Where rank (B) =m and (A, B) is a controllable pair and the switching function is represented as

Size: px
Start display at page:

Download "Where rank (B) =m and (A, B) is a controllable pair and the switching function is represented as"

Transcription

1 Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16 doi:1.1311/ Optimal Sliding Surface Design for a MIMO Distillation System Senthil Kumar B 1 *, K.Suresh Manic 1 Research Scholar, Faculty of Electrical Engineering, nna University,Chennai-65,India. *Corresponding uthor ( vlrsenthil@yahoo.co.in) Department of Electrical and Electronics Engineering, Sriram Engineering College, Chennai-64, India. ( ksureshmanic@gmail.com) bstract his Paper presents the Sliding Mode Control of a Distillation Colum System which are influenced by external disturbances or perturbations that are act along the input. he issue of robustness property to be studied only for Matched Perturbation or for Matched uncertainty. he switching surface design for Sliding Mode control to be interpreted as a static feedback selection problem.he switching hyper plane design to be carried out along with pole placement technique and Quadratic Minimization technique for regulatory problems or disturbance rejection problem.he results are obtained for the distillation column where the Eigen values are to be placed arbitrarily and optimally and the robustness property to be measured using the condition number or by obtaining norm of the state variables about where the Eigen values are placed in closed Loop Key words: Sliding Mode Control, Pole Placement Problem, Optimal Control, Matched Uncertainty, Distillation Colum 1. INRODUCION Sliding Mode Control is a robust control for both linear and nonlinear systems. In this paper perturbations or external bounded disturbances are to be studied only a vanishing disturbance. he vanishing disturbance is one where the disturbance vanishes about the equilibrium point. (nusha Rani, Kumar, and Manic 16)(Kumar and Manic 14). By properly applying high gain feedback in the closed loop the disturbance vanishes. State feedback controller can be used to place the closed loop Eigen values. For Multi Input the design of gain is not unique for a given set of Eigen values and for the same Eigen values different gain matrix will produce different Eigen vectors and different performance. Optimal values to be found for better performance. Switching hyper plane is designed by minimising cost functions in which quadratic terms of the states are used (KUSKY, NICHOLS, and DOOREN 1985). his method ensures weightings placed such that the control surface follows the modes(moore 1975). he Eigen vectors contain information about the interaction between the states which is arbitrarily fixed by pole placement design techniques. Optimal Eigen structure assignment design offers to minimise the effects of unmatched perturbations on the sliding mode dynamics optimally by designing the surface design(edwards andspurgeon 1988).. CONROL DESIGN.1. Regular Form regular form to be used for reduced order sliding mode dynamics Consider the system considered of the form x =x+bu (1) Where rank (B) =m and (, B) is a controllable pair and the switching function is represented as he system represented in regular form as [1] S ( =Sx ( () z1( 11z1( 1z( (3a) z( 1z1( z( Bu( nd the linear sliding manifold as S ( =S 1 z 1 ( +S z ( (4) nd the change of coordinates by an orthogonal matrix r Z( = r *x ( (5) (3b) 1

2 Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16 1 I nm BB r = 1 (6) B and change of coordinate is Input matrix B in () may be partitioned (after reordering the state vector components) as B1 B (7) B Where B 1 ϵr n-m m, B ϵr m n with det B 11 1 r r = 1 (8) and nd the switching function after transformation r B= B S r = [S 1 S ] (1) he switching function S( to be identically equal to zero during sliding motion From Equation (4) S 1 z 1 ( +S Z ( =. Z ( =-S -1 S 1 z 1 ( (11) Z ( =-M Z 1 ( where M=S -1 S 1 which is the existence problem to give (n-m) negative poles to the closed loop system. he sliding system can become totally insensitive to matched uncertainty but it will be affected by unmatched uncertainty. he sliding mode is governed by the above equation (11). On substituting Z in the first equation it becomes a closed with feedback form Z1( ( 11 1M) Z1( the techniques for switching surface selection or M to found by using pole placement technique or by using quadratic minimisation technique. S r = [M I m ] which is same as Equation (1) he designed manifold can be linear or nonlinear, depends on our specified goal. he simplest manifold is a linear one..1.1 Existence Condition of Sliding Mode Objective here is to design a control input u. Such that, the sliding motion occurs infinite time. where K> (9) (1) v=-ksign(s) (13).. Pole Placement echnique Full state feedback (FSF), or pole placement, is a method employed in feedback control system theory to place the closed-loop poles of a plant in pre-determined locations in the s-plane. Placing poles is desirable because the location of the poles corresponds directly to the eigenvalues of the system, which control the characteristics of the response of the system. he system must be considered controllable in order to implement this method...1.principle If the closed-loop input-output transfer function can be represented by a state space equation hen the poles of the system are the roots of the characteristic equation given by x =x+bu (14 a) Y=Cx+Du (14 b)

3 Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, 16 ǀSI-ǀ= (15) Full state feedback is utilized by commanding the input vector u. Consider an input proportional (in the matrix sense) to the state vector, System with state feedback (closed-loop) u=-kx (16) Substituting into the state space equations above, X ( BK) x (16a) Y=(C-DK) x he roots of the FSF system are given by the characteristic equation, det [SI-(-BK)].Comparing the terms of this equation with those of the desired characteristic equation yields the values of the feedback matrix K which force the closed-loop eigenvalues to the pole locations specified by the desired characteristic equation..3. Linear Quadratic Regulator Method he problem of minimising the quadratic performance index is given by positive definite Identity Matrix and t s is the time at which the sliding motion starts. Q11 Q1 r Qr 1 = Q Q Where Q 1 =Q 1, then the system is represented in the coordinate transformation as 1 J Z Q Z Z Q Z Z Q Z dt ts Z 1 determines the system dynamics and the effective control input is determined by Z 1 [ Z1 ( Q11 Q1 Q Q1 ) z1 1 ( ) ( J 1 1 ) Z Q Q Z Q Z Q Q z ts 1 Z Q Q Q Z ] Q Q Q Q Q (16b) 1 J ( ) ( ) x t Qx t dt where Q is a ts (17) (18) (19) () v Z Q Q Z nd the equation written as (1) 1 J 1 1 Z QZ v Q vdt ts Q Q () (3) he problem of Minimization of the standard linear quadratic optimal regulator problem ensures the positive definiteness of Q identity matrix which also ensures Q >, so that Q -1 exists and also Q >. he controllability of (, B) ensures ( 11, 1 ) Controllable and in turn ensures (, ) controllable Z MZ1 Q 1( 1 P1 Q1 ) where P 1 is a unique positive definite solution obtained from lgebraic Matrix Ricatti Equation. 3. NUMERICL EXMPLE Considering the distillation column (Kautsky, Nichols, and Dooren, 1985) Where the closed loop poles has been placed for n x n matrix. In this example the matrix considered was a 5 x 5 matrix.by reducing it into a 3 x 3 matrix and also the disturbance considered where the input appears. So the system has been formulated as a reduced order system. he condition numbers given in the example taken for full order matrix which is higher rather than for a reduced order system. 1 3

4 Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, B Converting it into regular form where obtained from Equation (6) B = 1 1 he system is obtained as in equation (3) he value obtained from equation (3) and Kopt hus the optimal value obtained is used for closed loop poles assignment. 4. RESUL ND DISCUSSION he Eigen values of matrix are [-5.3,-.914,-.8576,-.69,-.785].By pole placement technique the poles are placed as [-1,-,-3,-4,-5] and the condition number which is obtained as [ , , , ][5].By suing optimal quadratic minimization problem M is calculated. 4

5 Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, M, which is a reduced order matrix and the gain k which is optimally placed for the Eigen values 11 =[,-6.7,6] and the gain computed isk opt =.In the Input Matrix B=B it has no significance on the external bounded disturbance applied to the system. It becomes invariant to the input and when the system is in sliding phase. he condition number gives what will the maximum perturbation that an Eigen value can withstand. Figure 1 Control Input Figure Reduced order States Figure 3 Sliding Surface 5

6 Rev. éc. Ing. Univ. Zulia. Vol. 39, Nº 8, 1-6, CONCLUSION For MIMO system considered the system has been transformed into a reduced order system and robustness properties has been obtained using sliding mode controller for matched disturbance acting along the input. he optimal gain calculated using linear quadratic method gives for the weighted values considered for anidentity matrix.he daptive Sliding Mode controller using Fuzzy and Heuristic lgorithms for nonlinear systems has been discussed in the preceding papers (Dinesh Kumar and Meenakshipriya 16)(Li et al. 16).he other papers deals with only parametric robustness and not on the unmodelled dynamics which has not been considered(vijayan and Panda, 1a)(Vijayan and Panda, 1b). REFERENCES nusha Rani, V., B. Senthil Kumar, and K. Suresh Manic(16) Sliding Mode Controlfor Robust Regulationof Chemical Processes,Indian Journal of Science and echnology, 9(1), 1-1. doi: /ijst/16/v9i1/ C.Edwards, and S.K.Spurgeon(1988) Sliding Mode Control: heory and pplications, Crc Pres. Dinesh Kumar, D, and B Meenakshipriya(16) Design of Heuristic lgorithm for Non-Linear System,Rev. éc. Ing. Univ. Zulia, 39(6),pp Kautsky, J, N K Nichols, and P V N Dooren(1985) Robust Pole ssignment in Linear State Feedback,International Journal of Control, 41 (5), pp Kumar, B Senthil, and K.Suresh Manic(14) Sliding Mode Control for a Minimum Phase Unstable Second Order System. pplied Mechanics and Materials, pp Li, Mengmeng, Dehui Qiu, Jinwen Zheng, Yuan Li, and Qinglin Wang(16) daptive Fuzzy Sliding Mode Control Based on a DEP Flexible ctuator,rev. éc. Ing. Univ. Zulia, 39(3),pp Moore, B. (1975) On the Flexibility Offered by State Feedback in Multivariable Systems beyond Closed Loop Eigenvalue ssignment,ieee Conference on Decision and Control Including the 14th Symposium on daptive Processes, pp Vijayan, V., and Rames C. Panda (1a) Design of a Simple Setpoint Filter for Minimizing Overshoot for Low Order Processes,IS ransactions, 51(), pp V Vijayan, RC Panda(1b) Design of PID Controllers in Double Feedback Loops for SISO Systems with Set-Point Filters,IS ransactions, 51 (4), pp

Design of Nonlinear Control Systems with the Highest Derivative in Feedback

Design of Nonlinear Control Systems with the Highest Derivative in Feedback SERIES ON STAB1UTY, VIBRATION AND CONTROL OF SYSTEMS SeriesA Volume 16 Founder & Editor: Ardeshir Guran Co-Editors: M. Cloud & W. B. Zimmerman Design of Nonlinear Control Systems with the Highest Derivative

More information

Research Article State-PID Feedback for Pole Placement of LTI Systems

Research Article State-PID Feedback for Pole Placement of LTI Systems Mathematical Problems in Engineering Volume 211, Article ID 92943, 2 pages doi:1.1155/211/92943 Research Article State-PID Feedback for Pole Placement of LTI Systems Sarawut Sujitjorn and Witchupong Wiboonjaroen

More information

Modeling and Control Overview

Modeling and Control Overview Modeling and Control Overview D R. T A R E K A. T U T U N J I A D V A N C E D C O N T R O L S Y S T E M S M E C H A T R O N I C S E N G I N E E R I N G D E P A R T M E N T P H I L A D E L P H I A U N I

More information

Linear Quadratic Regulator (LQR) Design II

Linear 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 information

Digital Control Engineering Analysis and Design

Digital 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 information

Chap 8. State Feedback and State Estimators

Chap 8. State Feedback and State Estimators Chap 8. State Feedback and State Estimators Outlines Introduction State feedback Regulation and tracking State estimator Feedback from estimated states State feedback-multivariable case State estimators-multivariable

More information

Improved Identification and Control of 2-by-2 MIMO System using Relay Feedback

Improved Identification and Control of 2-by-2 MIMO System using Relay Feedback CEAI, Vol.17, No.4 pp. 23-32, 2015 Printed in Romania Improved Identification and Control of 2-by-2 MIMO System using Relay Feedback D.Kalpana, T.Thyagarajan, R.Thenral Department of Instrumentation Engineering,

More information

Feedback Control of Linear SISO systems. Process Dynamics and Control

Feedback Control of Linear SISO systems. Process Dynamics and Control Feedback Control of Linear SISO systems Process Dynamics and Control 1 Open-Loop Process The study of dynamics was limited to open-loop systems Observe process behavior as a result of specific input signals

More information

5. Observer-based Controller Design

5. Observer-based Controller Design EE635 - Control System Theory 5. Observer-based Controller Design Jitkomut Songsiri state feedback pole-placement design regulation and tracking state observer feedback observer design LQR and LQG 5-1

More information

TRACKING AND DISTURBANCE REJECTION

TRACKING AND DISTURBANCE REJECTION TRACKING AND DISTURBANCE REJECTION Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 13 General objective: The output to track a reference

More information

Intermediate Process Control CHE576 Lecture Notes # 2

Intermediate Process Control CHE576 Lecture Notes # 2 Intermediate Process Control CHE576 Lecture Notes # 2 B. Huang Department of Chemical & Materials Engineering University of Alberta, Edmonton, Alberta, Canada February 4, 2008 2 Chapter 2 Introduction

More information

Closed-loop system 2/1/2016. Generally MIMO case. Two-degrees-of-freedom (2 DOF) control structure. (2 DOF structure) The closed loop equations become

Closed-loop system 2/1/2016. Generally MIMO case. Two-degrees-of-freedom (2 DOF) control structure. (2 DOF structure) The closed loop equations become Closed-loop system enerally MIMO case Two-degrees-of-freedom (2 DOF) control structure (2 DOF structure) 2 The closed loop equations become solving for z gives where is the closed loop transfer function

More information

Control Systems. State Estimation.

Control Systems. State Estimation. State Estimation chibum@seoultech.ac.kr Outline Dominant pole design Symmetric root locus State estimation We are able to place the CLPs arbitrarily by feeding back all the states: u = Kx. But these may

More information

State Regulator. Advanced Control. design of controllers using pole placement and LQ design rules

State 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 information

Chapter 3. LQ, LQG and Control System Design. Dutch Institute of Systems and Control

Chapter 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 information

I. D. Landau, A. Karimi: A Course on Adaptive Control Adaptive Control. Part 9: Adaptive Control with Multiple Models and Switching

I. D. Landau, A. Karimi: A Course on Adaptive Control Adaptive Control. Part 9: Adaptive Control with Multiple Models and Switching I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 1 Adaptive Control Part 9: Adaptive Control with Multiple Models and Switching I. D. Landau, A. Karimi: A Course on Adaptive Control - 5 2 Outline

More information

Full State Feedback for State Space Approach

Full State Feedback for State Space Approach Full State Feedback for State Space Approach State Space Equations Using Cramer s rule it can be shown that the characteristic equation of the system is : det[ si A] 0 Roots (for s) of the resulting polynomial

More information

CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER

CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER 114 CHAPTER 5 ROBUSTNESS ANALYSIS OF THE CONTROLLER 5.1 INTRODUCTION Robust control is a branch of control theory that explicitly deals with uncertainty in its approach to controller design. It also refers

More information

Suppose that we have a specific single stage dynamic system governed by the following equation:

Suppose 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 information

Course Outline. Higher Order Poles: Example. Higher Order Poles. Amme 3500 : System Dynamics & Control. State Space Design. 1 G(s) = s(s + 2)(s +10)

Course 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 information

H-Infinity Controller Design for a Continuous Stirred Tank Reactor

H-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 information

Robust exact pole placement via an LMI-based algorithm

Robust exact pole placement via an LMI-based algorithm Proceedings of the 44th EEE Conference on Decision and Control, and the European Control Conference 25 Seville, Spain, December 12-15, 25 ThC5.2 Robust exact pole placement via an LM-based algorithm M.

More information

POLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 19

POLE PLACEMENT. Sadegh Bolouki. Lecture slides for ECE 515. University of Illinois, Urbana-Champaign. Fall S. Bolouki (UIUC) 1 / 19 POLE PLACEMENT Sadegh Bolouki Lecture slides for ECE 515 University of Illinois, Urbana-Champaign Fall 2016 S. Bolouki (UIUC) 1 / 19 Outline 1 State Feedback 2 Observer 3 Observer Feedback 4 Reduced Order

More information

FEL3210 Multivariable Feedback Control

FEL3210 Multivariable Feedback Control FEL3210 Multivariable Feedback Control Lecture 5: Uncertainty and Robustness in SISO Systems [Ch.7-(8)] Elling W. Jacobsen, Automatic Control Lab, KTH Lecture 5:Uncertainty and Robustness () FEL3210 MIMO

More information

Linear State Feedback Controller Design

Linear State Feedback Controller Design Assignment For EE5101 - Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University

More information

The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems

The ϵ-capacity of a gain matrix and tolerable disturbances: Discrete-time perturbed linear systems IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X. Volume 11, Issue 3 Ver. IV (May - Jun. 2015), PP 52-62 www.iosrjournals.org The ϵ-capacity of a gain matrix and tolerable disturbances:

More information

Didier HENRION henrion

Didier HENRION   henrion POLYNOMIAL METHODS FOR ROBUST CONTROL PART I.1 ROBUST STABILITY ANALYSIS: SINGLE PARAMETER UNCERTAINTY Didier HENRION www.laas.fr/ henrion henrion@laas.fr Pont Neuf over river Garonne in Toulouse June

More information

APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN

APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN APPLICATION OF D-K ITERATION TECHNIQUE BASED ON H ROBUST CONTROL THEORY FOR POWER SYSTEM STABILIZER DESIGN Amitava Sil 1 and S Paul 2 1 Department of Electrical & Electronics Engineering, Neotia Institute

More information

Robust Control 5 Nominal Controller Design Continued

Robust 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 information

APPLICATION OF ADAPTIVE CONTROLLER TO WATER HYDRAULIC SERVO CYLINDER

APPLICATION OF ADAPTIVE CONTROLLER TO WATER HYDRAULIC SERVO CYLINDER APPLICAION OF ADAPIVE CONROLLER O WAER HYDRAULIC SERVO CYLINDER Hidekazu AKAHASHI*, Kazuhisa IO** and Shigeru IKEO** * Division of Science and echnology, Graduate school of SOPHIA University 7- Kioicho,

More information

CONTROL DESIGN FOR SET POINT TRACKING

CONTROL 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 information

OUTPUT REGULATION OF RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL

OUTPUT REGULATION OF RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL International Journal in Foundations of Computer Science & Technology (IJFCST),Vol., No., March 01 OUTPUT REGULATION OF RÖSSLER PROTOTYPE-4 CHAOTIC SYSTEM BY STATE FEEDBACK CONTROL Sundarapandian Vaidyanathan

More information

Computation of Stabilizing PI and PID parameters for multivariable system with time delays

Computation of Stabilizing PI and PID parameters for multivariable system with time delays Computation of Stabilizing PI and PID parameters for multivariable system with time delays Nour El Houda Mansour, Sami Hafsi, Kaouther Laabidi Laboratoire d Analyse, Conception et Commande des Systèmes

More information

H-infinity Model Reference Controller Design for Magnetic Levitation System

H-infinity Model Reference Controller Design for Magnetic Levitation System H.I. Ali Control and Systems Engineering Department, University of Technology Baghdad, Iraq 6043@uotechnology.edu.iq H-infinity Model Reference Controller Design for Magnetic Levitation System Abstract-

More information

CHAPTER 6 CLOSED LOOP STUDIES

CHAPTER 6 CLOSED LOOP STUDIES 180 CHAPTER 6 CLOSED LOOP STUDIES Improvement of closed-loop performance needs proper tuning of controller parameters that requires process model structure and the estimation of respective parameters which

More information

Control Systems Theory and Applications for Linear Repetitive Processes

Control Systems Theory and Applications for Linear Repetitive Processes Eric Rogers, Krzysztof Galkowski, David H. Owens Control Systems Theory and Applications for Linear Repetitive Processes Springer Contents 1 Examples and Representations 1 1.1 Examples and Control Problems

More information

Exam. 135 minutes, 15 minutes reading time

Exam. 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 information

Multi-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures

Multi-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures Preprints of the 19th World Congress The International Federation of Automatic Control Multi-Model Adaptive Regulation for a Family of Systems Containing Different Zero Structures Eric Peterson Harry G.

More information

Lecture 7 LQG Design. Linear Quadratic Gaussian regulator Control-estimation duality SRL for optimal estimator Example of LQG design for MIMO plant

Lecture 7 LQG Design. Linear Quadratic Gaussian regulator Control-estimation duality SRL for optimal estimator Example of LQG design for MIMO plant L7: Lecture 7 LQG Design Linear Quadratic Gaussian regulator Control-estimation duality SRL for optimal estimator Example of LQG design for IO plant LQG regulator L7:2 If the process and measurement noises

More information

Robust Optimal Sliding Mode Control of Twin Rotor MIMO System

Robust Optimal Sliding Mode Control of Twin Rotor MIMO System Robust Optimal Sliding Mode Control of Twin Rotor MIMO System Chithra R. Department of Electrical and Electronics Engineering, TKM college of Engineering, Kollam, India Abstract The twin rotor MIMO system

More information

Chapter 3. State Feedback - Pole Placement. Motivation

Chapter 3. State Feedback - Pole Placement. Motivation Chapter 3 State Feedback - Pole Placement Motivation Whereas classical control theory is based on output feedback, this course mainly deals with control system design by state feedback. This model-based

More information

CBE507 LECTURE III Controller Design Using State-space Methods. Professor Dae Ryook Yang

CBE507 LECTURE III Controller Design Using State-space Methods. Professor Dae Ryook Yang CBE507 LECTURE III Controller Design Using State-space Methods Professor Dae Ryook Yang Fall 2013 Dept. of Chemical and Biological Engineering Korea University Korea University III -1 Overview States What

More information

Design of Decentralised PI Controller using Model Reference Adaptive Control for Quadruple Tank Process

Design of Decentralised PI Controller using Model Reference Adaptive Control for Quadruple Tank Process Design of Decentralised PI Controller using Model Reference Adaptive Control for Quadruple Tank Process D.Angeline Vijula #, Dr.N.Devarajan * # Electronics and Instrumentation Engineering Sri Ramakrishna

More information

3.1 Overview 3.2 Process and control-loop interactions

3.1 Overview 3.2 Process and control-loop interactions 3. Multivariable 3.1 Overview 3.2 and control-loop interactions 3.2.1 Interaction analysis 3.2.2 Closed-loop stability 3.3 Decoupling control 3.3.1 Basic design principle 3.3.2 Complete decoupling 3.3.3

More information

OPTIMAL CONTROL AND ESTIMATION

OPTIMAL 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 information

Process Modelling, Identification, and Control

Process 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 information

CDS 101/110a: Lecture 10-1 Robust Performance

CDS 101/110a: Lecture 10-1 Robust Performance CDS 11/11a: Lecture 1-1 Robust Performance Richard M. Murray 1 December 28 Goals: Describe how to represent uncertainty in process dynamics Describe how to analyze a system in the presence of uncertainty

More information

Design of Sliding Mode Control for Nonlinear Uncertain System

Design of Sliding Mode Control for Nonlinear Uncertain System Design of Sliding Mode Control for Nonlinear Uncertain System 1 Yogita Pimpale, 2 Dr.B.J.Parvat ME student,instrumentation and Control Engineering,P.R.E.C. Loni,Ahmednagar, Maharashtra,India Associate

More information

Advanced Control Theory

Advanced Control Theory State Feedback Control Design chibum@seoultech.ac.kr Outline State feedback control design Benefits of CCF 2 Conceptual steps in controller design We begin by considering the regulation problem the task

More information

Variable Structure Control ~ Disturbance Rejection. Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University

Variable Structure Control ~ Disturbance Rejection. Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Variable Structure Control ~ Disturbance Rejection Harry G. Kwatny Department of Mechanical Engineering & Mechanics Drexel University Outline Linear Tracking & Disturbance Rejection Variable Structure

More information

(Continued on next page)

(Continued on next page) (Continued on next page) 18.2 Roots of Stability Nyquist Criterion 87 e(s) 1 S(s) = =, r(s) 1 + P (s)c(s) where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic

More information

Decentralized control with input saturation

Decentralized 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 information

Robust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room

Robust Control. 2nd class. Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room Robust Control Spring, 2018 Instructor: Prof. Masayuki Fujita (S5-303B) 2nd class Tue., 17th April, 2018, 10:45~12:15, S423 Lecture Room 2. Nominal Performance 2.1 Weighted Sensitivity [SP05, Sec. 2.8,

More information

Control Systems Design

Control Systems Design ELEC4410 Control Systems Design Lecture 18: State Feedback Tracking and State Estimation Julio H. Braslavsky julio@ee.newcastle.edu.au School of Electrical Engineering and Computer Science Lecture 18:

More information

Sliding Mode Controller for Parallel Rotary Double Inverted Pendulum: An Eigen Structure Assignment Approach

Sliding Mode Controller for Parallel Rotary Double Inverted Pendulum: An Eigen Structure Assignment Approach IJCTA, 9(39), 06, pp. 97-06 International Science Press Closed Loop Control of Soft Switched Forward Converter Using Intelligent Controller 97 Sliding Mode Controller for Parallel Rotary Double Inverted

More information

Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08

Fall 線性系統 Linear Systems. Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian. NTU-EE Sep07 Jan08 Fall 2007 線性系統 Linear Systems Chapter 08 State Feedback & State Estimators (SISO) Feng-Li Lian NTU-EE Sep07 Jan08 Materials used in these lecture notes are adopted from Linear System Theory & Design, 3rd.

More information

MIMO analysis: loop-at-a-time

MIMO analysis: loop-at-a-time MIMO robustness MIMO analysis: loop-at-a-time y 1 y 2 P (s) + + K 2 (s) r 1 r 2 K 1 (s) Plant: P (s) = 1 s 2 + α 2 s α 2 α(s + 1) α(s + 1) s α 2. (take α = 10 in the following numerical analysis) Controller:

More information

10/8/2015. Control Design. Pole-placement by state-space methods. Process to be controlled. State controller

10/8/2015. Control Design. Pole-placement by state-space methods. Process to be controlled. State controller Pole-placement by state-space methods Control Design To be considered in controller design * Compensate the effect of load disturbances * Reduce the effect of measurement noise * Setpoint following (target

More information

University of Petroleum & Energy Studies, Dehradun Uttrakhand, India

University of Petroleum & Energy Studies, Dehradun Uttrakhand, India International Journal of Scientific & Engineering Research Volume 9, Issue 1, January-2018 891 Control of Inverted Pendulum System Using LabVIEW Devendra Rawat a, Deepak Kumar a*, Deepali Yadav a a Department

More information

EL 625 Lecture 10. Pole Placement and Observer Design. ẋ = Ax (1)

EL 625 Lecture 10. Pole Placement and Observer Design. ẋ = Ax (1) EL 625 Lecture 0 EL 625 Lecture 0 Pole Placement and Observer Design Pole Placement Consider the system ẋ Ax () The solution to this system is x(t) e At x(0) (2) If the eigenvalues of A all lie in the

More information

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review

Dr Ian R. Manchester Dr Ian R. Manchester AMME 3500 : Review Week Date Content Notes 1 6 Mar Introduction 2 13 Mar Frequency Domain Modelling 3 20 Mar Transient Performance and the s-plane 4 27 Mar Block Diagrams Assign 1 Due 5 3 Apr Feedback System Characteristics

More information

Return Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems

Return 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 information

A unified double-loop multi-scale control strategy for NMP integrating-unstable systems

A unified double-loop multi-scale control strategy for NMP integrating-unstable systems Home Search Collections Journals About Contact us My IOPscience A unified double-loop multi-scale control strategy for NMP integrating-unstable systems This content has been downloaded from IOPscience.

More information

CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION Vol. III Controller Design - Boris Lohmann

CONTROL SYSTEMS, ROBOTICS, AND AUTOMATION Vol. III Controller Design - Boris Lohmann CONROL SYSEMS, ROBOICS, AND AUOMAION Vol. III Controller Design - Boris Lohmann CONROLLER DESIGN Boris Lohmann Institut für Automatisierungstechnik, Universität Bremen, Germany Keywords: State Feedback

More information

Control Systems. Design of State Feedback Control.

Control Systems. Design of State Feedback Control. Control Systems Design of State Feedback Control chibum@seoultech.ac.kr Outline Design of State feedback control Dominant pole design Symmetric root locus (linear quadratic regulation) 2 Selection of closed-loop

More information

Controls Problems for Qualifying Exam - Spring 2014

Controls Problems for Qualifying Exam - Spring 2014 Controls Problems for Qualifying Exam - Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function

More information

Extensions and applications of LQ

Extensions and applications of LQ Extensions and applications of LQ 1 Discrete time systems 2 Assigning closed loop pole location 3 Frequency shaping LQ Regulator for Discrete Time Systems Consider the discrete time system: x(k + 1) =

More information

Control System Design

Control System Design ELEC ENG 4CL4: Control System Design Notes for Lecture #36 Dr. Ian C. Bruce Room: CRL-229 Phone ext.: 26984 Email: ibruce@mail.ece.mcmaster.ca Friday, April 4, 2003 3. Cascade Control Next we turn to an

More information

Autonomous Mobile Robot Design

Autonomous Mobile Robot Design Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:

More information

Control Design Techniques in Power Electronics Devices

Control Design Techniques in Power Electronics Devices Hebertt Sira-Ramfrez and Ramön Silva-Ortigoza Control Design Techniques in Power Electronics Devices With 202 Figures < } Spri inger g< Contents 1 Introduction 1 Part I Modelling 2 Modelling of DC-to-DC

More information

SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015

SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 FACULTY OF ENGINEERING AND SCIENCE SAMPLE SOLUTION TO EXAM in MAS501 Control Systems 2 Autumn 2015 Lecturer: Michael Ruderman Problem 1: Frequency-domain analysis and control design (15 pt) Given is a

More information

Zeros and zero dynamics

Zeros and zero dynamics CHAPTER 4 Zeros and zero dynamics 41 Zero dynamics for SISO systems Consider a linear system defined by a strictly proper scalar transfer function that does not have any common zero and pole: g(s) =α p(s)

More information

Switching H 2/H Control of Singular Perturbation Systems

Switching H 2/H Control of Singular Perturbation Systems Australian Journal of Basic and Applied Sciences, 3(4): 443-45, 009 ISSN 1991-8178 Switching H /H Control of Singular Perturbation Systems Ahmad Fakharian, Fatemeh Jamshidi, Mohammad aghi Hamidi Beheshti

More information

An Adaptive LQG Combined With the MRAS Based LFFC for Motion Control Systems

An Adaptive LQG Combined With the MRAS Based LFFC for Motion Control Systems Journal of Automation Control Engineering Vol 3 No 2 April 2015 An Adaptive LQG Combined With the MRAS Based LFFC for Motion Control Systems Nguyen Duy Cuong Nguyen Van Lanh Gia Thi Dinh Electronics Faculty

More information

H CONTROL AND SLIDING MODE CONTROL OF MAGNETIC LEVITATION SYSTEM

H CONTROL AND SLIDING MODE CONTROL OF MAGNETIC LEVITATION SYSTEM 333 Asian Journal of Control, Vol. 4, No. 3, pp. 333-340, September 2002 H CONTROL AND SLIDING MODE CONTROL OF MAGNETIC LEVITATION SYSTEM Jing-Chung Shen ABSTRACT In this paper, H disturbance attenuation

More information

Lecture 18 : State Space Design

Lecture 18 : State Space Design UCSI University Kuala Lumpur, Malaysia Faculty of Engineering Department of Mechatronics Lecture 18 State Space Design Mohd Sulhi bin Azman Lecturer Department of Mechatronics UCSI University sulhi@ucsi.edu.my

More information

Robust Speed Controller Design for Permanent Magnet Synchronous Motor Drives Based on Sliding Mode Control

Robust Speed Controller Design for Permanent Magnet Synchronous Motor Drives Based on Sliding Mode Control Available online at www.sciencedirect.com ScienceDirect Energy Procedia 88 (2016 ) 867 873 CUE2015-Applied Energy Symposium and Summit 2015: ow carbon cities and urban energy systems Robust Speed Controller

More information

FUZZY CONTROL CONVENTIONAL CONTROL CONVENTIONAL CONTROL CONVENTIONAL CONTROL CONVENTIONAL CONTROL CONVENTIONAL CONTROL

FUZZY CONTROL CONVENTIONAL CONTROL CONVENTIONAL CONTROL CONVENTIONAL CONTROL CONVENTIONAL CONTROL CONVENTIONAL CONTROL Eample: design a cruise control system After gaining an intuitive understanding of the plant s dynamics and establishing the design objectives, the control engineer typically solves the cruise control

More information

EE C128 / ME C134 Feedback Control Systems

EE 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 information

Design Methods for Control Systems

Design 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 information

EEE582 Homework Problems

EEE582 Homework Problems EEE582 Homework Problems HW. Write a state-space realization of the linearized model for the cruise control system around speeds v = 4 (Section.3, http://tsakalis.faculty.asu.edu/notes/models.pdf). Use

More information

Design and Comparative Analysis of Controller for Non Linear Tank System

Design and Comparative Analysis of Controller for Non Linear Tank System Design and Comparative Analysis of for Non Linear Tank System Janaki.M 1, Soniya.V 2, Arunkumar.E 3 12 Assistant professor, Department of EIE, Karpagam College of Engineering, Coimbatore, India 3 Associate

More information

MODERN CONTROL DESIGN

MODERN 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 information

Uncertainty and Robustness for SISO Systems

Uncertainty and Robustness for SISO Systems Uncertainty and Robustness for SISO Systems ELEC 571L Robust Multivariable Control prepared by: Greg Stewart Outline Nature of uncertainty (models and signals). Physical sources of model uncertainty. Mathematical

More information

LOC-PSS Design for Improved Power System Stabilizer

LOC-PSS Design for Improved Power System Stabilizer Journal of pplied Dynamic Systems and Control, Vol., No., 8: 7 5 7 LOCPSS Design for Improved Power System Stabilizer Masoud Radmehr *, Mehdi Mohammadjafari, Mahmoud Reza GhadiSahebi bstract power system

More information

Chapter 8 Stabilization: State Feedback 8. Introduction: Stabilization One reason feedback control systems are designed is to stabilize systems that m

Chapter 8 Stabilization: State Feedback 8. Introduction: Stabilization One reason feedback control systems are designed is to stabilize systems that m Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of echnology c Chapter 8 Stabilization:

More information

Chapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o

Chapter 7 Interconnected Systems and Feedback: Well-Posedness, Stability, and Performance 7. Introduction Feedback control is a powerful approach to o Lectures on Dynamic Systems and Control Mohammed Dahleh Munther A. Dahleh George Verghese Department of Electrical Engineering and Computer Science Massachuasetts Institute of Technology c Chapter 7 Interconnected

More information

EE C128 / ME C134 Fall 2014 HW 9 Solutions. HW 9 Solutions. 10(s + 3) s(s + 2)(s + 5) G(s) =

EE C128 / ME C134 Fall 2014 HW 9 Solutions. HW 9 Solutions. 10(s + 3) s(s + 2)(s + 5) G(s) = 1. Pole Placement Given the following open-loop plant, HW 9 Solutions G(s) = 1(s + 3) s(s + 2)(s + 5) design the state-variable feedback controller u = Kx + r, where K = [k 1 k 2 k 3 ] is the feedback

More information

LQG/LTR CONTROLLER DESIGN FOR ROTARY INVERTED PENDULUM QUANSER REAL-TIME EXPERIMENT

LQG/LTR CONTROLLER DESIGN FOR ROTARY INVERTED PENDULUM QUANSER REAL-TIME EXPERIMENT LQG/LR CONROLLER DESIGN FOR ROARY INVERED PENDULUM QUANSER REAL-IME EXPERIMEN Cosmin Ionete University of Craiova, Faculty of Automation, Computers and Electronics Department of Automation, e-mail: cosmin@automation.ucv.ro

More information

Video 6.1 Vijay Kumar and Ani Hsieh

Video 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 information

A DISCRETE-TIME SLIDING MODE CONTROLLER WITH MODIFIED FUNCTION FOR LINEAR TIME- VARYING SYSTEMS

A DISCRETE-TIME SLIDING MODE CONTROLLER WITH MODIFIED FUNCTION FOR LINEAR TIME- VARYING SYSTEMS http:// A DISCRETE-TIME SLIDING MODE CONTROLLER WITH MODIFIED FUNCTION FOR LINEAR TIME- VARYING SYSTEMS Deelendra Pratap Singh 1, Anil Sharma 2, Shalabh Agarwal 3 1,2 Department of Electronics & Communication

More information

Chap. 3. Controlled Systems, Controllability

Chap. 3. Controlled Systems, Controllability Chap. 3. Controlled Systems, Controllability 1. Controllability of Linear Systems 1.1. Kalman s Criterion Consider the linear system ẋ = Ax + Bu where x R n : state vector and u R m : input vector. A :

More information

State Feedback and State Estimators Linear System Theory and Design, Chapter 8.

State Feedback and State Estimators Linear System Theory and Design, Chapter 8. 1 Linear System Theory and Design, http://zitompul.wordpress.com 2 0 1 4 2 Homework 7: State Estimators (a) For the same system as discussed in previous slides, design another closed-loop state estimator,

More information

Robust LQR Control Design of Gyroscope

Robust LQR Control Design of Gyroscope Robust LQR Control Design of Gyroscope Ashok S. Chandak 1, Anil J. Patil 2 Abstract The basic problem in designing control systems is the ability to achieve good performance in the presence of uncertainties

More information

Adaptive Pole Assignment Control for Generic Elastic Hypersonic Vehicle

Adaptive Pole Assignment Control for Generic Elastic Hypersonic Vehicle Adaptive Pole Assignment Control for Generic Elastic Hypersonic Vehicle Yan Binbin he College of Astronautics, Northwestern Polytechnical University, Xi an, Shanxi, China Email: yanbinbin@ nwpu.edu.cn

More information

CHAPTER 4 STATE FEEDBACK AND OUTPUT FEEDBACK CONTROLLERS

CHAPTER 4 STATE FEEDBACK AND OUTPUT FEEDBACK CONTROLLERS 54 CHAPTER 4 STATE FEEDBACK AND OUTPUT FEEDBACK CONTROLLERS 4.1 INTRODUCTION In control theory, a controller is a device which monitors and affects the operational conditions of a given dynamic system.

More information

Here represents the impulse (or delta) function. is an diagonal matrix of intensities, and is an diagonal matrix of intensities.

Here 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 information

X 2 3. Derive state transition matrix and its properties [10M] 4. (a) Derive a state space representation of the following system [5M] 1

X 2 3. Derive state transition matrix and its properties [10M] 4. (a) Derive a state space representation of the following system [5M] 1 QUESTION BANK 6 SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 5758 QUESTION BANK (DESCRIPTIVE) Subject with Code :SYSTEM THEORY(6EE75) Year &Sem: I-M.Tech& I-Sem UNIT-I

More information

INVERSE MODEL APPROACH TO DISTURBANCE REJECTION AND DECOUPLING CONTROLLER DESIGN. Leonid Lyubchyk

INVERSE 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 information

ẋ 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)

ẋ 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 information