EECS 562AE 551 NONLINEAR SYSTEMS AND CONTROL Instructor: Prof. Jessy W. Grizzle, 4421 EECS Bldg., (734) 763-3598,
[email protected] Text: Hassan Khalil, Nonlinear Systems, Third Edition, Prentice Hall, 2002 Assumed background: EECS 560 and Matlab. Lectures: Tu-Th 12:10 PM to 1:30 PM (no break) Final Exam: 4:10 PM to 6 PM, Thursday, April 21, 2016. By CoE policy, the time of the final exam cannot be changed. Outline: The course will be roughly 20% mathematical basics and fundamentals of differential equations, 50% stability theory, and 30% feedback design. The field of nonlinear control system analysis and design is extremely vast, and is still very much incomplete. A lot is known, and much, much more is unknown. We will go through many proofs. I will try my best to illustrate ideas on examples. Many of the examples will seem simplistic as they will be chosen to illustrate concepts or computations AND to be doable on a whiteboard or a HW set. You will be able to find many papers in the literature where the material we cover is used in real engineering systems. My web site has many examples in automotive systems and robotics, for example. Everything we cover will have practical implications, even though you may not see this right away. 1. Fundamental Theory (Existence of solutions of ODEs, uniqueness, continuity, real analysis, etc.) 2. State Space Stability Theory (First and Second Methods of Lyapunov) 3. Nonlinear State Variable Feedback Design (Control Lyapunov Functions, Backstepping, Feedback Linearization) 4. Classic Input Output Techniques (Absolute Stability, Small Gain Theorem, Describing Functions.) Joshi. Nonlinear Systems Analysis. You will have at least two weeks warning of when the midterm exam will take place. Prentice Hall S. 1996. Khalil. Pearson. There is no oral report.. H. Slotine and W. A. Kanellakopoulos and P. . Pend. The final is at the time set by the registrar. Wiley Interscience. Nonlinear Systems Analysis. third edition. Applied Nonlinear Control. R. Nonlinear and Adaptive Control Design. Project 10% Exams 80% (equally weighted) There will be one (1) midterm exam and the final. and a written report to turn in. E.] M. Stability. Springer Lecture Notes in Control and Information Sciences 221.Grading: Weekly homework 10% Inv. we will design nonlinear controllers that perform much better! Other references for nonlinear control design and analysis: these are not on reserve. Control of Nonlinear Multibody Flexible Structures. see below. Robust Control of Nonlinear Systems. Prentice Hall. No late HW accepted. Vidyasagar. Kelkar and S. you will have very precise instructions on what to do. Two lowest HW set scores will be dropped. It uses linearization methods to control the classic inverted pendulum on a cart. Li. J. The project is like an extended homework problem. EECS 560) to a nonlinear system. Nonlinear Control. 1995. The midterm exam will be of two hours duration and will be given at night. Freeman and P. Isidori. 1995. and Control. 1996.e. Kokotovic. It focuses much more on feedback control and less on stability analysis.-J. Second Edition. Springer-Verlag. Kokotovic. Some of you may like this book. Nonlinear Control Systems. 2015 [A newer textbook by our textbook’s author. In later HW sets. Kristic. Birkhauser. 1999. Springer. Springer. The goal is for you to apply what you learned in linear systems theory (i. M. A. Sastry. I. 1993. Boston. It takes repetition and practice. 1996. G. Springer. Here is one very good method: a) Read the book before coming to class. Wiley Interscience. P. M.J. b) Take good notes in class. van der Schaft. Just scan the material to see what are the general concepts and which part is the most difficult for you. or dance. it’s your money. Kokotovic. Decorum: a) Arrive on time. Entering late bothers me and other students in the class. 1997. My expectations: This is included due to apparent confusion in past classes. Learning the material is not so different than learning a new game. c) Sitting in the back of the room and exchanging comments with your neighbor is very visible to me. put away your cell phones. Tao and P. you can do as you please. as long as you are not disrupting the class. Khalil and J. Jankovic and P. Kokotovic. Academic Press. 1990. Remark: Various studies have shown that sitting in the front of the room increases the odds of you focusing on the material at hand. Constructive Nonlinear Control. H. Sepulchre. b) Once class has started. O’Reilly. 1986. Kokotovic. sport. This will give you a high probability of storing the material in your long term memory instead of in your short term memory. the instructor at the front of the room. Nijmeijer and A. Nonlinear Dynamical Control Systems. Singular Perturbation Methods in Control: Analysis and Design. Following the above gives you a minimum of four passes through the material. Another good idea is to summarize the key parts of the lecture in half a page or less. c) Re-read the appropriate part of the textbook after attending the lecture. Re-read your notes before going to the next class. This time. Some students find it advantageous to re-copy their notes after class and to add remarks in the margins. spend extra effort to learn the main ideas. . New York. But hey. R. H. Adaptive Control of Systems with Actuator and Sensor Nonlinearities. Springer-Verlag. if you do not finish one or two HW sets on time. HW solutions will be posted promptly on the day HW is due. 3) Because of (2) above. d) zero (0) points only if the problem was not attempted. a class trip. c) one (1) point if the problem was attempted. your goal is to understand the reasoning employed in the solution of the problem. b) two (2) points if there are several minor errors or at least one major error. but the reasoning is quite flawed. Each week you should compare your solutions to the handed out solutions. due to travel or illness. no late HW will be accepted. or they are just extra busy that week. each problem will be worth 3 points. extremely incomplete. though it may. a conference. 2) I will drop your two lowest HW scores. Each HW set will have a varying number of problems and hence will be worth a varying number of points. Scoring: Each problem is graded as follows: a) three (3) points if the problem is perfectly correct or nearly so. "nearly so" is a subjective evaluation. in the long run. clear. Everyone has the same opportunity to miss two HW sets and it should not matter whether they have an ill family member. Hence. complete and correct. or if the solution was unreadable. it will not adversely affect your HW score. have an effect on your exam performance if you do not do the HW problems for yourself. . Further remarks: 1) As noted above. They do not have to match the solutions provided by the professor. no matter how good your excuse. I don't consider a single numerical mistake to be important if it doesn't change the basic problem nor lead to greatly simplified reasoning. Of course. but it is clear that you had a good idea of how to work the problem.EECS 562 HW GRADING SCHEME Policy: Homework solutions should be legible. you will need to construct a (full-order) Luenberger observer for a linear system: x˙ = Ax + Bu y = Cx This knowledge will not be used on any exam. That course does a lot of theory. Part of what EECS 560 gives you is mathematical maturity. and that complete controllabilty can be checked via the Kalman controllability rank condition: rank [B | AB | · · · | A(n−1) B] = n. 2. when it is asymptotically stable in the sense of Lyapunov. and Nyquist plots. With this many students. transfer matrices. it is going to be really hard to get special help from me.EECS 562 Handout: Grizzle Required Background 1. So does EECS 562. with x0 ∈ Rn . 8. with x ∈ Rn and u ∈ Rm . 4. We do not use much from undergrad control. If you have not had that course. I will do my best. you will be ne. At the end of the term. Then in the middle of the term. Late in the course. The enrollment this term is huge. the HW at a medium high level. You have to earn it. If you have completed the graduate linear systems course at Michigan. I am not going to police your background. I do lots of proofs in lecture. 5. When does there exist a linear state variable feedback u = F x such that the closed-loop system x˙ = (A + BF )x is asymptotically stable? It is enough to know that a su cient condition is for the system to be completely controllable. and the exams. then you have even more mathematical maturity than is acquired in EECS 560. For the PROJECT. You need to learn when a linear time-invariant model is stable in the sense of Lyapunov. So. but no matter how much I may wish to help you. If you have taken ROB 501. 7. then I may be empathetic to your case. there are a couple more hard things where proofs are required. do not require that you provide proofs. Then toward the end. Also. The rst half of the term. If it is used in HW. it will not possible to spend much time with any one person. then you have adequate background and do not need to read further. during a two week period. The conditions are given in terms of the e-values of A and its Jordan canonical form. The list is not so long. we do more design. Stability of a linear time-invariant state variable model: x˙ = Ax. while challenging. I have never had more than 60 students in the course before. I will try to list the key things you should learn on your own. I keep the lectures at a high level. There will be Multiple Choice or T/F questions that test your 1 . and when it is unstable. it is a very minor part of a HW. 6. if you say that you are failing because you do not have adequate background. Stabilizability of a linear time-invariant control system: x˙ = Ax + Bu. but I will not be able to be sympathetic! That is a sad thing to say in writing!! I cannot just give you a passing grade. EECS 560 = ME 564 = AERO 550. we will use transfer functions. you can compute the feedback using the place command in Matlab. so in that regard. it is very important that you have adequate background to take the course. 3. There will also be work-out problems where you have to demonstrate the ability to apply the concepts of the course to simple examples. you need to learn on your own the topics listed above. I welcome you to take it.knowledge of the theory. 9. I think this is a great course. HOWEVER. if you have not yet completed the graduate linear systems course. It is NOT enough to be taking the course concurrently with EECS 562. 2 .