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lect2 - EL 625 Lecture 2 1 EL 625 Lecture 2 State equations...

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EL 625 Lecture 2 1 EL 625 Lecture 2 State equations of finite dimensional linear systems Continuous-time : ˙ x ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) Discrete-time : x ( t k +1 ) = A ( t k ) x ( t k ) + B ( t k ) u ( t k ) y ( t k ) = C ( t k ) x ( t k ) + D ( t k ) u ( t k ) state x ( t ) - vector of length n × 1 input u ( t ) - r × 1 output y ( t ) - m × 1 A,B,C and D are matrices of sizes n × n, n × r, m × n and m × r respectively. If A,B,C,D are functions of time timevarying If these matrices are constant with time time-invariant
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EL 625 Lecture 2 2 State Differential equations of circuits basic elements resistor, capacitor, inductor Resistor : v R ( t ) = R ( t ) i R ( t ) v R ( t ) : the voltage across the resistor i R ( t ) : the current through the resistor R ( t ) : the resistance A resistor is a ‘zero-memory’ element. B B B B B B B B B B B B B B R + v R - i R Any circuit with only resistors (a purely resistive network) has zero-memory and is of zero order (needs no states to describe it).
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EL 625 Lecture 2 3 Capacitor : i C ( t ) = ˙ q C ( t ) , q C ( t ) = C ( t ) v C ( t ) where: i C ( t ) : the current through the capacitor q C ( t ) : the electrical charge in the capacitor v C ( t ) : the voltage across the capacitor C ( t ) : the capacitance C ( t ) dv C ( t ) dt = i C ( t ) - dC ( t ) dt v C ( t ) If the capacitance does not change with time, C dv C ( t ) dt = i C ( t ) C + v C - i C
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EL 625 Lecture 2 4 Inductor : v L ( t ) = ˙ φ ( t ) , φ ( t ) = L ( t ) i L ( t ) where: i L ( t ) : the current through the inductor v L ( t ) : the voltage across the inductor φ ( t ) : the flux stored in the inductor L ( t ) : the inductance L ( t ) di L ( t ) dt = v L ( t ) - dL ( t ) dt i L ( t ) If the inductance does not change with time, L di L ( t ) dt = v L ( t ) L + v L - i L
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EL 625 Lecture 2 5 Example : L C 1 P P P P P P P P P P P R 1 C 2 P P P P P P P P P P P R 2 + e 1 ( t ) e 2 ( t ) + + + + - i L ( t ) v L ( t ) ? ? i 1 ( t ) i 2 ( t ) v 2 ( t ) v 1 ( t ) Applying Kirchoff’s current and voltage laws, i L = i 1 + i 2 e 1 = v L + i 1 R 1 + v 1 e 1 = v L + v 2 + i 2 R 2 + e 2 From the terminal relationships of the capacitors and the inductor, ˙ v 1 = 1 C 1 i 1 ˙ v 2 = 1 C 2 i 2 ˙ i L = 1 L v L
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EL 625 Lecture 2 6 The inputs are e 1 and e 2 . The output is the voltage across the inductor, v L . Choosing as our states, i L , v 1 and v 2 , x = i L v 1 v 2 , u = e 1 e 2 , y = [ v L ] Need to express ˙ x in terms of x and the inputs, e 1 and e 2 ,. . . ˙ i L = 1 L v L But, v L is not a state variable. . . we need to express v L in terms of the state variables and the inputs. . . v L = i L - R 1 R 2 R 1 + R 2 + v 1 - R 2 R 1 + R 2 + v 2 - R 1 R 1 + R 2 + e 1 + e 2 - R 1 R 1 + R 2 ˙ i L = i L - R 1 R 2 L ( R 1 + R 2 ) + v 1 - R 2 L ( R 1 + R 2 ) + v 2 - R 1 L ( R 1 + R 2 ) + e 1 1 L + e 2 - R 1 L ( R 1 + R 2 )
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EL 625 Lecture 2 7 Similarly, i 1 = i L R 2 R 1 + R 2 + v 1 - 1 R 1 + R 2 + v 2 1 R 1 + R 2 + e 2 1 R 1 + R 2 ˙ v 1 = 1 C 1 i 1 = i L R 2 C 1 ( R 1 + R 2 ) + v 1 - 1 C 1 ( R 1 + R 2 )
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