Control Systems! Copyright 2017 by Robert Stengel. All rights reserved. For educational use only.
|
|
- Alaina Ball
- 6 years ago
- Views:
Transcription
1 Control Systems Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 2017 Analog vs. digital systems Continuous- and Discretetime Dynamic Models Frequency Response Transfer Functions Bode Plots Root Locus Proportional-Integral- Derivative (PID) Control Copyright 2017 by Robert Stengel. All rights reserved. For educational use only. 1 Analog vs. Digital Signals Signal: A physical indicator that conveys information, e.g., position, voltage, temperature, or pressure Analog signal: A signal that is continuous in time, with infinite precision and infinitesimal time spacing between indication points Discrete-time signal: A signal with infinite precision and discrete spacing between indication points Digital signal: A signal with finite precision and discrete spacing between indication points 2
2 Analog vs. Digital Systems System: Assemblage of parts with structure, connectivity, and behavior that responds to input signals and produces output signals Analog system: A system that operates continuously, with infinite precision and infinitesimal time spacing between signaling points Discrete-time system: A system that operates continuously, with infinite precision and discrete spacing between signaling points Digital system: A system that operates continuously, with finite precision and discrete spacing between signaling points 3 Frequency Response of a Continuous- Time Dynamic System Sinusoidal command input (i.e., Desired rotational angle) Long-term output of the dynamic system: Sinusoid with same frequency as input Output/input amplitude ratio dependent on input frequency Output/input phase shift dependent on input frequency Bandwidth: Input frequency below which amplitude and phase angle variations are negligible < n = n > n Output Angle, deg Output Rate, deg/s Input Angle, deg : Input frequency, rad/s n : Natural frequency of system, rad/s Hz Hz : Hertz, frequency, cycles per sec = ( rad / s) 2 4
3 Amplitude Ratio and Phase Angle Amplitude Ratio (AR) = y Output Peak y Input Peak Phase Angle = 360 t Input Peak t Output Peak Period of Input, deg 5 Amplitude-Ratio and Phase-Angle Frequency Response of a 2 nd -Order System Output Angle AR Straight-line asymptotes at high and low frequency 6 db Bandwidth, Hz Output Rate AR 20log AR and vs. log Straight-line asymptotes at high and low frequency Logarithm of transfer function Amplitude ratio and phase angle are orthogonal Decibels (db): 20 log AR Decade: power of 10 6
4 Asymptotes of 2 nd -Order Frequency Response from Bode Plot Natural Frequency 20 log 10 ( n ) has slope of 20n db/decade on Bode plot 20log 10 H CLi (j ) = 20log 10 AR i ( ) e j i ( ) = 20 log 10 AR i + log 10 e j i ( ) = 20log 10 AR i + j i ( ) 20log 10 e 7 F x(t) Fourier Transform of a Scalar Variable [ ] = x( j ) = x(t)e jt dt = frequency, rad / s j i 1 x(t) x( j) = a() + jb() x(t) :real variable x( j ) :complex variable = a( ) + jb( ) j ( ) = A( )e A :amplitude :phase angle 8
5 Laplace Transforms of Scalar Variables Laplace transform of a scalar variable is a complex number s is the Laplace operator, a complex variable [ ] = x = x(t)e st dt L x(t), s = + j 0 x(t) :real variable x : complex variable = a( ) + jb( ) j ( ) = A( )e Multiplication by a constant L a x(t) Sum of Laplace transforms [ ] = x 1 + x 2 L x 1 (t) + x 2 (t) [ ] = a x 9 Laplace Transforms of Vectors and Matrices Laplace transform of a vector variable Laplace transform of a matrix variable L[ x(t) ] = x = x 1 x 2... L[ A(t) ] = A = a 11 a a 21 a Laplace transform of a derivative w.r.t. time L dx(t) dt = sx x(0) Laplace transform of an integral over time L x( )d ( = x s
6 Laplace Transforms of the System Equations Time-Domain System Equations x(t) = Fx(t) + Gu(t) y(t) = H x x(t) + H u u(t) Dynamic Equation Output Equation Laplace Transforms of System Equations sx x(0) = Fx + Gu y = H x x + H u u Dynamic Equation Output Equation 11 Laplace Transform of State Response to Initial Condition and Control Adj si F Rearrange sx Fx = x(0) + Gu [ si F]x = x(0) + Gu [ si F] 1 = [ ] 1 [ x(0) + Gu ] x = si F Matrix inverse si F Adj si F (n n) : Adjoint matrix (n n) : Transpose of matrix of cofactors si F = det( si F) : Determinant ( 11) 12
7 Laplace Transform of System Output Transfer function relates control input to system output y = H x { x } + H u u { 1 [ Gu + x(0) ]} + H u u = H x si F Let: x(0) = 0 [initial condition, not control] H u = 0 [control input not contained in the output measurement] y = H x ( si F) 1 G u H u 13 Transfer Function Matrix, H [from u to y] Dimension of y = r 1 H H x ( si F) 1 G G 1 Adj si F = H x si F Dimension of the Matrix ( r n) n n 11 Dimension of x = n 1 Dimension of u = m 1 ( n m) = ( r m) 14
8 Characteristic Polynomial of a Dynamic System Characteristic matrix ( si F) = ( s f 11 ) f f 1n f 21 s f f 2n f n1 f n2... s f nn (n ( n) Characteristic polynomial, si F = det( si F) = s n + a n1 s n a 1 s + a 0 15 Eigenvalues (or Roots) of the Dynamic System Characteristic equation = s n + a n1 s n a 1 s + a 0 = 0 ( s 2 )... = s 1 = 0 s n i are eigenvalues of F or roots of the characteristic polynomial, s is the denominator of the transfer function matrix s 16
9 Numerator of the Matrix Inverse [ si F] 1 = Matrix Inverse si F Adj si F (n n) Adjoint matrix is the transpose of the matrix of cofactors * Adj( si F) = C T (n n) ( si F) = 2 x 2 example ( s f 11 ) f 12 f 21 s f 22 Adj( si F) = = ( s f 22 ) f 21 f 12 s f 11 ( s f 22 ) f 12 f 21 s f 11 T * Cofactors = signed minor determinants of the matrix 17 Example of System Transformation DC Motor Equations x 1 (t) x 2 (t) = 0 1 x 1 (t) 0 0 x 2 (t) + 0 u(t) 1/ J y 1 (t) y 2 (t) = 1 0 x 1 (t) 0 1 x 2 (t) + 0 x 1 (t) u(t) = 0 x 2 (t) Laplace Transforms of Motor Equations Dynamic Equation Output Equation sx 1 x 1 (0) sx 2 x 2 (0) y 1 y 2 = = x 1 x 2 x 1 x / J u u = x 1 x 2 Dynamic Equation Output Equation 18
10 2 x 2 Eigenvalue Example Characteristic matrix ( si F) = ( s f 11 ) f 12 f 21 s f 22 Determinant of characteristic matrix si F = ( s f 11 ) f 12 f 21 s f 22 ( s f 22 ) f 12 f 21 = s f 11 = s 2 ( f 11 + f 22 )s + f 11 f 22 f 12 f Factors of the 2 nd - Degree Characteristic Equation x = s 2 f 12 + f 21 s s + f 11 f 22 f 12 f 21 = 0 x x = cos 1 = ( s 1 )( s 2 ) = 0 Solutions of the equation are eigenvalues of the system, either 2 real roots, or Complex-conjugate pair x s Plane 1 = 1 + j 1, 2 = 1 j 1 1 = 1, 2 = 2 ( s) = s n s + n 20
11 Eigenvalues Determine the Stability of the LTI System Positive real part represents instability Envelope of time response converges or diverges x x x 1 (t) = e nt cos n 1 2 t + 1 ( ) x 1(0) s Plane Same criterion for real roots x 1 (t) = e nt x 1 (0) 21 Open-Loop Transfer Function Matrix for DC Motor Transfer Function Matrix H OL = H x [ si F OL ] 1 G (r m) where F OL = H x = ; [ si F OL ] 1 = Adj si F OL = si F OL [ si F OL ] = = I 2 ; G = Matrix Inverse s 1 0 s J = s 1 0 s = s 2 s 1 0 s 1/ s 1/ s 2 0 1/ s 22
12 Simplify H OL = H x [ si F OL ] 1 Adj si F G = H OL x si F OL = s 1 0 s 0 s 2 1/ J = G Dimension = 2 x 1 1/ Js 2 1/ Js y 1 u y 2 u H 1 H 2 ( s) ( s) = 1 Js 2 1 Js Angle = Double integral of input torque, u Angular rate = Integral of input torque, u 23 Closed-Loop System x 1 (t) x 2 (t) Closed-loop dynamic equation = 0 1 x 1 (t) c 1 / J c 2 / J x 2 (t) 0 1 2)( n ( n 2 x 1 (t) x 2 (t) + 0 c 1 / J + 0 ( n 2 y C y C Natural Frequency: n = c 1 J Damping Ratio: = ( c 2 J ) 2 n = ( c 2 J ) 2 c 1 J 24
13 Closed-Loop Transfer Function Matrix H CL = H x [ si F CL ] 1 Adj si F G = H CL x si F CL = s + c 2 J c 1 J s s 2 + c 2 J s + c ( 1 + ) * J, - 0 c 1 / J Scalar transfer functions differ only in numerators G H 1 H 2 ( s) ( s) = y 1 y C y 2 y C c 1 J s( c 1 J ) = s 2 + c 2 J s + c 1 * ( ) J +, n 1 n 2 ( s) ( s) - s n 1 n 2 ( s) ( s) d s 25 H 1 ( j ) = Angle Frequency Response = Substitute s = j in transfer functions Output Angle Commanded Angle = y ( j ) ( c 2 J )( j ) + c 1 J y C ( j ) = c 1 J j 2 n n ( j ) + a 2 1( ) + jb 1 ( ) AR 1 ( ) e j 1( ) n 26
14 Angular Rate Frequency Response H 2 ( j ) = Output Rate Commanded Angle = y 2( j ) 2 ( j ) = n n j c 1 J 2 + ( c 2 J )( j ) + c 1 J y C ( j ) = j j + n 2 AR 2( ) e j 2 ( ) 27 y 1 y C y 2 y C Root (Eigenvalue) Locus c 1 J ( c 1 J )s = s 2 + c 2 J s + c = 1 * ( ) J +, 2 - n - 2 n s s n s + - n = ( s n s +1) Variation of roots as scalar gain, k, goes from 0 to With nominal gains, c 1 and c 2, 1 s n = c 1 J = 1 rad / s, Natural frequency = c 2 J 2 n = 0.707, Damping ratio Root Locus of DC Motor Angle Control F = [0 1; ]; G = [0;1]; Hx1 = [1 0]; Angle Output Hx2 = [0 1]; Angular Rate Output Sys1 Sys2 = ss(f,g,hx1,0); = ss(f,g,hx2,0); rlocus(sys1), grid figure rlocus(sys2), grid 28
15 Root (Eigenvalue) Locus Increase Angle Feedback Gain, Sys1 Increase Rate Feedback Gain, Sys2 Initial Root Initial Root Zero Initial Root Initial Root See for rules of root locus construction 29 Classical Control System Design Criteria Step Response Frequency Response Eigenvalue Location 30
16 Single-Axis Angular Control of Non-Spinning Spacecraft Pitching motion (about the y axis) is to be controlled (t) q(t) = (t) q(t) / I yy M (t) y (t) q(t) = Pitch Angle Pitch Rate Identical to the DC Motor control problem 31 Proportional-Integral- Derivative (PID) Controller Control Error Control Command Control Law Transfer Function (w/common denominator) e = C e u = c P e + c I + c D se s u e = c Ps + c I + c D s 2 s Proportional term weights control error directly Integrator compensates for persistent (bias) disturbance Differentiator produces rate term for damping Ziegler-Nichols PID Tuning Method Nichols_method 32
17 Open-Loop Angle Transfer Function g A = Actuator gain H OL ( s) = e = c + c s + c I P D s2 s g A I yy s 2 33 Closed-Loop Spacecraft Control Transfer Function w/pid Control = H OL ( s)e = H OL s ( s) +1 = H OL ( s) c H OL H CL ( s) = = c = H OL s 1+H OL s [ ] c e 1+ e 34
18 Closed-Loop Frequency Response w/pid Control c I + c P s + c D s 2 g A * ) s I yy s 2, H CL ( s) = () +, c = D s 2 + c P s + c I 1+ c + c s + c I P D s2 g A * I yy s 3 g A + c D s 2 + c P s + c I ) s I yy s 2, () +, Let s = j. As 0 As H CL ( j ) = H CL ( j ) = ( j ) c ( j ) c I c I = 1 Steady-state output = desired steady-state input ( j ) c ( j ) c D 2 ji yy 3 g A = c D ji yy g A = jc D I yy g A AR c D I yy g A; 90 deg High-frequency response rolls off and lags input 35 Digital Control Stengel, Optimal Control and Estimation, 1994, pp , E-Reserve 36
19 From Continuous- to Discrete- Time Systems Continuous-time systems are described by differential equations, e.g., x(t) = Fx(t) + Gu(t) Discrete-time systems are described by difference equations, e.g., x(t k +1 ) = x(t k ) + u(t k ) Discrete-time systems that are meant to describe continuous-time systems are called sampled-data systems = fcn( F); = fcn( F,G) 37 Integration of Linear, Time-Invariant (LTI) Systems x(t) = Fx(t) + Gu(t) t x(t) = x(0) + xd = x(0) + [ Fx + Gu]d 0 Homogeneous solution (no input), u = 0 x(t) = x(0) + [ Fx]d = e F(t0) x(0) = ( t,0)x(0) t 0 ( t,0) = e F(t0) = e Ft = State transition matrix from 0 to t t 0 38
20 Equivalence of 1 st -Order Continuousand Discrete-Time Systems Continuous- and discrete-time values are identical at time instants, kt x(t) = ax(t), x(0) given t x(t) = x(0) + x(t)dt = e at x(0) (t)x(0) 0 Choose small time interval, t Propagate state to time, t + nt 39 Equivalence of 1 st -Order Continuous- and Discrete-Time Systems x 0 = x(0) x 1 = x(t) = e at x 0 = (t)x 0 x 2 = x(2t) = e 2at x 0 = e at x 1 = (t)x 1... x k = x(kt) = e kat x 0 = e at x k1 = (t)x k1... x n = x(nt) = x(t) = e nat x 0 = (t)x n1 40
21 Input Response of LTI System Inhomogeneous (forced) solution t ( x(t) = e Ft x(0) + e F(t ) Gu d = ( t 0)x(0) + t 0 0 t t ) 0 Gu( ) ( d 41 Sampled-Data System (Stepwise application of prior results) t k+1 t k Assume (t k t k+1 ) = t = constant = e F(t k+1t k ) = e F(t ) = t Assume u = constant from t k to t k+1 x(t k+1 ) = (t)x(t k ) + t t t Evaluate the integral ) G ( d u k x(t k+1 ) = (t)x(t k ) + (t) I 1 (t) F 1 Gu t k = (t)x(t k ) + ((t)u( t k ) 0 42
22 Digital Feedback Control System x(t k+1 ) = (t)x(t k ) + (t)u( t k ) Linear feedback control law with command input u( t k ) = u c ( t k ) C k x( t k ) = t x t k+1 Closed-loop system x(t k ) + t u c ( t k ) C k x( t k ) ( = (t) (t)c k ( x(t k ) + (t)u c t k 43 Digital Flight Control Historical notes: NASA Fly-By-Wire F-8C (Apollo GNC system, 1972) 1 st conventional aircraft with digital flight control NASA Armstrong (Dryden) Research Center Princetons Variable-Response Research Aircraft (Z-80 microprocessor, Spring, 1978) 2 nd conventional aircraft with digital flight control Princeton University s Flight Research Laboratory 44
23 Next Time: Sensors and Actuators 45 SupplementaL Material 46
24 Princetons Variable- Response Research Aircraft 47 Eigenvalues, Damping Ratio, and Natural Frequency Eigenvalues, Damping Ratio, and Natural Frequency of Angle Control F1 = [0 1;-1 0]; G1 = [0;1]; F1a = [0 1; ]; F1b = [0 1; ]; Hx = [1 0;0 1]; Eigenvalues 0 + 1i 0-1i Damping Ratio, Natural Frequency 1, 2 n 0 1 Sys1 Sys2 Sys3 eig(f1) eig(f1a) eig(f1b) damp(sys1) damp(sys2) = ss(f1,g1,hx,0); = ss(f1a,g1,hx,0); = ss(f1b,g1,hx,0); i i Overdamped 48
25 Bode Plots of Proportional, Integral, and Derivative Compensation H( j) =1 H( j) =10 H( j) =100 H( j) = 1 j H( j) = 10 j H( j) = j H( j) =10 j Amplitude ratio, AR(dB) = Slope of AR(dB) = Slope of AR(dB) = constant 20 db/decade +20 db/decade Phase Angle, ( deg) = 0 Phase Angle, = 90 deg Phase Angle, = +90 deg 49 Equilibrium Response of Linear Discrete-Time Model Dynamic model x k +1 = x k + u k + w k At equilibrium x k +1 = x k = x* = constant Equilibrium response ( I )x* = u*+w * 1 ( u*+w *) x* = I (.)* = constant 50
26 Discrete-Time and Sampled-Data Models Linear, time-invariant, discrete-time model x k +1 = x k + u k + w k Discrete-time model is a sampled-data model if it represents a continuous-time system 51 Sampled-Data (Digital) Feedback Control Digital-to-Analog Conversion Analog-to-Digital Conversion 52
27 Sampling Effect on Continuous Signal Two waves, different frequencies, indistinguishable in periodically sampled data Frequencies above the sampling frequency aliased to appear at lower frequency (frequency folding) Solutions: Either Sampling at higher rate or Analog low-pass filtering Folding Frequency Aliased Spectrum Actual Spectrum 53 Quantization and Delay Effects Continuous signal sampled with finite precision (quantized) Solution: More bits in A/D and D/A conversion Effective delay of sampled signal Described as phase shift or lag Solution: Higher sampling rate or Lead compensation Analog-to-Digital Converter r h illustrates zero-order hold effect 54
28 Time Delay Effect Control command delayed by sec Laplace transform of pure time delay L u( t ) = e s u( s) Delay introduces frequency-dependent phase lag with no change in amplitude AR e j = 1 = e j Phase lag reduces closed-loop stability 55 Effect of Closed-Loop Time Delay on Step Response With no delay n = 1 rad / s, = With varying delays 56
29 Discrete-Time Frequency Response can be Evaluated Using the z Transform x k+1 = x k + u k z transform is the Laplace transform of a periodic sequence System z transform is zx(z) x(0) = x(z) + u(z) which leads to [ ] x(z) = ( zi ) 1 x(0) + u(z) Further details are beyond the present scope 57 Factors That Complicate Precise Control 58
30 Error, Uncertainty, and Incompleteness May Have Significant Effects Scale factors and biases: y = kx + b Disturbances: Constant/variable forces Measurement error: Noise, bias Modeling error Parameters: e.g., Inertia Structure: e.g., Higher-order modes 59 Saturation (limiting) Displacement limit, maximum force or rate Threshold Dead zone, slop Preload Breakout force Friction Sliding surface resistance Rectifier Hard constraint, absolute value (double) Hysteresis Backlash, slop Higher-degree terms Cubic spring, separation Nonlinearity Complicates Response 60
31 Second-Order Example: Airplane Roll Motion p ( ( = Roll rate, rad/s ( Roll angle, rad ( ) A = Aileron deflection, rad L p : Rolling motion of an airplane, continuous-time p ( ( = L p 0 p ( ( 1 0 ( ( + L ) A 0 roll damping coefficient L A : aileron control effect ( ) A ( Rolling motion of an airplane, discrete-time p k k ( ( = e L pt 0 ( e LpT )1) 1 L p ( ( ( ( p k)1 k)1 ( ( + ~ L T * A 0 ( * A k)1 ( T = sampling interval, s 61
Time Response of Linear, Time-Invariant (LTI) Systems Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018
Time Response of Linear, Time-Invariant LTI Systems Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018 Learning Objectives Methods of time-domain analysis Continuous- and discrete-time models Transient
More informationTime Response of Dynamic Systems! Multi-Dimensional Trajectories Position, velocity, and acceleration are vectors
Time Response of Dynamic Systems Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Multi-dimensional trajectories Numerical integration Linear and nonlinear systems Linearization
More informationTranslational and Rotational Dynamics!
Translational and Rotational Dynamics Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Copyright 217 by Robert Stengel. All rights reserved. For educational use only.
More informationFirst-Order Low-Pass Filter!
Filters, Cost Functions, and Controller Structures! Robert Stengel! Optimal Control and Estimation MAE 546! Princeton University, 217!! Dynamic systems as low-pass filters!! Frequency response of dynamic
More informationReturn Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems
Spectral Properties of Linear- Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of single-input/singleoutput (SISO) systems! Characterizations
More informationFirst-Order Low-Pass Filter
Filters, Cost Functions, and Controller Structures Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 218! Dynamic systems as low-pass filters! Frequency response of dynamic systems!
More informationStochastic Optimal Control!
Stochastic Control! Robert Stengel! Robotics and Intelligent Systems, MAE 345, Princeton University, 2015 Learning Objectives Overview of the Linear-Quadratic-Gaussian (LQG) Regulator Introduction to Stochastic
More informationLinearized Equations of Motion!
Linearized Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 216 Learning Objectives Develop linear equations to describe small perturbational motions Apply to aircraft dynamic equations
More informationSAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015
FACULTY OF ENGINEERING AND SCIENCE SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 Lecturer: Michael Ruderman Problem 1: Frequency-domain analysis and control design (15 pt) Given is a
More informationIndex. Index. More information. in this web service Cambridge University Press
A-type elements, 4 7, 18, 31, 168, 198, 202, 219, 220, 222, 225 A-type variables. See Across variable ac current, 172, 251 ac induction motor, 251 Acceleration rotational, 30 translational, 16 Accumulator,
More informationDr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review
Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics
More informationVideo 5.1 Vijay Kumar and Ani Hsieh
Video 5.1 Vijay Kumar and Ani Hsieh Robo3x-1.1 1 The Purpose of Control Input/Stimulus/ Disturbance System or Plant Output/ Response Understand the Black Box Evaluate the Performance Change the Behavior
More informationOptimal Control and Estimation MAE 546, Princeton University Robert Stengel, Preliminaries!
Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, 2018 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/mae546.html
More informationFATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY
FATIMA MICHAEL COLLEGE OF ENGINEERING & TECHNOLOGY Senkottai Village, Madurai Sivagangai Main Road, Madurai - 625 020. An ISO 9001:2008 Certified Institution DEPARTMENT OF ELECTRONICS AND COMMUNICATION
More informationD(s) G(s) A control system design definition
R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure
More informationFeedback Control of Linear SISO systems. Process Dynamics and Control
Feedback Control of Linear SISO systems Process Dynamics and Control 1 Open-Loop Process The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals
More informationMechatronics Assignment # 1
Problem # 1 Consider a closed-loop, rotary, speed-control system with a proportional controller K p, as shown below. The inertia of the rotor is J. The damping coefficient B in mechanical systems is usually
More informationTime-Invariant Linear Quadratic Regulators!
Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 17 Asymptotic approach from time-varying to constant gains Elimination of cross weighting
More informationAppendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2)
Appendix A: Exercise Problems on Classical Feedback Control Theory (Chaps. 1 and 2) For all calculations in this book, you can use the MathCad software or any other mathematical software that you are familiar
More informationExam. 135 minutes + 15 minutes reading time
Exam January 23, 27 Control Systems I (5-59-L) Prof. Emilio Frazzoli Exam Exam Duration: 35 minutes + 5 minutes reading time Number of Problems: 45 Number of Points: 53 Permitted aids: Important: 4 pages
More informationOptimal Control and Estimation MAE 546, Princeton University Robert Stengel, Preliminaries!
Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, 2017 Copyright 2017 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/mae546.html
More informationLearn2Control Laboratory
Learn2Control Laboratory Version 3.2 Summer Term 2014 1 This Script is for use in the scope of the Process Control lab. It is in no way claimed to be in any scientific way complete or unique. Errors should
More informationOverview of the Seminar Topic
Overview of the Seminar Topic Simo Särkkä Laboratory of Computational Engineering Helsinki University of Technology September 17, 2007 Contents 1 What is Control Theory? 2 History
More informationLinear-Quadratic-Gaussian Controllers!
Linear-Quadratic-Gaussian Controllers! Robert Stengel! Optimal Control and Estimation MAE 546! Princeton University, 2017!! LTI dynamic system!! Certainty Equivalence and the Separation Theorem!! Asymptotic
More informationStochastic and Adaptive Optimal Control
Stochastic and Adaptive Optimal Control Robert Stengel Optimal Control and Estimation, MAE 546 Princeton University, 2018! Nonlinear systems with random inputs and perfect measurements! Stochastic neighboring-optimal
More informationControl for. Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e
Control for Maarten Steinbuch Dept. Mechanical Engineering Control Systems Technology Group TU/e Motion Systems m F Introduction Timedomain tuning Frequency domain & stability Filters Feedforward Servo-oriented
More informationAcceleration Feedback
Acceleration Feedback Mechanical Engineer Modeling & Simulation Electro- Mechanics Electrical- Electronics Engineer Sensors Actuators Computer Systems Engineer Embedded Control Controls Engineer Mechatronic
More informationDynamic Effects of Feedback Control!
Dynamic Effecs of Feedback Conrol! Rober Sengel! Roboics and Inelligen Sysems MAE 345, Princeon Universiy, 2017 Inner, Middle, and Ouer Feedback Conrol Loops Sep Response of Linear, Time- Invarian (LTI)
More informationRaktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Preliminaries
. AERO 632: of Advance Flight Control System. Preliminaries Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. Preliminaries Signals & Systems Laplace
More informationINTRODUCTION TO DIGITAL CONTROL
ECE4540/5540: Digital Control Systems INTRODUCTION TO DIGITAL CONTROL.: Introduction In ECE450/ECE550 Feedback Control Systems, welearnedhow to make an analog controller D(s) to control a linear-time-invariant
More informationRecitation 11: Time delays
Recitation : Time delays Emilio Frazzoli Laboratory for Information and Decision Systems Massachusetts Institute of Technology November, 00. Introduction and motivation. Delays are incurred when the controller
More informationTime-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2015
Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 15 Asymptotic approach from time-varying to constant gains Elimination of cross weighting
More informationControl of Manufacturing Processes
Control of Manufacturing Processes Subject 2.830 Spring 2004 Lecture #18 Basic Control Loop Analysis" April 15, 2004 Revisit Temperature Control Problem τ dy dt + y = u τ = time constant = gain y ss =
More informationAPPLICATIONS FOR ROBOTICS
Version: 1 CONTROL APPLICATIONS FOR ROBOTICS TEX d: Feb. 17, 214 PREVIEW We show that the transfer function and conditions of stability for linear systems can be studied using Laplace transforms. Table
More informationProblem Set 3: Solution Due on Mon. 7 th Oct. in class. Fall 2013
EE 56: Digital Control Systems Problem Set 3: Solution Due on Mon 7 th Oct in class Fall 23 Problem For the causal LTI system described by the difference equation y k + 2 y k = x k, () (a) By first finding
More informationContents. PART I METHODS AND CONCEPTS 2. Transfer Function Approach Frequency Domain Representations... 42
Contents Preface.............................................. xiii 1. Introduction......................................... 1 1.1 Continuous and Discrete Control Systems................. 4 1.2 Open-Loop
More informationÜbersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 9. 8. 2 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -
More informationAnswers to multiple choice questions
Answers to multiple choice questions Chapter 2 M2.1 (b) M2.2 (a) M2.3 (d) M2.4 (b) M2.5 (a) M2.6 (b) M2.7 (b) M2.8 (c) M2.9 (a) M2.10 (b) Chapter 3 M3.1 (b) M3.2 (d) M3.3 (d) M3.4 (d) M3.5 (c) M3.6 (c)
More informationAircraft Dynamics First order and Second order system
Aircraft Dynamics First order and Second order system Prepared by A.Kaviyarasu Assistant Professor Department of Aerospace Engineering Madras Institute Of Technology Chromepet, Chennai Aircraft dynamic
More informationControls Problems for Qualifying Exam - Spring 2014
Controls Problems for Qualifying Exam - Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function
More informationR a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage analogies.
SET - 1 II B. Tech II Semester Supplementary Examinations Dec 01 1. a) Compare open loop and closed loop control systems. b) Clearly bring out, from basics, Force-current and Force-Voltage analogies..
More informationDynamic Systems. Modeling and Analysis. Hung V. Vu. Ramin S. Esfandiari. THE McGRAW-HILL COMPANIES, INC. California State University, Long Beach
Dynamic Systems Modeling and Analysis Hung V. Vu California State University, Long Beach Ramin S. Esfandiari California State University, Long Beach THE McGRAW-HILL COMPANIES, INC. New York St. Louis San
More informationMAS107 Control Theory Exam Solutions 2008
MAS07 CONTROL THEORY. HOVLAND: EXAM SOLUTION 2008 MAS07 Control Theory Exam Solutions 2008 Geir Hovland, Mechatronics Group, Grimstad, Norway June 30, 2008 C. Repeat question B, but plot the phase curve
More informationAutonomous Mobile Robot Design
Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:
More informationLecture 1: Introduction to System Modeling and Control. Introduction Basic Definitions Different Model Types System Identification
Lecture 1: Introduction to System Modeling and Control Introduction Basic Definitions Different Model Types System Identification What is Mathematical Model? A set of mathematical equations (e.g., differential
More informationMech 6091 Flight Control System Course Project. Team Member: Bai, Jing Cui, Yi Wang, Xiaoli
Mech 6091 Flight Control System Course Project Team Member: Bai, Jing Cui, Yi Wang, Xiaoli Outline 1. Linearization of Nonlinear F-16 Model 2. Longitudinal SAS and Autopilot Design 3. Lateral SAS and Autopilot
More informationEC CONTROL SYSTEM UNIT I- CONTROL SYSTEM MODELING
EC 2255 - CONTROL SYSTEM UNIT I- CONTROL SYSTEM MODELING 1. What is meant by a system? It is an arrangement of physical components related in such a manner as to form an entire unit. 2. List the two types
More informationAircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2018
Aircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2018 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2018 by Robert Stengel. All rights reserved. For
More informationDr Ian R. Manchester
Week Content Notes 1 Introduction 2 Frequency Domain Modelling 3 Transient Performance and the s-plane 4 Block Diagrams 5 Feedback System Characteristics Assign 1 Due 6 Root Locus 7 Root Locus 2 Assign
More informationToday (10/23/01) Today. Reading Assignment: 6.3. Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10
Today Today (10/23/01) Gain/phase margin lead/lag compensator Ref. 6.4, 6.7, 6.10 Reading Assignment: 6.3 Last Time In the last lecture, we discussed control design through shaping of the loop gain GK:
More informationStability of CL System
Stability of CL System Consider an open loop stable system that becomes unstable with large gain: At the point of instability, K( j) G( j) = 1 0dB K( j) G( j) K( j) G( j) K( j) G( j) =± 180 o 180 o Closed
More informationsc Control Systems Design Q.1, Sem.1, Ac. Yr. 2010/11
sc46 - Control Systems Design Q Sem Ac Yr / Mock Exam originally given November 5 9 Notes: Please be reminded that only an A4 paper with formulas may be used during the exam no other material is to be
More informationTopic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback
Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall
More informationAutomatique. A. Hably 1. Commande d un robot mobile. Automatique. A.Hably. Digital implementation
A. Hably 1 1 Gipsa-lab, Grenoble-INP ahmad.hably@grenoble-inp.fr Commande d un robot mobile (Gipsa-lab (DA)) ASI 1 / 25 Outline 1 2 (Gipsa-lab (DA)) ASI 2 / 25 of controllers Signals must be sampled and
More informationOutline. Classical Control. Lecture 1
Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction
More informationNoise - irrelevant data; variability in a quantity that has no meaning or significance. In most cases this is modeled as a random variable.
1.1 Signals and Systems Signals convey information. Systems respond to (or process) information. Engineers desire mathematical models for signals and systems in order to solve design problems efficiently
More informationLecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
Lecture 6 Classical Control Overview IV Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lead Lag Compensator Design Dr. Radhakant Padhi Asst.
More informationControl Systems Lab - SC4070 Control techniques
Control Systems Lab - SC4070 Control techniques Dr. Manuel Mazo Jr. Delft Center for Systems and Control (TU Delft) m.mazo@tudelft.nl Tel.:015-2788131 TU Delft, February 16, 2015 (slides modified from
More informationR10 JNTUWORLD B 1 M 1 K 2 M 2. f(t) Figure 1
Code No: R06 R0 SET - II B. Tech II Semester Regular Examinations April/May 03 CONTROL SYSTEMS (Com. to EEE, ECE, EIE, ECC, AE) Time: 3 hours Max. Marks: 75 Answer any FIVE Questions All Questions carry
More informationNonlinear State Estimation! Extended Kalman Filters!
Nonlinear State Estimation! Extended Kalman Filters! Robert Stengel! Optimal Control and Estimation, MAE 546! Princeton University, 2017!! Deformation of the probability distribution!! Neighboring-optimal
More informationCDS 101/110a: Lecture 8-1 Frequency Domain Design
CDS 11/11a: Lecture 8-1 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve
More informationManufacturing Equipment Control
QUESTION 1 An electric drive spindle has the following parameters: J m = 2 1 3 kg m 2, R a = 8 Ω, K t =.5 N m/a, K v =.5 V/(rad/s), K a = 2, J s = 4 1 2 kg m 2, and K s =.3. Ignore electrical dynamics
More informationSTABILITY OF CLOSED-LOOP CONTOL SYSTEMS
CHBE320 LECTURE X STABILITY OF CLOSED-LOOP CONTOL SYSTEMS Professor Dae Ryook Yang Spring 2018 Dept. of Chemical and Biological Engineering 10-1 Road Map of the Lecture X Stability of closed-loop control
More informationIC6501 CONTROL SYSTEMS
DHANALAKSHMI COLLEGE OF ENGINEERING CHENNAI DEPARTMENT OF ELECTRICAL AND ELECTRONICS ENGINEERING YEAR/SEMESTER: II/IV IC6501 CONTROL SYSTEMS UNIT I SYSTEMS AND THEIR REPRESENTATION 1. What is the mathematical
More informationOPTIMAL CONTROL AND ESTIMATION
OPTIMAL CONTROL AND ESTIMATION Robert F. Stengel Department of Mechanical and Aerospace Engineering Princeton University, Princeton, New Jersey DOVER PUBLICATIONS, INC. New York CONTENTS 1. INTRODUCTION
More informationLecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design.
ISS0031 Modeling and Identification Lecture 5: Linear Systems. Transfer functions. Frequency Domain Analysis. Basic Control Design. Aleksei Tepljakov, Ph.D. September 30, 2015 Linear Dynamic Systems Definition
More informationTheory and Problems of Signals and Systems
SCHAUM'S OUTLINES OF Theory and Problems of Signals and Systems HWEI P. HSU is Professor of Electrical Engineering at Fairleigh Dickinson University. He received his B.S. from National Taiwan University
More informationPositioning Servo Design Example
Positioning Servo Design Example 1 Goal. The goal in this design example is to design a control system that will be used in a pick-and-place robot to move the link of a robot between two positions. Usually
More informationÜbersetzungshilfe / Translation aid (English) To be returned at the end of the exam!
Prüfung Regelungstechnik I (Control Systems I) Prof. Dr. Lino Guzzella 3.. 24 Übersetzungshilfe / Translation aid (English) To be returned at the end of the exam! Do not mark up this translation aid -
More informationControl Systems. EC / EE / IN. For
Control Systems For EC / EE / IN By www.thegateacademy.com Syllabus Syllabus for Control Systems Basic Control System Components; Block Diagrammatic Description, Reduction of Block Diagrams. Open Loop
More informationAircraft Flight Dynamics!
Aircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2016 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2016 by Robert Stengel. All rights reserved. For
More informationMAE 142 Homework #5 Due Friday, March 13, 2009
MAE 142 Homework #5 Due Friday, March 13, 2009 Please read through the entire homework set before beginning. Also, please label clearly your answers and summarize your findings as concisely as possible.
More informationState will have dimension 5. One possible choice is given by y and its derivatives up to y (4)
A Exercise State will have dimension 5. One possible choice is given by y and its derivatives up to y (4 x T (t [ y(t y ( (t y (2 (t y (3 (t y (4 (t ] T With this choice we obtain A B C [ ] D 2 3 4 To
More informationMassachusetts Institute of Technology
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.11: Introduction to Communication, Control and Signal Processing QUIZ 1, March 16, 21 QUESTION BOOKLET
More informationRichiami di Controlli Automatici
Richiami di Controlli Automatici Gianmaria De Tommasi 1 1 Università degli Studi di Napoli Federico II detommas@unina.it Ottobre 2012 Corsi AnsaldoBreda G. De Tommasi (UNINA) Richiami di Controlli Automatici
More informationMAE143A Signals & Systems, Final Exam - Wednesday March 16, 2005
MAE13A Signals & Systems, Final Exam - Wednesday March 16, 5 Instructions This quiz is open book. You may use whatever written materials you choose including your class notes and the textbook. You may
More informationECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67
1/67 ECEN 420 LINEAR CONTROL SYSTEMS Lecture 6 Mathematical Representation of Physical Systems II State Variable Models for Dynamic Systems u 1 u 2 u ṙ. Internal Variables x 1, x 2 x n y 1 y 2. y m Figure
More information(Refer Slide Time: 00:01:30 min)
Control Engineering Prof. M. Gopal Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture - 3 Introduction to Control Problem (Contd.) Well friends, I have been giving you various
More informationProcess Control & Instrumentation (CH 3040)
First-order systems Process Control & Instrumentation (CH 3040) Arun K. Tangirala Department of Chemical Engineering, IIT Madras January - April 010 Lectures: Mon, Tue, Wed, Fri Extra class: Thu A first-order
More informationExample: Modeling DC Motor Position Physical Setup System Equations Design Requirements MATLAB Representation and Open-Loop Response
Page 1 of 5 Example: Modeling DC Motor Position Physical Setup System Equations Design Requirements MATLAB Representation and Open-Loop Response Physical Setup A common actuator in control systems is the
More informationRadar Dish. Armature controlled dc motor. Inside. θ r input. Outside. θ D output. θ m. Gearbox. Control Transmitter. Control. θ D.
Radar Dish ME 304 CONTROL SYSTEMS Mechanical Engineering Department, Middle East Technical University Armature controlled dc motor Outside θ D output Inside θ r input r θ m Gearbox Control Transmitter
More informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010
[E2.5] IMPERIAL COLLEGE LONDON DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING EXAMINATIONS 2010 EEE/ISE PART II MEng. BEng and ACGI SIGNALS AND LINEAR SYSTEMS Time allowed: 2:00 hours There are FOUR
More informationAlternative Expressions for the Velocity Vector Velocity restricted to the vertical plane. Longitudinal Equations of Motion
Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flig Dynamics MAE 33, 008 Separate solutions for nominal and perturbation flig paths Assume that nominal path is steady and in the vertical
More informationSAMPLE EXAMINATION PAPER (with numerical answers)
CID No: IMPERIAL COLLEGE LONDON Design Engineering MEng EXAMINATIONS For Internal Students of the Imperial College of Science, Technology and Medicine This paper is also taken for the relevant examination
More informationMAE143a: Signals & Systems (& Control) Final Exam (2011) solutions
MAE143a: Signals & Systems (& Control) Final Exam (2011) solutions Question 1. SIGNALS: Design of a noise-cancelling headphone system. 1a. Based on the low-pass filter given, design a high-pass filter,
More informationGATE EE Topic wise Questions SIGNALS & SYSTEMS
www.gatehelp.com GATE EE Topic wise Questions YEAR 010 ONE MARK Question. 1 For the system /( s + 1), the approximate time taken for a step response to reach 98% of the final value is (A) 1 s (B) s (C)
More information(a) Find the transfer function of the amplifier. Ans.: G(s) =
126 INTRDUCTIN T CNTR ENGINEERING 10( s 1) (a) Find the transfer function of the amplifier. Ans.: (. 02s 1)(. 001s 1) (b) Find the expected percent overshoot for a step input for the closed-loop system
More informationBIBO STABILITY AND ASYMPTOTIC STABILITY
BIBO STABILITY AND ASYMPTOTIC STABILITY FRANCESCO NORI Abstract. In this report with discuss the concepts of bounded-input boundedoutput stability (BIBO) and of Lyapunov stability. Examples are given to
More informationECE557 Systems Control
ECE557 Systems Control Bruce Francis Course notes, Version.0, September 008 Preface This is the second Engineering Science course on control. It assumes ECE56 as a prerequisite. If you didn t take ECE56,
More informationTHE subject of the analysis is system composed by
MECHANICAL VIBRATION ASSIGNEMENT 1 On 3 DOF system identification Diego Zenari, 182160, M.Sc Mechatronics engineering Abstract The present investigation carries out several analyses on a 3-DOF system.
More informationTime Varying Circuit Analysis
MAS.836 Sensor Systems for Interactive Environments th Distributed: Tuesday February 16, 2010 Due: Tuesday February 23, 2010 Problem Set # 2 Time Varying Circuit Analysis The purpose of this problem set
More information2.004 Dynamics and Control II Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 2.004 Dynamics and Control II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Massachusetts Institute
More informationTime Response Analysis (Part II)
Time Response Analysis (Part II). A critically damped, continuous-time, second order system, when sampled, will have (in Z domain) (a) A simple pole (b) Double pole on real axis (c) Double pole on imaginary
More informationMAE 143B - Homework 7
MAE 143B - Homework 7 6.7 Multiplying the first ODE by m u and subtracting the product of the second ODE with m s, we get m s m u (ẍ s ẍ i ) + m u b s (ẋ s ẋ u ) + m u k s (x s x u ) + m s b s (ẋ s ẋ u
More informationReview of Linear Time-Invariant Network Analysis
D1 APPENDIX D Review of Linear Time-Invariant Network Analysis Consider a network with input x(t) and output y(t) as shown in Figure D-1. If an input x 1 (t) produces an output y 1 (t), and an input x
More informationControl Systems I. Lecture 7: Feedback and the Root Locus method. Readings: Jacopo Tani. Institute for Dynamic Systems and Control D-MAVT ETH Zürich
Control Systems I Lecture 7: Feedback and the Root Locus method Readings: Jacopo Tani Institute for Dynamic Systems and Control D-MAVT ETH Zürich November 2, 2018 J. Tani, E. Frazzoli (ETH) Lecture 7:
More informationChapter 2. Classical Control System Design. Dutch Institute of Systems and Control
Chapter 2 Classical Control System Design Overview Ch. 2. 2. Classical control system design Introduction Introduction Steady-state Steady-state errors errors Type Type k k systems systems Integral Integral
More informationECSE 4962 Control Systems Design. A Brief Tutorial on Control Design
ECSE 4962 Control Systems Design A Brief Tutorial on Control Design Instructor: Professor John T. Wen TA: Ben Potsaid http://www.cat.rpi.edu/~wen/ecse4962s04/ Don t Wait Until The Last Minute! You got
More informationControl System Design
ELEC4410 Control System Design Lecture 19: Feedback from Estimated States and Discrete-Time Control Design Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science
More informationThe Relation Between the 3-D Bode Diagram and the Root Locus. Insights into the connection between these classical methods. By Panagiotis Tsiotras
F E A T U R E The Relation Between the -D Bode Diagram and the Root Locus Insights into the connection between these classical methods Bode diagrams and root locus plots have been the cornerstone of control
More information