1 / 8

# State Equations - PowerPoint PPT Presentation

State Equations. BIOE 4200. Processes. A process transforms input to output States are variables internal to the process that determine how this transformation occurs. u 1 (t). N state variables x 1 (t) x 2 (t) . . . x n (t). y 1 (t). u 2 (t). M inputs. y 1 (t). P outputs. . .

Related searches for State Equations

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about 'State Equations' - shasta

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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

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

Matrix Form of State Equations