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State Equations

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State Equations

BIOE 4200

- A process transforms input to output
- States are variables internal to the process that determine how this transformation occurs

u1(t)

N state variables

x1(t)

x2(t)

.

.

.

xn(t)

y1(t)

u2(t)

M inputs

y1(t)

P outputs

...

...

um(t)

yp(t)

- Inputs u(t) and outputs y(t) evolve with time t
- Inputs u(t) are known, states x(t) determine how outputs y(t) evolve with time
- States x(t) represent dynamics internal to the process
- Knowledge of all current states and inputs is required to calculate future output values
- Examples of states include velocities, voltages, temperatures, pressures, etc.

- Derive mathematical equations based on physical properties to find a quantity of interest
- Find the velocity of the first mass in a two-mass system
- Find the voltage across a resistor in an electrical circuit with 3 nodes

- Should have same number of equations and unknowns
- Two mass system should yield two differential equations based on Newton’s 2nd law
- Three node circuit should yield three differential equations based on Kirchoff’s Current Law

- Constants k1, k2, ... are known values that describe the physical properties of the system
- Inputs u1, u2, ... are variables representing known quantities that vary with time
- Known force or displacements on elements of the mechanical system
- Voltage and or current sources in circuit

- State variables x1, x2, ... are remaining unknown quantities that vary with time
- Velocities of each mass in a two-mass system
- Voltages at each node of the electrical circuit

- Express original equations as 1st order differential equations of with state variables: dx/dt = f(x, u)
- Additional states must be added if higher order derivatives are present
- Outputs y1, y2, ... are quantities you originally wanted to find
- Output can be expressed as a combination of states and/or inputs: y = g(x, u)

- Obtain necessary equations to solve problem
- Identify constants ki, inputs ui and states xi
- Rearrange equations into the form dx/dt = f(x, u)
- Introduce additional states to eliminate higher order derivatives

- Express output as a function of states and input
- y = g(x, u)
- Outputs y(t) can equal individual states x(t) by setting some elements of C = 1 and all elements of D = 0
- Input u(t) can also be directly incorporated into the output if D 0

- Equations can be represented in matrix form if state derivatives and outputs are linear combinations of states and inputs

State equation

x(t) is N x 1 state vector

u(t) is M x 1 input vector

A is N x N state transition matrix

B is N x M matrix

Output equation

y(t) is P x 1 output vector

C is P x N matrix

D is P x M matrix