Problem 1: (50 points)
Consider a discrete-time LTI system with state-space model
x(k + 1) = Ax(k) + Bu(k) (1)
y(k) = cx(k) , (2)
where we assume that the output y(k) is scalar and generated by a single sensor, so that c is
a 1×n row vector. If this sensor operates under its own power, to save power it is sometimes
convenient to take observations less frequently, so observations are obtained only at times k
= Lℓ with L integer. In other words, we record only one out of every L observations of system
(1)–(2). We consider the problem of determining whether the dynamical system is
observable from the periodically subsampled sequence
z(ℓ) = y(Lℓ)) = cx(Lℓ) .
Note that to study observability, we can assume without loss of generality that u(k) ≡ 0, so
the problem reduces to one of finding the initial state x(0) = x0 from observations z(ℓ) with ℓ
≥ 0.
a) Show that the periodically subsampled system is observable if and only if the pair(c,AL)
is observable.
b) Prove that if (c,AL) is observable, the pair (c,A) must be observable.
c) In the remainder of this problem it is assumed that the pair (c,A) corresponding to the
non-subsampled dynamics is observable. To determine whether the observability of
(c,A) implies the observability of (c,AL), consider the case where
and .
Verify that (c,A) is observable, and that for L = 2, (c,AL) is not observable.
d) To gain a better understanding of why observability may be lost by subsampling, it is
useful to use the PBH test to prove that if A has two independent right eigenvectors p1
and p2 corresponding to the same eigenvalue λ, then for any choice of c, the pair (c,A) is
not observable. In other words, a system cannot be observed with a single output if it
has two different Jordan blocks corresponding to the same eigenvalue. To prove this
result, note that if p1 and p2 are right eigenvectors of A corresponding to λ, so is the
linear combination
p = αp1 + βp2
where α and β are two arbitrary real numbers. Verify that we can always select α and β
such that cp = 0, so that (c,A) is not observable.
e) Starting from an observable pair (c,A), to see why observability may be lost by
subsampling, note first that if p is a right eigenvector of A corresponding to eigenvalue
λ, it is also an eigenvector of AL corresponding to eigenvalue µ = λL. Then if λ1 =6 λ2 are
two distinct eigenvalues of A corresponding to eigenvectors p1 and p2, find the
conditions that λ1 and λ2 must satisfy in order to be mapped onto the same eigenvalue
of AL. In this case, the result of part d) ensures that observability is lost by subsampling,
since p1 and p2 are two independent eigenvectors of AL. Obtain a necessary and
sufficient condition on the eigenvalues λi of A which ensures that (c,AL) is observable if
(c,A) is observable.
Problem 2: (50 points)
Consider two minimal realizations
of dimensions n1 and n2 of strictly proper SISO rational transfer functions H1(s) and H2(s). In
the above representations, the polynomials bi(s) and ai(s) are relatively prime for i = 1, 2 and
it is assumed that H1(s) and H2(s) do not have common poles, i.e., the roots of
a1(s) = det(sIn1 − A1) and a2(s) = det(sIn2 − A2)
no not overlap. Obviously, if the realizations (A1,b1,c1) and (A2,b2,c2) are minimal, the pairs
(Ai,bi) and (ci,A1) are reachable and observable, respectively, for i = 1, 2. We are interested in
examining the effect of parallel, series, and feedback connection of these two systems, as
described in parts a)-c)
Figure 1: Block diagrams of a) parallel, b) series, and c) feedback connections of systems H1(s)
and H2(s).
a) If x1(t) and x2(t) denote the state vectors of systems 1 and 2, and
is the state vector of the combined system, the combined system has a state-space
representation of the form.
x˙(t) = Ax(t) + bu(t) , y(t) = cx(t) .
For the parallel connection check that (A,b,c) = (Ak,bk,ck) with
and .
For the series connection check that (A,b,c) = (AS,bS,cS) with
.
Finally, for the feedback connection, verify that (A,b,c) = (AF,bF,cF) where
.
b) Use the PBH tests to verify that (Ak,bk) is reachable and (ck,Ak) is observable, so that the
parallel connection is always minimal.
c) By using the PBH test, prove that (AS,bS) is reachable if and only if b1(s) and a2(s) have
no common zero, i.e. iff there is no pole-zero cancellation between the numerator of
H1(s) and the denominator of H2(s). Likewise, show that (cS,AS) is observable if and only
if b2(s) and a1(s) have no common zero, i.e. iff there is no cancellation between the
numerator of H2(s) and the denominator of H1(s).
d) By using the PBH test, prove that (AF,bF) is reachable if and only b1(s) and a2(s) have no
common zero. Similarly, show that (cF,AF) is observable if and only if b1(s) and a2(s)
have no common zero.
e) The tests obtained in part d) indicate that the feedback connection of H2(s) and H1(s) of
part c) of Fig. 1 is reachable if the series connection of H2(s) and H1(s) shown in part b)
is reachable. Likewise, the feedback connection is observable if the series connection
of H1(s) and H2(s) (the order of H1(s) and H2(s) is exchanged in part b) of Fig. 1) is
reachable. Use the definitions of reachability and observability to interpret these
results. Problem 3: (50 points)
Consider the electrical circuit shown in Fig. 2Figure 2: Electrical circuit.
a) By selecting the capacitor voltages and inductor current as state variables, obtain a
state-space model for this circuit.
b) Perform. a four part Kalman decomposition of this system into parts which are reach-
able and observable, reachable and unobservable, unreachable and observable,
unreachable and unobservable.
c) Specify which physical variables in the circuit are unreachable or unobservable,
andprovide a physical interpretation for their lack of reachability and observability.
d) Compoute the transfer function H(s) = Y (s)/U(s) of the system. What would be the
dimension of a minimal realization of this transfer function?
Problem 4: (50 points)
Spring-mass models play a major role in the study of large mechanical structures such as
satellites, antennas, off-shore platforms, etc. They consist of coupled second-order
differential equations of the form.
where x, u and y are n-, m-, and p-dimensional vectors of position coordinates, inputs, and
outputs, respectively. M is a diagonal matrix of positive constants representing the masses or
moments of intertia of the system, and we will assume here that the measurement matrix
units are selected in such a way that M is normalized to the identity matrix, so that M = In. K
is typically a symmetric non-negative definite matrix, i.e. KT = K ≥ 0. The matrix L can be
decomposed as
L = D + G
where
D = (L + LT)/2 and G = (L − LT)/2
represent respectively the symmetric and skew-symmetric parts of L, since DT = D and GT =
−G. D represents dampling forces and is typically non-negative definite, and G represents
gyroscopic forces. The matrices P, Q and R describe the coupling of sensors and actuators to
the system and do not have a specific structure.
a) What is the matrix transfer function of this system? Construct a state-space realiza-tion
z˙(t) = Az(t) + Bu(t) (3)
y(t) = Cz(t) (4)
of this system. Give a physical interpretation of the states of this realization.
b) The stability of spring-mass systems is usually determined by considering the so-
calledlatent values and latent vectors of the matrix polynomial
W(s) = s2In + sL + K
which are the complex numbers λ ∈C and nonzero complex vectors p ∈Cn such that
W(λ)p = 0 .
They can be viewed in some sense as generalized eigenvalues and eigenvectors
associated with the matrix polynomial W(s). The latent values are just the zeros of the
characteristic polynomial w(s) = det(W(s)) .
Show that w(s) is also the chracteristic polynomial of the matrix A appearing in the state
space-realization (3) of part a). Can you interpret the latent vectors p in function of the
eigenvectors v of A?
Hint: A good starting point is to note that if v = 06 is an eigenvector of A corresponding
to eigenvalue λ, we have
Av = λv .
Then, partitioning v in accordance with the block structure of A and eliminating
variables should lead you to the desired answer.
c) Show that the state-space realization that you have obtained in part a) is reachableif
and only if there is no left latent vector q associated to a latent value λ, i.e,
qTW(λ) = 0 with qT =6 0
such that
qTP = 0 .
d) Similarly, show that the state-space realization obtained in part a) is observable ifand
only if there is no right latent vector p associated to latent root λ, i.e.
W(λ)p = 0 with p =6 0
such that
(λQ + R)p = 0 .
e) Consider the case of a conservative system with D = 0 (no dampling). Show that the
nonzero latent values of such a system occur in pairs ±jω, and the zero latent values
correspond to latent vectors in the null space of K.
f) To illustrate the results of parts a)-e), consider a system consisting of three carts
withmass m coupled by springs with constant k, rolling frictionlessly in the horizontal
direction, as shown in Fig. 3.
The displacement of the i-th cart with 1 ≤ i ≤ 3 with respect to its rest position is denoted
by xi and ui is the external force applied to cart i. Then the equations of motion of the
carts are given by
mx¨1 = u1 + k(x2− x1)
mx¨2 = u2 + k(x3− x2) − k(x2− x1)
mx¨3 = u3− k(x3− x2) .
Figure 3: System of three elastically coupled carts.
Find all the latent values and latent right and left vectors of this spring mass system.
Show that the system is unstable. Give a physical interpretation of the lack of stability
of the system. to do so, you may want to derive the equation of motion satisfied by the
center of gravity of the system.
g) Determine whether the spring-mass system of part f) is reachable with the inputu1
alone, or u2 alone, or u3 alone. Suppose that we seek to observe the system by measuring
the displacement xi, 1 ≤ i ≤ 3 of one cart. Is the system observable from x1 alone, from x2
alone, or from x3 alone? If the system of part f) is viewed as an idealization of a train
with three cars (including the locomotive), where do you think the locomotive should
be placed in the train?
h) A MEMS z-axis gyroscope which is intended to estimate the angular velocity Ωz of a
system rotating about its z-axis is shown in Fig. 4.
Under the effect of the Coriolis force induced by the rotation about the z axis, a mass m
oscillates in the x and y directions. Spring and dampling forces are present in the x and
y direction. To keep the model simple, we assume that the spring and damping
constants k and d are the same in the x and y directions. If ux and uy are the control forces
applied in the x and y directions, the dynamics of the gyroscope are described by
mx¨ + dx˙ + kx = ux + 2mΩzy˙
my¨+ dy˙ + ky = uy − 2mΩzx ,˙
which can be rewritten in vector form. as
Typically the constant d/m is very small, so it can be neglected in first approximation.
With this assumption, verify that this system has four latent purely imaginary values
±jω1 and ±jω2, and that the gyroscope is reachable with ux alone, or uy alone.