Suppose that we have a specific single stage dynamic system governed by the following equation:
|
|
- Crystal McDaniel
- 5 years ago
- Views:
Transcription
1 Dynamic Optimisation Discrete Dynamic Systems A single stage example Suppose that we have a specific single stage dynamic system governed by the following equation: x 1 = ax 0 + bu 0, x 0 = x i (1) where x is a scalar state and u is a scalar control input We wish to minimise the following objective, subject to the dynamics of the system: min u 0 J = x 2 1 (2) 1
2 Substituting x 1 to include the system constraint: min u 0 J = (ax 0 +bu 0 ) 2 = a 2 x 2 0 +b2 u abx 0u 0 (3) We can find J/ u 0 and make it zero: J u 0 = 2b 2 u 0 + 2abx 0 = 0 (4) Therefore: u(0) = (a/b)x 0 (5) which gives: x 1 = ax 0 + b( a/b)x 0 = 0 (6) 2
3 Therefore the optimal control u 0 = (a/b)x 0 gives the minimun value of the objective function J = 0 A two stage example Now, take the same system over two stages: x 0 = x i x 1 = ax 0 + bu 0 (7) x 2 = ax 1 + bu 1 = a 2 x 0 + abu 0 + bu 1 And assume that we wish to minimise the following objective function: min J = x 2 u 0,u 2 + u2 1 + u2 0 (8) 1 3
4 Substituting x 2, we have min J = (a 2 x u 0,u 0 + abu 0 + bu 1 ) 2 + u u2 0 (9) 1 The partial derivatives of J with respect to u 0 and u 1 are: J u 0 = 2ab(a 2 x 0 +abu 0 +bu 1 )+2u 0 = 0 (10) J u 1 = 2b(a 2 x 0 + abu 0 + bu 1 ) + 2u 1 = 0 (11) We have two linear equations with two unknows: u 0 and u 1 The solution is: u 0 = ba3 ba 2 +b 2 +1 x 0 u 1 = ba2 ba 2 +b 2 +1 x 0 (12) 4
5 Suppose that we know that a = 05, b = 1 and x 0 = 1 Then we have: u 0 = 011 u 1 = x 1 = x 2 = J = (13) u x
6 General discrete case Suppose that a dynamic system is described by the following equation which determines the transition from the n-dimensional state x k to state x k+1, given the m-dimensional control vector u k : x k+1 = f(x k, u k, k), x 0 = x i (14) 6
7 A fairly general optimisation problem for such systems is to find the sequence of controls u k, k = 0, N 1 to minimise a performance index of the form: J = φ (x N ) + N 1 k=0 L (x k, u k, k) (15) subject to x k+1 = f(x k, u k, k), x 0 = x i (16) This is an optimisation problem with equality constraints 7
8 Necessary optimality conditions Adjoin the constraints to the performance index with a sequence of Lagrange multiplier vectors λ k as follows: + N 1 k=0 J = φ (x N ) + λ T 0 [x i x 0 ] { Lk + λ T k+1 [ fk x k+1 ]} (17) Define the Hamiltonian as follows: H k = L(x k, u k, k) + λ T k+1 f (x k, u k, k) (18) 8
9 So that J = φ (x N ) λ T N x N + λ T 0 x i + N 1 k=0 { Hk λ T k x k} (19) The optimality conditions are then found using optimisation theory, which involves calculating the increment d J and making it zero 9
10 The optimality conditions are: x k+1 = f (x k, u k, k) (20) λ k = H T x k (21) H u k T = 0 (22) The boundary conditions are: x 0 = x i (23) λ N = φ T x N (24) We have a two point boundary value problem 10
11 Discrete Linear-Quadratic Regulator Let the plant to be controlled be described by the linear equation x k+1 = Ax k + Bu k (25) Suppose that we wish to minimise the following quadratic performance index: J = 1 2 xt N Sx N N 1 k=0 [ x T k Qx k + u T k Ru k] (26) We assume that Q and S are positive semidefinite matrices and that R is positive definite 11
12 In this case, the Hamiltonian is given by: H k = 1 2 ( x T k Qx k + u T k Ru k +λ T k+1 (Ax k + Bu k ) ) (27) From the necessary optimality conditions, we have: x k+1 = Ax k + Bu k (28) λ k = H T x k = Qx k + A T λ k+1 (29) H u k T = Ru k + B T λ k+1 = 0 (30) 12
13 From (30) we can obtain the optimal control: u k = R 1 B T λ k+1 (31) The boundary conditions are: x 0 specified (32) λ N = Sx N We have a linear two point boundary value problem 13
14 Riccati Solution The solution to the linear two point boundary value problem can be found by solving backwards from S N = S, the following Riccati equation: S k = A T [S k+1 S k+1 B(B T S k+1 B + R) 1 B T S k+1 ]A + Q (33) The state feedback gain matrix is given by: K k = ( B T S k+1 B + R k ) 1 B T S k+1 A (34) The optimal control is: u k = K k x k (35) and the optimal state is: x k+1 = (A BK k ) x k (36) 14
15 Steady State Solutions When the number of samples N approaches infinity, then under certain conditions the Riccati solution converges to a fixed values of S and K S = A T [ S SB(B T SB + R) 1 B T S ] A + Q (37) K = ( B T SB + R ) 1 B T SA (38) Equation (37) is known as the discrete Riccati Algebraic Equation (RAE) In this case, the optimal control is: u k = Kx k (39) and the optimal state is: x k+1 = (A BK) x k (40) 15
16 Example: LQ regulation of an unstable scalar system Consider the unstable system: x k+1 = 2x k + u k (41) Assume that we wish to regulate this system using steady state LQ control, given the following performance index: J = 1 2 k=0 [ x 2 k + 2u 2 k] (42) The algebraic Riccati equation becomes: S 2 7S 2 = 0 (43) which has the solutions S 1 S 2 = = and 16
17 Taking the positive solution, gives the following state feedback law: u k = 15687x k (44) and the closed loop system becomes stable: x k+1 = 2x k + ( 15687x k ) = 04313x k (45) 17
18 Example: DC motor under state feedback + i I a = constant u R L ω J The state equations for a dc motor with constant armature current are: di(t) dt = R L i(t) + 1 u(t) (46) L dω(t) dt = K J i(t) (47) where K = k t i a, and J is the moment of inertia Assuming R = L = K = 1, find a discrete time state feedback controller with sampling time h = 1 to minimise the following performance index: J = k=0 i 2 k + ω2 k + u2 k (48) 18
19 Solution: Continuous time model: [ ] [ ] d i 1 0 = dt ω 1 0 } {{ } Ā [ i ω ] + [ 1 0 ] }{{} B u (49) Discrete time model using zero order hold discretisation: [ ] [ ] [ ] [ ] ik ik 0632 = + u k ω k+1 }{{} eāh ω k }{{} h 0 eāh dt B (50) Riccati solution using Matlab: K = dlqr( A, B, Q, R ) K = [ ] (51) Control law: u k = 0615i k ω k (52) 19
20 Suppose that the system starts from the initial condition i(0) = 1 A and ω(0) = 2 rad/s x time (s) u time (s) 20
Homework Solution # 3
ECSE 644 Optimal Control Feb, 4 Due: Feb 17, 4 (Tuesday) Homework Solution # 3 1 (5%) Consider the discrete nonlinear control system in Homework # For the optimal control and trajectory that you have found
More informationState Regulator. Advanced Control. design of controllers using pole placement and LQ design rules
Advanced Control State Regulator Scope design of controllers using pole placement and LQ design rules Keywords pole placement, optimal control, LQ regulator, weighting matrixes Prerequisites Contact state
More informationQuadratic Stability of Dynamical Systems. Raktim Bhattacharya Aerospace Engineering, Texas A&M University
.. Quadratic Stability of Dynamical Systems Raktim Bhattacharya Aerospace Engineering, Texas A&M University Quadratic Lyapunov Functions Quadratic Stability Dynamical system is quadratically stable if
More informationEE C128 / ME C134 Final Exam Fall 2014
EE C128 / ME C134 Final Exam Fall 2014 December 19, 2014 Your PRINTED FULL NAME Your STUDENT ID NUMBER Number of additional sheets 1. No computers, no tablets, no connected device (phone etc.) 2. Pocket
More information9 Controller Discretization
9 Controller Discretization In most applications, a control system is implemented in a digital fashion on a computer. This implies that the measurements that are supplied to the control system must be
More informationLecture 4 Continuous time linear quadratic regulator
EE363 Winter 2008-09 Lecture 4 Continuous time linear quadratic regulator continuous-time LQR problem dynamic programming solution Hamiltonian system and two point boundary value problem infinite horizon
More information4F3 - Predictive Control
4F3 Predictive Control - Lecture 2 p 1/23 4F3 - Predictive Control Lecture 2 - Unconstrained Predictive Control Jan Maciejowski jmm@engcamacuk 4F3 Predictive Control - Lecture 2 p 2/23 References Predictive
More informationMODERN CONTROL DESIGN
CHAPTER 8 MODERN CONTROL DESIGN The classical design techniques of Chapters 6 and 7 are based on the root-locus and frequency response that utilize only the plant output for feedback with a dynamic controller
More informationLinear Quadratic Regulator (LQR) Design I
Lecture 7 Linear Quadratic Regulator LQR) Design I Dr. Radhakant Padhi Asst. Proessor Dept. o Aerospace Engineering Indian Institute o Science - Bangalore LQR Design: Problem Objective o drive the state
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 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 informationOptimal Control. Quadratic Functions. Single variable quadratic function: Multi-variable quadratic function:
Optimal Control Control design based on pole-placement has non unique solutions Best locations for eigenvalues are sometimes difficult to determine Linear Quadratic LQ) Optimal control minimizes a quadratic
More informationTopic # Feedback Control Systems
Topic #17 16.31 Feedback Control Systems Deterministic LQR Optimal control and the Riccati equation Weight Selection Fall 2007 16.31 17 1 Linear Quadratic Regulator (LQR) Have seen the solutions to the
More informationLecture 2: Discrete-time Linear Quadratic Optimal Control
ME 33, U Berkeley, Spring 04 Xu hen Lecture : Discrete-time Linear Quadratic Optimal ontrol Big picture Example onvergence of finite-time LQ solutions Big picture previously: dynamic programming and finite-horizon
More informationCourse Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)
Amme 35 : System Dynamics Control State Space Design Course Outline Week Date Content Assignment Notes 1 1 Mar Introduction 2 8 Mar Frequency Domain Modelling 3 15 Mar Transient Performance and the s-plane
More informationOPTIMAL CONTROL. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 28
OPTIMAL CONTROL Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 28 (Example from Optimal Control Theory, Kirk) Objective: To get from
More informationRobust Control 5 Nominal Controller Design Continued
Robust Control 5 Nominal Controller Design Continued Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University 4/14/2003 Outline he LQR Problem A Generalization to LQR Min-Max
More informationLinear-Quadratic-Gaussian (LQG) Controllers and Kalman Filters
Linear-Quadratic-Gaussian (LQG) Controllers and Kalman Filters Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 204 Emo Todorov (UW) AMATH/CSE 579, Winter
More informationOutline. 1 Linear Quadratic Problem. 2 Constraints. 3 Dynamic Programming Solution. 4 The Infinite Horizon LQ Problem.
Model Predictive Control Short Course Regulation James B. Rawlings Michael J. Risbeck Nishith R. Patel Department of Chemical and Biological Engineering Copyright c 217 by James B. Rawlings Outline 1 Linear
More informationLinear Quadratic Regulator (LQR) Design II
Lecture 8 Linear Quadratic Regulator LQR) Design II Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Outline Stability and Robustness properties
More informationMATH4406 (Control Theory) Unit 6: The Linear Quadratic Regulator (LQR) and Model Predictive Control (MPC) Prepared by Yoni Nazarathy, Artem
MATH4406 (Control Theory) Unit 6: The Linear Quadratic Regulator (LQR) and Model Predictive Control (MPC) Prepared by Yoni Nazarathy, Artem Pulemotov, September 12, 2012 Unit Outline Goal 1: Outline linear
More informationLinear State Feedback Controller Design
Assignment For EE5101 - Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University
More information1. Find the solution of the following uncontrolled linear system. 2 α 1 1
Appendix B Revision Problems 1. Find the solution of the following uncontrolled linear system 0 1 1 ẋ = x, x(0) =. 2 3 1 Class test, August 1998 2. Given the linear system described by 2 α 1 1 ẋ = x +
More informationChapter 2 Optimal Control Problem
Chapter 2 Optimal Control Problem Optimal control of any process can be achieved either in open or closed loop. In the following two chapters we concentrate mainly on the first class. The first chapter
More informationModel Predictive Control Short Course Regulation
Model Predictive Control Short Course Regulation James B. Rawlings Michael J. Risbeck Nishith R. Patel Department of Chemical and Biological Engineering Copyright c 2017 by James B. Rawlings Milwaukee,
More informationControl Systems. Design of State Feedback Control.
Control Systems Design of State Feedback Control chibum@seoultech.ac.kr Outline Design of State feedback control Dominant pole design Symmetric root locus (linear quadratic regulation) 2 Selection of closed-loop
More informationEE221A Linear System Theory Final Exam
EE221A Linear System Theory Final Exam Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2016 12/16/16, 8-11am Your answers must be supported by analysis,
More informationLecture 9: Discrete-Time Linear Quadratic Regulator Finite-Horizon Case
Lecture 9: Discrete-Time Linear Quadratic Regulator Finite-Horizon Case Dr. Burak Demirel Faculty of Electrical Engineering and Information Technology, University of Paderborn December 15, 2015 2 Previous
More informationESC794: Special Topics: Model Predictive Control
ESC794: Special Topics: Model Predictive Control Nonlinear MPC Analysis : Part 1 Reference: Nonlinear Model Predictive Control (Ch.3), Grüne and Pannek Hanz Richter, Professor Mechanical Engineering Department
More informationTopic # Feedback Control
Topic #5 6.3 Feedback Control State-Space Systems Full-state Feedback Control How do we change the poles of the state-space system? Or,evenifwecanchangethepolelocations. Where do we put the poles? Linear
More informationLinear-Quadratic Optimal Control: Full-State Feedback
Chapter 4 Linear-Quadratic Optimal Control: Full-State Feedback 1 Linear quadratic optimization is a basic method for designing controllers for linear (and often nonlinear) dynamical systems and is actually
More information6 OUTPUT FEEDBACK DESIGN
6 OUTPUT FEEDBACK DESIGN When the whole sate vector is not available for feedback, i.e, we can measure only y = Cx. 6.1 Review of observer design Recall from the first class in linear systems that a simple
More informationSYSTEMTEORI - KALMAN FILTER VS LQ CONTROL
SYSTEMTEORI - KALMAN FILTER VS LQ CONTROL 1. Optimal regulator with noisy measurement Consider the following system: ẋ = Ax + Bu + w, x(0) = x 0 where w(t) is white noise with Ew(t) = 0, and x 0 is a stochastic
More informationLinear Quadratic Regulator (LQR) I
Optimal Control, Guidance and Estimation Lecture Linear Quadratic Regulator (LQR) I Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore Generic Optimal Control Problem
More informationEE363 homework 2 solutions
EE363 Prof. S. Boyd EE363 homework 2 solutions. Derivative of matrix inverse. Suppose that X : R R n n, and that X(t is invertible. Show that ( d d dt X(t = X(t dt X(t X(t. Hint: differentiate X(tX(t =
More informationAustralian Journal of Basic and Applied Sciences, 3(4): , 2009 ISSN Modern Control Design of Power System
Australian Journal of Basic and Applied Sciences, 3(4): 4267-4273, 29 ISSN 99-878 Modern Control Design of Power System Atef Saleh Othman Al-Mashakbeh Tafila Technical University, Electrical Engineering
More informationRobotics. Control Theory. Marc Toussaint U Stuttgart
Robotics Control Theory Topics in control theory, optimal control, HJB equation, infinite horizon case, Linear-Quadratic optimal control, Riccati equations (differential, algebraic, discrete-time), controllability,
More informationSome New Results on Linear Quadratic Regulator Design for Lossless Systems
Some New Results on Linear Quadratic Regulator Design for Lossless Systems Luigi Fortuna, Giovanni Muscato Maria Gabriella Xibilia Dipartimento Elettrico Elettronico e Sistemistico Universitá degli Studi
More informationSUCCESSIVE POLE SHIFTING USING SAMPLED-DATA LQ REGULATORS. Sigeru Omatu
SUCCESSIVE POLE SHIFING USING SAMPLED-DAA LQ REGULAORS oru Fujinaka Sigeru Omatu Graduate School of Engineering, Osaka Prefecture University, 1-1 Gakuen-cho, Sakai, 599-8531 Japan Abstract: Design of sampled-data
More informationLQR, Kalman Filter, and LQG. Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin
LQR, Kalman Filter, and LQG Postgraduate Course, M.Sc. Electrical Engineering Department College of Engineering University of Salahaddin May 2015 Linear Quadratic Regulator (LQR) Consider a linear system
More informationLecture 5 Linear Quadratic Stochastic Control
EE363 Winter 2008-09 Lecture 5 Linear Quadratic Stochastic Control linear-quadratic stochastic control problem solution via dynamic programming 5 1 Linear stochastic system linear dynamical system, over
More informationDenis ARZELIER arzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.2 LMIs IN SYSTEMS CONTROL STATE-SPACE METHODS PERFORMANCE ANALYSIS and SYNTHESIS Denis ARZELIER www.laas.fr/ arzelier arzelier@laas.fr 15
More information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 24: H2 Synthesis Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology May 4, 2011 E. Frazzoli (MIT) Lecture 24: H 2 Synthesis May
More informationTheoretical Justification for LQ problems: Sufficiency condition: LQ problem is the second order expansion of nonlinear optimal control problems.
ES22 Lecture Notes #11 Theoretical Justification for LQ problems: Sufficiency condition: LQ problem is the second order expansion of nonlinear optimal control problems. J = φ(x( ) + L(x,u,t)dt ; x= f(x,u,t)
More informationLecture 2 and 3: Controllability of DT-LTI systems
1 Lecture 2 and 3: Controllability of DT-LTI systems Spring 2013 - EE 194, Advanced Control (Prof Khan) January 23 (Wed) and 28 (Mon), 2013 I LTI SYSTEMS Recall that continuous-time LTI systems can be
More informationState Feedback and State Estimators Linear System Theory and Design, Chapter 8.
1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 State Estimator In previous section, we have discussed the state feedback, based on the assumption that all state variables are
More informationEN Applied Optimal Control Lecture 8: Dynamic Programming October 10, 2018
EN530.603 Applied Optimal Control Lecture 8: Dynamic Programming October 0, 08 Lecturer: Marin Kobilarov Dynamic Programming (DP) is conerned with the computation of an optimal policy, i.e. an optimal
More informationAdvanced Control Theory
State Feedback Control Design chibum@seoultech.ac.kr Outline State feedback control design Benefits of CCF 2 Conceptual steps in controller design We begin by considering the regulation problem the task
More information1 Steady State Error (30 pts)
Professor Fearing EECS C28/ME C34 Problem Set Fall 2 Steady State Error (3 pts) Given the following continuous time (CT) system ] ẋ = A x + B u = x + 2 7 ] u(t), y = ] x () a) Given error e(t) = r(t) y(t)
More information6. Linear Quadratic Regulator Control
EE635 - Control System Theory 6. Linear Quadratic Regulator Control Jitkomut Songsiri algebraic Riccati Equation (ARE) infinite-time LQR (continuous) Hamiltonian matrix gain margin of LQR 6-1 Algebraic
More informationExtensions and applications of LQ
Extensions and applications of LQ 1 Discrete time systems 2 Assigning closed loop pole location 3 Frequency shaping LQ Regulator for Discrete Time Systems Consider the discrete time system: x(k + 1) =
More informationLecture 9. Introduction to Kalman Filtering. Linear Quadratic Gaussian Control (LQG) G. Hovland 2004
MER42 Advanced Control Lecture 9 Introduction to Kalman Filtering Linear Quadratic Gaussian Control (LQG) G. Hovland 24 Announcement No tutorials on hursday mornings 8-9am I will be present in all practical
More information2 The Linear Quadratic Regulator (LQR)
2 The Linear Quadratic Regulator (LQR) Problem: Compute a state feedback controller u(t) = Kx(t) that stabilizes the closed loop system and minimizes J := 0 x(t) T Qx(t)+u(t) T Ru(t)dt where x and u are
More informationTopic # /31 Feedback Control Systems. Analysis of Nonlinear Systems Lyapunov Stability Analysis
Topic # 16.30/31 Feedback Control Systems Analysis of Nonlinear Systems Lyapunov Stability Analysis Fall 010 16.30/31 Lyapunov Stability Analysis Very general method to prove (or disprove) stability of
More informationMathematical Modelling, Stability Analysis and Control of Flexible Double Link Robotic Manipulator: A Simulation Approach
IOSR Journal of Engineering (IOSRJEN) e-issn: 50-301, p-issn: 78-8719 Vol 3, Issue 4 (April 013), V3 PP 9-40 Mathematical Modelling, Stability Analysis and Control of Flexible Double Link Robotic Manipulator:
More informationPole Placement (Bass Gura)
Definition: Open-Loop System: System dynamics with U =. sx = AX Closed-Loop System: System dynamics with U = -Kx X sx = (A BK x )X Characteristic Polynomial: Pole Placement (Bass Gura) a) The polynomial
More informationEE C128 / ME C134 Feedback Control Systems
EE C128 / ME C134 Feedback Control Systems Lecture Additional Material Introduction to Model Predictive Control Maximilian Balandat Department of Electrical Engineering & Computer Science University of
More informationHere represents the impulse (or delta) function. is an diagonal matrix of intensities, and is an diagonal matrix of intensities.
19 KALMAN FILTER 19.1 Introduction In the previous section, we derived the linear quadratic regulator as an optimal solution for the fullstate feedback control problem. The inherent assumption was that
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 informationChapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control
Chapter 3 LQ, LQG and Control System H 2 Design Overview LQ optimization state feedback LQG optimization output feedback H 2 optimization non-stochastic version of LQG Application to feedback system design
More informationNumerical Methods for Model Predictive Control. Jing Yang
Numerical Methods for Model Predictive Control Jing Yang Kongens Lyngby February 26, 2008 Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800 Kongens Lyngby, Denmark
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 informationFuzzy modeling and control of rotary inverted pendulum system using LQR technique
IOP Conference Series: Materials Science and Engineering OPEN ACCESS Fuzzy modeling and control of rotary inverted pendulum system using LQR technique To cite this article: M A Fairus et al 13 IOP Conf.
More informationSteady State Kalman Filter
Steady State Kalman Filter Infinite Horizon LQ Control: ẋ = Ax + Bu R positive definite, Q = Q T 2Q 1 2. (A, B) stabilizable, (A, Q 1 2) detectable. Solve for the positive (semi-) definite P in the ARE:
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 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 information5. Observer-based Controller Design
EE635 - Control System Theory 5. Observer-based Controller Design Jitkomut Songsiri state feedback pole-placement design regulation and tracking state observer feedback observer design LQR and LQG 5-1
More informationCDS 110b: Lecture 2-1 Linear Quadratic Regulators
CDS 110b: Lecture 2-1 Linear Quadratic Regulators Richard M. Murray 11 January 2006 Goals: Derive the linear quadratic regulator and demonstrate its use Reading: Friedland, Chapter 9 (different derivation,
More informationOutline. Linear regulation and state estimation (LQR and LQE) Linear differential equations. Discrete time linear difference equations
Outline Linear regulation and state estimation (LQR and LQE) James B. Rawlings Department of Chemical and Biological Engineering 1 Linear Quadratic Regulator Constraints The Infinite Horizon LQ Problem
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 informationOptimal control and estimation
Automatic Control 2 Optimal control and estimation Prof. Alberto Bemporad University of Trento Academic year 2010-2011 Prof. Alberto Bemporad (University of Trento) Automatic Control 2 Academic year 2010-2011
More informationNumerics and Control of PDEs Lecture 1. IFCAM IISc Bangalore
1/1 Numerics and Control of PDEs Lecture 1 IFCAM IISc Bangalore July 22 August 2, 2013 Introduction to feedback stabilization Stabilizability of F.D.S. Mythily R., Praveen C., Jean-Pierre R. 2/1 Q1. Controllability.
More information1. LQR formulation 2. Selection of weighting matrices 3. Matlab implementation. Regulator Problem mm3,4. u=-kx
MM8.. LQR Reglator 1. LQR formlation 2. Selection of weighting matrices 3. Matlab implementation Reading Material: DC: p.364-382, 400-403, Matlab fnctions: lqr, lqry, dlqr, lqrd, care, dare 3/26/2008 Introdction
More informationRegional Solution of Constrained LQ Optimal Control
Regional Solution of Constrained LQ Optimal Control José DeDoná September 2004 Outline 1 Recap on the Solution for N = 2 2 Regional Explicit Solution Comparison with the Maximal Output Admissible Set 3
More informationStatic and Dynamic Optimization (42111)
Static and Dynamic Optimization (421) Niels Kjølstad Poulsen Build. 0b, room 01 Section for Dynamical Systems Dept. of Applied Mathematics and Computer Science The Technical University of Denmark Email:
More informationProblem Set 4 Solution 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback
More informationSubject: Optimal Control Assignment-1 (Related to Lecture notes 1-10)
Subject: Optimal Control Assignment- (Related to Lecture notes -). Design a oil mug, shown in fig., to hold as much oil possible. The height and radius of the mug should not be more than 6cm. The mug must
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 informationSemidefinite Programming Duality and Linear Time-invariant Systems
Semidefinite Programming Duality and Linear Time-invariant Systems Venkataramanan (Ragu) Balakrishnan School of ECE, Purdue University 2 July 2004 Workshop on Linear Matrix Inequalities in Control LAAS-CNRS,
More informationLinear control of inverted pendulum
Linear control of inverted pendulum Deep Ray, Ritesh Kumar, Praveen. C, Mythily Ramaswamy, J.-P. Raymond IFCAM Summer School on Numerics and Control of PDE 22 July - 2 August 213 IISc, Bangalore http://praveen.cfdlab.net/teaching/control213
More informationLinear Quadratic Optimal Control Topics
Linear Quadratic Optimal Control Topics Finite time LQR problem for time varying systems Open loop solution via Lagrange multiplier Closed loop solution Dynamic programming (DP) principle Cost-to-go function
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 informationLaboratory 11 Control Systems Laboratory ECE3557. State Feedback Controller for Position Control of a Flexible Joint
Laboratory 11 State Feedback Controller for Position Control of a Flexible Joint 11.1 Objective The objective of this laboratory is to design a full state feedback controller for endpoint position control
More informationDynamic Optimal Control!
Dynamic Optimal Control! Robert Stengel! Robotics and Intelligent Systems MAE 345, Princeton University, 2017 Learning Objectives Examples of cost functions Necessary conditions for optimality Calculation
More informationControl Systems Design
ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:
More informationLinear Quadratic Regulator (LQR) II
Optimal Control, Guidance and Estimation Lecture 11 Linear Quadratic Regulator (LQR) II Pro. Radhakant Padhi Dept. o Aerospace Engineering Indian Institute o Science - Bangalore Outline Summary o LQR design
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 informationAn LQR Controller Design Approach For Pitch Axis Stabilisation Of 3-DOF Helicopter System
International Journal of Scientific & Engineering Research, Volume 4, Issue 4, April-2013 1398 An LQR Controller Design Approach For Pitch Axis Stabilisation Of 3-DOF Helicopter System Mrs. M. Bharathi*Golden
More informationThe norms can also be characterized in terms of Riccati inequalities.
9 Analysis of stability and H norms Consider the causal, linear, time-invariant system ẋ(t = Ax(t + Bu(t y(t = Cx(t Denote the transfer function G(s := C (si A 1 B. Theorem 85 The following statements
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 informationIdentification Methods for Structural Systems
Prof. Dr. Eleni Chatzi System Stability Fundamentals Overview System Stability Assume given a dynamic system with input u(t) and output x(t). The stability property of a dynamic system can be defined from
More informationLinear Systems Theory
ME 3253 Linear Systems Theory Review Class Overview and Introduction 1. How to build dynamic system model for physical system? 2. How to analyze the dynamic system? -- Time domain -- Frequency domain (Laplace
More informationEC Control Engineering Quiz II IIT Madras
EC34 - Control Engineering Quiz II IIT Madras Linear algebra Find the eigenvalues and eigenvectors of A, A, A and A + 4I Find the eigenvalues and eigenvectors of the following matrices: (a) cos θ sin θ
More informationChapter 8 Stabilization: State Feedback 8. Introduction: Stabilization One reason feedback control systems are designed is to stabilize systems that m
Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of echnology c Chapter 8 Stabilization:
More informationChap 8. State Feedback and State Estimators
Chap 8. State Feedback and State Estimators Outlines Introduction State feedback Regulation and tracking State estimator Feedback from estimated states State feedback-multivariable case State estimators-multivariable
More informationReverse Order Swing-up Control of Serial Double Inverted Pendulums
Reverse Order Swing-up Control of Serial Double Inverted Pendulums T.Henmi, M.Deng, A.Inoue, N.Ueki and Y.Hirashima Okayama University, 3-1-1, Tsushima-Naka, Okayama, Japan inoue@suri.sys.okayama-u.ac.jp
More informationContents lecture 6 2(17) Automatic Control III. Summary of lecture 5 (I/III) 3(17) Summary of lecture 5 (II/III) 4(17) H 2, H synthesis pros and cons:
Contents lecture 6 (7) Automatic Control III Lecture 6 Linearization and phase portraits. Summary of lecture 5 Thomas Schön Division of Systems and Control Department of Information Technology Uppsala
More informationmoments of inertia at the center of gravity (C.G.) of the first and second pendulums are I 1 and I 2, respectively. Fig. 1. Double inverted pendulum T
Real-Time Swing-up of Double Inverted Pendulum by Nonlinear Model Predictive Control Pathompong Jaiwat 1 and Toshiyuki Ohtsuka 2 Abstract In this study, the swing-up of a double inverted pendulum is controlled
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 informationICS 6N Computational Linear Algebra Matrix Algebra
ICS 6N Computational Linear Algebra Matrix Algebra Xiaohui Xie University of California, Irvine xhx@uci.edu February 2, 2017 Xiaohui Xie (UCI) ICS 6N February 2, 2017 1 / 24 Matrix Consider an m n matrix
More information