1. State Space Representation
Linear/nonlinear
Order
Model Classification
Time-Varying
Linear models
Time-Invariant

2. Linear or Nonlinear, Mathematic Definition: A linear model admits the principle of superposition, a nonlinear model does no.
Practical ways to distinguish
A model is nonlinear if there is at least one occurrence of a nonlinear function of the dependent variable(s) or their derivatives, .
A model is linear if the dependent variable(s) are multiplied by constant coefficients, or time-varying coefficients

3. Examples

4. Examples

5. Order for a Model with a Single Equation

6. Order for a Model with a Multiple Equations

7. Exceptions: Internal Variables and Equations without Derivatives

9. Linear Time-Varying (LTV), or Linear-Time Invarient (LTIV)

10. Alternate Terminology
Time-Invariant  (LTIV)  Autonomous
Time-Varying  (LTV)  Non-autonomous

11. State Space Representation- Concept Applied to Coupled Mass-Spring-Damper
Physical Formulation
State-Space Representation
2 Differential Equations
Each Differential Equation 2nd Order
2 Unknowns
4th Order Model
4 Differential Equations
Each Differential Equation 1st Order
4 Unknowns (State Variables)
4th Order Model 

13. Procedure Applied to Mass-Spring-Damper

19. Procedure Applied to Coupled Mass-Spring-Damper

25. General State Space Representation
Example: Mass-Spring-Damper

27. Definition of An Output….applied to the Mass-Spring-Damper

29. If a model is linear, and the output is defined as a linear combination of the states, then the general state space representation may be placed into ABCD format
Output Linear Combination of the States
y(t)=c1q1(t)+c2q2(t)+…+cnqn(t), n=order of model=# of states

30. Example: Mass-Spring-Damper

33. Example: Coupled Mass-Spring-Damper