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State Equations

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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  1. State Equations BIOE 4200

  2. 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)

  3. 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.

  4. 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

  5. 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

  6. 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)

  7. 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

  8. 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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