I: Linear model. mse=0.0281. training(static) mse=0.0059. II: NARMA model. mse=0.0082. training(static) mse=0.0021. III. Neural network. mse=3.6034e-4. training(static) mse=0.0016.N IV. NLPN. mse=7.2765e-4. training(static) mse=2.6622e-4.

  45. Simulation Results I: Linear model. mse=0.0271. II: NARMA model. mse=0.0067. III. Neural network. mse=5.3790e-4. IV. NLPN. mse=7.1835e-4.

  46. Conclusions • A number of data driven modeling techniques are suitable for an observable state space transformation • Rough guidelines were given for when and how to use NARMA, neural network and NLPN models • NLPN modifications make it an easily trainable option with excellent capabilities • Substantial training & design issues include data sampling rate and input repetition due to the reduced coupling restriction

  47. Fluid Power Application

  48. INTRODUCTION APPLICATIONS: • Robotics • Manufacturing • Automobile industry • Hydraulics EXAMPLE: EHPV control (electro-hydraulic poppet valve) • Highly nonlinear • Time varying characteristics • Control schemes needed to open two or more valves simultaneously

  49. Motivation • The valve opening is controlled by means of the solenoid input current • The standard approach is to calibrate of the current-opening relationship for each valve • Manual calibration is time consuming and inefficient

  50. Research Goals • Precisely control the conductivity of each valve using a nominal input-output relationship. • Auto-calibrate the input-output relationship • Use the auto-calibration for precise control without requiring the exact input-output relationship

More Related