UNIVERSITY OF CALIFORNIA AT BERKELEY
Department of Mechanical Engineering
ME233 Advanced Control Systems IISpring 2016

Homework #4

Assigned: Mar. 30 (Wed)Due:Apr. 5 (Tu)

1. Consider the state-space realization G(z) = C(zI −A)−1 B+D, where DT D is invertible.
Assume that the dimensions of the matrices are given by A ∈ Rnx ×nx , B ∈ Rnx ×nu ,C ∈ Rny ×nx , and D ∈ Rny ×nu . In this problem, we will establish the relationshipbetween the transmission zeros of this realization and the unobservable modes of (Ĉ, Â),
whereÂ = A − B(DT D)−1 DT C
Ĉ = C − D(DT D)−1 DT C .
(a) Show that, for any matrix M , the columns of the matrix X are linearly independent if and only if the columns of the matrix
 IZ :=XM
are linearly independent
Hint: A good way to start is by showing that the null space of X is equal to thenull space of Z.
(b) Using the result from part (a), prove that the following conditions are equivalent:Â − λI•
X = 0 and the columns of X are linearly independentĈ  A − λI B XX• ∃Y such that
= 0 and the columns ofare linearly
CD YYindependent(c) Using the result from part (b), prove thatA − λI BÂ − λI,nullity= nullityCDĈ∀λ ∈ C .

(d) Using the result from part (c), find the relationship between the transmission zerosof the state-space realization G(z) = C(zI − A)−1 B + D and the unobservable
modes of (Ĉ, Â).
Hint: First convert the condition in part (b) into a condition relating the rank of
the two matrices. Then show thatA − λI B
A − λI B
Â − λIrank< normalrank⇔ rank< nx .
CDCDĈ
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2. When designing an infinite-horizon LQR for the discrete-time systemx(k + 1) = Ax(k) + Bu(k)under the cost functionJ=∞X

pT (k)p(k)k=0where p(k) = Cx(k) + Du(k) and DT D is invertible, we can guarantee that the optimal solution exists provided that (A, B) is stabilizable and a condition involving thetransmission zeros of the state space realization C(zI − A)−1 B + D holds. Using theresult of problem 1, show that when C T D = 0, the transmission zeros of the state spacerealization C(zI − A)−1 B + D are the unobservable modes of (C, A).
3. Consider a discrete-time process described byx(k + 1) = Ax(k) + Bu(k) + Bw w(k)where the stationary, Gaussian white noise w ∈ Rs satisfies
E{w(k + j)wT (k)} = W δ(j) .

E{w(k)} = 0,The signals x and u respectively have dimension n and m.
There are two sensors configurations we will be considering.
Sensor Configuration A: In this configuration, the output equation isy(k) = Cx(k) + vA (k)where y ∈ Rr is output vector and vA ∈ Rr is a stationary Gaussian measurement
noise. The measurement noise is independent from the initial state and input
noise, and the following quantities are given
E{vA (k)} = 0,

E{vA (k + j)vAT (k)} = VA δ(j) .

Sensor Configuration B: For the purpose of preparing for any sensor failures, thisconfiguration uses two identical sets of sensors and measures the output twice.
With this sensor configuration the measurement vector is 2r dimensional, and it
is given by  
y1 (k)C
vB1 (k)=x(k) +y2 (k)CvB2 (k)where the measurement noises are independent from the initial state and the input
noise, and the following quantities are given for i = 1, 2:
E{vBi (k)} = 0,

T
E{vBi (k + j)vBi(k)} = VB δ(j),2T
E{vB1 (k + j)vB2(k)} = 0

While the duplication of the output measurement increases the hardware cost in Sensor
Configuration B, it is also true that the specification for each sensor may be relaxed if
the same output is measured twice and the two measurements are used in the Kalmanfilter.
(a) List a set of conditions that guarantee that the stationary Kalman filter exists
for Sensor Configuration A. Show that these conditions also guarantee that thestationary Kalman filter exists for Sensor Configuration B. (Do not assume that
A is Schur.)(b) Let the assumptions you listed in part (a) hold and assume that you design and use
a Kalman filter for each sensor configuration. Determine a relationship betweenthe sensor noise covariances VA and VB so that MA = MB , where MA is thesteady state a-priori state estimation error covariance for Sensor Configuration A,and MB is the steady state a-priori state estimation error covariance for Sensor
Configuration B.
Hint: The following relationships for stationary Kalman filters are helpful insolving this problem:• (I + M C T V −1 C)−1 = I − M C T (CM C T + V )−1 C• A − LC = A I − M C T (CM C T + V )−1 C• M = (A − LC)M AT + Bw W BwT
The first relationship comes from the matrix inversion lemma, the second relationship expresses A − LC as a function of M , and the third relationship is arestatement of the discrete algebraic Riccati equation.
4. Define the matrices
BBe :=  0 B2 0De :=D2



A 0 0
Ae := B1 A1 0 0 0 A2


D1 C1 0Ce :=0 0 C2
For the frequency-shaped linear quadratic regulator (FSLQR) problem considered in
the first half of Lecture 15, we derived the following set of conditions that guarantee
the existence of the optimal FSLQR:
A1: (Ae , Be ) is stabilizable
A2: The state space realization Ce (zI − Ae )−1 Be + De has no transmission zeros
on the unit circle.
It was subsequently stated (but not proved) that the following conditions imply that
(A1)–(A2) hold:3B1: (A, B) is stabilizable
B2: A1 and A2 are SchurA − λI B2
B3: nullity 2= 0 whenever |λ| = 1D2  C2A − λI B1
B4: nullity 1= 0 whenever λ is a unit circle eigenvalue of AC1
D1
Prove that conditions (B1)–(B4) imply conditions (A1)–(A2).
Hint: You will find it useful to use the following characterizations: T
A − λI• (A, B) is stabilizable if and only if nullity= 0 whenever |λ| ≥ 1
BT• λ is a transmission zero of the realization Ce (zI − Ae )−1 Be + De if and only if
A − λI Be
nullity e>0Ce
DeThe second characterization is a result of DeT De = D2T D2  0 holding for the FSLQRproblem.

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