Contents lecture 5. Automatic Control III. Summary of lecture 4 (II/II) Summary of lecture 4 (I/II) u y F r. Lecture 5 H 2 and H loop shaping
|
|
- Hubert Stephens
- 5 years ago
- Views:
Transcription
1 Contents lecture 5 Automatic Control III Lecture 5 H 2 and H loop shaping Thomas Schön Division of Systems and Control Department of Information Technology Uppsala University. thomas.schon@it.uu.se, www: user.it.uu.se/~thosc Summary of lecture 4 2. Specifications closed loop system 3. An extended system 4. H 2 control 5. H control 6. Design example 1 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 2 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping Summary of lecture 4 (I/II) Summary of lecture 4 (II/II) The key difficulty in controlling multivariable systems is that there are cross couplings between the input and the output signals. The relative gain array (RGA) is a way of measuring the amount of cross couplings in a system RGA(G) G. (G ) T. Decentralized control means that we let every input be determined by feedback from one single output. The pairing problem is to select which input-output pairs that should be used for the feedback. Decoupled control makes use of a change of variables such that suitable pairings of measurements and control signals becomes easier to see. If there were no model errors and no disturbances we can make y follow r perfectly by choosing Q = G 1 in y = GQr. r + u y F r Q G 0 The idea in Internal Model Control (IMC) is to choose Q G 1 and feedback only using the new information y Gu. G + 3 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 4 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping
2 Linear quadratic synthesis pros and cons Problem formulation (+) All reasonable (stabilizable and detectable, Q 1 0, Q 2 > 0, R 1 0, R 2 > 0) choices of Q 1, Q 2, R 1, R 2 results in a closed loop system with poles strictly in the left half plane. (+) Handles the trade-offs between the sizes of the different components in x and u well. (+) It is easy to adjust Q 1, Q 2 such that we obtain nice results in the time domain. (+) Some possibilities of handling robustness. (-) Hard to see how Q 1, Q 2, R 1, R 2 affects S, T, G wu, etc. For the system find the feedback control law y = G(p)u + w, u = F y (p)y, such that the closed loop system has good properties in terms of the sensitivity functions S and T, and the transfer function from disturbance to input response G wu. 5 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 6 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping Direct synthesis of S, T and G wu Example of requirements: S We typically want the sensitivity S to be small for low frequencies where we are likely to have process disturbances (e.g. not to be sensitive to the slope of a hill). Classical methods (lead-lag) scalar systems. Intuitive design using Bode diagrams. Finding a compromize using optimization: H 2 design Fulfilling bounds on S, T and G wu : H design log S Today: H 2 design and H design. log ω Typically encoded using a function W S (iω) S(iω) W 1 S (iω), ω W SS 1. 7 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 8 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping
3 Example of requirements: T We typically want the complementary sensitivity T to be small for high frequencies where we are likely to have e.g. measurement noise. log T Example of requirements: G wu Recall that G wu (or sometimes G ru ) is the gain from a disturbance (or reference) to the control signal. This gain is important in order to make sure that the control signals does not become too big. Analogously to what we did before, we obtain Typically encoded using a function W T (iω) log ω G wu (iω) W 1 u ω W u G wu 1 It is important to note that the requirements on S, T and G wu can collide, implying that a compromize must be made. T (iω) < 1 G (iω) ω W T T 1. 9 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 10 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping Design spec. in the frequency domain Recall the H -norm Weightings of S, T and G wu : W S (iω)s(iω) W T (iω)t (iω) W u (iω)g wu (iω) Design criterion H 2 -design: Choose the controller such that V = W S S W T T W u G wu 2 2 is minimized. Design criterion H -design: Choose the controller such that Definition: The H -norm of the system y = G(p)u is given by G = sup u y 2 = sup σ(g(iω)) u 2 ω For scalar systems G = sup ω G(iω). The input u that maximizes the size of the outputs is a sinusoid with the frequency ω for which G(iω) attains its maximum. Interpretation: Highest peak/ worst case. W S S 1, W T T 1, W u G wu / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 12 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping
4 Recall the H 2 -norm An extended imaginary system G W (I/II) Definition: The H 2 -norm of the system y = G(p)u is given by G 2 2 = 1 2π G(iω) 2 2dω W u z 1 where G(iω) 2 2 = tr(g (iω)g(iω)). For a square matrix A = [a ij ] the trace is tr(a) = n a kk = k=1 n λ k (A). k=1 Interpretations due to Parseval s theorem: Deterministic: G 2 = g(t) 2, i.e. the 2-norm of the impulse response g(t). Stochastic: If u is a stochastic process with Φ u (ω) = I, then G 2 2 = 1 2π Φy (ω) 2 2 dω = tr(r y) = y / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping An extended imaginary system G W (II/II) G W 0 G w u F y y W T z 2 W S z 3 When we close the system using u = F y y we have: z 1 = W u G wu w, z 2 = W T T w, z 3 = W S Sw 14 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping Compact representation Introduce the performance variables z 1 W u G wu z = z 2 = W T T w G ec w z 3 W S S as (artificial) outputs of an extended closed loop system G ec with w as input. This way z 1, z 2 and z 3 represent the requirements for the closed loop system. w u G W F y y z State space realization of G W (i.e. state space model of the open loop system (u, w) (z, y)): ẋ = Ax + Bu + Nw z = Mx + Du y = Cx + w A technical assumption: Assume D T ( M D ) = ( 0 I ) Closed loop system: z = G ec w where z = z 2 z 3 z 1 15 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 16 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping
5 Digital Signal Processing, TSRT78! T. Schön! Digital Signal Processing, TSRT78! T. Schön! Structural property MSc thesis example 1 General Wiener Filter Algorithm (Alg. 7.1)! 7 8 MSc Thesis Example 1! Vision and GPS Based Autonomous Landing of an Unmanned Aerial by Joel Hermansson, performed at Vision and GPS BasedVehicle, Autonomous Landing of an Unmanned Aerial Vehicle, Cybaero. by Joel Hermansson, performed at Cybaero.! Themsystem Select > 0 for prediction, m < 0 for smoothing and m = 0 for filtering.! 1. Spectral factorization!x = Ax + Bu + N w z = M x + Du y = Cx + w 2. Compute the filter (let the direct term be part of the causal term)! C x ),, where! if A N C has all its eigenvalues strictly in the LHP. Lots of signal processing (modelling, filtering, Modelling, filtering, Kalman filter, computer Kalman filter, computer vision, ) and some control(pid).! (several PIDs). control Lecture 8! 3. The Wiener filterx is= now by + N (y Ax given + Bu Digital Signal Processing, TSRT78! T. Schön! 17 / 33 T. Schön, / 33 Recall the H2 norm and the design criterion V (Fy ) = kgec k22 = kws Sk22 + kwt T k22 + kwu Gwu k22 is equivalent to minimizing kzk22 = km xk22 + kuk22. V = kws Sk22 + kwt T k22 + kwu Gwu k22 This is the LQG problem! (with z 0 = M x, Q1 = I and Q2 = I) is minimized. Automatic Control III, Lecture 5 H2 and H loop shaping Automatic Control III, Lecture 5 H2 and H loop shaping Minimizing the criterion Design criterion H2 -design: Choose the controller such that T. Schön, 2014 T. Schön, 2014 Optimal H2 control the problem Definition: The H2 -norm of the system y = G(p)u is given by Z Z 1 1 kgk22 = G(iω) 22 dω = tr (G (iω)g(iω)) dω. 2π 2π 19 / 33 Show movie!!! Show movie! Digital Signal Processing, TSRT78! T. Schön! Automatic Control III, Lecture 5 H2 and H loop shaping vision and of course Lecture 8! is on innovation form since v1 = v2 = w, which implies that the system is its own observer and the Kalman filter (K = N ) is simply 20 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H2 and H loop shaping
6 Optimal H 2 control the solution Solution provided by Theorem 9.1 (page 270). The optimal H 2 controller is given by (recall that the system is given on innovations form K = N in the Kalman filter) Recall the H norm and the design objective Definition: The H -norm of the system y = G(p)u is given by with Hence, ˆx = Aˆx + Bu + N(y C ˆx), u = Lˆx, L = B T S, 0 = A T S + SA + M T M SBB T S. F y (s) = L(SI A + BB T S + NC) 1 N G = sup u y 2 = sup σ(g(iω)) u 2 ω Design objective H -design: Find the controller that minimize G ec = max σ(g ec(iω)). ω This is a hard (non-convex) problem, instead we search for controllers that satisfies G ec < γ (If A NC not stable, compute and use the Kalman filter.) 21 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping Optimal H control the solution 22 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping H control design steps Assume A T S + SA + M T M + S(γ 2 NN T BB T )S = 0 has a positive semidefinite solution S = S γ, and that A BB T S γ is stable. Consider the controller with L = B T S γ. Then ˆx = Aˆx + Bu + N(y C ˆx), u = L ˆx, 1. G is given. 2. Choose weights W u, W S, W T. 3. Form the extended system. 4. Choose a constant γ. 5. Solve the Riccati equation and compute L. 5.1 If no solution exists, increase γ and go to step If a solution exists, accept it, or decrease γ and go to step Check the properties of the closed loop system. If not acceptable, go to step 2. F y (s) = L (si A + BB T S γ + NC) 1 N 23 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 24 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping
7 H 2, H synthesis pros and cons Linear multivariable controller synthesis (+) Directly handles the specifications on S, T and G wu (+) Let us know when certain specifications are impossible to achieve (via γ). (+) Easy to handle several different specifications (in the frequency domain) (-) Can be hard to control the behaviour in the time domain in detail. (-) Often results in complex controllers (number of states in the controller = number of states in G, W u, W S, W T ). Summary: 1. Perform an RGA analysis 2. Employ simple PID controllers if the RGA indicates that it is possible. 3. Otherwise make use of LQ, MPC or H 2 /H -synthesis. 25 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 26 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping Design example (I/V) ³ DESIGN EXAMPLE Control the system G(s) = 0.2s+1. Bode plots s s+1 Control the system ( ) = , with Bode plot below: G(iω) arg G(iω) Design example (II/V) Specifications: 1. G wu (iω) < 4, ω G wu < T (iω) < 1.25, ω T < Integral action S(0) = Bandwidth ω G 2 rad/s Rule of thumb: ω B < z/2 for RHP zero z ω B < 2.5 rad/s here! Difficulties: Difficulties: Higly Highly resonant resonant system, system, non-minimum non-minimum phase phase zero zero in in Try with the following weighting functions W u = 1 4 = 0.25, W T = = 0.8, W S(s) = (0.5s) 1 = 2 s 27 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping ± ² Fall 2012 Automatic Control III:3 part 2 Page 8 28 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping
8 Design example (III/V) Design example (IV/V) DESIGN EXAMPLE, cont d Four control strategies were employed: 1. PID control/lead-lag compensation: F (s) = K τ Ds + 1 τ I s + 1 βτ d s + 1 τ I s 2. IMC, with ( ) 1 0.2s Q(s) = 0.5s + 1 s s s + 1 T (s) = (0.2s + 1)(0.5s + 1) 3. H 2 control 4. H control with γ = 2.9. Results: the sensitivity ³ Results: functions The sensitivity functions. S(iω) / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping ² PID IMC H 2 H T(iω) PID IMC H 2 H Fall 2012 Automatic Control III:3 part 2 Page / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping Design example (V/V) DESIGN EXAMPLE, cont d Outline entire course ± Results: Results: G w u and comparisons with the ³ ÛÙ and comparisons with the specifications. specifications. G wu (iω) PID IMC H 2 Ì ½ ÛÙ ½ H À ½ À PID T G1 64 wu ω B IMC PID IMC H H All controllers have integral action. All controllers have integral action. 1. Lecture 1: Introduction and linear multivariable systems 2. Lecture 2 5: Multivariable linear control theory a) systems theory, closed loop system b) Basic limitations c) Controller structures and control design d) H 2 and H loop shaping 1. Lecture 6 8: Nonlinear control theory a) Linearization and phase portraits b) Lyapunov theory and the circle criterion c) Describing functions 2. Lecture 9: Optimal control 3. Lecture 10: Summary and repetition ± Fall 2012 Automatic Control III:3 part 2 Page 18 ² 31 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping 32 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping
9 A few concepts to summarize lecture 5 Extended system: Allows us to systematically incorporate design specifications in the frequency domain by extending the system G. This is done by introducing three performance variables z 1, z 2 and z 3 which are viewed as artificial outputs of the system. H -norm: The H -norm of the system y = G(p)u is given by y G = sup 2 u u 2 = sup ω σ(g(iω)). Interpretation: highest peak/ worst case. H 2 -norm: The H 2 -norm of the system y = G(p)u is given by G 2 2 = 1 2π G(iω) 2 2dω. Interpretation: total energy. 33 / 33 T. Schön, 2014 Automatic Control III, Lecture 5 H 2 and H loop shaping
Contents lecture 6 2(17) Automatic Control III. Summary of lecture 5 (I/III) 3(17) Summary of lecture 5 (II/III) 4(17) H 2, H synthesis pros and cons:
Contents lecture 6 (7) Automatic Control III Lecture 6 Linearization and phase portraits. Summary of lecture 5 Thomas Schön Division of Systems and Control Department of Information Technology Uppsala
More 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 informationEL2520 Control Theory and Practice
So far EL2520 Control Theory and Practice r Fr wu u G w z n Lecture 5: Multivariable systems -Fy Mikael Johansson School of Electrical Engineering KTH, Stockholm, Sweden SISO control revisited: Signal
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 informationFRTN10 Multivariable Control, Lecture 13. Course outline. The Q-parametrization (Youla) Example: Spring-mass System
FRTN Multivariable Control, Lecture 3 Anders Robertsson Automatic Control LTH, Lund University Course outline The Q-parametrization (Youla) L-L5 Purpose, models and loop-shaping by hand L6-L8 Limitations
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 informationCourse Outline. FRTN10 Multivariable Control, Lecture 13. General idea for Lectures Lecture 13 Outline. Example 1 (Doyle Stein, 1979)
Course Outline FRTN Multivariable Control, Lecture Automatic Control LTH, 6 L-L Specifications, models and loop-shaping by hand L6-L8 Limitations on achievable performance L9-L Controller optimization:
More informationExercises Automatic Control III 2015
Exercises Automatic Control III 205 Foreword This exercise manual is designed for the course "Automatic Control III", given by the Division of Systems and Control. The numbering of the chapters follows
More informationAutomatic Control (TSRT15): Lecture 7
Automatic Control (TSRT15): Lecture 7 Tianshi Chen Division of Automatic Control Dept. of Electrical Engineering Email: tschen@isy.liu.se Phone: 13-282226 Office: B-house extrance 25-27 Outline 2 Feedforward
More informationDenis ARZELIER arzelier
COURSE ON LMI OPTIMIZATION WITH APPLICATIONS IN CONTROL PART II.2 LMIs IN SYSTEMS CONTROL STATE-SPACE METHODS PERFORMANCE ANALYSIS and SYNTHESIS Denis ARZELIER www.laas.fr/ arzelier arzelier@laas.fr 15
More informationThe Q-parametrization (Youla) Lecture 13: Synthesis by Convex Optimization. Lecture 13: Synthesis by Convex Optimization. Example: Spring-mass System
The Q-parametrization (Youla) Lecture 3: Synthesis by Convex Optimization controlled variables z Plant distubances w Example: Spring-mass system measurements y Controller control inputs u Idea for lecture
More informationIntro. Computer Control Systems: F9
Intro. Computer Control Systems: F9 State-feedback control and observers Dave Zachariah Dept. Information Technology, Div. Systems and Control 1 / 21 dave.zachariah@it.uu.se F8: Quiz! 2 / 21 dave.zachariah@it.uu.se
More informationExam in Automatic Control II Reglerteknik II 5hp (1RT495)
Exam in Automatic Control II Reglerteknik II 5hp (1RT495) Date: August 4, 018 Venue: Bergsbrunnagatan 15 sal Responsible teacher: Hans Rosth. Aiding material: Calculator, mathematical handbooks, textbooks
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 informationMAE143a: Signals & Systems (& Control) Final Exam (2011) solutions
MAE143a: Signals & Systems (& Control) Final Exam (2011) solutions Question 1. SIGNALS: Design of a noise-cancelling headphone system. 1a. Based on the low-pass filter given, design a high-pass filter,
More informationAutonomous Mobile Robot Design
Autonomous Mobile Robot Design Topic: Guidance and Control Introduction and PID Loops Dr. Kostas Alexis (CSE) Autonomous Robot Challenges How do I control where to go? Autonomous Mobile Robot Design Topic:
More information6 OUTPUT FEEDBACK DESIGN
6 OUTPUT FEEDBACK DESIGN When the whole sate vector is not available for feedback, i.e, we can measure only y = Cx. 6.1 Review of observer design Recall from the first class in linear systems that a simple
More informationFeedback Control of Turbulent Wall Flows
Feedback Control of Turbulent Wall Flows Dipartimento di Ingegneria Aerospaziale Politecnico di Milano Outline Introduction Standard approach Wiener-Hopf approach Conclusions Drag reduction A model problem:
More informationTopic # Feedback Control Systems
Topic #20 16.31 Feedback Control Systems Closed-loop system analysis Bounded Gain Theorem Robust Stability Fall 2007 16.31 20 1 SISO Performance Objectives Basic setup: d i d o r u y G c (s) G(s) n control
More informationModern Optimal Control
Modern Optimal Control Matthew M. Peet Arizona State University Lecture 22: H 2, LQG and LGR Conclusion To solve the H -optimal state-feedback problem, we solve min γ such that γ,x 1,Y 1,A n,b n,c n,d
More informationLecture 5: Control Over Lossy Networks
Lecture 5: Control Over Lossy Networks Yilin Mo July 2, 2015 1 Classical LQG Control The system: x k+1 = Ax k + Bu k + w k, y k = Cx k + v k x 0 N (0, Σ), w k N (0, Q), v k N (0, R). Information available
More informationProblem Set 4 Solution 1
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Problem Set 4 Solution Problem 4. For the SISO feedback
More informationAutomatic Control II Computer exercise 3. LQG Design
Uppsala University Information Technology Systems and Control HN,FS,KN 2000-10 Last revised by HR August 16, 2017 Automatic Control II Computer exercise 3 LQG Design Preparations: Read Chapters 5 and 9
More informationLecture 3. Chapter 4: Elements of Linear System Theory. Eugenio Schuster. Mechanical Engineering and Mechanics Lehigh University.
Lecture 3 Chapter 4: Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 3 p. 1/77 3.1 System Descriptions [4.1] Let f(u) be a liner operator, u 1 and u
More informationStability of Parameter Adaptation Algorithms. Big picture
ME5895, UConn, Fall 215 Prof. Xu Chen Big picture For ˆθ (k + 1) = ˆθ (k) + [correction term] we haven t talked about whether ˆθ(k) will converge to the true value θ if k. We haven t even talked about
More informationModern Control Systems
Modern Control Systems Matthew M. Peet Illinois Institute of Technology Lecture 18: Linear Causal Time-Invariant Operators Operators L 2 and ˆL 2 space Because L 2 (, ) and ˆL 2 are isomorphic, so are
More informationRaktim Bhattacharya. . AERO 632: Design of Advance Flight Control System. Norms for Signals and Systems
. AERO 632: Design of Advance Flight Control System Norms for Signals and. Raktim Bhattacharya Laboratory For Uncertainty Quantification Aerospace Engineering, Texas A&M University. Norms for Signals ...
More informationThe norms can also be characterized in terms of Riccati inequalities.
9 Analysis of stability and H norms Consider the causal, linear, time-invariant system ẋ(t = Ax(t + Bu(t y(t = Cx(t Denote the transfer function G(s := C (si A 1 B. Theorem 85 The following statements
More 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 informationTopic # Feedback Control. State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback
Topic #17 16.31 Feedback Control State-Space Systems Closed-loop control using estimators and regulators. Dynamics output feedback Back to reality Copyright 21 by Jonathan How. All Rights reserved 1 Fall
More informationLinear State Feedback Controller Design
Assignment For EE5101 - Linear Systems Sem I AY2010/2011 Linear State Feedback Controller Design Phang Swee King A0033585A Email: king@nus.edu.sg NGS/ECE Dept. Faculty of Engineering National University
More informationRECURSIVE ESTIMATION AND KALMAN FILTERING
Chapter 3 RECURSIVE ESTIMATION AND KALMAN FILTERING 3. The Discrete Time Kalman Filter Consider the following estimation problem. Given the stochastic system with x k+ = Ax k + Gw k (3.) y k = Cx k + Hv
More informationA FEEDBACK STRUCTURE WITH HIGHER ORDER DERIVATIVES IN REGULATOR. Ryszard Gessing
A FEEDBACK STRUCTURE WITH HIGHER ORDER DERIVATIVES IN REGULATOR Ryszard Gessing Politechnika Śl aska Instytut Automatyki, ul. Akademicka 16, 44-101 Gliwice, Poland, fax: +4832 372127, email: gessing@ia.gliwice.edu.pl
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 2: Drawing Bode Plots, Part 2 Overview In this Lecture, you will learn: Simple Plots Real Zeros Real Poles Complex
More informationECE317 : Feedback and Control
ECE317 : Feedback and Control Lecture : Routh-Hurwitz stability criterion Examples Dr. Richard Tymerski Dept. of Electrical and Computer Engineering Portland State University 1 Course roadmap Modeling
More informationEL1820 Modeling of Dynamical Systems
EL1820 Modeling of Dynamical Systems Lecture 10 - System identification as a model building tool Experiment design Examination and prefiltering of data Model structure selection Model validation Lecture
More informationState estimation and the Kalman filter
State estimation and the Kalman filter PhD, David Di Ruscio Telemark university college Department of Technology Systems and Control Engineering N-3914 Porsgrunn, Norway Fax: +47 35 57 52 50 Tel: +47 35
More informationChapter Eleven. Frequency Domain Design Sensitivity Functions
Feedback Systems by Astrom and Murray, v2.11b http://www.cds.caltech.edu/~murray/fbswiki Chapter Eleven Frequency Domain Design Sensitivity improvements in one frequency range must be paid for with sensitivity
More informationDisturbance modelling
Lecture 3 Disturbance modelling This section reviews the main aspects in disturbance modelling and the corresponding relations of descriptions in the time and frequency domain, respectively. We will also
More informationOutline. Classical Control. Lecture 1
Outline Outline Outline 1 Introduction 2 Prerequisites Block diagram for system modeling Modeling Mechanical Electrical Outline Introduction Background Basic Systems Models/Transfers functions 1 Introduction
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 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 informationH 2 Optimal State Feedback Control Synthesis. Raktim Bhattacharya Aerospace Engineering, Texas A&M University
H 2 Optimal State Feedback Control Synthesis Raktim Bhattacharya Aerospace Engineering, Texas A&M University Motivation Motivation w(t) u(t) G K y(t) z(t) w(t) are exogenous signals reference, process
More informationIntroduction. Performance and Robustness (Chapter 1) Advanced Control Systems Spring / 31
Introduction Classical Control Robust Control u(t) y(t) G u(t) G + y(t) G : nominal model G = G + : plant uncertainty Uncertainty sources : Structured : parametric uncertainty, multimodel uncertainty Unstructured
More informationEL 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 informationMultivariable Control Exam
Department of AUTOMATIC CONTROL Multivariable Control Exam Exam 4 8 Grading All answers must include a clear motivation and a well-formulated answer. AnswersmaybegiveninEnglishorSwedish.Thetotalnumberofpointsis5.The
More informationControl Systems Lab - SC4070 Control techniques
Control Systems Lab - SC4070 Control techniques Dr. Manuel Mazo Jr. Delft Center for Systems and Control (TU Delft) m.mazo@tudelft.nl Tel.:015-2788131 TU Delft, February 16, 2015 (slides modified from
More informationEECS C128/ ME C134 Final Wed. Dec. 14, am. Closed book. One page, 2 sides of formula sheets. No calculators.
Name: SID: EECS C128/ ME C134 Final Wed. Dec. 14, 211 81-11 am Closed book. One page, 2 sides of formula sheets. No calculators. There are 8 problems worth 1 points total. Problem Points Score 1 16 2 12
More informationD(s) G(s) A control system design definition
R E Compensation D(s) U Plant G(s) Y Figure 7. A control system design definition x x x 2 x 2 U 2 s s 7 2 Y Figure 7.2 A block diagram representing Eq. (7.) in control form z U 2 s z Y 4 z 2 s z 2 3 Figure
More informationLecture 9 Nonlinear Control Design. Course Outline. Exact linearization: example [one-link robot] Exact Feedback Linearization
Lecture 9 Nonlinear Control Design Course Outline Eact-linearization Lyapunov-based design Lab Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.] and [Glad-Ljung,ch.17] Lecture
More informationPole Placement Design
Department of Automatic Control LTH, Lund University 1 Introduction 2 Simple Examples 3 Polynomial Design 4 State Space Design 5 Robustness and Design Rules 6 Model Reduction 7 Oscillatory Systems 8 Summary
More informationContents. PART I METHODS AND CONCEPTS 2. Transfer Function Approach Frequency Domain Representations... 42
Contents Preface.............................................. xiii 1. Introduction......................................... 1 1.1 Continuous and Discrete Control Systems................. 4 1.2 Open-Loop
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 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 informationIntroduction to Nonlinear Control Lecture # 4 Passivity
p. 1/6 Introduction to Nonlinear Control Lecture # 4 Passivity È p. 2/6 Memoryless Functions ¹ y È Ý Ù È È È È u (b) µ power inflow = uy Resistor is passive if uy 0 p. 3/6 y y y u u u (a) (b) (c) Passive
More informationLecture 15: H Control Synthesis
c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control for Linear Systems: Lecture 15 p. 1/14 Lecture 15: H Control Synthesis Example c A. Shiriaev/L. Freidovich. March 12, 2010. Optimal Control
More informationEE 380. Linear Control Systems. Lecture 10
EE 380 Linear Control Systems Lecture 10 Professor Jeffrey Schiano Department of Electrical Engineering Lecture 10. 1 Lecture 10 Topics Stability Definitions Methods for Determining Stability Lecture 10.
More informationStructured State Space Realizations for SLS Distributed Controllers
Structured State Space Realizations for SLS Distributed Controllers James Anderson and Nikolai Matni Abstract In recent work the system level synthesis (SLS) paradigm has been shown to provide a truly
More information6.241 Dynamic Systems and Control
6.241 Dynamic Systems and Control Lecture 12: I/O Stability Readings: DDV, Chapters 15, 16 Emilio Frazzoli Aeronautics and Astronautics Massachusetts Institute of Technology March 14, 2011 E. Frazzoli
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 informationRobust Multivariable Control
Lecture 2 Anders Helmersson anders.helmersson@liu.se ISY/Reglerteknik Linköpings universitet Today s topics Today s topics Norms Today s topics Norms Representation of dynamic systems Today s topics Norms
More informationControl Design. Lecture 9: State Feedback and Observers. Two Classes of Control Problems. State Feedback: Problem Formulation
Lecture 9: State Feedback and s [IFAC PB Ch 9] State Feedback s Disturbance Estimation & Integral Action Control Design Many factors to consider, for example: Attenuation of load disturbances Reduction
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 21: Stability Margins and Closing the Loop Overview In this Lecture, you will learn: Closing the Loop Effect on Bode Plot Effect
More informationLecture 4: Analysis of MIMO Systems
Lecture 4: Analysis of MIMO Systems Norms The concept of norm will be extremely useful for evaluating signals and systems quantitatively during this course In the following, we will present vector norms
More informationControls Problems for Qualifying Exam - Spring 2014
Controls Problems for Qualifying Exam - Spring 2014 Problem 1 Consider the system block diagram given in Figure 1. Find the overall transfer function T(s) = C(s)/R(s). Note that this transfer function
More informationA Comparative Study on Automatic Flight Control for small UAV
Proceedings of the 5 th International Conference of Control, Dynamic Systems, and Robotics (CDSR'18) Niagara Falls, Canada June 7 9, 18 Paper No. 13 DOI: 1.11159/cdsr18.13 A Comparative Study on Automatic
More informationExercise 1 (A Non-minimum Phase System)
Prof. Dr. E. Frazzoli 5-59- Control Systems I (Autumn 27) Solution Exercise Set 2 Loop Shaping clruch@ethz.ch, 8th December 27 Exercise (A Non-minimum Phase System) To decrease the rise time of the system,
More informationControl Systems Design, SC4026. SC4026 Fall 2010, dr. A. Abate, DCSC, TU Delft
Control Systems Design, SC426 SC426 Fall 2, dr A Abate, DCSC, TU Delft Lecture 5 Controllable Canonical and Observable Canonical Forms Stabilization by State Feedback State Estimation, Observer Design
More informationStep Response Analysis. Frequency Response, Relation Between Model Descriptions
Step Response Analysis. Frequency Response, Relation Between Model Descriptions Automatic Control, Basic Course, Lecture 3 November 9, 27 Lund University, Department of Automatic Control Content. Step
More informationME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ
ME 234, Lyapunov and Riccati Problems. This problem is to recall some facts and formulae you already know. (a) Let A and B be matrices of appropriate dimension. Show that (A, B) is controllable if and
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 informationControl Systems 2. Lecture 4: Sensitivity function limits. Roy Smith
Control Systems 2 Lecture 4: Sensitivity function limits Roy Smith 2017-3-14 4.1 Input-output controllability Control design questions: 1. How well can the plant be controlled? 2. What control structure
More informationAn LQ R weight selection approach to the discrete generalized H 2 control problem
INT. J. CONTROL, 1998, VOL. 71, NO. 1, 93± 11 An LQ R weight selection approach to the discrete generalized H 2 control problem D. A. WILSON², M. A. NEKOUI² and G. D. HALIKIAS² It is known that a generalized
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 24: Compensation in the Frequency Domain Overview In this Lecture, you will learn: Lead Compensators Performance Specs Altering
More informationCDS 101/110: Lecture 10.3 Final Exam Review
CDS 11/11: Lecture 1.3 Final Exam Review December 2, 216 Schedule: (1) Posted on the web Monday, Dec. 5 by noon. (2) Due Friday, Dec. 9, at 5: pm. (3) Determines 3% of your grade Instructions on Front
More information6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski. Solutions to Problem Set 1 1. Massachusetts Institute of Technology
Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.245: MULTIVARIABLE CONTROL SYSTEMS by A. Megretski Solutions to Problem Set 1 1 Problem 1.1T Consider the
More informationAchievable performance of multivariable systems with unstable zeros and poles
Achievable performance of multivariable systems with unstable zeros and poles K. Havre Λ and S. Skogestad y Chemical Engineering, Norwegian University of Science and Technology, N-7491 Trondheim, Norway.
More informationStabilization with Disturbance Attenuation over a Gaussian Channel
Proceedings of the 46th IEEE Conference on Decision and Control New Orleans, LA, USA, Dec. 1-14, 007 Stabilization with Disturbance Attenuation over a Gaussian Channel J. S. Freudenberg, R. H. Middleton,
More informationTopic # Feedback Control Systems
Topic #19 16.31 Feedback Control Systems Stengel Chapter 6 Question: how well do the large gain and phase margins discussed for LQR map over to DOFB using LQR and LQE (called LQG)? Fall 2010 16.30/31 19
More informationLecture 9 Nonlinear Control Design
Lecture 9 Nonlinear Control Design Exact-linearization Lyapunov-based design Lab 2 Adaptive control Sliding modes control Literature: [Khalil, ch.s 13, 14.1,14.2] and [Glad-Ljung,ch.17] Course Outline
More informationDesign and Tuning of Fractional-order PID Controllers for Time-delayed Processes
Design and Tuning of Fractional-order PID Controllers for Time-delayed Processes Emmanuel Edet Technology and Innovation Centre University of Strathclyde 99 George Street Glasgow, United Kingdom emmanuel.edet@strath.ac.uk
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 informationExercise 1 (A Non-minimum Phase System)
Prof. Dr. E. Frazzoli 5-59- Control Systems I (HS 25) Solution Exercise Set Loop Shaping Noele Norris, 9th December 26 Exercise (A Non-minimum Phase System) To increase the rise time of the system, we
More information3 Gramians and Balanced Realizations
3 Gramians and Balanced Realizations In this lecture, we use an optimization approach to find suitable realizations for truncation and singular perturbation of G. It turns out that the recommended realizations
More informationFRTN10 Multivariable Control Lecture 1
FRTN10 Multivariable Control Lecture 1 Anton Cervin Automatic Control LTH, Lund University Department of Automatic Control Founded 1965 by Karl Johan Åström (IEEE Medal of Honor) Approx. 50 employees Education
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Illinois Institute of Technology Lecture 23: Drawing The Nyquist Plot Overview In this Lecture, you will learn: Review of Nyquist Drawing the Nyquist Plot Using
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 informationSystem identification and sensor fusion in dynamical systems. Thomas Schön Division of Systems and Control, Uppsala University, Sweden.
System identification and sensor fusion in dynamical systems Thomas Schön Division of Systems and Control, Uppsala University, Sweden. The system identification and sensor fusion problem Inertial sensors
More informationEECE 460. Decentralized Control of MIMO Systems. Guy A. Dumont. Department of Electrical and Computer Engineering University of British Columbia
EECE 460 Decentralized Control of MIMO Systems Guy A. Dumont Department of Electrical and Computer Engineering University of British Columbia January 2011 Guy A. Dumont (UBC EECE) EECE 460 - Decentralized
More informationSystems Analysis and Control
Systems Analysis and Control Matthew M. Peet Arizona State University Lecture 6: Generalized and Controller Design Overview In this Lecture, you will learn: Generalized? What about changing OTHER parameters
More informationEEE582 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 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 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 informationLecture 6 Classical Control Overview IV. Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore
Lecture 6 Classical Control Overview IV Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Lead Lag Compensator Design Dr. Radhakant Padhi Asst.
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 informationPlan of the Lecture. Goal: wrap up lead and lag control; start looking at frequency response as an alternative methodology for control systems design.
Plan of the Lecture Review: design using Root Locus; dynamic compensation; PD and lead control Today s topic: PI and lag control; introduction to frequency-response design method Goal: wrap up lead and
More informationNumerical atmospheric turbulence models and LQG control for adaptive optics system
Numerical atmospheric turbulence models and LQG control for adaptive optics system Jean-Pierre FOLCHER, Marcel CARBILLET UMR6525 H. Fizeau, Université de Nice Sophia-Antipolis/CNRS/Observatoire de la Côte
More informationEngineering Tripos Part IIB Nonlinear Systems and Control. Handout 4: Circle and Popov Criteria
Engineering Tripos Part IIB Module 4F2 Nonlinear Systems and Control Handout 4: Circle and Popov Criteria 1 Introduction The stability criteria discussed in these notes are reminiscent of the Nyquist criterion
More informationLecture 7 (Weeks 13-14)
Lecture 7 (Weeks 13-14) Introduction to Multivariable Control (SP - Chapters 3 & 4) Eugenio Schuster schuster@lehigh.edu Mechanical Engineering and Mechanics Lehigh University Lecture 7 (Weeks 13-14) p.
More informationLecture 1: Feedback Control Loop
Lecture : Feedback Control Loop Loop Transfer function The standard feedback control system structure is depicted in Figure. This represend(t) n(t) r(t) e(t) u(t) v(t) η(t) y(t) F (s) C(s) P (s) Figure
More information