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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

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Introduction to System Modeling and Control

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  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

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