Advanced Mechatronics Engineering
|
|
- Melina Chambers
- 5 years ago
- Views:
Transcription
1 Advanced Mechatronics Engineering German University in Cairo 21 December, 2013
2 Outline Necessary conditions for optimal input Example Linear regulator problem Example
3 Necessary conditions for optimal input The problem is to find an optimal input (u ) that causes the system ẋ(t) = f(x(t), u(t), t), (1) to follow a trajectory (x ) that minimizes the performance measure tf J(x(t), t) = h(x(t f ), t f ) + g(x(τ), u(τ), τ)dτ. (2) t 0
4 Necessary conditions for optimal input The Hamiltonian is given by H(x(t), u(t), p(t), t) g(x(t), u(t), t) + p T (t) [f(x(t), u(t), t)]. (3) The necessary conditions of optimality are ẋ (t) = H p (x (t), u (t), p (t), t), (4) ṗ (t) = H x (x (t), u (t), p (t), t), (5) 0 = H u (x (t), u (t), p (t), t). (6)
5 Example The system ẋ 1 (t) = x 2 (t) (7) ẋ 2 (t) = x 2 (t) + u(t) (8) is to be controlled so that its input is conserved. Therefore, the performance measure is given by J(u) = tf The Hamiltonian (36) is given by t u2 (t)dt. (9) H(x(t), u(t), p(t), t) = 1 2 u2 (t)+p 1 (t)x 2 (t) p 2 (t)x 2 (t)+p 2 (t)u(t). (10)
6 Example H(x(t), u(t), p(t), t) = 1 2 u2 (t)+p 1 (t)x 2 (t) p 2 (t)x 2 (t)+p 2 (t)u(t). The necessary conditions for optimality are (11) ṗ 1(t) = H x 1 = 0 (12) ṗ 2(t) = H x 2 = p 1(t) + p 2(t), (13) and 0 = H u = u (t) + p 2(t). (14)
7 Linear Regulator Problems The plant is described by the linear state equations ẋ(t) = A(t)x(t) + B(t)u(t) (15) which may have time-varying coefficients. The performance measure to be minimized is J = 1 2 xt (t f )Hx(t f ) + tf t [ x T Qx + u T Ru ] dt, (16) where the final time t f is fixed. Further, H and Q are real symmetric positive semi-definite matrices. Finally, R is a real symmetric positive definite matrix. The Hamiltonian is H(x(t), u(t), p(t), t) g(x(t), u(t), t) + p T (t) [f(x(t), u(t), t)]. (17)
8 Linear Regulator Problems H(x(t), u(t), p(t), t) = 1 2 xt Qx+ 1 2 ut Ru+p T A(t)x(t)+p T B(t)u(t), (18) and the necessary conditions for optimality are ẋ (t) = A(t)x (t) + B(t)u (t), (19) ṗ (t) = Q(t)x (t) A T (t)p (t), (20) 0 = R(t)u (t) + B T (t)p (t). (21) Solving (54) for u (t) yields u (t) = R 1 (t)b T (t)p (t). (22) Substitution of (55) into (52) yields ẋ (t) = A(t)x (t) B(t)R 1 (t)b T (t)p (t). (23)
9 Linear Regulator Problems Putting (53) and (56) into the following matrix format ẋ (t) A(t) B(t)R 1 (t)b T (t) x (t) =. (24) ṗ (t) Q(t) p (t) p (t) The solution of these equations has the following form x (t f ) x (t) = ϕ(t f, t), (25) p (t f ) p (t) where ϕ(t f, t) is the state-transition matrix of the system (57). x (t f ) ϕ 11 (t f, t) ϕ 12 (t f, t) x (t) =, (26) p (t f ) ϕ 21 (t f, t) ϕ 22 (t f, t) p (t)
10 Linear Regulator Problems From the boundary condition, the final co-states are related to the final states using p (t f ) = Hx (t f ). (27) Solving for p (t f ), we obtain p (t) = [ϕ 22 (t f, t) Hϕ 12 (t f, t)] 1 [Hϕ 11 (t f, t) ϕ 21 (t f, t)] x (t). (28) p (t) = K(t)x (t). (29) The optimal input is given by u (t) = R 1 (t)b T (t)k(t)x (t). (30)
11 Example It is desired to determine the input (using the principle of optimality and the Hamilton-Jacobi-Bellman equation) that causes the plant ẋ 1 = x 2 (t) (31) ẋ 2 = x 1 (t) 2x 2 (t) + u(t) (32) to minimize the performance measure J = 10x 2 1 (T ) T 0 [ x 2 1 (t) + 2x 2 2 (t) + u 2 (t) ]. (33)
12 State Transition Matrix Consider the scalar case ẋ(t) = ax(t). (34) Taking the Laplace transform of (34), we obtain sx (s) x(0) = ax (s), (35) X (s) = x(0) s a = (s a) 1 x(0). (36) Finally, inverse Laplace transform of (36) yields x(t) = e at x(0). (37)
13 State Transition Matrix Now consider the following homogenous state equation ẋ(t) = Ax(t). (38) sx(s) x(0) = AX(s), (39) X(s) = (si A) 1 x(0). (40) The inverse Laplace transform yields x(t) = L 1 [ (si A) 1] x(0) = e At x(0). (41) Therefore, the state transition matrix (e At ) is given by e At = L 1 [ (si A) 1]. (42)
14 State Transition Matrix Calculate the state transition matrix of the following system ] [ẋ1 = ẋ 2 [si A] = [si A] 1 = [ ] [ ] x1 x 2 [ ] (s + 1) 0 2 (s + 3) (43) (44) [ (s+3) (s+1)(s+3) 0 2 (s+1) (s+1)(s+3) (s+1)(s+3) ] = [ 1 (s+1) 0 ( 1 (s+1) 1 (s+1) ) 1 (s+3) e At = L 1 [ (si A) 1], (45) [ e At e = t ] 0 (e t e 3t ) e 3t. ]
15 State Transition Matrix Calculate the state transition matrix of the following system ] [ ] [ ] [ẋ1 0 1 x1 = 2 3 ẋ 2 [si A] = x 2 [ ] s 1 2 (s + 3) (46) (47) e At = L 1 [ (si A) 1], (48) [ 2e = t e 2t e t e 2t ] 2e t + 2e 2t e t + 2e 2t. [si A] 1 = [ (s+3) (s+1)(s+2) 2 (s+1)(s+2) 1 (s+1)(s+2) s (s+1)(s+2) ]
16 State Transition Matrix If the matrix A can be transformed into a diagonal form, then the state transition matrix e At is given by e λ 1t e At = Pe Dt P 1 0 e λ 2t... 0 = P P 1, (49) e λnt where P is a digonalizing matrix for A. Further, λ i is the ith eigenvalue of the matrix A, for i = 1,..., n.
17 State Transition Matrix Derivation: Consider the following homogenous state equation ẋ = Ax, (50) and the following similarity transformation: x = Pξ, ẋ = P ξ. (51) Substituting (51) in (50) yields ξ = P 1 APξ = Dξ. (52) Solution of (52) is using (51) ξ(t) = e Dt ξ(0), (53) x(t) = Pξ(t) = Pe Dt ξ(0), x(0) = Pξ(0). (54) Therefore x(t) = Pe Dt P 1 x(0) = e At x(0). (55)
18 State Transition Matrix Calculate the state transition matrix of the following system ] [ẋ1 = ẋ 2 [ ] [ ] x1 x 2 (56) The eigenvalues of A are λ 1 = 0 and λ 2 = 2. A similarity transformation matrix P is [ ] 1 1 P =. (57) 0 2 Using (49) to calculate the state transition matrix e At = Pe Dt P 1 (58) [ ] [ ] [ ] 1 1 e = e 2t [ e At 1 1 = 2 (1 ] e 2t ) 0 e 2t. (59)
19 Thanks Questions please
OPTIMAL 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 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 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 informationAdvanced Control Theory
State Space Solution and Realization chibum@seoultech.ac.kr Outline State space solution 2 Solution of state-space equations x t = Ax t + Bu t First, recall results for scalar equation: x t = a x t + b
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 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 informationDeterministic Dynamic Programming
Deterministic Dynamic Programming 1 Value Function Consider the following optimal control problem in Mayer s form: V (t 0, x 0 ) = inf u U J(t 1, x(t 1 )) (1) subject to ẋ(t) = f(t, x(t), u(t)), x(t 0
More informationOptimal Control. Lecture 18. Hamilton-Jacobi-Bellman Equation, Cont. John T. Wen. March 29, Ref: Bryson & Ho Chapter 4.
Optimal Control Lecture 18 Hamilton-Jacobi-Bellman Equation, Cont. John T. Wen Ref: Bryson & Ho Chapter 4. March 29, 2004 Outline Hamilton-Jacobi-Bellman (HJB) Equation Iterative solution of HJB Equation
More informationState Variable Analysis of Linear Dynamical Systems
Chapter 6 State Variable Analysis of Linear Dynamical Systems 6 Preliminaries In state variable approach, a system is represented completely by a set of differential equations that govern the evolution
More informationLinear Quadratic Optimal Control
156 c Perry Y.Li Chapter 6 Linear Quadratic Optimal Control 6.1 Introduction In previous lectures, we discussed the design of state feedback controllers using using eigenvalue (pole) placement algorithms.
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 informationSYSTEMTEORI - ÖVNING 1. In this exercise, we will learn how to solve the following linear differential equation:
SYSTEMTEORI - ÖVNING 1 GIANANTONIO BORTOLIN AND RYOZO NAGAMUNE In this exercise, we will learn how to solve the following linear differential equation: 01 ẋt Atxt, xt 0 x 0, xt R n, At R n n The equation
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 informationHamilton-Jacobi-Bellman Equation Feb 25, 2008
Hamilton-Jacobi-Bellman Equation Feb 25, 2008 What is it? The Hamilton-Jacobi-Bellman (HJB) equation is the continuous-time analog to the discrete deterministic dynamic programming algorithm Discrete VS
More informationModule 02 CPS Background: Linear Systems Preliminaries
Module 02 CPS Background: Linear Systems Preliminaries Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html August
More informationModule 05 Introduction to Optimal Control
Module 05 Introduction to Optimal Control Ahmad F. Taha EE 5243: Introduction to Cyber-Physical Systems Email: ahmad.taha@utsa.edu Webpage: http://engineering.utsa.edu/ taha/index.html October 8, 2015
More informationProject Optimal Control: Optimal Control with State Variable Inequality Constraints (SVIC)
Project Optimal Control: Optimal Control with State Variable Inequality Constraints (SVIC) Mahdi Ghazaei, Meike Stemmann Automatic Control LTH Lund University Problem Formulation Find the maximum of the
More informationPrinciples of Optimal Control Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.323 Lecture
More informationChap 4. State-Space Solutions and
Chap 4. State-Space Solutions and Realizations Outlines 1. Introduction 2. Solution of LTI State Equation 3. Equivalent State Equations 4. Realizations 5. Solution of Linear Time-Varying (LTV) Equations
More informationChapter III. Stability of Linear Systems
1 Chapter III Stability of Linear Systems 1. Stability and state transition matrix 2. Time-varying (non-autonomous) systems 3. Time-invariant systems 1 STABILITY AND STATE TRANSITION MATRIX 2 In this chapter,
More informationSolution via Laplace transform and matrix exponential
EE263 Autumn 2015 S. Boyd and S. Lall Solution via Laplace transform and matrix exponential Laplace transform solving ẋ = Ax via Laplace transform state transition matrix matrix exponential qualitative
More informationLinear System Theory
Linear System Theory Wonhee Kim Chapter 6: Controllability & Observability Chapter 7: Minimal Realizations May 2, 217 1 / 31 Recap State space equation Linear Algebra Solutions of LTI and LTV system Stability
More informationECEEN 5448 Fall 2011 Homework #5 Solutions
ECEEN 5448 Fall 211 Homework #5 Solutions Professor David G. Meyer December 8, 211 1. Consider the 1-dimensional time-varying linear system ẋ t (u x) (a) Find the state-transition matrix, Φ(t, τ). Here
More informationCONTROL DESIGN FOR SET POINT TRACKING
Chapter 5 CONTROL DESIGN FOR SET POINT TRACKING In this chapter, we extend the pole placement, observer-based output feedback design to solve tracking problems. By tracking we mean that the output is commanded
More informationObservability. Dynamic Systems. Lecture 2 Observability. Observability, continuous time: Observability, discrete time: = h (2) (x, u, u)
Observability Dynamic Systems Lecture 2 Observability Continuous time model: Discrete time model: ẋ(t) = f (x(t), u(t)), y(t) = h(x(t), u(t)) x(t + 1) = f (x(t), u(t)), y(t) = h(x(t)) Reglerteknik, ISY,
More informationLecture Note 13:Continuous Time Switched Optimal Control: Embedding Principle and Numerical Algorithms
ECE785: Hybrid Systems:Theory and Applications Lecture Note 13:Continuous Time Switched Optimal Control: Embedding Principle and Numerical Algorithms Wei Zhang Assistant Professor Department of Electrical
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 information1. The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ = Ax + Bu is
ECE 55, Fall 2007 Problem Set #4 Solution The Transition Matrix (Hint: Recall that the solution to the linear equation ẋ Ax + Bu is x(t) e A(t ) x( ) + e A(t τ) Bu(τ)dτ () This formula is extremely important
More informationSequential State Estimation (Crassidas and Junkins, Chapter 5)
Sequential State Estimation (Crassidas and Junkins, Chapter 5) Please read: 5.1, 5.3-5.6 5.3 The Discrete-Time Kalman Filter The discrete-time Kalman filter is used when the dynamics and measurements are
More informationarxiv: v1 [math.oc] 29 Apr 2017
BALANCED TRUNCATION MODEL ORDER REDUCTION FOR QUADRATIC-BILINEAR CONTROL SYSTEMS PETER BENNER AND PAWAN GOYAL arxiv:175.16v1 [math.oc] 29 Apr 217 Abstract. We discuss balanced truncation model order reduction
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 informationIterative methods to compute center and center-stable manifolds with application to the optimal output regulation problem
Iterative methods to compute center and center-stable manifolds with application to the optimal output regulation problem Noboru Sakamoto, Branislav Rehak N.S.: Nagoya University, Department of Aerospace
More informationFormula Sheet for Optimal Control
Formula Sheet for Optimal Control Division of Optimization and Systems Theory Royal Institute of Technology 144 Stockholm, Sweden 23 December 1, 29 1 Dynamic Programming 11 Discrete Dynamic Programming
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 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 informationECSE.6440 MIDTERM EXAM Solution Optimal Control. Assigned: February 26, 2004 Due: 12:00 pm, March 4, 2004
ECSE.6440 MIDTERM EXAM Solution Optimal Control Assigned: February 26, 2004 Due: 12:00 pm, March 4, 2004 This is a take home exam. It is essential to SHOW ALL STEPS IN YOUR WORK. In NO circumstance is
More informationACM/CMS 107 Linear Analysis & Applications Fall 2016 Assignment 4: Linear ODEs and Control Theory Due: 5th December 2016
ACM/CMS 17 Linear Analysis & Applications Fall 216 Assignment 4: Linear ODEs and Control Theory Due: 5th December 216 Introduction Systems of ordinary differential equations (ODEs) can be used to describe
More informationToward nonlinear tracking and rejection using LPV control
Toward nonlinear tracking and rejection using LPV control Gérard Scorletti, V. Fromion, S. de Hillerin Laboratoire Ampère (CNRS) MaIAGE (INRA) Fondation EADS International Workshop on Robust LPV Control
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 informationSuppose that we have a specific single stage dynamic system governed by the following equation:
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
More informationBoyce - DiPrima 7.9, Nonhomogeneous Linear Systems
Boyce - DiPrima 7.9, Nonhomogeneous Linear Systems Section 7.9, pp. 49--440:, 7, 11, 15 (vop) Additional problems: 1 (diagonalization) Initialization In[77]:= In[78]:= Import"ColorNames.m" DiffEqs` Example
More informationProfessor Fearing EE C128 / ME C134 Problem Set 7 Solution Fall 2010 Jansen Sheng and Wenjie Chen, UC Berkeley
Professor Fearing EE C8 / ME C34 Problem Set 7 Solution Fall Jansen Sheng and Wenjie Chen, UC Berkeley. 35 pts Lag compensation. For open loop plant Gs ss+5s+8 a Find compensator gain Ds k such that the
More informationME Fall 2001, Fall 2002, Spring I/O Stability. Preliminaries: Vector and function norms
I/O Stability Preliminaries: Vector and function norms 1. Sup norms are used for vectors for simplicity: x = max i x i. Other norms are also okay 2. Induced matrix norms: let A R n n, (i stands for induced)
More informationTutorial on Control and State Constrained Optimal Control Pro. Control Problems and Applications Part 3 : Pure State Constraints
Tutorial on Control and State Constrained Optimal Control Problems and Applications Part 3 : Pure State Constraints University of Münster Institute of Computational and Applied Mathematics SADCO Summer
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 information21 Linear State-Space Representations
ME 132, Spring 25, UC Berkeley, A Packard 187 21 Linear State-Space Representations First, let s describe the most general type of dynamic system that we will consider/encounter in this class Systems may
More informationECEN 605 LINEAR SYSTEMS. Lecture 7 Solution of State Equations 1/77
1/77 ECEN 605 LINEAR SYSTEMS Lecture 7 Solution of State Equations Solution of State Space Equations Recall from the previous Lecture note, for a system: ẋ(t) = A x(t) + B u(t) y(t) = C x(t) + D u(t),
More informationMath Ordinary Differential Equations
Math 411 - Ordinary Differential Equations Review Notes - 1 1 - Basic Theory A first order ordinary differential equation has the form x = f(t, x) (11) Here x = dx/dt Given an initial data x(t 0 ) = x
More informationIntro. Computer Control Systems: F8
Intro. Computer Control Systems: F8 Properties of state-space descriptions and feedback Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 22 dave.zachariah@it.uu.se F7: Quiz! 2
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 informationUCLA Chemical Engineering. Process & Control Systems Engineering Laboratory
Constrained Innite-Time Nonlinear Quadratic Optimal Control V. Manousiouthakis D. Chmielewski Chemical Engineering Department UCLA 1998 AIChE Annual Meeting Outline Unconstrained Innite-Time Nonlinear
More informationTheorem 1. ẋ = Ax is globally exponentially stable (GES) iff A is Hurwitz (i.e., max(re(σ(a))) < 0).
Linear Systems Notes Lecture Proposition. A M n (R) is positive definite iff all nested minors are greater than or equal to zero. n Proof. ( ): Positive definite iff λ i >. Let det(a) = λj and H = {x D
More informationSolution of Linear State-space Systems
Solution of Linear State-space Systems Homogeneous (u=0) LTV systems first Theorem (Peano-Baker series) The unique solution to x(t) = (t, )x 0 where The matrix function is given by is called the state
More informationModeling and Analysis of Dynamic Systems
Modeling and Analysis of Dynamic Systems Dr. Guillaume Ducard Fall 2017 Institute for Dynamic Systems and Control ETH Zurich, Switzerland G. Ducard c 1 / 57 Outline 1 Lecture 13: Linear System - Stability
More informationApplied Differential Equation. November 30, 2012
Applied Differential Equation November 3, Contents 5 System of First Order Linear Equations 5 Introduction and Review of matrices 5 Systems of Linear Algebraic Equations, Linear Independence, Eigenvalues,
More informationOn Controllability of Linear Systems 1
On Controllability of Linear Systems 1 M.T.Nair Department of Mathematics, IIT Madras Abstract In this article we discuss some issues related to the observability and controllability of linear systems.
More informationUNIVERSITY OF OSLO. Please make sure that your copy of the problem set is complete before you attempt to answer anything.
UNIVERSITY OF OSLO Faculty of mathematics and natural sciences Examination in MAT2440 Differential equations and optimal control theory Day of examination: 11 June 2015 Examination hours: 0900 1300 This
More informationMath 331 Homework Assignment Chapter 7 Page 1 of 9
Math Homework Assignment Chapter 7 Page of 9 Instructions: Please make sure to demonstrate every step in your calculations. Return your answers including this homework sheet back to the instructor as a
More informationPrinciples of Optimal Control Spring 2008
MIT OpenCourseWare http://ocw.it.edu 16.323 Principles of Optial Control Spring 2008 For inforation about citing these aterials or our Ters of Use, visit: http://ocw.it.edu/ters. 16.323 Lecture 10 Singular
More informationINTRODUCTION TO OPTIMAL CONTROL
INTRODUCTION TO OPTIMAL CONTROL Fernando Lobo Pereira, João Borges de Sousa Faculdade de Engenharia da Universidade do Porto 4200-465 Porto, Portugal {flp,jtasso}@fe.up.pt C4C Autumn School Verona, October
More informationThird In-Class Exam Solutions Math 246, Professor David Levermore Thursday, 3 December 2009 (1) [6] Given that 2 is an eigenvalue of the matrix
Third In-Class Exam Solutions Math 26, Professor David Levermore Thursday, December 2009 ) [6] Given that 2 is an eigenvalue of the matrix A 2, 0 find all the eigenvectors of A associated with 2. Solution.
More informationEML Spring 2012
Home http://vdol.mae.ufl.edu/eml6934-spring2012/ Page 1 of 2 1/10/2012 Search EML6934 - Spring 2012 Optimal Control Home Instructor Anil V. Rao Office Hours: M, W, F 2:00 PM to 3:00 PM Office: MAE-A, Room
More informationLINEAR-CONVEX CONTROL AND DUALITY
1 LINEAR-CONVEX CONTROL AND DUALITY R.T. Rockafellar Department of Mathematics, University of Washington Seattle, WA 98195-4350, USA Email: rtr@math.washington.edu R. Goebel 3518 NE 42 St., Seattle, WA
More informationRobotics. Islam S. M. Khalil. November 15, German University in Cairo
Robotics German University in Cairo November 15, 2016 Fundamental concepts In optimal control problems the objective is to determine a function that minimizes a specified functional, i.e., the performance
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 informationIntroduction to Modern Control MT 2016
CDT Autonomous and Intelligent Machines & Systems Introduction to Modern Control MT 2016 Alessandro Abate Lecture 2 First-order ordinary differential equations (ODE) Solution of a linear ODE Hints to nonlinear
More informationLinear dynamical systems with inputs & outputs
EE263 Autumn 215 S. Boyd and S. Lall Linear dynamical systems with inputs & outputs inputs & outputs: interpretations transfer function impulse and step responses examples 1 Inputs & outputs recall continuous-time
More informationPeriodic Linear Systems
Periodic Linear Systems Lecture 17 Math 634 10/8/99 We now consider ẋ = A(t)x (1) when A is a continuous periodic n n matrix function of t; i.e., when there is a constant T>0 such that A(t + T )=A(t) for
More informationME8281-Advanced Control Systems Design
ME8281 - Advanced Control Systems Design Spring 2016 Perry Y. Li Department of Mechanical Engineering University of Minnesota Spring 2016 Lecture 4 - Outline 1 Homework 1 to be posted by tonight 2 Transition
More informationZ i Q ij Z j. J(x, φ; U) = X T φ(t ) 2 h + where h k k, H(t) k k and R(t) r r are nonnegative definite matrices (R(t) is uniformly in t nonsingular).
2. LINEAR QUADRATIC DETERMINISTIC PROBLEM Notations: For a vector Z, Z = Z, Z is the Euclidean norm here Z, Z = i Z2 i is the inner product; For a vector Z and nonnegative definite matrix Q, Z Q = Z, QZ
More informationGeometric Optimal Control with Applications
Geometric Optimal Control with Applications Accelerated Graduate Course Institute of Mathematics for Industry, Kyushu University, Bernard Bonnard Inria Sophia Antipolis et Institut de Mathématiques de
More informationSub-Riemannian geometry in groups of diffeomorphisms and shape spaces
Sub-Riemannian geometry in groups of diffeomorphisms and shape spaces Sylvain Arguillère, Emmanuel Trélat (Paris 6), Alain Trouvé (ENS Cachan), Laurent May 2013 Plan 1 Sub-Riemannian geometry 2 Right-invariant
More informationCost-extended control systems on SO(3)
Cost-extended control systems on SO(3) Ross M. Adams Mathematics Seminar April 16, 2014 R.M. Adams (RU) Cost-extended control systems on SO(3) RU Maths Seminar 1 / 34 Outline 1 Introduction 2 Control systems
More informationProblem 1 Cost of an Infinite Horizon LQR
THE UNIVERSITY OF TEXAS AT SAN ANTONIO EE 5243 INTRODUCTION TO CYBER-PHYSICAL SYSTEMS H O M E W O R K # 5 Ahmad F. Taha October 12, 215 Homework Instructions: 1. Type your solutions in the LATEX homework
More informationMay 9, 2014 MATH 408 MIDTERM EXAM OUTLINE. Sample Questions
May 9, 24 MATH 48 MIDTERM EXAM OUTLINE This exam will consist of two parts and each part will have multipart questions. Each of the 6 questions is worth 5 points for a total of points. The two part of
More informationA Tutorial on Recursive methods in Linear Least Squares Problems
A Tutorial on Recursive methods in Linear Least Squares Problems by Arvind Yedla 1 Introduction This tutorial motivates the use of Recursive Methods in Linear Least Squares problems, specifically Recursive
More informationAn Application of Pontryagin s Maximum Principle in a Linear Quadratic Differential Game
An Application of Pontryagin s Maximum Principle in a Linear Quadratic Differential Game Marzieh Khakestari (Corresponding author) Institute For Mathematical Research, Universiti Putra Malaysia, 43400
More informationMathematical Economics. Lecture Notes (in extracts)
Prof. Dr. Frank Werner Faculty of Mathematics Institute of Mathematical Optimization (IMO) http://math.uni-magdeburg.de/ werner/math-ec-new.html Mathematical Economics Lecture Notes (in extracts) Winter
More informationChapter Introduction. A. Bensoussan
Chapter 2 EXPLICIT SOLUTIONS OFLINEAR QUADRATIC DIFFERENTIAL GAMES A. Bensoussan International Center for Decision and Risk Analysis University of Texas at Dallas School of Management, P.O. Box 830688
More informationGame Theory Extra Lecture 1 (BoB)
Game Theory 2014 Extra Lecture 1 (BoB) Differential games Tools from optimal control Dynamic programming Hamilton-Jacobi-Bellman-Isaacs equation Zerosum linear quadratic games and H control Baser/Olsder,
More informationCalculus C (ordinary differential equations)
Calculus C (ordinary differential equations) Lesson 9: Matrix exponential of a symmetric matrix Coefficient matrices with a full set of eigenvectors Solving linear ODE s by power series Solutions to linear
More informationKalman Filter and Parameter Identification. Florian Herzog
Kalman Filter and Parameter Identification Florian Herzog 2013 Continuous-time Kalman Filter In this chapter, we shall use stochastic processes with independent increments w 1 (.) and w 2 (.) at the input
More informationControl Systems Design, SC4026. SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft
Control Systems Design, SC4026 SC4026 Fall 2009, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) vs Observability Algebraic Tests (Kalman rank condition & Hautus test) A few
More informationLyapunov Theory for Discrete Time Systems
Università degli studi di Padova Dipartimento di Ingegneria dell Informazione Nicoletta Bof, Ruggero Carli, Luca Schenato Technical Report Lyapunov Theory for Discrete Time Systems This work contains a
More informationA Numerical Scheme for Generalized Fractional Optimal Control Problems
Available at http://pvamuedu/aam Appl Appl Math ISSN: 1932-9466 Vol 11, Issue 2 (December 216), pp 798 814 Applications and Applied Mathematics: An International Journal (AAM) A Numerical Scheme for Generalized
More informationDiscrete and continuous dynamic systems
Discrete and continuous dynamic systems Bounded input bounded output (BIBO) and asymptotic stability Continuous and discrete time linear time-invariant systems Katalin Hangos University of Pannonia Faculty
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 informationControl Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft
Control Systems Design, SC4026 SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft Lecture 4 Controllability (a.k.a. Reachability) and Observability Algebraic Tests (Kalman rank condition & Hautus test) A few
More informationMA Ordinary Differential Equations and Control Semester 1 Lecture Notes on ODEs and Laplace Transform (Parts I and II)
MA222 - Ordinary Differential Equations and Control Semester Lecture Notes on ODEs and Laplace Transform (Parts I and II Dr J D Evans October 22 Special thanks (in alphabetical order are due to Kenneth
More informationPrinciples of Optimal Control Spring 2008
MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 16.323 Lecture
More informationModal Decomposition and the Time-Domain Response of Linear Systems 1
MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING.151 Advanced System Dynamics and Control Modal Decomposition and the Time-Domain Response of Linear Systems 1 In a previous handout
More informationEE291E/ME 290Q Lecture Notes 8. Optimal Control and Dynamic Games
EE291E/ME 290Q Lecture Notes 8. Optimal Control and Dynamic Games S. S. Sastry REVISED March 29th There exist two main approaches to optimal control and dynamic games: 1. via the Calculus of Variations
More informationIngegneria dell Automazione - Sistemi in Tempo Reale p.1/30
Ingegneria dell Automazione - Sistemi in Tempo Reale Selected topics on discrete-time and sampled-data systems Luigi Palopoli palopoli@sssup.it - Tel. 050/883444 Ingegneria dell Automazione - Sistemi in
More informationME 132, Fall 2015, Quiz # 2
ME 132, Fall 2015, Quiz # 2 # 1 # 2 # 3 # 4 # 5 # 6 Total NAME 14 10 8 6 14 8 60 Rules: 1. 2 sheets of notes allowed, 8.5 11 inches. Both sides can be used. 2. Calculator is allowed. Keep it in plain view
More informationNumerical Optimal Control Overview. Moritz Diehl
Numerical Optimal Control Overview Moritz Diehl Simplified Optimal Control Problem in ODE path constraints h(x, u) 0 initial value x0 states x(t) terminal constraint r(x(t )) 0 controls u(t) 0 t T minimize
More informationLinear Algebra. P R E R E Q U I S I T E S A S S E S S M E N T Ahmad F. Taha August 24, 2015
THE UNIVERSITY OF TEXAS AT SAN ANTONIO EE 5243 INTRODUCTION TO CYBER-PHYSICAL SYSTEMS P R E R E Q U I S I T E S A S S E S S M E N T Ahmad F. Taha August 24, 2015 The objective of this exercise is to assess
More informationStochastic optimal control theory
Stochastic optimal control theory Bert Kappen SNN Radboud University Nijmegen the Netherlands July 5, 2008 Bert Kappen Introduction Optimal control theory: Optimize sum of a path cost and end cost. Result
More informationNumerical Computational Techniques for Nonlinear Optimal Control
Seminar at LAAS Numerical Computational echniques for Nonlinear Optimal Control Yasuaki Oishi (Nanzan University) July 2, 2018 * Joint work with N. Sakamoto and. Nakamura 1. Introduction Optimal control
More informationOptimal Linear Feedback Control for Incompressible Fluid Flow
Optimal Linear Feedback Control for Incompressible Fluid Flow Miroslav K. Stoyanov Thesis submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the
More informationLinear System Theory
Linear System Theory Wonhee Kim Lecture 3 Mar. 21, 2017 1 / 38 Overview Recap Nonlinear systems: existence and uniqueness of a solution of differential equations Preliminaries Fields and Vector Spaces
More information