Lecture 13 Linear dynamical systems with inputs & outputs
EE263 Autumn 2007-08
Stephen Boyd
Lecture 13 Linear dynamical systems with inputs &
outputs
? inputs & outputs: interpretations ? transfer matrix ? impulse and step matrices ? examples
13?1
Inputs & outputs
recall continuous-time time-invariant LDS has form
x = Ax + Bu, y = Cx + Du
? Ax is called the drift term (of x ) ? Bu is called the input term (of x ) picture, with B R2?1:
x (t) (with u(t) = 1)
x (t) (with u(t) = -1.5) Ax(t) x(t) B
Linear dynamical systems with inputs & outputs
13?2
Interpretations
write x = Ax + b1u1 + ? ? ? + bmum, where B = [b1 ? ? ? bm] ? state derivative is sum of autonomous term (Ax) and one term per
input (biui) ? each input ui gives another degree of freedom for x (assuming columns
of B independent)
write x = Ax + Bu as x i = a~Ti x + ~bTi u, where a~Ti , ~bTi are the rows of A, B ? ith state derivative is linear function of state x and input u
Linear dynamical systems with inputs & outputs
13?3
Block diagram
u(t)
D
B
x (t) 1/s
x(t) C
A
? Aij is gain factor from state xj into integrator i ? Bij is gain factor from input uj into integrator i ? Cij is gain factor from state xj into output yi ? Dij is gain factor from input uj into output yi
Linear dynamical systems with inputs & outputs
y(t)
13?4
interesting when there is structure, e.g., with x1 Rn1, x2 Rn2:
d dt
x1 x2
=
A11 A12 0 A22
x1 x2
+
B1 0
u,
y = C1 C2
x1 x2
u
B1
1/s
x1 C1
y
A11
A12
x2
1/s
C2
A22
? x2 is not affected by input u, i.e., x2 propagates autonomously ? x2 affects y directly and through x1
Linear dynamical systems with inputs & outputs
13?5
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