linear system and state space design methods
DESCRIPTION
The short introduction to Linear System and State Space Design methods. Several examples have also been included.TRANSCRIPT
Linear Dynamical Systems and State Space Design
Methods-1
“Text Book”
Feedback Systems: An Introduction for Scientists and Engineers
Karl J. Åström and Richard M. Murray
Why State Space Design Method are popular? What it might mean to you?
[4,5,6] • Recognized importance in study of any physical
processes where time behaviour is of interest.• Mathematicians have known the underlying
methods for many years • Commonly popular among engineering, physics,
medicine and economics. • Applicable to all systems that can be analyzed by
integral transforms in time, and where such transforms breaks down.
So what are Dynamical Systems?
• System – major understanding comes from input/output systems.
• Describes, and freezes our understanding of the interactions with environment. Overly simplified, but true.
• From this, we derive a mathematical model of the system
Different types (know till yet)[7]
1. Linear and non-linear systems2. Constant parameter and time-varying parameters
systems3. Instantaneous (memory less) and dynamic (with
memory) systems 4. Causal and non-causal systems 5. Continuous-time and discrete-time systems 6. Analogue and digital systems 7. Invertible and non-invertible systems 8. Stable and unstable systems
Properties of Linear systems
• Scaling property (also called Homogeneity property)
• Additivity property• Superposition property
Linear Dynamical System
• The division of Linear Dynamical Systems with respect to Non-linear Dynamical System is done based on the input/output system.
• A properties that qualifies systems as linear dynamical system:
1. Are systems with or with out memory2. Causality3. Stability (BIBO)
Process and Tools for investigating Dynamical Systems [1]
1. Dynamical System Modelling2. Investigating Dynamic Behaviour
Dynamical System Modelling
• A model is a mathematical representation of a system that can be used to answer question about the system. The choice of the model depends on the questions one wants to ask. eg. model of Pendulum, model of circuit etc.
• The state of a system is a collection of variables that summarizes the past history of the system for the purpose of predicting the future. Thus, a state space model is one that describes how the state of a system evolves over time.
• The given the ordinary systems (governed either just by continuous-time or discrete
time dynamics) modelling uses ordinary differential equations. • For example, Linear dynamical systems are
represented by following general form • Why this form? Partially, because of system’s inherent
structure after linearization, and also because of the mathematical and computational ease in implementing. (use of digital computer also had a great impact on arriving at this general form).
(Interesting) Physically Releavant Model [4]
• Volterra- Lotka Predator-Prey Equations where are a,b,c,d are positive co-efficient. The state is x is Prey and y is Predator.
Exercises (State Space model)
• Free-hanging Pendulum http://underactuated.csail.mit.edu/underactuated.html?chapter=2 [2]• Modelling the behaviour of the diode (?)[6]• Van Der Pol Oscillator • Double Integrator• Objectives: 1. Learn to recognize, and put into States Space form from the
ordinary differential equation.2. Learn to appreciate the information (stability, observability and
controllability) state space form givens us.
Investigating Dynamic Behaviour
• Classical form of analysis: frequency domain methods- Nyquist plot, Bode plot, Input/output stability analysis
• Modern Dynamical System Analysis- Eigenvalues analysis, analysis, Lyapunov Direct and Indirect Methods
Exercise Stability Analysis • Types of stability – 1. Lyapunov stability In order to determine whether the solution is stable, or unstable, we need to define stabilityLet denote the solution with initial conditions . The Solution is Lyapunov stable if all solutions that start near stay close to . 2. Asymptotic stability3. Exponential stability
Next time-
• Topics that were in color in this slide• Proofs and theorems (fun!)• More back-of -envelope exercises
Thank you. Questions, and constructive criticism!
References
[1]http://www.cds.caltech.edu/~murray/amwiki/index.php/System_Modeling[2] Russ Tedrake. Underactuated Robotics: Algorithms for Walking, Running, Swimming, Flying, and Manipulation (Course Notes for MIT 6.832). Downloaded in April, 2015 from http://people.csail.mit.edu/russt/underactuated/[3]http://www.cds.caltech.edu/~murray/amwiki/index.php/Dynamic_Behavior
[4] Sastry, S. (2013). Nonlinear systems: analysis, stability, and control (Vol. 10). Springer Science & Business Media.[5] Strogatz, S. H. (2014). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview press.[6] Khalil, H. K., & Grizzle, J. W. (1996). Nonlinear systems (Vol. 3). New Jersey: Prentice hall[7]http://www.ee.ic.ac.uk/pcheung/teaching/ee2_signals/Lecture%202%20-%20Introduction%20to%20Systems.pdf