UCLA Chemical Engineering. Process & Control Systems Engineering Laboratory
|
|
- Ethelbert Cole
- 5 years ago
- Views:
Transcription
1 Constrained Innite-time Optimal Control Donald J. Chmielewski Chemical Engineering Department University of California Los Angeles February 23, 2000
2 Stochastic Formulation - Min Max Formulation - UCLA Chemical Engineering Outline Background and Motivation - Objectives of Constrained Control - Finite-time Predictive Control Constrained Innite-time Optimal Control - Advantages of Innite-time Formulation - Finite-time Solution Method Constrained Disturbance Attenuation
3 From FCC Feed Rate (MV1+CV1) Vent Position (MV3) O 2 out (CV) CO out (CV8) Naphtha Hydrotreating Process Reactor Temperature (CV4) From Column Recycle Ratio (MV2 + CV2) Heat Exchanger Furnace Hydrotreating Reactors Fuel Feed Feed Temp Loop (MV2) MV4 Hydrogen Feed H2 Sep Clean Product
4 Modern Control System Architecture Real Time Optimizer (Steady State Calculations) Predictive Control Predictive Control Predictive Control PID Servo Loops Servo Loops Servo Loops PI PI PID PI PID PI PID PI Process 1 Process 2 Process 3
5 Objectives of Real-time Optimization Min Feed Rate Operator Optimum Max * Reactor Temperature Target Steady State Expected Domain of Operation Min Reactor Temperature * * Max CO Emissions Economic Optimum Max Feed Rate
6 Constrained Set Point Tracking Min Feed Rate Max Reactor Temperature Min Reactor Temperature Old Target * * OldOptimum Max CO Emissions Max Feed Rate NewTarget * * NewOptimum
7 Design a Feedback Controller so that the process follows - given Set Point Trajectory the UCLA Chemical Engineering Set Point Tracking y k (sp) y k + - C u k P
8 PI Control System Design time manipulated variable manipulated variable control variable control variable time time time Time Constant () : Small Time Constant () : Large PI Gain (K): Small PI Gain (K): Large
9 control variable control variable Quadratic Optimal Control Design time time manipulated variable manipulated variable time time Output Concern (Q oc ) : Small Output Concern (Q oc ) : Large Move Suppression (R ms ) : Large Move Suppression (R ms ) : Small
10 umin y k x k k 8 >< >: NX Model: x k+1 = Ax k + Bu k 0 < k < N Process k = Cx k 0 k N y 9 >= > UCLA Chemical Engineering Linear Quadratic Optimal Control Problem Formulation k y ) T N;1 X Q oc (y k ; y tgt ) + tgt ; (y (u k ; u tgt ) T R ms (u k ; u tgt ) k=0 k=0 subject to Input Constraints: u u k u 0 k < N Output Constraints: y y k y 0 < k N
11 min k ~x k ~u 8 >< >: ~x T k ~ Q oc ~x k + ~u T k R ms ~u k! 9 > T = Q oc ~x N > ~ N ~x + UCLA Chemical Engineering Constrained LQ Optimal Control X N;1 k=0 subject to Model: ~x k+1 = A~x k + B ~u k 0 < k < N Process 0 = ;x tgt ~x Input Constraints: u ~u k u 0 k < N Output Constraints: y C ~x k y 0 < k N where: ~ Q oc = C T Q oc C, x tgt = Ax tgt + Bu tgt, y tgt = Cx tgt, and k = x k ; x tgt y = y ; y tgt u = u ; u tgt ~x k = u k ; u tgt y = y ; y tgt u = u ; u tgt ~u
12 Trajectory Predictions vs. Actual Present Past Future Predicted Output y k Actual Output Δt 2Δt 3Δt 4Δt 5Δt time u k Predicted Input
13 ^P - Process Model P - Actual Process P = P ; ^P UCLA Chemical Engineering Model Mismatch y k (sp) + - P C + + u k ΔP y k
14 Predictive Control Present Past Future New Predicted Output y k Δt 2Δt 3Δt 4Δt 5Δt time u k New Predicted Input
15 Set Point Change in Hydrotreating Process From FCC Feed Rate (MV1+CV1) Vent Position (MV3) O 2 out (CV) CO out (CV8) Reactor Temperature (CV4) From Column Recycle Ratio (MV2 + CV2) Heat Exchanger Furnace Hydrotreating Reactors Fuel Feed Feed Temp Loop (MV2) MV4 Hydrogen Feed H2 Sep Clean Product
16 0 0 0 b b b a c13 UCLA Chemical Engineering State Space Model of Hydrotreating Unit 2 3 a a a32 a a a a a a A = a a a a1110 a a b b c c c c33 c34 c35 c B = c C = c b81 b c11 c b c b b133 0
17 Set Point Change in Hydrotreating Process New Target Point: Reactor Exit Temperature: Reduce by 4 degrees Reactor Inlet Temperature: Reduce by 5 degrees Reactant Feed Rate: Unchanged Quadratic Program Statistics: Size: N = 3 Horizon of Optimization Variables: 1 N = 48 # of Equality Constraints: 13 N = 39 # of Inequality Constraints: 12 N = 3 #
18 u min x k k 8 >< 9 > Q T + u T k R = msu k > ocx k k x UCLA Chemical Engineering Constrained Innite-time LQ Optimal Control 1X >: k=0 subject to Process Model: x k+1 = Ax k + Bu k k > 0 Input Constraints: u u k u k 0 Output Constraints: y Cx k y k > 0
19 LQ Optimal Control The Unconstrained LQ Problem, both Finite and Innite-time versions 190) (Kalman, Introduced Constrained Predictive Control to the ChE Community Group - Cutler, Prett, et. al., 1980) (Shell Guaranteed Closed-loop Stability with a Constrained Predictive Control (Rawlings and Muske, 1993) Algorithm The Constrained Innite-time LQ Problem (Sznaier and Damborg, 198 and Manousiouthakis, 199 and Scokaert and Rawlings, 1998 ) Chmielewski
20 If there exists any constrained stabilizing input trajectories, UCLA Chemical Engineering Main Result 1 - Constrained Stability then the solution to Innite-time Problem will be stabilizing If there does not exist any constrained stabilizing inputs, then the Innite-time Problem will be infeasible
21 T 0 Px 0 = min u x x k k 8 >< 9 > Q x T + u T = k R ms u k > k oc k x x k+1 = Ax k + Bu k UCLA Chemical Engineering Unconstrained Innite-time LQ Optimal Control Let P satisfy the Riccati Equation: = T 0 ; PB R ms + B T ;1 PB B T 1 P A A + Q A P Then the Solution to the Unconstrained Problem is 1X >: k=0 subject to
22 >: >: >< >: >< >: 1X k+1 = Ax k + Bu k k 0 x u k u k 0 u x k+1 = Ax k + Bu k > x k+1 = Ax k + Bu k x k+1 = Ax k + Bu k >= > >= > >= > UCLA Chemical Engineering Finite-time Solution to the Innite-time Problem 8 9 >< x T k Q ocx k + u T k R msu k >= 1X min s.t. k=0 y Cx k y k > 0 8 x T k Q ocx k + u T k R msu k >< X N;1 0 k < N = min s.t. u u k u 0 k < N k=0 y Cx k y 0 < k N k N x T k Q ocx k + u T k R msu k k N + min s.t. k=n u u k u k > N y Cx k y 9 8 x T k Q ocx k + u T k R msu k 0 k < N X N;1 0 k < N? =? min u u k u + x T N Px N s.t. k=0 y Cx k y 0 < k N
23 x Finite-time Solution to Innite-time Problem k Region of Constrained Stabilizability Region of Constraint Inactivity N * time N-1 * Σ (x k=0 2 +u k 2 k ) + 8 Σ (x k=n * 2 k 2 +u ) k P x 2 N*
24 umin x k k 8 >< T oc x k + u T k R ms u T k + x NPx N Q k x x k+1 = Ax k + Bu k 9 >= UCLA Chemical Engineering Main Result 2 - Existence If there exists a constrained stabilizing input trajectory, then there always exists N (x0) < 1 such that X N;1 >: > k=0 subject to 0 k < N u u k u 0 k < N y Cx k y 0 < k N gives the solution to the Innite-time Problem for all N N (x0)
25 Z 1 1T A f(x) ; 1 4 1T A UCLA Chemical Engineering Constrained Innite-time Nonlinear Optimal Control x T Q oc (x)x + u T R ms (x)u dt min u(t) x(t) 0 subject to Process Model: _x = f(x) +g(x)u x(0) = Input Constraints: u u(t) u t 0 Output Constraints: y Cx(t) y t > 0 Unconstrained Solution: Find J(x) to satisfy the Hamilton-Jacobi-Bellman @ 1 B(x)R ;1 (x)b T (x) 0 = x T @x
26 ( Z tf Input Constraints: u u(t) u 0 t t f Output Constraints: y Cx(t) y 0 t t f UCLA Chemical Engineering Constrained Innite-time Nonlinear Optimal Control ) x T Q oc (x)x + u T R ms (x)u dt + J (x (t f )) min u(t) x(t) 0 subject to Process Model: _x = f(x) + g(x)u x(0) =
27 Disturbance Rejection Process Disturbances Actual Output + - C P + + Measurement Noise Measured Output
28 Disturbance Types: Reactor Example CA in Possible Measurement of C Ain C Ain C Ain A k 1 B Possible Disturbance Models
29 >< >: w max v k k >: 8 >< >: NX T Qx k + T (y k ) R (y k ) k x w T k k w w T k k w v T k k v v T k k v = v 9 >= > >= v > = >= > UCLA Chemical Engineering Standard Disturbance Attenuation Formulations Min Max LQ Optimal Control - Assumes Worst Case Scenario 1 xt P Qx k + T (y k ) R (y k ) k=0 k P T 1k=0 w k + v T k v s.t. k k w k+1 = Ax k + B (y k )+w k x k = Cx k + v k y min ) ( Stochastic LQ Optimal Control - Assumes Most Likely Scenario 9 8 k+1 = Ax k + B (y k )+w k x k = Cx k + v k y 1 min ) ( s.t. lim N!1 k=0 N = w E E 9 8 >< x T k Qx k + T (y k ) R (y k ) E N!1 k+1 = Ax k + B (y k )+w x k = Cx k + v k y s.t. min = ) ( lim E = w E
30 Unconstrained Stochastic Optimal Control Separation Principle: A Cascaded Implementation of the Optimal Estimator followed by the Optimal Controller does not sacrice Overall Optimality. Certainty Equivalence: Feedback Implementation of the Disturbance Free Optimal Controller provides the Solution to the Stochastic Optimal Control Problem.
31 y k (sp) + Separation Principle Stochastic Optimal Controller w k P y k - u k + + v k y k (sp) + - Optimal State Estimator x k w k FSI Stoc Optimal Controller u k P + + y v k k
32 x T Q 0 ocx + u T R ms u + E [J (Ax + Bu + w)] u T R ms u + E [ J (Ax + Bu + w) ] UCLA Chemical Engineering Certainty Equivalence If J satises the Hamilton-Jacobi-Bellman Equation ( E w w T = ) (x) = J 0 (x) ; inf x J fj 0 (x)g J 0 (x) = u u u min Then the full information stochastic optimal feedback policy is (x) = arg min u u u In the Unconstrained Case, Certainty Equivalence holds ( i.e., (x) = 0 (x) ). In the Constrained Case, (x) 0 (x) only if the disturbances are small.
33 Stochastic Optimal Control Constrained Results Main UCLA Chemical Engineering Separation Principle Continues to hold in the Constrained Case Certainty Equivalence Does Not hold in the Constrained Case If the Open-Loop Process (i.e., the A matrix) is Stable, the Constrained Stochastic Optimal Policy Exists then If the Open-Loop Process (i.e., the A matrix) is Unstable, the Constrained Stochastic Optimal Policy Does Not Exist then Output Constraints can be Incorporated through a Soft Constraints
34 Disturbance Rejection: CSTR Example C A in Disturbance Model C A Measurement of (Relative Error: 0.2%) A k 1 B Operating Conditions: Flow Rate ( o ): 0.4 (m 3 /min) 0:325 o 0:45 Volumetric Rate Constant (k 1 ): 0.0 (m 3 /kmole min) Reaction Residence Time (): 5 (min) Concentration of A In (C Ain ): 10 (kmole/m 3 )
35 8 >< >: 8 >< >: 1X x 2 X max if if x =2 X max +1 9 >= > UCLA Chemical Engineering Constrained Worst Case Problem Formulation H(x k ) s.t. x k+1 = Ax k + B(x k ) +!(x k ) min ( ) u u max!( ) w w k=0 x T Qx + T (x)(x) ; 2! T (x)!(x) H(x) = where X max is the constrained stabilizable set for the disturbance free problem The Nonlinear Operators ( ) and!( ) are the Optimization Variables > 0 is a xed constant
36 4 (x) >< >: >< >: Ax + Bu + w 2 X max Ax + Bu + w 2 X max >= > >= > UCLA Chemical Engineering Constrained Min Max Optimal Control If J satises the Isaacs' Equation 8 9 x T Qx + u T u ; 2 w T w + J(Ax + Bu + w) min u u u max w w w J(x) = s.t. Then the Min Max Optimal Feedback policy is u T u ; 2 w T w + J(Ax + Bu + w) 5 = arg min u u u max w w w!(x) s.t.
37 = arg max 5 k w >< >: u min x k k >< >: N;1 P k=0 T Qx k + u T k u k ; 2 w T k w k + x T N Px N k x u k u u w k w w >= > >= > UCLA Chemical Engineering A Finite-time Max Min Problem (x 0) 3 s.t. k+1 = Ax k + Bu k + w k x Cx k y y!(x 0 ) where P = Q + A T P ;1 A and = I + BB T ; ;2 GG T P
38 4 Value Functions Min Max Example 3 2 Solution to Constrained Issacs' Equation Solution to Constrained Max Min Problem the One State Consider System: 1 0 x k+1 = 0:9x k + u k + w k x Control Policies: u(x) With Constraints: -0.2 ju k j 0:3 and jw k j 0: x 0.15 Disturbance Policies: w(x) 0.10 And Design Parameters: Q = 1 and 2 = p x
39 Conclusions The CITLQOC Controller has many Advantages over the Finite-time Controller (i.e., Tuning and Stability) Predictive The CITLQOC can be solved using a Finite-Dimensional Optimization Problem The Constrained Stochastic Optimal Controller has Performance Advantages, a Substantial Increase in Computational Cost with The Constrained Min Max Controller also has Performance Advantages, a only Moderate Increase in Computational Cost with
40 Acknowledgments Special Thanks to: Professor Vasilios Manousiouthakis and The Optimal Group Financial Support is Gratefully Acknowledged from: The Department of Education # P200A and The National Science Foundation #'s GER , CTS
UCLA 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 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 informationFAULT-TOLERANT CONTROL OF CHEMICAL PROCESS SYSTEMS USING COMMUNICATION NETWORKS. Nael H. El-Farra, Adiwinata Gani & Panagiotis D.
FAULT-TOLERANT CONTROL OF CHEMICAL PROCESS SYSTEMS USING COMMUNICATION NETWORKS Nael H. El-Farra, Adiwinata Gani & Panagiotis D. Christofides Department of Chemical Engineering University of California,
More informationA Tour of Reinforcement Learning The View from Continuous Control. Benjamin Recht University of California, Berkeley
A Tour of Reinforcement Learning The View from Continuous Control Benjamin Recht University of California, Berkeley trustable, scalable, predictable Control Theory! Reinforcement Learning is the study
More informationPart II: Model Predictive Control
Part II: Model Predictive Control Manfred Morari Alberto Bemporad Francesco Borrelli Unconstrained Optimal Control Unconstrained infinite time optimal control V (x 0 )= inf U k=0 [ x k Qx k + u k Ru k]
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 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 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 informationOptimizing Economic Performance using Model Predictive Control
Optimizing Economic Performance using Model Predictive Control James B. Rawlings Department of Chemical and Biological Engineering Second Workshop on Computational Issues in Nonlinear Control Monterey,
More informationA Globally Stabilizing Receding Horizon Controller for Neutrally Stable Linear Systems with Input Constraints 1
A Globally Stabilizing Receding Horizon Controller for Neutrally Stable Linear Systems with Input Constraints 1 Ali Jadbabaie, Claudio De Persis, and Tae-Woong Yoon 2 Department of Electrical Engineering
More informationNonlinear Model Predictive Control Tools (NMPC Tools)
Nonlinear Model Predictive Control Tools (NMPC Tools) Rishi Amrit, James B. Rawlings April 5, 2008 1 Formulation We consider a control system composed of three parts([2]). Estimator Target calculator Regulator
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 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 informationPrashant Mhaskar, Nael H. El-Farra & Panagiotis D. Christofides. Department of Chemical Engineering University of California, Los Angeles
HYBRID PREDICTIVE OUTPUT FEEDBACK STABILIZATION OF CONSTRAINED LINEAR SYSTEMS Prashant Mhaskar, Nael H. El-Farra & Panagiotis D. Christofides Department of Chemical Engineering University of California,
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 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 informationOPTIMAL CONTROL SYSTEMS
SYSTEMS MIN-MAX Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University OUTLINE MIN-MAX CONTROL Problem Definition HJB Equation Example GAME THEORY Differential Games Isaacs
More informationModel predictive control of industrial processes. Vitali Vansovitš
Model predictive control of industrial processes Vitali Vansovitš Contents Industrial process (Iru Power Plant) Neural networ identification Process identification linear model Model predictive controller
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 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 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 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 informationEnlarged terminal sets guaranteeing stability of receding horizon control
Enlarged terminal sets guaranteeing stability of receding horizon control J.A. De Doná a, M.M. Seron a D.Q. Mayne b G.C. Goodwin a a School of Electrical Engineering and Computer Science, The University
More informationReal-Time Optimization (RTO)
Real-Time Optimization (RTO) In previous chapters we have emphasized control system performance for disturbance and set-point changes. Now we will be concerned with how the set points are specified. In
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 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 informationMATH4406 (Control Theory) Unit 1: Introduction Prepared by Yoni Nazarathy, July 21, 2012
MATH4406 (Control Theory) Unit 1: Introduction Prepared by Yoni Nazarathy, July 21, 2012 Unit Outline Introduction to the course: Course goals, assessment, etc... What is Control Theory A bit of jargon,
More informationAn Investigation into the Nonlinearity of Polyethylene Reactor Operation. Department of Chemical Engineering Queen s University
An Investigation into the Nonlinearity of Polyethylene Reactor Operation M.-A. Benda,, K. McAuley and P.J. McLellan Department of Chemical Engineering Queen s University Outline Motivation Background:
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 information4F3 - Predictive Control
4F3 Predictive Control - Lecture 3 p 1/21 4F3 - Predictive Control Lecture 3 - Predictive Control with Constraints Jan Maciejowski jmm@engcamacuk 4F3 Predictive Control - Lecture 3 p 2/21 Constraints on
More informationLecture 10 Linear Quadratic Stochastic Control with Partial State Observation
EE363 Winter 2008-09 Lecture 10 Linear Quadratic Stochastic Control with Partial State Observation partially observed linear-quadratic stochastic control problem estimation-control separation principle
More informationDesign Methods for Control Systems
Design Methods for Control Systems Maarten Steinbuch TU/e Gjerrit Meinsma UT Dutch Institute of Systems and Control Winter term 2002-2003 Schedule November 25 MSt December 2 MSt Homework # 1 December 9
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 informationConstrained Linear Quadratic Optimal Control
5 Constrained Linear Quadratic Optimal Control 51 Overview Up to this point we have considered rather general nonlinear receding horizon optimal control problems Whilst we have been able to establish some
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 informationProcess Modelling, Identification, and Control
Jan Mikles Miroslav Fikar 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. Process Modelling, Identification, and
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 informationRobust control and applications in economic theory
Robust control and applications in economic theory In honour of Professor Emeritus Grigoris Kalogeropoulos on the occasion of his retirement A. N. Yannacopoulos Department of Statistics AUEB 24 May 2013
More informationIntroduction to Model Predictive Control. Dipartimento di Elettronica e Informazione
Introduction to Model Predictive Control Riccardo Scattolini Riccardo Scattolini Dipartimento di Elettronica e Informazione Finite horizon optimal control 2 Consider the system At time k we want to compute
More informationModel Predictive Control For Interactive Thermal Process
Model Predictive Control For Interactive Thermal Process M.Saravana Balaji #1, D.Arun Nehru #2, E.Muthuramalingam #3 #1 Assistant professor, Department of Electronics and instrumentation Engineering, Kumaraguru
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 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-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 informationSubject: Introduction to Process Control. Week 01, Lectures 01 02, Spring Content
v CHEG 461 : Process Dynamics and Control Subject: Introduction to Process Control Week 01, Lectures 01 02, Spring 2014 Dr. Costas Kiparissides Content 1. Introduction to Process Dynamics and Control 2.
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 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 informationTemperature Control of CSTR Using Fuzzy Logic Control and IMC Control
Vo1ume 1, No. 04, December 2014 936 Temperature Control of CSTR Using Fuzzy Logic Control and Control Aravind R Varma and Dr.V.O. Rejini Abstract--- Fuzzy logic controllers are useful in chemical processes
More informationarxiv: v1 [cs.sy] 2 Oct 2018
Non-linear Model Predictive Control of Conically Shaped Liquid Storage Tanks arxiv:1810.01119v1 [cs.sy] 2 Oct 2018 5 10 Abstract Martin Klaučo, L uboš Čirka Slovak University of Technology in Bratislava,
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 informationJournal of Process Control
Journal of Process Control 3 (03) 404 44 Contents lists available at SciVerse ScienceDirect Journal of Process Control j ourna l ho me pag e: www.elsevier.com/locate/jprocont Algorithms for improved fixed-time
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 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 informationIntroduction to Optimization!
Introduction to Optimization! Robert Stengel! Robotics and Intelligent Systems, MAE 345, Princeton University, 2017 Optimization problems and criteria Cost functions Static optimality conditions Examples
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 informationRiccati Equations and Inequalities in Robust Control
Riccati Equations and Inequalities in Robust Control Lianhao Yin Gabriel Ingesson Martin Karlsson Optimal Control LP4 2014 June 10, 2014 Lianhao Yin Gabriel Ingesson Martin Karlsson (LTH) H control problem
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 informationNonlinear Optimal Tracking Using Finite-Horizon State Dependent Riccati Equation (SDRE)
Nonlinear Optimal Tracking Using Finite-Horizon State Dependent Riccati Equation (SDRE) AHMED KHAMIS Idaho State University Department of Electrical Engineering Pocatello, Idaho USA khamahme@isu.edu D.
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 informationOptimal Control. McGill COMP 765 Oct 3 rd, 2017
Optimal Control McGill COMP 765 Oct 3 rd, 2017 Classical Control Quiz Question 1: Can a PID controller be used to balance an inverted pendulum: A) That starts upright? B) That must be swung-up (perhaps
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 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 informationA new low-and-high gain feedback design using MPC for global stabilization of linear systems subject to input saturation
A new low-and-high gain feedbac design using MPC for global stabilization of linear systems subject to input saturation Xu Wang 1 Håvard Fjær Grip 1; Ali Saberi 1 Tor Arne Johansen Abstract In this paper,
More informationIndustrial Model Predictive Control
Industrial Model Predictive Control Emil Schultz Christensen Kongens Lyngby 2013 DTU Compute-M.Sc.-2013-49 Technical University of Denmark DTU Compute Matematiktovet, Building 303B, DK-2800 Kongens Lyngby,
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 informationControlled Diffusions and Hamilton-Jacobi Bellman Equations
Controlled Diffusions and Hamilton-Jacobi Bellman Equations Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 2014 Emo Todorov (UW) AMATH/CSE 579, Winter
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 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 informationA Proof of the EOQ Formula Using Quasi-Variational. Inequalities. March 19, Abstract
A Proof of the EOQ Formula Using Quasi-Variational Inequalities Dir Beyer y and Suresh P. Sethi z March, 8 Abstract In this paper, we use quasi-variational inequalities to provide a rigorous proof of the
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 informationNonlinear Control Systems
Nonlinear Control Systems António Pedro Aguiar pedro@isr.ist.utl.pt 5. Input-Output Stability DEEC PhD Course http://users.isr.ist.utl.pt/%7epedro/ncs2012/ 2012 1 Input-Output Stability y = Hu H denotes
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 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 informationVisual feedback Control based on image information for the Swirling-flow Melting Furnace
Visual feedback Control based on image information for the Swirling-flow Melting Furnace Paper Tomoyuki Maeda Makishi Nakayama Hirokazu Araya Hiroshi Miyamoto Member In this paper, a new visual feedback
More informationNEW DEVELOPMENTS IN PREDICTIVE CONTROL FOR NONLINEAR SYSTEMS
NEW DEVELOPMENTS IN PREDICTIVE CONTROL FOR NONLINEAR SYSTEMS M. J. Grimble, A. Ordys, A. Dutka, P. Majecki University of Strathclyde Glasgow Scotland, U.K Introduction Model Predictive Control (MPC) is
More informationTaylor expansions for the HJB equation associated with a bilinear control problem
Taylor expansions for the HJB equation associated with a bilinear control problem Tobias Breiten, Karl Kunisch and Laurent Pfeiffer University of Graz, Austria Rome, June 217 Motivation dragged Brownian
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 informationState-Feedback Optimal Controllers for Deterministic Nonlinear Systems
State-Feedback Optimal Controllers for Deterministic Nonlinear Systems Chang-Hee Won*, Abstract A full-state feedback optimal control problem is solved for a general deterministic nonlinear system. The
More informationCourse on Model Predictive Control Part II Linear MPC design
Course on Model Predictive Control Part II Linear MPC design Gabriele Pannocchia Department of Chemical Engineering, University of Pisa, Italy Email: g.pannocchia@diccism.unipi.it Facoltà di Ingegneria,
More informationMarch Algebra 2 Question 1. March Algebra 2 Question 1
March Algebra 2 Question 1 If the statement is always true for the domain, assign that part a 3. If it is sometimes true, assign it a 2. If it is never true, assign it a 1. Your answer for this question
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 informationIn search of the unreachable setpoint
In search of the unreachable setpoint Adventures with Prof. Sten Bay Jørgensen James B. Rawlings Department of Chemical and Biological Engineering June 19, 2009 Seminar Honoring Prof. Sten Bay Jørgensen
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 informationH-Infinity Controller Design for a Continuous Stirred Tank Reactor
International Journal of Electronic and Electrical Engineering. ISSN 974-2174 Volume 7, Number 8 (214), pp. 767-772 International Research Publication House http://www.irphouse.com H-Infinity Controller
More informationLecture 18: Optimization Programming
Fall, 2016 Outline Unconstrained Optimization 1 Unconstrained Optimization 2 Equality-constrained Optimization Inequality-constrained Optimization Mixture-constrained Optimization 3 Quadratic Programming
More informationSolutions for Tutorial 10 Stability Analysis
Solutions for Tutorial 1 Stability Analysis 1.1 In this question, you will analyze the series of three isothermal CSTR s show in Figure 1.1. The model for each reactor is the same at presented in Textbook
More informationLocally optimal controllers and application to orbital transfer (long version)
9th IFAC Symposium on Nonlinear Control Systems Toulouse, France, September 4-6, 13 FrA1.4 Locally optimal controllers and application to orbital transfer (long version) S. Benachour V. Andrieu Université
More informationSuboptimality of minmax MPC. Seungho Lee. ẋ(t) = f(x(t), u(t)), x(0) = x 0, t 0 (1)
Suboptimality of minmax MPC Seungho Lee In this paper, we consider particular case of Model Predictive Control (MPC) when the problem that needs to be solved in each sample time is the form of min max
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 informationx Time
Trans. of the Society of Instrument and Control Engineers Vol.E-3, No., 66/72 (24) Nonlinear Model Predictive Control Using Successive Linearization - Application to Chemical Reactors - Hiroya Seki Λ,
More informationA Stable Block Model Predictive Control with Variable Implementation Horizon
American Control Conference June 8-,. Portland, OR, USA WeB9. A Stable Block Model Predictive Control with Variable Implementation Horizon Jing Sun, Shuhao Chen, Ilya Kolmanovsky Abstract In this paper,
More informationReal Time Stochastic Control and Decision Making: From theory to algorithms and applications
Real Time Stochastic Control and Decision Making: From theory to algorithms and applications Evangelos A. Theodorou Autonomous Control and Decision Systems Lab Challenges in control Uncertainty Stochastic
More informationStochastic Tube MPC with State Estimation
Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems MTNS 2010 5 9 July, 2010 Budapest, Hungary Stochastic Tube MPC with State Estimation Mark Cannon, Qifeng Cheng,
More informationCHAPTER 1. Introduction
CHAPTER 1 Introduction Linear geometric control theory was initiated in the beginning of the 1970 s, see for example, [1, 7]. A good summary of the subject is the book by Wonham [17]. The term geometric
More informationUNIVERSITY OF CALIFORNIA. Los Angeles. Distributed Model Predictive Control of Nonlinear. and Two-Time-Scale Process Networks
UNIVERSITY OF CALIFORNIA Los Angeles Distributed Model Predictive Control of Nonlinear and Two-Time-Scale Process Networks A dissertation submitted in partial satisfaction of the requirements for the degree
More informationEvery real system has uncertainties, which include system parametric uncertainties, unmodeled dynamics
Sensitivity Analysis of Disturbance Accommodating Control with Kalman Filter Estimation Jemin George and John L. Crassidis University at Buffalo, State University of New York, Amherst, NY, 14-44 The design
More informationOrder Reduction of a Distributed Parameter PEM Fuel Cell Model
Order Reduction of a Distributed Parameter PEM Fuel Cell Model María Sarmiento Carnevali 1 Carles Batlle Arnau 2 Maria Serra Prat 2,3 Immaculada Massana Hugas 2 (1) Intelligent Energy Limited, (2) Universitat
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 informationPOSITIVE CONTROLS. P.O. Box 513. Fax: , Regulator problem."
LINEAR QUADRATIC REGULATOR PROBLEM WITH POSITIVE CONTROLS W.P.M.H. Heemels, S.J.L. van Eijndhoven y, A.A. Stoorvogel y Technical University of Eindhoven y Dept. of Electrical Engineering Dept. of Mathematics
More informationThe Important State Coordinates of a Nonlinear System
The Important State Coordinates of a Nonlinear System Arthur J. Krener 1 University of California, Davis, CA and Naval Postgraduate School, Monterey, CA ajkrener@ucdavis.edu Summary. We offer an alternative
More information