State equations
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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. . .

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State Equations

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State equations

State Equations

BIOE 4200


Processes

Processes

  • 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)


State variables

State Variables

  • 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.


Equations and unknowns

Equations and Unknowns

  • 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


Finding state variables

Finding State Variables

  • 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


Obtaining state equations

Obtaining State Equations

  • 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)


Obtaining state equations1

Obtaining State Equations

  • 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


Matrix form of state equations

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

Matrix Form of State Equations


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