1. Let {w1,......, wr} be a linearly independent set of vectors in Rn. Suppose we are given integers p and q such that 1 ≤ q ≤ p ≤ r. Define the subspaces of Rn:
V1 = {V1,......,Vp}
V2 = {Vq,.....,Vr}
Show that V1 ∩ V2 = { Vq,........,VP}.
2. Prove the following lemma.
Lemma 1. Suppose (R1)-(R4) hold. Also suppose that B ∩ c(vr1) ⊂ sp{b1,......,br1-1}, where sp{b1, ......, br1-1} is the unique minimal subspace containing B ∩ c(vr1 and generated by the linearly independent vectors {b1,... ,br1-1, bm^+1 }. There exists br1 ∈ B ∩ C(vr1) such that
br1 = C1b1 + ........ + Cr1-1 - br1-1
3. Consider the affine control system
x. = Ax + Bu + a. (1)
Suppose Os = co{v1, ... , vk+1} and let IOs := {1, ... ,k+1}. Let B^ = sp{b1, ... ,bm^|bi ∈ B ∩ C(vi)} be a maximal linearly independent set with respect to Os, in the sense discussed in the lectures.
Define the affine feedback transformation
u = K1x + g1 + G1w
where w is a new exogenous input. We obtain the new system
= (A + BK1)x + BG1w + (Bg1 + a) =: A^x + B^u + a^. (2)
We assume K1, G1, and g1 are selected so that
A^vi + a^= 0, i ∈ IOs
A^vi + a^ ∈ C(vi) , i ∈ {k + 1,...,n}
B^ = [ b1, b2..... bm^] .
Prove or disprove the following statements:
(a) Os := {x ∈ Rn| A^x + a^∈ B^}.
(b) If there exists u = Kx + g such that S → Fo for system (1), then there exists w = K^x+g^ such that S → Fo for system (2).
4. Let S be determined by v0 = (0, 0), v1 = (0,1) and v2 = (1, 0), and consider the affine dynamics
(a) Find B, 0, and Os.
(b) Compute B ∩ C(v0).
(c) Show that the RCP is not solvable by continuous state feedback.
(d) Compute the reach control indices.
(e) Using the algorithm shown in lecture, solve the RCP by discontinuous PWA feedback.
Instructions: You are allowed to use all available sources and tools when solving this problem set. You are also allowed to discuss these problems with coworkers and classmates. However, all sources and tools you use (living, literature, code or otherwise) must be declared in the solutions you hand in. Additionally, while you are allowed to discuss the problems with others, the expectation is that the work performed will be your own, and you are solely responsible for your own work.