Report

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 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. Engineering Modeling Process Theory Data Engineering System f m dv bv dt c v T v x Numerical Solution f Solution Data Math. Model Model Reduction Control Design Example: Automobile • Engine Design and Control • Heat & Vibration Analysis • Structural Analysis Graphical Visualization/Animation Definition of System System: An aggregation or assemblage of things so combined by man or nature to form an integral and complex whole. From engineering point of view, a system is defined as an interconnection of many components or functional units act together to perform a certain objective, e.g., automobile, machine tool, robot, aircraft, etc. System Variables Every system is associated with 3 variables: y u System x 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. Mathematical Model Types Lumped-parameter discrete-event Most General x f ( x , u , t ) y h( x, u , t ) Input-Output Model y (n) f (y Linear-Time invariant (LTI) x Ax Bu y Cx Du distributed ( n 1 ) (n) , , y , y , u , , u , u , t ) LTI Input-Output Model y (n) a1 y ( n 1 ) a n 1 y a n y b0 u (n) b n 1u b n u Discrete-time model: Transfer Function Model Y ( s ) G ( s )U ( s ) x ( t ) x ( t 1) (i) y (t ) y (t i ) Example: Accelerometer (Text 6.6.1) Consider the mass-spring-damper (may be used as accelerometer or seismograph) system shown below: Free-Body-Diagram x x fs fs M M fd fd fs(y): position dependent spring force, y=u-x fd(y): velocity dependent spring force Newton’s 2nd law Linearizaed model: M u y f d ( y ) f s ( y ) Mx M y b y ky M u u Example II: Delay Feedback Consider the digital system shown below: u Input-Output Eq.: D e la y z -1 y y ( k ) y ( k 1) u ( k 1) k 1 Equivalent to an integrator: y (k ) u( j ) j 0 Transfer Function Transfer Function is the algebraic input-output relationship of a linear time-invariant system in the s (or z) domain U G Y Example: Accelerometer System m y b y ky u G ( s ) Y (s ) ms U (s ) ms 2 2 bs k ,s d dt Example: Digital Integrator y ( k ) y ( k 1) u ( k 1) G Y (z ) u(z ) z 1 1 z ,z 1 Forward shift Comments on TF Transfer function is a property of the system independent from input-output signal It is an algebraic representation of differential equations Systems from different disciplines (e.g., mechanical and electrical) may have the same transfer function Acceleromter Transfer Function Accelerometer Model: M y b y ky M u Transfer Function: Y/A=1/(s2+2ns+n2) n=(k/m)1/2, =b/2n Natural Frequency n, damping factor Model can be used to evaluate the sensitivity of the accelerometer Impulse Response Frequency Response Impulse Response Frequency Response Bode Diagrams From: U(1) 40 0 -20 -40 -60 0 -50 To: Y(1) Phase (deg); Magnitude (dB) 20 -100 -150 -200 -1 10 10 0 Frequency (rad/sec) 10 /n 1 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 Electro-Mechanical Example Input: voltage u Output: Angular velocity Ra u La ia B dc Elecrical Subsystem (loop method): u R a i a La di a dt e b , e b back - emf voltage Mechanical Subsystem B T motor J J Electro-Mechanical Example Power Transformation: Torque-Current: Voltage-Speed: Ra B T motor K t i a eb K b La u ia dc where Kt: torque constant, Kb: velocity constant For an Kt Kb ideal motor Combing previous equations results in the following mathematical model: di a L R aia K b u a dt J B K t ia 0 Brushless D.C. Motor A brushless PMSM has a wound stator, a PM rotor assembly and a position sensor. The combination of inner PM rotor and outer windings offers the advantages of low rotor inertia efficient heat dissipation, and reduction of the motor size. dq-Coordinates q b d e a c e=p + 0 offset Electrical angle Number of poles/2 Mathematical Model di d R dt L di q R dt L id p m iq iq p m id 1 L vd K em L 1 L vq Where p=number of poles/2, Ke=back emf constant m Te K t iq J 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. Electro-Mechanical Example Ra Transfer Function, La=0: Ω(s) U(s) K t Ra Js B K t K b R a La B k ia u Ts 1 Kt 12 u t ku 10 k=10, T=0.1 Amplitude 8 6 4 T 2 0 0 0.1 0.2 0.3 Time (secs) 0.4 0.5 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 Least Squares Estimation Given a linear system with uniformly sampled input output data, (u(k),y(k)), then y ( k ) a 1 y ( k 1) a n y ( k n ) b1u ( k 1) b n u ( k n ) noise Least squares curve-fitting technique may be used to estimate the coefficients of the above model called ARMA (Auto Regressive Moving Average) model. Frequency-Domain Identification Method I (Sweeping Sinusoidal): f Ai system A0 , M agnitude Ai d b Ao t>>0 P hase Method II (Random Input): system Transfer function is determined by analyzing the spectrum of the input and output Photo Receptor Drive Test Fixture Experimental Bode Plot System Models M a g n it u d e (d B ) 25 0 high order 25 low order 50 75 0.1 1 10 100 1 10 100 1 10 3 Frequency (Hz) 180 Phas e ( D eg) 90 0 90 180 0.1 1 10 Frequency (Hz) 3 Nonlinear System Modeling & Control Neural Network Approach Introduction Real world nonlinear systems often difficult to characterize by first principle modeling First principle models are often suitable for control design Modeling often accomplished with inputoutput maps of experimental data from the system Neural networks provide a powerful tool for data-driven modeling of nonlinear systems Input-Output (NARMA) Model u z -1 z -1 z -1 y g z -1 z -1 z -1 y [ k ] g ( y [ k m ],..., y [ k 1], u [ k m ],..., u [ k 1]) 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 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 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 Artificial Neural Networks They are used to learn a given inputoutput 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) Multilayer Perceptron MLP is used to learn, store, and produce input output relationships y x1 y x2 w i ( x j v ij ) i j weights activation function The activation function (x) is a suitable nonlinear function: Sigmidal: (x)=tanh(x) Gaussian: (x)=e-x2 Triangualr (to be described later) Sigmoidal and Gaussian Activation Functions 1 0.9 gaussian sigmoid 0.8 0.7 sig(x) 0.6 0.5 0.4 0.3 0.2 0.1 0 -5 -4 -3 -2 -1 0 x 1 2 3 4 5 Multilayer Netwoks y x W0 Wp Wk,ij: Weight from node i in layer k-1 to node j in layer k y W p σ W p 1σ σ W 1 σ W 0 x T T T T Universal Approximation Theorem (UAT) A single hidden layer perceptron network with a sufficiently large number of neurons can approximate any continuous function arbitrarily close. Comments: The UAT does not say how large the network should be Optimal design and training may be difficult Training Objective: Given a set of training inputoutput data (x,yt) FIND the network weights that minimize the expected L E( y y ) error Steepest Descent Method: Adjust weights in the direction of steepest descent of L to make dL as negative as possible. 2 t dL E ( e d y ) 0, T e y yt Neural Network Approximation of NARMA Model u[k-1] y y[k-m] Question: Is an arbitrary neural network model consistent with a physical system (i.e., one that has an internal realization)? State-Space Model u system States: x1,…,xn x [ k 1] f ( x [ k ], u [ k ]) y [ k ] h ( x [ k ]) y A Class of Observable State Space Realizable Models Consider the input-output model: y [ k ] g ( y [ k m ],..., y [ k 1], u [ k m ],..., u [ k 1]) When does the input-output model have a state-space realization? x [ k 1] f ( x [ k ], u [ k ]) y [ k ] h ( x [ k ]) 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 statespace 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 Fluid Power Application 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 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 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 INTRODUCTION EXAMPLE: Several EHPV’s were used to control the hydraulic piston Each EHPV is supplied with its own learning controller Learning Controller employs a Neural Network (NLPN) in the feedback Satisfactory results for single EHPV used for pressure control Control Design Nonlinear system (‘lifted’ to a square system) x k n F x k , u k Feedback Control Law ˆ (xd , xd ) K p u ˆ (xd , xd ) x d (x xd ) d ˆ ( x , x d ) is the neural network output The neural network controller is directly trained based on the time history of the tracking error Learning Control Block Diagram Experimental Results Experimental Results