State Equations BIOE 4200 Processes A process transforms input to output States are variables internal to the process that determine how this transformation occurs u1(t) M inputs u2(t) y1(t) y1(t) ... ... um(t)
N state variables x1(t) x2(t) . . . xn(t) yp(t) P outputs 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 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 Newtons 2nd law Three node circuit should yield three differential equations based on Kirchoffs Current Law
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
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 d2 d x kx let x1 x , x 2 x 2 dt dt d d then x1 x 2 and x 2 kx1 dt dt 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 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 dx(t) A x(t) B u(t) dt y(t) C x(t) D u(t)
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