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# ADV NONLINEAR CNTRL E C E 874

Clemson

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This 109 page Class Notes was uploaded by Eloy Ferry on Saturday September 26, 2015. The Class Notes belongs to E C E 874 at Clemson University taught by Staff in Fall. Since its upload, it has received 124 views. For similar materials see /class/214317/e-c-e-874-clemson-university in ELECTRICAL AND COMPUTER ENGINEERING at Clemson University.

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2E1262 2006 2E1262 Nonlinear Control Automatic Control 0 Disposition 5 credits Ip 2 28h lectures 28h exercises 3 homeworks o Instructors Bo Wahlberg professor boeekthse Krister Jacobsson teaching assistant kristerjacobssoneekthse STEX entrance floor in 83 building administration stexs3kthse Lecture 1 2E1262 2006 Course Goal To provide participants With a solid theoretical foundation of nonlinear control systems combined With a good engineering understanding You should after the course be able to o understand common nonlinear control phenomena 0 apply the most powerful nonlinear analysis methods 0 use some practical nonlinear design methods Lecture 1 2 2E1262 2006 2E1262 Nonlinear Control Lecture 1 0 Practical information 0 Course outline 0 Nonlinear control phenomena 0 Nonlinear differential equations Lecture 1 8 2E1262 2006 Today s Goal You should be able to o Recognize some phenomena in nonlinear systems a Transform differential equation to firstorder form 0 Mathematically describe saturation deadzone relay backlash o Derive equilibrium points Lecture 1 4 2E1262 2006 Course Information 0 All info and handouts are available at httpwwws3kthsecontrolkurser2E1262 0 Get an E computer account for the computer exercise see STEX o Homeworks are mandatory and have to be handed in on time Lecture 1 5 2E1262 2006 Material 0 Textbook Khalil Nonlinear Systems Prentice Hall 3rd ed 2002 Optional but highly recommended 0 Lecture notes Copies of transparencies o Exercises Class room and home exercises o Homeworks 3 computer exercises to hand in 0 Software Matlab provided by KTH Library Alternative textbooks decreasing mathematical brilliance Sastry Nonlinear Systems Analysis Stability and Control Vidyasagar Nonlinear Systems Analysis Slotine amp Li Applied Nonlinear Control Glad amp Ljung Regerteori flervariabla och olinja39ra metoder Only references to Khalil will be given Two course compendia sold by STEX Lecture 1 6 2E1262 2006 Course Outline 0 Introduction nonlinear models computer simulation L1L2 0 Feedback analysis linearization stability theory describing function L3L6 0 Control design compensation highgain design Lyapunov methods L7L10 0 Alternatives gain scheduling optimal control neural networks fuzzy control L11L13 0 Summary L14 Lecture 1 7 2E1262 2006 Linear Systems Definition Let M be a signal space The system S M gt M is linear if for all u v E M and 04 E R 8au aSu scaling Su v 81 superposition Example Linear timeinvariant systems A5130 But yt Cat 5130 O t W get W glt7gtultt new 0 Ys G8Us Notice the importance to have zero initial conditions Lecture 1 2E1262 2006 Linear Systems Have Nice Properties Local stabilityglobal stability Stability if all eigenvalues of A or poles of Gs are in the left halfplane Superposition Enough to know a step or impulse response Frequency analysis possible Sinusoidal inputs give sinusoidal outputs Yz w GiwUiw Lecture 1 2Et262 2006 Linear Models are not Enough ExamplePositioning of a ball on a beam Nonlinear model mit my sin 4575 Linear model 94505 Lecture t 2E1262 2006 Can the ball move 01 meter in 01 seconds from steady state Linear model step response with gb gbo gives t2 5130 m 105 m 005 so that O 1 0 2rad1140 Unrealistic answer Clearly outside linear region Linear model valid only if Sin gb m gb Must consider nonlinear model Possibly also include other nonlinearities such as centripetal force saturation friction etc Lecture 1 2E1262 2006 2 minute exercise Find a simple system 5139 f as u that is stable for a small input step but unstable for large input steps Lecture 1 12 2E1262 2006 Stability Can Depend on Reference Signal Example Control system with valve characteristic f U2 Motor Valve Process Lo l a a 1 y s s l l2 Simulink block diagram 1 s1s1 Lecture 1 2E1262 2006 STEP RESPONSES Tirnet 15 20 25 30 Time 1 Stability depends on amplitude of the reference signal The linearized gain of the valve increases with increasing amplitude Lecture 1 14 2E1262 2006 Stable Periodic Solutions Example Position control of motor with backlash Constant Sum P controller Motor Cs m Controller K 5 Lecture 1 2E1262 Backlash induces an oscillation Frequency and amplitude independent of initial conditions How predict and avoid oscillations Lecture 1 2006 2E1262 2006 Automatic Tuning of PID Controllers Relay induces a desired oscillation whose frequency and amplitude are used to choose PID parameters Time Lecture 1 17 2E1262 2006 Harmonic Distortion Example Sinusoidal response of saturation a sint yt 221 Ak sinkt Saturation gt 10 61 1 1 2 1 O 1 2 3 4 5 Time t Amplitude y 10 Frequency HZ 10 61 2 1 1 1111111 2 0 1 2 3 4 5 Time t Amplitude y O 1 m 10 Frequency HZ Lecture 1 2E1262 2006 Example Electrical power distribution Nonlinearities such as rectifiers switched electronics and transformers give rise to harmonic distortion 222 Energy in tone k Total Harmonic Distortion Energy In tone 1 Example Electrical amplifiers Effective amplifiers work in nonlinear region Introduces spectrum leakage which is a problem in cellular systems Tradeoff between effectivity and linearity Lecture 1 19 2E1262 2006 Subharmonics Example Duffing s equation y 3 y 33 a Sinwt 05 A o 05 o 5 1o 15 20 25 30 Time If 1 05 C1 0 rr 1 m 605 1 o 5 1o 15 20 25 30 Time 2 Lecture 1 20 2E1262 2006 Nonlinear Differential Equations Definition A solution to 9315 11930 930 030 1 over an interval 07 T is a C1 function a 07 T gt Rquot such that 1 iS fulfilled 0 When does there exists a solution 0 When is the solution unique Example 5139 Ax 5130 5130 gives 513t eXpAta0 Lecture 1 21 2E1262 2006 Existence Problems Example The differential equation 5139 5132 5130 5130 a 1 has solution 513t 0 0 g t lt 1 025 0 1 Solution not defined for if 5130 Solution interval depends on initial condition 2 da Recall the trick 1 a gt 2 dt 1 1 1 0 Integrate gt 15 gt 51315 1 270t WW Lecture 1 22 2006 2E1262 me te Escape T39 ini F Simulation for various initial conditions 5130 X2 Finite escape time of dxdt 28 Lecture 1 2E1262 2006 Uniqueness Problems Example 5139 5130 O has many solutions t C24 tgtC xt 0 th Lecture 1 24 2E1262 2006 Physical Interpretation Consider the reverse example ie the water tank lab process with d 93T 0 where a is the water level It is then impossible to know at what time t lt T the level was 513t 510 gt O Hint Reverse time s T 25 gt d5 dt and thus da da d5 dt Lecture 1 25 2E1262 2006 Lipschitz Continuity Definition f Rquot gt Rquot is Lipschitz continuous if there exists L 7 gt 0 such that for all 51331 6 B74930 z E Rquot 930 lt 7 f93 fy S Lllx yll Slope L Euclidean norm is given by 93293 95i Lecture 1 26 2E1262 2006 Local Existence and Uniqueness Theorem If f is Lipschitz continuous then there exists 5 gt 0 such that 9315 f93t7 930 I 930 has a unique solution in B74330 over 0 6 Proof See Khalil Appendix 01 Based on the contraction mapping theorem Remarks 0 6 67 L o f being 00 is not sufficient cf tank example 0 f being 01 implies Lipschitz continuity Lecture 1 27 2E1262 2006 StateSpace Models State as input u output y General fa u y 13 a y O Explicit d fa y Affine in u 51 y Linear 51 Ax Bu y Ca Lecture 1 28 2E1262 2006 Transformation to Autonomous System A nonautonomous system dfat is always possible to transform to an autonomous system by introducing xn1 t it fx7xn1 in l l 1 Lecture 1 29 2E1262 2006 Transformation to FirstOrder System Given a differential equation in y with highest derivative T n l express the equation In a y Example Pendulum MR2939 k MgRsin6 0 a 6 gives 33931 512 9 SlIl l71 x2 M R2x2 R Lecture 1 80 2E1262 2006 Equilibria Definition A point 5137 u y is an equilibrium if a solution starting in 513 u y stays there forever Corresponds to putting all derivatives to zero General Explicit Affine in u Linear fa3uy00 O 0 fx u y M95quot 0 fx 995u y M93quot 0 1433 Bu y 0513 Often the equilibrium is defined only through the state 513 Lecture 1 81 2E1262 2006 Multiple Equilibria Example Pendulum MR2939 k MgRsin6 0 59Ogivessin60andthu36 k7t Alternatively in firstorder form 171 272 is g a2 8m 5131 M R2 R 33931 33932 0 gives 513 O and sina f 0 51 32 Lecture 1 82 2E1262 2006 Some Static and Dynamic Nonlinearities gt u gt gt eu gt gt Abs F Math Saturation unctlon gt gt gt gt gt 3 gt Sign Dead Zone Loggglgp gt gt gt gt gt F gt Relay Backlash Coulomb amp Viscous Friction Lecture 1 88 2E1262 2006 2 minute exercise Construct a rate limiter ie a system that limit the rate of change of a signal using one of the previous models l s T Hint Try a saturation Lecture 1 84 2E1262 Next Lecture 0 Simulation in Matlab o Linearization Lecture 1 2006 85 2E1262 2006 2E1262 Nonlinear Control Lecture 14 o 2E1262 Nonlinear Control revisited 0 Spring courses in control 0 Master thesis projects Lecture 14 1 2E1262 2006 Exam 0 Regular written exam in English with five problems 0 Sign up on course web page 0 You may bring lecture notes Glad amp Ljung Reglerteknik and TEFYMA or BETA No other material textbooks exercises calculators etc Any other basic control book must be approved by me before the exam 0 See homepage for old exams Lecture 14 2 2E1262 2006 Question 1 What s on the exam 0 Nonlinear models and equilibria linearization and stability 0 Lyapunov functions local and global stability and LaSalle 0 Circle Criterion Small Gain Theorem Passivity Theorem 0 Describing functions 0 Sliding modes 0 Backstepping o Exact feedback linearization 0 Nonlinear controllability 0 Optimal control 0 Fuzzy control Warning Not necessarily the same distribution as earlier exams Lecture 14 8 2E1262 2006 Question 2 What design method should I use in practice Exists no simple answer see AK for design schemes Hints 0 Start with simplest approach Linear methods PID 0 Be critical Which are the assumptions Varying operating conditions Analyze and simulate with nonlinear model 0 Some nonlinearities to compensate for Saturations valves etc o Is the system generically nonlinear Example 13 am Lecture 14 2E1262 2006 Question 3 Can the system be stable if I use the Small Gain Theorem and unstable if I use the Circle Criterion o The system cannot be proved stable with one criterion and proved unstable with another a If the Small Gain Circle Criterion Passivity Theorems etc are not satisfied then we get no information from that particular method We then have to try some other method Lecture 14 5 2E1262 2006 Question 4 Can you review the circle criterion What about k1 lt 0 lt k2 Lecture 14 6 2E1262 2006 The Circle Criterion 29 y 19 i k U male 9 G Theorem Consider a feedback loop with y Gu and u Assume Cs is stable and that 761 S S k2 If the Nyquist curve of Cs stays on the correct side of the circle defined by the points 1k1 and 1k2 then the closedloop system is BIBO stable Lecture 14 7 2E1262 2006 The different cases Stable system G 1 O lt k1 lt k2 Stay outside circle 2 O k1 lt k2 Stay to the right of the line Re 5 1k2 3 k1 lt O lt k2 Stay inside the circle Other cases Multiply f and G with 1 Only Case 1 and 2 studied in lectures Only G stable studied Lecture 14 8 2E1262 2006 Question 5 Please repeat the most important facts about sliding modes Lecture 14 9 2E1262 2006 Sliding Modes The sliding mode is d af 1 Ozf where 04 satisfies 04fo 1 05 O for normal projections of f f Lecture 14 10 2E1262 2006 Sliding Mode Dynamics The dynamics along the sliding sun ace 023 0 is obtained by setting u ueq E 717 1 such that 71 0 ueq is called the equivalent control Fhase Plane Lecture 14 11 2E1262 2006 Example 9 31 1 714 3172 U u sgn 5132 Two approaches 1 Convex combination of fJr and f o 0szer 1 ozf Ogivesa 12 0 Hence 51 31 1 is the sliding dynamics 2 Equivalent control a x It gives ueq O 0 Hence 51 31 1 is the sliding dynamics Lecture 14 12 2E1262 Lecture 14 Question 6 Please repeat antiwindup 2006 2E1262 Lecture 14 Tracking PID m Actuator Actuator Actuator model 2006 2E1262 2006 Antiwindup General StateSpace Model Choose K such that F KC has stable eigenvalues Lecture 14 15 2E1262 Lecture 14 Question 7 Please repeat Lyapunov theory 2006 2E1262 2006 Stability Definitions An equilibrium point a O of 51 is locally stable if for every R gt 0 there exists 7 gt 0 such that 930 lt 7 gt 93t lt R7 152 0 locally asymptotically stable if locally stable and 930 lt 7 gt tag 031 I 0 globally asymptotically stable if asymptotically stable for all 5130 6 Rquot Lecture 14 2E1262 2006 Lyapunov Theorem for Local Stability Theorem Let d f0 O and 0 E Q C Rquot Assume that V 9 gt R is a 01 function If e VO O o Va gt Oforalla 6 251 7amp0 o g 0 along all trajectories in Q then a O is locally stable Furthermore if e lt Oforalla 6 251 75 0 then a O is locally asymptotically stable Lecture 14 2E1262 2006 Lyapunov Theorem for Global Stability Theorem Let d and f0 0 Assume that V Rquot gt R is a 01 function If VO O Vygtammm0 V mlt0mmm0 Va gt 00 as gt 00 then a O is globally asymptotically stable Lecture 14 19 2E1262 2006 LaSalle s Invariant Set Theorem Theorem Let Q E Rquot be a bounded and closed set that is invariant with respect to at f 9 Let V Rquot gt R be a 01 function such that g 0 for a E 2 Let E be the set of points in Q where 0 If M is the largest invariant set in E then every solution with 5130 6 Q approaches M ast gt 00 f N Lecture 14 2O 2E1262 2006 Question 8 Stability theorems in Lecture 5 all guarantee BIBO stability Are there ways to extend these results to guarantee local asymptotic stability of the system What extra conditions on the input signals and the system must be added Lecture 14 21 2E1262 2006 Yes there exist connections between BIBO and Lyapunov stability For example if the system Cs is controllable and observable there is a proof of the Circle Criterion using the Lyapunov function Va asTPas P gt O which proves global asymptotic stability Similar statement is true for the Small Gain Theorem Some further discussion in the end of Lecture 5 Lecture 14 22 2E1262 Lecture 14 Question 9 Repeat backlash compensation 2006 28 2E1262 2006 Backlash Compensation 0 Deadzone 0 Linear controller design 0 Backlash inverse Linear controller design Phase lead compensation 8ref 6 K18T2 u 1 6m 1 6in 7 gout 18T1 8T 4 Lecture 14 24 2E1262 2006 0 Choose compensation F s such that the intersection with the describing function is removed Fs 13 with T1 05 T2 20 Nyquisl Diagrams 1 8 1 6 4 0395 y withwithout filter 9 2 i 0 5 10 15 20 g 0 E 7 2 4 with filter 6 8 without filter u withwithout filter 40 5 0 5 1 o 5 1o 15 20 Oscillation removed 25 Lecture 14 2E1262 2006 Question 10 What should we know about input output stability You should understand and be able to deriveapply y2 u2 a System gain yS SUpue 2 BIBO stability 0 Small Gain Theorem Circle Criterion Passivity Theorem Lecture 14 26 2E1262 Lecture 14 Question 11 What about describing functions 2006 27 2E1262 2006 Idea Behind Describing Function Method T e NL UgtGs y 82 A sin wt gives ut Z a b sinnwt arctananbn n21 If ltlt for n 2 2 then n 1 suffices so that yt a b sinwt arctana1b1 erg Giw Lecture 14 28 2E1262 2006 Definition of Describing Function The describing function is NAw A 803 NL J05 gtNAw gt1t If G is low pass and a0 0 then 105 NAwAsinwt argNAw ut Lecture 14 29 2E1262 2006 Existence of Periodic Solutions Giw 0 i 1NA 64 y Gz wu Gz39wNAy CW The intersections of the curves Ciw and 1NA give w and A for a possible periodic solution Lecture 14 80 2E1262 2006 Stability of Periodic Solutions J 9 am J 4 1N 4 Assume Cs is stable 0 If CQ encircles the point 1NA then the oscillation amplitude is increasing 0 If CQ does not encircle the point 1NA then the oscillation amplitude is decreasing Lecture 14 81 2E1262 2006 Spring Courses in Control 0 2E1245 Hybrida och inbyggda reglersystem p3 Hybrid and embedded control systems a 2E1252 Reglerteknik Forts ttningskurs Ip 3 Advanced control a 2E1241 Reglerteknik Projektkurs Ip 4 Project course a 2E128O Modellering av Dynamiska System Ip 4 System modeling Lecture 14 82 2E1262 2006 2E1245 Hybrida och inbyggda reglersystem Embedded control AimNew course on analysis design and implementation of control algorithms in networked and embedded systems 0 Period 3 5 p o How is control implemented in reality Computerimplementation of control algorithms Scheduling of realtime software Control over communication networks 0 Lectures exercises homework computer exercises Contact Karl Henrik Johansson kallej s3 kth se Lecture 14 88 2E1262 2006 2E1252 Reglerteknik fortsattningskurs Advanced control Aim Provide an introduction to ideas and methods in advanced control especially multivariable feedback systems 0 Period 3 5 p o Multivariable control Linear multivariable systems Robustness and performance Design of multivariable controllers LQG H00 0 Lectures exercises labs computer exercises Contact Elling Jacobsen jacobsens3 kth se Lecture 14 34 2E1262 2006 2E1241 Reglerteknik projektkurs Project Aim Provide practical knowledge about modeling analysis design and implementation of control systems Give some experience in project management and presentation 0 Period 4 5 p c From start to goal apply the theory from other courses 0 Team work 0 Preparation for Master thesis project a Project management lecturers from industry a No regular lectures or labs Contact Karl Henrik Johansson kallej s3 kth se Lecture 14 85 2E1262 2006 2E1280 Modellering av dynamiska system System modeling Aim Teach how to systematically build mathematical models of technical systems from physical laws and from measured signals 0 Period 4 4 p 0 Model dynamical systems from physics lagrangian mechanics electrical circuits etc experiments parametric identification frequency response 0 Computer tools for modeling identification and simulation 0 Lectures exercises labs computer exercises Contact Bo Wahlberg bos3 kth se Lecture 14 86 2E1262 2006 Doing Master Thesis Project at S3 0 Theory and practice 0 Crossdisciplinary O The research edge 0 Collaboration with leading industry and universities 0 Get insight in research and development Hints o The topic and the results of your thesis are up to you 0 Discuss with professors lecturers PhD and MS students 0 Check old projects Lecture 14 87 2E1262 2006 2E1262 Nonlinear Control Lecture 14 o 2E1262 Nonlinear Control revisited 0 Spring courses in control 0 Master thesis projects Lecture 14 1 2E1262 2006 Exam 0 Regular written exam in English with five problems 0 Sign up on course web page 0 You may bring lecture notes Glad amp Ljung Reglerteknik and TEFYMA or BETA No other material textbooks exercises calculators etc Any other basic control book must be approved by me before the exam 0 See homepage for old exams Lecture 14 2 2E1262 2006 Question 1 What s on the exam 0 Nonlinear models and equilibria linearization and stability 0 Lyapunov functions local and global stability and LaSalle 0 Circle Criterion Small Gain Theorem Passivity Theorem 0 Describing functions 0 Sliding modes 0 Backstepping o Exact feedback linearization 0 Nonlinear controllability 0 Optimal control 0 Fuzzy control Warning Not necessarily the same distribution as earlier exams Lecture 14 8 2E1262 2006 Question 2 What design method should I use in practice Exists no simple answer see AK for design schemes Hints 0 Start with simplest approach Linear methods PID 0 Be critical Which are the assumptions Varying operating conditions Analyze and simulate with nonlinear model 0 Some nonlinearities to compensate for Saturations valves etc o Is the system generically nonlinear Example 13 am Lecture 14 2E1262 2006 Question 3 Can the system be stable if I use the Small Gain Theorem and unstable if I use the Circle Criterion o The system cannot be proved stable with one criterion and proved unstable with another a If the Small Gain Circle Criterion Passivity Theorems etc are not satisfied then we get no information from that particular method We then have to try some other method Lecture 14 5 2E1262 2006 Question 4 Can you review the circle criterion What about k1 lt 0 lt k2 Lecture 14 6 2E1262 2006 The Circle Criterion 29 y 19 i k U male 9 G Theorem Consider a feedback loop with y Gu and u Assume Cs is stable and that 761 S S k2 If the Nyquist curve of Cs stays on the correct side of the circle defined by the points 1k1 and 1k2 then the closedloop system is BIBO stable Lecture 14 7 2E1262 2006 The different cases Stable system G 1 O lt k1 lt k2 Stay outside circle 2 O k1 lt k2 Stay to the right of the line Re 5 1k2 3 k1 lt O lt k2 Stay inside the circle Other cases Multiply f and G with 1 Only Case 1 and 2 studied in lectures Only G stable studied Lecture 14 8 2E1262 2006 Question 5 Please repeat the most important facts about sliding modes Lecture 14 9 2E1262 2006 Sliding Modes The sliding mode is d af 1 Ozf where 04 satisfies 04fo 1 05 O for normal projections of f f Lecture 14 10 2E1262 2006 Sliding Mode Dynamics The dynamics along the sliding sun ace 023 0 is obtained by setting u ueq E 717 1 such that 71 0 ueq is called the equivalent control Fhase Plane Lecture 14 11 2E1262 2006 Example 9 31 1 714 3172 U u sgn 5132 Two approaches 1 Convex combination of fJr and f o 0szer 1 ozf Ogivesa 12 0 Hence 51 31 1 is the sliding dynamics 2 Equivalent control a x It gives ueq O 0 Hence 51 31 1 is the sliding dynamics Lecture 14 12 2E1262 Lecture 14 Question 6 Please repeat antiwindup 2006 2E1262 Lecture 14 Tracking PID m Actuator Actuator Actuator model 2006 2E1262 2006 Antiwindup General StateSpace Model Choose K such that F KC has stable eigenvalues Lecture 14 15 2E1262 Lecture 14 Question 7 Please repeat Lyapunov theory 2006 2E1262 2006 Stability Definitions An equilibrium point a O of 51 is locally stable if for every R gt 0 there exists 7 gt 0 such that 930 lt 7 gt 93t lt R7 152 0 locally asymptotically stable if locally stable and 930 lt 7 gt tag 031 I 0 globally asymptotically stable if asymptotically stable for all 5130 6 Rquot Lecture 14 2E1262 2006 Lyapunov Theorem for Local Stability Theorem Let d f0 O and 0 E Q C Rquot Assume that V 9 gt R is a 01 function If e VO O o Va gt Oforalla 6 251 7amp0 o g 0 along all trajectories in Q then a O is locally stable Furthermore if e lt Oforalla 6 251 75 0 then a O is locally asymptotically stable Lecture 14 2E1262 2006 Lyapunov Theorem for Global Stability Theorem Let d and f0 0 Assume that V Rquot gt R is a 01 function If VO O Vygtammm0 V mlt0mmm0 Va gt 00 as gt 00 then a O is globally asymptotically stable Lecture 14 19 2E1262 2006 LaSalle s Invariant Set Theorem Theorem Let Q E Rquot be a bounded and closed set that is invariant with respect to at f 9 Let V Rquot gt R be a 01 function such that g 0 for a E 2 Let E be the set of points in Q where 0 If M is the largest invariant set in E then every solution with 5130 6 Q approaches M ast gt 00 f N Lecture 14 2O 2E1262 2006 Question 8 Stability theorems in Lecture 5 all guarantee BIBO stability Are there ways to extend these results to guarantee local asymptotic stability of the system What extra conditions on the input signals and the system must be added Lecture 14 21 2E1262 2006 Yes there exist connections between BIBO and Lyapunov stability For example if the system Cs is controllable and observable there is a proof of the Circle Criterion using the Lyapunov function Va asTPas P gt O which proves global asymptotic stability Similar statement is true for the Small Gain Theorem Some further discussion in the end of Lecture 5 Lecture 14 22 2E1262 Lecture 14 Question 9 Repeat backlash compensation 2006 28 2E1262 2006 Backlash Compensation 0 Deadzone 0 Linear controller design 0 Backlash inverse Linear controller design Phase lead compensation 8ref 6 K18T2 u 1 6m 1 6in 7 gout 18T1 8T 4 Lecture 14 24 2E1262 2006 0 Choose compensation F s such that the intersection with the describing function is removed Fs 13 with T1 05 T2 20 Nyquisl Diagrams 1 8 1 6 4 0395 y withwithout filter 9 2 i 0 5 10 15 20 g 0 E 7 2 4 with filter 6 8 without filter u withwithout filter 40 5 0 5 1 o 5 1o 15 20 Oscillation removed 25 Lecture 14 2E1262 2006 Question 10 What should we know about input output stability You should understand and be able to deriveapply y2 u2 a System gain yS SUpue 2 BIBO stability 0 Small Gain Theorem Circle Criterion Passivity Theorem Lecture 14 26 2E1262 Lecture 14 Question 11 What about describing functions 2006 27 2E1262 2006 Idea Behind Describing Function Method T e NL UgtGs y 82 A sin wt gives ut Z a b sinnwt arctananbn n21 If ltlt for n 2 2 then n 1 suffices so that yt a b sinwt arctana1b1 erg Giw Lecture 14 28 2E1262 2006 Definition of Describing Function The describing function is NAw A 803 NL J05 gtNAw gt1t If G is low pass and a0 0 then 105 NAwAsinwt argNAw ut Lecture 14 29 2E1262 2006 Existence of Periodic Solutions Giw 0 i 1NA 64 y Gz wu Gz39wNAy CW The intersections of the curves Ciw and 1NA give w and A for a possible periodic solution Lecture 14 80 2E1262 2006 Stability of Periodic Solutions J 9 am J 4 1N 4 Assume Cs is stable 0 If CQ encircles the point 1NA then the oscillation amplitude is increasing 0 If CQ does not encircle the point 1NA then the oscillation amplitude is decreasing Lecture 14 81 2E1262 2006 Spring Courses in Control 0 2E1245 Hybrida och inbyggda reglersystem p3 Hybrid and embedded control systems a 2E1252 Reglerteknik Forts ttningskurs Ip 3 Advanced control a 2E1241 Reglerteknik Projektkurs Ip 4 Project course a 2E128O Modellering av Dynamiska System Ip 4 System modeling Lecture 14 82 2E1262 2006 2E1245 Hybrida och inbyggda reglersystem Embedded control AimNew course on analysis design and implementation of control algorithms in networked and embedded systems 0 Period 3 5 p o How is control implemented in reality Computerimplementation of control algorithms Scheduling of realtime software Control over communication networks 0 Lectures exercises homework computer exercises Contact Karl Henrik Johansson kallej s3 kth se Lecture 14 88 2E1262 2006 2E1252 Reglerteknik fortsattningskurs Advanced control Aim Provide an introduction to ideas and methods in advanced control especially multivariable feedback systems 0 Period 3 5 p o Multivariable control Linear multivariable systems Robustness and performance Design of multivariable controllers LQG H00 0 Lectures exercises labs computer exercises Contact Elling Jacobsen jacobsens3 kth se Lecture 14 34 2E1262 2006 2E1241 Reglerteknik projektkurs Project Aim Provide practical knowledge about modeling analysis design and implementation of control systems Give some experience in project management and presentation 0 Period 4 5 p c From start to goal apply the theory from other courses 0 Team work 0 Preparation for Master thesis project a Project management lecturers from industry a No regular lectures or labs Contact Karl Henrik Johansson kallej s3 kth se Lecture 14 85 2E1262 2006 2E1280 Modellering av dynamiska system System modeling Aim Teach how to systematically build mathematical models of technical systems from physical laws and from measured signals 0 Period 4 4 p 0 Model dynamical systems from physics lagrangian mechanics electrical circuits etc experiments parametric identification frequency response 0 Computer tools for modeling identification and simulation 0 Lectures exercises labs computer exercises Contact Bo Wahlberg bos3 kth se Lecture 14 86 2E1262 2006 Doing Master Thesis Project at S3 0 Theory and practice 0 Crossdisciplinary O The research edge 0 Collaboration with leading industry and universities 0 Get insight in research and development Hints o The topic and the results of your thesis are up to you 0 Discuss with professors lecturers PhD and MS students 0 Check old projects Lecture 14 87

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