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Problem solutions for a lti (linear time-invariant) systems course. Topics covered include finding the transfer function from input to output, system stability, controllability, and reachability. Students are required to find matrices, calculate eigenvalues, and determine system behaviors.
Typology: Assignments
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Hw 4 Nicola Elia Out : 11/14/ Due in class :11/27/
Consider teh block diagram shown in the figure below.
P(s)
P 0 (s)
Q(s)
r + y
a ) Suppose P s a (^) s +^2 1, P 0 s a (^) s?^11 , and Q a 2. Calculate the transfer function from r to y. b ) Is the system in part a) stable? c ) Now suppose that P s a P 0 s a H s for some H s . Under what conditions is the system stable for any stable Q s ?
We are given the following information about the response of a single input LTI continuous time system in the form dxdt a Ax C bu.
c 1
is x t a
c 1
e^2 t.
The step response (with zero initial condition) is x t a
1 c e? t
a ) ( 25pt ) Find the matrix A. b ) ( 10pt ) BONUS Is the system controllable? Justify.
Consider the system with state-space description 1. dx dt
a Ax C Bu
Assume the system is controllable. Let u be given by 1. u a Fx C v for some constant matrix F , and any input v. This describe a state feedback. a ) ( 10pt ) Write down that state space description for the system mapping v to x. b ) ( 15pt ) Is the new system controllable? Prove it or show a counter example. Hint : use PBH test. Also recall that rank GH a rank G , if H is invertible
Consider now a new system S obtained by connecting in series S 1 and S 2 as show in the figure below.
S 1 S 2
u y
Let S 1 : A 1 a 1 ; b 1 a 1 ; c 1 a 1
S 2 : A 2 a
; b 2 a
; c 2 a (^0 )
a ) Is the system S reachable? b ) Is it observable?
Consider the system S obtained by connecting two SISO Continuous-Time systems as shown in the figure below. The system S 1 has a state space representation A 1 , b 1 , c 1 , 0. The system S 2 has a state space representation A 2 , b 2 , c 2 , 0. Both systems are reachable and observable.
Show that: S is controllable if and only if A 1 and A 2 have no common eigenvalues.
