#### Transcript Minimax Controllers

Continuous Control MAE 443/543 Closed Loop Control system Control input Reference Input + Output Process (Plant) Controller - Sensor Positive vs Negative Feedback Story “The distinction between the stabilizing and destabilizing character of negative and positive feedback loops is neatly captured in the story of the misconnected electric blanket. The newlyweds were given an electric blanket for their queensize double bed. The blanket had separate temperature settings for the two sides of the bed, one for him and one for her. Properly connected, there should have been two separate negative feedback systems, each attempting to control the temperature of the blanket for the comfort of each individual. The story goes that the newlyweds misconnected the blanket so that his setting controlled her blanket temperature and hers controlled his. The result…was a nasty positive feedback system. She felt cold, turned up her setting, making his side too warm for him so he turned down his setting, making her even colder, so she raised her setting even further, and so on. How such a scenario would end is left up to the fertile imagination of the reader. -Richardson, G. and Pugh, A. Introduction to System Dynamics Modeling with Dynamo, MIT Press, Cambridge, MA 1981, pp 11-12 Centrifugal Governor http://en.wikipedia.org/wiki/Centrifugal_governor Centrifugal governor, Boulton & Watt, 1798 http://en.wikipedia.org/wiki/James_Watt Controlled Flight The key innovation, which had no clear predecessor, was the use of wing warping to effect lateral control; that is, control for turning (Fig. 7). This innovation provided a full complement of movable aerodynamic surfaces to allow control over all three axes of rotational motion. This innovation was critical, since it made controlled flight possible. Feedback Control: An Invisible Thread in the History of Technology, Dennis Bernstein IEEE Control System Magazine, April 2002 MagLev Train http://www.o-keating.com/hsr/mlx01.htm The principal of a Magnet train is that floats on a magnetic field and is propelled by a linear induction motor. They follow guidance tracks with magnets. These trains are often referred to as Magnetically Levitated trains which is abbreviated to MagLev. The MLX01 ML stands for maglev, and X for experimental. The Aerodynamic brakes on the MLX01 Control of Epidemics • I(t): infective class • S(t): susceptive class • R(t): removed class • V(t): vaccinated individuals, is the vaccination rate Control of Epidemics 0.2 Infective Class I(t) 0.18 0.16 0.14 0.12 0.1 Without Vaccination With Vaccination 0.08 0.06 0.04 0.02 0 0 5 10 15 20 25 30 Time Introduction to Optimal Control, (pp 17-18) Jack Macki and Aaron Strauss Springer-Verlag, 1982 35 40 45 50 Tuned Mass Damped (TMD) http://www.oiles.co.jp/en/menshin/building/control/amd.html Hydrocephalus Therapy http://www.medit.hia.rwthaachen.de/en/research/hydrocephalus_implant/index.html The human brain is immersed in a fluid (cerebrospinal fluid, CSF), which, among other things, protects the brain from mechanical stress (e.g. concussion) and helps support its weight through buoyancy. In normal situations, the production and reabsorption of this fluid are equal. However, a constant overproduction, blockage (i.e. tumor) or reabsorption difficulty can upset this natural balance, resulting in a build-up of fluid in the skull (Hydrocephalus). In adults, this excess fluid causes large pressures to develop rapidly in the skull, and impairs brain function. The most common solution today is the implantation of a passive pressure-control valve and catheter system (shunt). Once the pressure in the skull exceeds a certain critical value, the excess fluid is released through the open valve and typically drained into the stomach cavity (Fig. 1). Unfortunately, these passive valves encounter many problems, including over- and under- drainage, occlusions, and system failure. These problems may be avoided through the use of a mechatronic valve, which could monitor the patient's health and properly regulate the amount of fluid in the skull through pressure, flow and inclination sensors. Stages in Control System Design Modeling Physics based model derivation Non-Physics based modeling System Identification mode structure/class selection parameter estimation model validation Analysis Stability, Controllability, … Control System Design Performance specification Synthesis of controller Simulation and testing System Models Dynamic Systems Distributed Parameter Lumped Parameter Stochastic Deterministic Continuous Time Discrete Time Nonlinear Time Varying Linear Time Invariant Online examples http://www.engin.umich.edu/group/ctm/ http://www.mathworks.com/applications/controldesign/ http://lorien.ncl.ac.uk/ming/Dept/Swot/connotes.htm http://www-control.eng.cam.ac.uk/extras/Virtual_Library/Control_VL.html http://www.jhu.edu/~signals/explore/index.html Example For the spring-mass dashpot system u m k y Force balance leads to the free body diagram m At initial time (t=0), we need the values of and to solve the differential equation which are referred to as initial conditions. Overview of Control Approaches • Classical Control: employs primarily frequency domain tools to achieve control objectives •MAE 443/543 (Continuous Control) •MAE 444/544 (Digital Control) • Modern Control: employs primarily time-domain tools • MAE 571 (System Analysis) • MAE 672 (Optimal Control) • MAE 670 (Nonlinear Control) •Post-Modern Control: integrate time-domain and frequency domain tools. Software Tools • MATLAB (MATrix LABoratory) powerful numerical software package with toolboxes for control, optimization, system identification, etc. http://www.engin.umich.edu/group/ctm/ •MAPLE (MAniPulator LanguagE) symbolic manipulator for analytically solving algebraic, differential equation and for linear algebra besides other functionality http://www.mapleapps.com/categories/whatsnew/html/SCCCmapletutorial.shtml •MATLAB is now bundled with MAPLE permitting the user to exploit the strengths of both packages Laplace Transform Let f(t) be a function of time t, such that f(t) = 0, for t < 0, then is the Laplace transform of f(t). s is a complex variable Example: Laplace Transform Example: Example (Step function): Laplace Transform of the Derivative of a Function Consider Let Laplace Transform of the Derivative of a Function Similarly Final and Initial Value Theorem Final Value Theorem: If f(t) and f(t), and if Consider Since are Laplace transformable, if F(s) is the Laplace transform of exists, then Final and Initial Value Theorem we have Thus, Initial Value Theorem: If f(t) and f(t), and if are Laplace transformable, if F(s) is the Laplace transform of exists, then Inverse Laplace Transform where Not easy. We use table to find the Inverse Laplace Transforms Partial Fraction method for Inverse Laplace Transform Let B(s), A(s) are polynomials in s with the order of A(s) > B(s). If F(s) can be represented as then Inverse Laplace Transform Example: (Distinct Poles) We can write where ak – residue at the pole s=pk. Example: Inverse Laplace Transform therefore, Example: This has to be written as Inverse Laplace Transform therefore, Example: (Multiple Poles) Inverse Laplace Transform Consider, Similiarly Matlab function Matlab command for determining residue for the transfer function G(s) = B/A: [R,P,K] = RESIDUE(B,A) [R,P,K] = residue([1 2 3],[1 3 3 1]) R= 1.0000 0.0000 2.0000 P= -1.0000 -1.0000 -1.0000 K= []