# Optimal control system notes

By | 11.10.2020

Optimal control theory is a branch of applied mathematics that deals with finding a control law for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in both science and engineering. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure. Optimal control is an extension of the calculus of variationsand is a mathematical optimization method for deriving control policies.

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost function.

The optimal control can be derived using Pontryagin's maximum principle a necessary condition also known as Pontryagin's minimum principle or simply Pontryagin's Principle[4] or by solving the Hamilton—Jacobi—Bellman equation a sufficient condition.

We begin with a simple example. Consider a car traveling in a straight line on a hilly road. The question is, how should the driver press the accelerator pedal in order to minimize the total traveling time?

In this example, the term control law refers specifically to the way in which the driver presses the accelerator and shifts the gears.

The system consists of both the car and the road, and the optimality criterion is the minimization of the total traveling time. Control problems usually include ancillary constraints. For example, the amount of available fuel might be limited, the accelerator pedal cannot be pushed through the floor of the car, speed limits, etc. A proper cost function will be a mathematical expression giving the traveling time as a function of the speed, geometrical considerations, and initial conditions of the system.

Constraints are often interchangeable with the cost function. Another related optimal control problem may be to find the way to drive the car so as to minimize its fuel consumption, given that it must complete a given course in a time not exceeding some amount. Yet another related control problem may be to minimize the total monetary cost of completing the trip, given assumed monetary prices for time and fuel.

Furthermore, it is noted that the path constraints are in general inequality constraints and thus may not be active i.

It is also noted that the optimal control problem as stated above may have multiple solutions i. A special case of the general nonlinear optimal control problem given in the previous section is the linear quadratic LQ optimal control problem. The LQ problem is stated as follows. Minimize the quadratic continuous-time cost functional. A particular form of the LQ problem that arises in many control system problems is that of the linear quadratic regulator LQR where all of the matrices i.

The LQR problem is stated as follows.

## Lecture Notes in Control and Information Sciences

Minimize the infinite horizon quadratic continuous-time cost functional. Note that the LQ or LQR cost functional can be thought of physically as attempting to minimize the control energy measured as a quadratic form. The infinite horizon problem i. This is indeed correct. However the problem of driving the output to a desired nonzero level can be solved after the zero output one is.Advanced Control Systems Notes — Topics covered.

Direct method of Lypanov for the Linear and Nonlinear continuous time autonomous systems. Full order observer and reduced order observer. Minimum principle. Control variable inequality constraints.

Control and state variable inequality constraints. Euler Lagrangine Equation. Minimum time, Minimum energy, minimum fuel problems.

State regulator problem. Output regulator problem. Tracking problem, Continuous-Time Linear Regulators. Modern Control Engineering — by K. Ogata, Prentice Hall of India, 3rd edition, 2.

Control Systems Engineering by I. Nagarath and M. Systems and Control by Stainslaw H. ZakOxford Press, To Support us please like our fb page: fb. Average rating 4. Vote count: No votes so far! Be the first to rate this post. Tags acs note acs note advanced control systems pdf advance control system notes Advanced Control Systems. Your email address will not be published. Table of Contents.

How useful was this post? Click on a star to rate it! Leave this field empty.The following lecture notes are made available for students in AGEC and other interested readers. An updated version of the notes is created each time the course is taught and will be available at least 48 hours before each class. Check the date at the top of each set of notes; you may be looking at last year's version.

If taking the course, you will want to wait for an updated version to be created before printing the notes. If they are not available in time, printed copies will be provided in class. Here's the tentative calendar for the current semester. These notes provide an introduction to optimal control and numerical dynamic programming. For a more complete treatment of these topics, please consult the books listed on the syllabus.

Lecture 1. Lecture 2. Lecture 3. Lecture 4. Lecture 5. Lecture 6. Lecture Lecture 7 Video of example. Lecture 8. Lecture 9. Matlab Tutorial. VB Tutorial :. Supplementary notes on Eigen values Nice Khan Academy video on eigenvalues. Dorfman The meaning of lambda video 11 minutes. Hotelling Hartwick Optimal control with constraints, bang-bang and most rapid approach path MRAP problems. Richard Bellman on the Birth of Dynamic Programming.

Operations Research 50 1 Powell especially chapter 4, and Hull Computer Programming Tutorials.Cite This Course. Don't show me this again. This is one of over 2, courses on OCW. Find materials for this course in the pages linked along the left. No enrollment or registration. Freely browse and use OCW materials at your own pace.

There's no signup, and no start or end dates. Knowledge is your reward. Use OCW to guide your own life-long learning, or to teach others. We don't offer credit or certification for using OCW. Made for sharing. Download files for later. Send to friends and colleagues. Modify, remix, and reuse just remember to cite OCW as the source. This course studies basic optimization and the principles of optimal control. It considers deterministic and stochastic problems for both discrete and continuous systems.

The course covers solution methods including numerical search algorithms, model predictive control, dynamic programming, variational calculus, and approaches based on Pontryagin's maximum principle, and it includes many examples and applications of the theory. Archived versions:. Jonathan How. Spring For more information about using these materials and the Creative Commons license, see our Terms of Use.

Principles of Optimal Control. Graphical representation of a line search equation. Graph by Jonathan How. Instructor s Prof. Jonathan P. Some Description Instructor s Prof.

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Need help getting started? Don't show me this again Welcome! Course Description Course Features Lecture notes Assignments: problem sets no solutions Assignments: programming no examples Exams no solutions Course Description This course studies basic optimization and the principles of optimal control.Teaching Format and Key Resources. Lecture Meetings.

Home Assignments. Relevant Literature. Comments and Additional Resources. Presentation 2h. The various aspects of control are be outlined. You Tube Watt's fly ball governor. WolframDemo : InvertedPendulumControls. Related on-line lectures from Stanford.

Presentation 4h. The objectives facing control engineers are taught together with an illustration of PID controllers as well as the Nyquist stability criterion. WolframDemo : SecondOrder. Control Systems with Mathematica. Linear State-Space Systems and Control. MIMO Linear Systems are studied together with controllability, observability and their generalizations. State feedback controllers and the Luenberger observers are studied in depth.

Note: it is recommended to do HW5 first for those who are doing Quiz 3. WolframDemo : 2 state system. WolframDemo : Phase Portraits. Board 3h. Lyapunov stability theory. This unit deals with both the notion of linearization about an equilibrium point and equally or more important with the notion of Lyapunov functions.

The best practice for it is HW5!!! Bruce Hajek's Notes book - see Section 6. Presentation 3h. This unit only presents the results of LQR optimal control problems without proving. Wolfram Demo: LQR. A highly cited survey paper on MPC. No one took photos during 9. Dynamic Programming.

Richard Weber's Optimization and Control Course useful notes in pdf. Wolfram Demo: Moon Landing. Board 2h. Link to a full course on this subject and generalizations by Nahum Shimkin.

Presentation 2h SlidesUnit Summary and outlook to further theory and practice. Some nice course summaries of students:. A workshop to attend The current course web-site is here.ISSN: You are not logged in! Please log in to edit your catalogs. Get the table of contents of every new volume published in Lecture Notes in Control and Information Sciences. About this series Titles in this series Titles also in This series reports new developments in the fields of control and information sciences—quickly, informally and at a high level.

### Principles of Optimal Control

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Hamiltonian Formulation for Solution of optimal control problem and numerical example

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Import List. Read this Series on Springerlink. Look inside Available online in SpringerLink. For authors and editors. Services for the Series. Contacts Contacts. Responsible Editor, Oliver Jackson. Alerts for this Series.The art is to pull all that together in a way that benefits you the most, without giving Google a reason to place less trust in your site. Back To Table Of Contents Any link that sends you traffic could be seen as a good link, but what are quality links, what are good links and what are crap links, when it comes to improving the actual ranking of your pages in Google SERPs.

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