ESE 601: Hybrid Systems - Rensselaer Polytechnic Institute

ESE 601: Hybrid Systems - Rensselaer Polytechnic Institute

ESE 601: Hybrid Systems Review material on continuous systems I Spring semester 2006 References Kwakernaak, H. and Sivan, R. Modern signal and systems, Prentice Hall, 1991. Brogan, W., Modern control theory, Prentice Hall Intl,

1991. Textbooks or lecture notes on linear systems or systems theory. Contents

Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concept Simulation and numerical methods State space representation

Stability Reachability Physical systems Resistor Damper Inductor

Mass Capacitor Spring Electric circuit I(t) I(t) 1

+ V L t V(t) L 0

t More electric circuit L R + I(t)

V C A pendulum r Mg

Contents

Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concept Simulation and numerical methods State space representation Stability Reachability

Linear vs nonlinear Linear systems: if the set of solutions is closed under linear operation, i.e. scaling and addition. All the examples are linear systems, except for the pendulum. Time invariant vs time varying Time invariant: the set of solutions is closed under time shifting.

Time varying: the set of solutions is not closed under time shifting. Autonomous vs non-autonomous Autonomous systems: given the past of the signals, the future is already fixed. Non-autonomous systems: there is possibility for input, non-determinism.

Contents

Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concept Simulation and numerical methods State space representation Stability Reachability

Techniques for autonomous systems Techniques for non-autonomous systems Techniques for non-autonomous systems Example: u(t) y(t)

1 1 t t Contents

Modeling with differential equations Taxonomy of systems

Solution to linear ODEs General solution concepts Simulation and numerical methods State space representation Stability Reachability Solution concepts Example of weak solution

Contents

Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concepts Simulation and numerical methods State space representation Stability Reachability

Simulation methods x[1] x[2] x(t) x[3] Simulation methods

Contents

Modeling with differential equations Taxonomy of systems Solution to linear ODEs General solution concepts Simulation and numerical methods State space representation Stability Reachability

State space representation One of the most important representations of linear time invariant systems. State space representation Solution to state space rep. Solution:

Exact discretization of autonomous systems x[3] x[1] x(t) x[2]

t Contents

Modeling with differential equations Taxonomy of systems Solution to linear ODEs Simulation and numerical methods State space representation Stability Reachability Discrete time systems

Stability of LTI systems Stability of nonlinear systems p stable p

Stability of nonlinear systems p Asymptotically stable Lyapunov functions Contents

Modeling with differential equations

Taxonomy of systems Solution to linear ODEs General solution concept Simulation and numerical methods State space representation Stability Reachability Reachability

Reachability of linear systems

Recently Viewed Presentations

  • Industry Weighs in on the Value/Impact of Operational Test ...

    Industry Weighs in on the Value/Impact of Operational Test ...

    Industry Weighs in on the Value/Impact of Operational Test & Evaluation/Developmental Operational Test & Evaluation W. John Stoddart Executive Vice President and President - Defense Oshkosh Truck Corporation Benefits of OT&E Objective and benefits of Operational Test and Evaluation/Developmental Operational...
  • Science and Creationism 6. Abiogenesis  Colin Frayn, 2008-2011

    Science and Creationism 6. Abiogenesis Colin Frayn, 2008-2011

    RNA is (probably) too complex to have arisen in one step Plausible precursors are required Benefits of RNA Can replicate itself Is simpler than DNA Can self-catalyse Cell replication apparatus is RNA Formation of RNA through ribonucleotides Problematic as ribose...
  • スライド 1 - Ustc

    スライド 1 - Ustc

    Large SUSY breaking from 10d nonsupersymmetric heterotic string. superstring is a UV finite quantum mechanics (rather than QFT) living in 10d. unique 10d tachyon free nonsupersymmetric string. how to compactify down to 4dim, giving a very small cosmological constant. formula...
  • Fire Protection Systems in the International Existing Building

    Fire Protection Systems in the International Existing Building

    NFPA 13 and NFPA 13R. Requires a paper trail to justify the sprinklers. Use as a handout: D:\Programs\EBC. 803.6. Level 2 Alteration with fire sprinklers throughout could reduce the fire rating of a two-hour fire barrier to a one-hour fire...
  • 4.2: Experiments Section 4.2 Experiments After this section,

    4.2: Experiments Section 4.2 Experiments After this section,

    Three Principles of Experimental Design. Control . for lurking variables that might affect the response: Use a comparative design and ensure that the only systematic difference between the groups is the treatment administered. Random assignment: Use impersonal chance to assign...
  • The Role of Metadata in Machine Learning for

    The Role of Metadata in Machine Learning for

    Amanda Jones Marzieh Bazrafshan Fernando Delgado Tania Lihatsh Tami Schuyler. [email protected] [email protected] [email protected] [email protected] [email protected] ... All machine learning models developed using an open source Support Vector Machine (SVM) implementation. ... PowerPoint Presentation
  • Instructions for the WG Chair The IEEE-SA strongly

    Instructions for the WG Chair The IEEE-SA strongly

    The IEEE-SA strongly recommends that at each WG meeting the chair or a designee: Show slides #1 through #4 of this presentation Advise the WG attendees that: IEEE's patent policy is described in Clause 6 of the IEEE-SA Standards Board...
  • Finn Clinic Vallensbæk

    Finn Clinic Vallensbæk

    Cunningham åbner sejlet i toppen, så brug den til at kunne hænge jollen ned (udveksling 1:4) Indhal justerer sejlets indgangsvinkel og åbner/lukker midten af sejlet. Er typisk først halet helt ind til masten når bommen er i dækket (udveksling 1:4)...