linear system and state space design methods

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Linear Dynamical Systems and State Space Design Methods-1

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The short introduction to Linear System and State Space Design methods. Several examples have also been included.

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Page 1: Linear System and State Space Design Methods

Linear Dynamical Systems and State Space Design

Methods-1

Page 3: Linear System and State Space Design Methods

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.

Page 4: Linear System and State Space Design Methods

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

Page 5: Linear System and State Space Design Methods

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

Page 6: Linear System and State Space Design Methods

Properties of Linear systems

• Scaling property (also called Homogeneity property)

• Additivity property• Superposition property

Page 7: Linear System and State Space Design Methods

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)

Page 8: Linear System and State Space Design Methods

Process and Tools for investigating Dynamical Systems [1]

1. Dynamical System Modelling2. Investigating Dynamic Behaviour

Page 9: Linear System and State Space Design Methods

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.

Page 10: Linear System and State Space Design Methods

• 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).

Page 11: Linear System and State Space Design Methods

(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.

Page 12: Linear System and State Space Design Methods

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.

Page 13: Linear System and State Space Design Methods

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

Page 14: Linear System and State Space Design 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

Page 15: Linear System and State Space Design Methods

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!

Page 16: Linear System and State Space Design Methods

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

Page 17: Linear System and State Space Design Methods

[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