Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2015
|
|
- Elizabeth Goodwin
- 5 years ago
- Views:
Transcription
1 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 in cost function Controllability and observability of an LTI system Requirements for closed-loop stability Algebraic Riccati equation Equilibrium response to commands Copyright 15 by Robert Stengel. All rights reserved. For educational use only Continuous-Time, Linear, Time-Invariant System Model x = Fx + Gu + Lw, x(t o ) given y = H x x + H u u + H w w
2 Linear-Quadratic Regulator: Finite Final Time x = Fx + Gu J = 1 xt (t f )P(t f )x(t f ) u = R 1 M T + G T P t x t = Cx t t + 1 t f x T u T Q M T M R x u dt P = t F GR 1 M T T P t PF t GR 1 M T + PGR t 1 G T P+ t MR 1 M T Q P( t f )= P f 3 Transformation of Variables to Eliminate Cost Function Cross Weighting Original LTI minimization problem min J 1 = 1 u 1 t f x T 1 Q 1 x 1 + x T 1 M 1 u 1 + u 1 R 1 u 1 dt subject to x 1 = F 1 x 1 + G 1 u 1 min u J = 1 t f x T Q x + u T R u dt subject to x = F x + G u = min u 1 J 1 4
3 Artful Manipulation Rewrite integrand of J 1 to eliminate cross weighting of state and control x T 1 Q 1 x 1 + x T 1 M 1 u 1 + u 1 R 1 u 1 = x T 1 Q 1 M 1 R 1 T ( 1 M 1 )x 1 + u 1 + R 1 1 M T T 1 x 1 R1 u 1 + R 1 1 M T 1 x 1 x T 1 Q x 1 + u T R 1 u The transformation produces the following equivalences x = x 1 Q = Q 1 M 1 R 1 T 1 M 1 u = u 1 + R 1 1 M T 1 x 1 R = R 1 5 (Q,R) and (Q,M,R) LQ Problems are Equivalent x = x 1 x = x 1 u = u 1 + R 1 1 M 1 T x 1 Q = Q 1 M 1 R 1 1 M 1 T R = R 1 x = F x + G u x = F x 1 + G u 1 + R 1 1 M T 1 x 1 = F + R 1 T ( 1 M 1 )x 1 + G u 1 = x 1 = F 1 x 1 + G 1 u 1 G = G 1 F = F 1 G R 1 1 M 1 T = F 1 G 1 R 1 1 M 1 T Therefore, the forms are equivalent Whatever we prove for a (Q,R) cost function pertains to a (Q,M,R) cost function 6
4 Recall: LQ Optimal Control of an Unstable First-Order System x = x + u; x( )= 1 p= t 1 pt + p t pt ( f )= 1 f = 1; g = 1 Control gain = p t u = pt x x = 1 pt x 7 Riccati Solution and Control Gain for Open-Loop Stable and Unstable 1 st -Order Systems P( t f )= last 1-% of the illustrated time interval As time interval increases, percentage decreases 8
5 P() Approaches Steady State as t f -> With M =, P( )= { Q F T P t PF t + PGR t 1 G T P t }dt t f from t f to Progression of initial Riccati matrix is monotonic with for t f > t f1 P ( ) P 1 ( ) Rate of change approaches zero with dp( ) dt t f 9 Algebraic Riccati Equation and Constant Control Gain Matrix Steady-state Riccati solution Q F T P( ) P( )F + P( )GR 1 G T P( )= Q F T P SS P SS F + P SS GR 1 G T P SS = Steady-state control gain matrix C ss = R 1 G T P( t f )= R 1 G T P ss 1
6 Controllability of a LTI System Controllability: All elements of the state can be brought from arbitrary initial conditions to zero in x = Fx + Gu x() = x x(t finite ) = System is Completely Controllable if Controllability Matrix = G FG F n1 G n nm has Rank n 11 Controllability Examples F = G 1 n n ; G = n FG = n Rank = 3 n n F = G FG 1 n n = n ; G = n 4 n Rank = G F = 1 b ; G = b FG = b Rank = 1 G F = FG 1 b = b b b ; G = b Rank = 1
7 Requirements for Guaranteed Closed-Loop Stability 13 Optimal Cost with Feedback Control = 1 J *( t f )= 1 t f = 1 With u= t Cx t = R 1 G T Px t t f x * T Qx * + u* T Ru* dt x * T T Qx * + Cx t * R Cx t * dt t f With terminal cost = Substitute optimal control law in cost function x * T Qx * + x * T C t T RC t x t * t dt 14
8 J *( t f )= 1 Optimal Cost with LQ Feedback Control t f Consolidate terms x * T Q + C T RC t t From eq , OCE, optimal cost depends only on the initial condition J( t f )= 1 xt ()P( )x() x * t dt 15 Optimal Quadratic Cost Function is Bounded J *( t f )= 1 t f x * T Q + C T RC t t x * t dt J *( )= lim 1 t f 1 t f x * T Q + C T RC t t x * T Q + C T RC x * t J is bounded and positive provided that Q > R > Because J is bounded, C is a stabilizing gain matrix x * t dt dt = 1 xt ()Px() 16
9 Requirements for Guaranteeing Stability of the LQ Regulator x = Fx + Gu = [ F GC]x Closed-loop system is stable whether or not open-loop system is stable if... Q > R >... and (F,G) is a controllable pair Rank G FG F n1 G = n 17 Lyapunov Stability of the LQ Regulator = x T t x = [ F GC]x = F GR 1 G T P x Lyapunov function V x t = xt Px t t P F GR 1 G T P Rate of change of Lyapunov function Px t + t x T Px t t V = x T { + F GR 1 G T T P P } x t 18
10 Lyapunov Stability of the LQ Regulator Algebraic Riccati equation Q F T P PF + PGR 1 G T P = Substituting in rate equation { } x t V = x T P t F GR 1 G T P + F GR 1 G T T P P = x T Q t { + PGR 1 G T P}x t Therefore, closed-loop system is stable 19 Less Restrictive Stability Requirements Q may be if (F,D) is an observable pair, where Q D T D, where D may not be ( n n) Observability requirement Rank D T F T D T F T ( ) n1 D T = n
11 Observability Example x 1 x = 1 n n y = 1 x 1 x x 1 x = Hx t = Fx t H T F T H T = n 1 n Rank = 1 Even Less Restrictive Stability Requirements If F contains stable modes, closed-loop stability is guaranteed if (F,G) is a stabilizable pair (F,D) is a detectable pair
12 Stability Requirements with Cross Weighting If F contains stable modes, closed-loop stability is guaranteed if [(F GR -1 M T ),G] is a stabilizable pair [(F GR -1 M T ),D] is a detectable pair (Q GR -1 M T ) R > 3 Example: LQ Optimal Control of a First-Order LTI System Cost Function J = 1 ( )x (t f ) + lim 1 t f t f t o ( qx + ru )dt Open-Loop System x = f x + gu Algebraic Riccati Equation q fp + g p r = p fr qr p g g = u = gp r x = cx Choose positive solution of p = fr g ± Control Law fr g + qr g = fr 1± 1+ g g fr qr 4
13 Example: LQ Optimal Control of a First-Order LTI System Closed-Loop System x = f g p r x = ( f c)x Stability requires that ( f c)< If f <, then system is stable with no control ( c = ) 5 Example: LQ Optimal Control of a First-Order LTI System If f > (unstable), and r >, then fr >, and g p = fr g g fr qr If q, and g, then p fr 1+ 1 q g = fr g and closed-loop system is, as q, f g p r = g f r fr g = ( f f )= f Stable closed - loop system is "mirror image" of unstable open - loop system when q = 6
14 Solution of the Algebraic Riccati Equation 7 Solution Methods for the Continuous- Time Algebraic Riccati Equation Q F T P PF + PGR 1 G T P = 1) Integrate Riccati differential equation to steady state ) Explicit scalar equations for elements of P a)n > 3 b) May use symbolic math (MATLAB Symbolic Math Toolbox, Mathematica,...) 8
15 Example: Scalar Solution for the Algebraic Riccati Equation Q F T P PF + PGR 1 G T P = Second-order example q 11 q f 11 f 1 f 1 f T p 11 p 1 g 11 g 1 + p 1 p g 1 g p 11 p 1 p 1 p p 11 p 1 p 1 p r 11 r 1 g 11 g 1 g 1 g T f 11 f 1 f 1 f p 11 p 1 p 1 p = Solve three scalar equations for p 11, p 1, and p 9 More Solutions for the Algebraic Riccati Equation Q F T P PF + PGR 1 G T P = See OCE, Section 6.1 for Kalman-Englar method Kleinmans method MacFarlane-Potter method Laubs method [used in MATLAB] 3
16 Equilibrium Response to a Command Input 31 Steady-State Response to Commands x = Fx + Gu + Lw, x(t o ) given y = H x x + H u u + H w w State equilibrium with constant inputs... = Fx *+Gu*+Lw * ( ) x* = F 1 Gu*+Lw *... constrained by requirement to satisfy command input y* = H x x * +H u u * +H w w * 3
17 Steady-State Response to Commands y C = Fx * +Gu * +Lw * y* = H x x * +H u u * +H w w * Combine equations y C = F H x G H u x * u * + L H w w * (n + r) x (n + m) 33 Equilibrium Values of State and Control to Satisfy Commanded Input y C x * u* = F H x G H u 1 Lw * A 1 y C H w w * Lw * y C H w w * A must be square for inverse to exist Then, number of commands = number of controls 34
18 Inverse of the Matrix F H x G H u 1 A 1 = B = B 11 B 1 B 1 B x * u* = B 11 B 1 B 1 B B ij have same dimensions as equivalent blocks of A y C Lw * y C H w w * ( ) ( ) x* = B 11 Lw * +B 1 y C H w w * u* = B 1 Lw * +B y C H w w * 35 Elements of Matrix Inverse and Solutions for Open-Loop Equilibrium Substitution and elimination (see Supplement) B 11 B 1 B 1 B = F 1 ( GB 1 + I n ) F 1 GB B H x F 1 ( H x F 1 G + H u ) 1 Solve for B, then B 1 and B 1, then B 1 x* = B 1 y C ( B 11 L + B 1 H w )w * u* = B y C ( B 1 L + B H w )w * 36
19 LQ Regulator with Command Input (Proportional Control Law) u = u C Cx t u C? 37 Non-Zero Steady-State Regulation with LQ Regulator Command input provides equivalent state and control values for the LQ regulator Control law with command input u = u* C xx t * t B 1 y * = B y *C x t = ( B + CB 1 )y *Cx t 38
20 LQ Regulator with Forward Gain Matrix u = u * C xx t * t where = C F y * C B x t C F B + CB 1 C B C Disturbance affects the system, whether or not it is measured If measured, disturbance effect of can be countered by C D 39 Next Time: Cost Functions and Controller Structures 4
21 Supplemental Material 41 Square-Root Solution for the Algebraic Riccati Equation Q F T P PF + PGR 1 G T P = Square root of P: P DD T ; D P Integrate D to steady state ( )D T ( t f )= P t f t f D = t D T M LT, t D t f = D 1 F t T D t D T F t T D T u = R 1 where M t M LT + t M UT t G T T D SS D SS = C SS x t t D 1 QD t T x t where d 11 d D = 11 d 11 d 11 d 11 d 11 ( ) + t D T GR t 1 G T D T and ( m ij ) LT = t 1 m ij m ij, i < j i = j i > j 4 t
22 Matrix Inverse Identity OCE, eq..-57 to -67 B 11 B 1 B 1 B A 11 A 1 A 1 A I m+n = I n I m B 11 B 1 B 1 B A 11 A 1 A 1 A ( ) ( B 11 A 1 + B 1 A ) ( ) ( B 1 A 1 + B A ) = B A + B A B 1 A 11 + B A 1 ( B 11 A 11 + B 1 A 1 ) = I n ( B 11 A 1 + B 1 A ) = ( ) = B 1 A 11 + B A 1 ( B 1 A 1 + B A ) = I m Solve for B, then B 1 and B 1, then B 1 43
Time-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 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 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 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 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 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 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 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 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 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 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 informationUCLA Chemical Engineering. Process & Control Systems Engineering Laboratory
Constrained Innite-time Optimal Control Donald J. Chmielewski Chemical Engineering Department University of California Los Angeles February 23, 2000 Stochastic Formulation - Min Max Formulation - UCLA
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 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 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 information7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system
7 Stability 7.1 Linear Systems Stability Consider the Continuous-Time (CT) Linear Time-Invariant (LTI) system ẋ(t) = A x(t), x(0) = x 0, A R n n, x 0 R n. (14) The origin x = 0 is a globally asymptotically
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 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 informationHomework 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 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 informationEL2520 Control Theory and Practice
EL2520 Control Theory and Practice Lecture 8: Linear quadratic control Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden Linear quadratic control Allows to compute the controller
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 informationTheory in Model Predictive Control :" Constraint Satisfaction and Stability!
Theory in Model Predictive Control :" Constraint Satisfaction and Stability Colin Jones, Melanie Zeilinger Automatic Control Laboratory, EPFL Example: Cessna Citation Aircraft Linearized continuous-time
More informationMATH 235: Inner Product Spaces, Assignment 7
MATH 235: Inner Product Spaces, Assignment 7 Hand in questions 3,4,5,6,9, by 9:3 am on Wednesday March 26, 28. Contents Orthogonal Basis for Inner Product Space 2 2 Inner-Product Function Space 2 3 Weighted
More informationContents. 1 State-Space Linear Systems 5. 2 Linearization Causality, Time Invariance, and Linearity 31
Contents Preamble xiii Linear Systems I Basic Concepts 1 I System Representation 3 1 State-Space Linear Systems 5 1.1 State-Space Linear Systems 5 1.2 Block Diagrams 7 1.3 Exercises 11 2 Linearization
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 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 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 informationand the nite horizon cost index with the nite terminal weighting matrix F > : N?1 X J(z r ; u; w) = [z(n)? z r (N)] T F [z(n)? z r (N)] + t= [kz? z r
Intervalwise Receding Horizon H 1 -Tracking Control for Discrete Linear Periodic Systems Ki Baek Kim, Jae-Won Lee, Young Il. Lee, and Wook Hyun Kwon School of Electrical Engineering Seoul National University,
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 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 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 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 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 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 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 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 informationMS-E2133 Systems Analysis Laboratory II Assignment 2 Control of thermal power plant
MS-E2133 Systems Analysis Laboratory II Assignment 2 Control of thermal power plant How to control the thermal power plant in order to ensure the stable operation of the plant? In the assignment Production
More informationControl Systems! Copyright 2017 by Robert Stengel. All rights reserved. For educational use only.
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
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 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 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 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 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 informationCDS 101/110: Lecture 3.1 Linear Systems
CDS /: Lecture 3. Linear Systems Goals for Today: Describe and motivate linear system models: Summarize properties, examples, and tools Joel Burdick (substituting for Richard Murray) jwb@robotics.caltech.edu,
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 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 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 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 informationOptimization of Linear Systems of Constrained Configuration
Optimization of Linear Systems of Constrained Configuration Antony Jameson 1 October 1968 1 Abstract For the sake of simplicity it is often desirable to restrict the number of feedbacks in a controller.
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 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 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 informationLyapunov Stability Analysis: Open Loop
Copyright F.L. Lewis 008 All rights reserved Updated: hursday, August 8, 008 Lyapunov Stability Analysis: Open Loop We know that the stability of linear time-invariant (LI) dynamical systems can be determined
More informationVideo 6.1 Vijay Kumar and Ani Hsieh
Video 6.1 Vijay Kumar and Ani Hsieh Robo3x-1.6 1 In General Disturbance Input + - Input Controller + + System Output Robo3x-1.6 2 Learning Objectives for this Week State Space Notation Modeling in the
More informationCDS Solutions to Final Exam
CDS 22 - Solutions to Final Exam Instructor: Danielle C Tarraf Fall 27 Problem (a) We will compute the H 2 norm of G using state-space methods (see Section 26 in DFT) We begin by finding a minimal state-space
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 informationMCE693/793: Analysis and Control of Nonlinear Systems
MCE693/793: Analysis and Control of Nonlinear Systems Introduction to Nonlinear Controllability and Observability Hanz Richter Mechanical Engineering Department Cleveland State University Definition of
More informationINVERSE MODEL APPROACH TO DISTURBANCE REJECTION AND DECOUPLING CONTROLLER DESIGN. Leonid Lyubchyk
CINVESTAV Department of Automatic Control November 3, 20 INVERSE MODEL APPROACH TO DISTURBANCE REJECTION AND DECOUPLING CONTROLLER DESIGN Leonid Lyubchyk National Technical University of Ukraine Kharkov
More informationLinear-Quadratic Control System Design
Linear-Quadratic Control Sytem Deign Robert Stengel Optimal Control and Etimation MAE 546 Princeton Univerity, 218! Control ytem configuration! Proportional-integral! Proportional-integral-filtering! Model
More informationEML5311 Lyapunov Stability & Robust Control Design
EML5311 Lyapunov Stability & Robust Control Design 1 Lyapunov Stability criterion In Robust control design of nonlinear uncertain systems, stability theory plays an important role in engineering systems.
More information1 Controllability and Observability
1 Controllability and Observability 1.1 Linear Time-Invariant (LTI) Systems State-space: Dimensions: Notation Transfer function: ẋ = Ax+Bu, x() = x, y = Cx+Du. x R n, u R m, y R p. Note that H(s) is always
More informationLecture 7 : Generalized Plant and LFT form Dr.-Ing. Sudchai Boonto Assistant Professor
Dr.-Ing. Sudchai Boonto Assistant Professor Department of Control System and Instrumentation Engineering King Mongkuts Unniversity of Technology Thonburi Thailand Linear Quadratic Gaussian The state space
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 Zero-Sum Two-Person Differential Games Pierre Bernhard June 15, 2013
Linear Quadratic Zero-Sum Two-Person Differential Games Pierre Bernhard June 15, 2013 Abstract As in optimal control theory, linear quadratic (LQ) differential games (DG) can be solved, even in high dimension,
More informationESC794: Special Topics: Model Predictive Control
ESC794: Special Topics: Model Predictive Control Discrete-Time Systems Hanz Richter, Professor Mechanical Engineering Department Cleveland State University Discrete-Time vs. Sampled-Data Systems A continuous-time
More informationDecentralized control with input saturation
Decentralized control with input saturation Ciprian Deliu Faculty of Mathematics and Computer Science Technical University Eindhoven Eindhoven, The Netherlands November 2006 Decentralized control with
More informationComputational Issues in Nonlinear Dynamics and Control
Computational Issues in Nonlinear Dynamics and Control Arthur J. Krener ajkrener@ucdavis.edu Supported by AFOSR and NSF Typical Problems Numerical Computation of Invariant Manifolds Typical Problems Numerical
More information4F3 - Predictive Control
4F3 Predictive Control - Discrete-time systems p. 1/30 4F3 - Predictive Control Discrete-time State Space Control Theory For reference only Jan Maciejowski jmm@eng.cam.ac.uk 4F3 Predictive Control - Discrete-time
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 informationLyapunov Stability Theory
Lyapunov Stability Theory Peter Al Hokayem and Eduardo Gallestey March 16, 2015 1 Introduction In this lecture we consider the stability of equilibrium points of autonomous nonlinear systems, both in continuous
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 informationMCE/EEC 647/747: Robot Dynamics and Control. Lecture 8: Basic Lyapunov Stability Theory
MCE/EEC 647/747: Robot Dynamics and Control Lecture 8: Basic Lyapunov Stability Theory Reading: SHV Appendix Mechanical Engineering Hanz Richter, PhD MCE503 p.1/17 Stability in the sense of Lyapunov A
More informationAppendix A Solving Linear Matrix Inequality (LMI) Problems
Appendix A Solving Linear Matrix Inequality (LMI) Problems In this section, we present a brief introduction about linear matrix inequalities which have been used extensively to solve the FDI problems described
More informationonly nite eigenvalues. This is an extension of earlier results from [2]. Then we concentrate on the Riccati equation appearing in H 2 and linear quadr
The discrete algebraic Riccati equation and linear matrix inequality nton. Stoorvogel y Department of Mathematics and Computing Science Eindhoven Univ. of Technology P.O. ox 53, 56 M Eindhoven The Netherlands
More informationSingular Value Analysis of Linear- Quadratic Systems!
Singular Value Analyi of Linear- Quadratic Sytem! Robert Stengel! Optimal Control and Etimation MAE 546! Princeton Univerity, 2017!! Multivariable Nyquit Stability Criterion!! Matrix Norm and Singular
More informationExam. 135 minutes, 15 minutes reading time
Exam August 6, 208 Control Systems II (5-0590-00) Dr. Jacopo Tani Exam Exam Duration: 35 minutes, 5 minutes reading time Number of Problems: 35 Number of Points: 47 Permitted aids: 0 pages (5 sheets) A4.
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 informationOptimization-Based Control
Optimization-Based Control Richard M. Murray Control and Dynamical Systems California Institute of Technology DRAFT v1.7a, 19 February 2008 c California Institute of Technology All rights reserved. This
More informationLinear Systems. Manfred Morari Melanie Zeilinger. Institut für Automatik, ETH Zürich Institute for Dynamic Systems and Control, ETH Zürich
Linear Systems Manfred Morari Melanie Zeilinger Institut für Automatik, ETH Zürich Institute for Dynamic Systems and Control, ETH Zürich Spring Semester 2016 Linear Systems M. Morari, M. Zeilinger - Spring
More informationẋ n = f n (x 1,...,x n,u 1,...,u m ) (5) y 1 = g 1 (x 1,...,x n,u 1,...,u m ) (6) y p = g p (x 1,...,x n,u 1,...,u m ) (7)
EEE582 Topical Outline A.A. Rodriguez Fall 2007 GWC 352, 965-3712 The following represents a detailed topical outline of the course. It attempts to highlight most of the key concepts to be covered and
More informationLimit Theorems. MATH 464/506, Real Analysis. J. Robert Buchanan. Summer Department of Mathematics. J. Robert Buchanan Limit Theorems
Limit s MATH 464/506, Real Analysis J. Robert Buchanan Department of Mathematics Summer 2007 Bounded Functions Definition Let A R, let f : A R, and let c R be a cluster point of A. We say that f is bounded
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 informationOn a Nonlinear Riccati Matrix Eigenproblem
Applied Mathematical Sciences, Vol. 11, 2017, no. 33, 1619-1650 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.74152 On a Nonlinear Riccati Matrix Eigenproblem L. Kohaupt Beuth University
More informationRoots and Coefficients Polynomials Preliminary Maths Extension 1
Preliminary Maths Extension Question If, and are the roots of x 5x x 0, find the following. (d) (e) Question If p, q and r are the roots of x x x 4 0, evaluate the following. pq r pq qr rp p q q r r p
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 informationRobotics: Science & Systems [Topic 6: Control] Prof. Sethu Vijayakumar Course webpage:
Robotics: Science & Systems [Topic 6: Control] Prof. Sethu Vijayakumar Course webpage: http://wcms.inf.ed.ac.uk/ipab/rss Control Theory Concerns controlled systems of the form: and a controller of the
More informationOptimal Polynomial Control for Discrete-Time Systems
1 Optimal Polynomial Control for Discrete-Time Systems Prof Guy Beale Electrical and Computer Engineering Department George Mason University Fairfax, Virginia Correspondence concerning this paper should
More informationA Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra
A Control Methodology for Constrained Linear Systems Based on Positive Invariance of Polyhedra Jean-Claude HENNET LAAS-CNRS Toulouse, France Co-workers: Marina VASSILAKI University of Patras, GREECE Jean-Paul
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 informationControl of Mobile Robots
Control of Mobile Robots Regulation and trajectory tracking Prof. Luca Bascetta (luca.bascetta@polimi.it) Politecnico di Milano Dipartimento di Elettronica, Informazione e Bioingegneria Organization and
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 informationOn Stochastic Adaptive Control & its Applications. Bozenna Pasik-Duncan University of Kansas, USA
On Stochastic Adaptive Control & its Applications Bozenna Pasik-Duncan University of Kansas, USA ASEAS Workshop, AFOSR, 23-24 March, 2009 1. Motivation: Work in the 1970's 2. Adaptive Control of Continuous
More informationFunctions of Several Variables
Functions of Several Variables The Unconstrained Minimization Problem where In n dimensions the unconstrained problem is stated as f() x variables. minimize f()x x, is a scalar objective function of vector
More informationDigital Control Engineering Analysis and Design
Digital Control Engineering Analysis and Design M. Sami Fadali Antonio Visioli AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO Academic Press is
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 informationLecture 1 From Continuous-Time to Discrete-Time
Lecture From Continuous-Time to Discrete-Time Outline. Continuous and Discrete-Time Signals and Systems................. What is a signal?................................2 What is a system?.............................
More informationECE7850 Lecture 7. Discrete Time Optimal Control and Dynamic Programming
ECE7850 Lecture 7 Discrete Time Optimal Control and Dynamic Programming Discrete Time Optimal control Problems Short Introduction to Dynamic Programming Connection to Stabilization Problems 1 DT nonlinear
More information