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ECE4501/6501: Matrix Analysis Project 2 (Spring 2022)

发布时间：2022-05-12

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ECE4501/6501: Matrix Analysis Project 2 (Spring 2022)

Problem 1 (20 %): Consider the system equation

x(k + 1) = Ax(k), A = aλ20 ∈ R3×3 .

(a) Find all possible cases ofλ1 , λ2 , λ3 and a which result in an asymptotically stable system; (b) ﬁnd all possible cases of λ1 , λ2 , λ3 and a for a stable (but not asymptotically stable) system;

(d) (i) (for ECE4501) ﬁnd all possible cases of λ1 = λ2 λ3, and a which result in a stable (but not

asymptotically stable) system, for A =

aλ10 0aλ3 ∈ R3×3 .

(d) (ii) (for ECE6501) ﬁnd all possible cases ofλ1 = λ2 λ3 , a and b which result in a stable (but not

asymptotically stable) system, for A = aλ1 ∈ R3×3 .

Problem 2 (20 %): Consider the system equation

x(k + 1) =Ax(k) +Bu(k)

− 1

1 1 1 − 1 1 0 1 1

, B =

− 1

(a) Using Matlab, check the stability ofx(k + 1) =Ax(k);

(b) using Matlab, check the controllability ofx(k + 1) =Ax(k) +Bu(k);

(c) ﬁnd u(k), k = 0,1,2,3, which make x(k) from x(0) = [1,0,0,0]T to x(4) = [0,0,0,1]T, and calculate E4 = (u(0))2 + (u(1))2 + (u(2))2 + (u(3))2 (as the control energy which is in fact minimal, as compared with any other controls u(0), u(1), u(2), u(3) to do the same job); and

(d) (i) (for ECE4501) formulate the equation for solving u(k), k = 0, 1,2,3,4, which can control the system state vector from x(0) = [1,0,0,0]T to x(5) = [0,0,0,1]T, and ﬁnd a set of such solutions u(i),i = 0,1,2,3,4 (in this case, the solutions are not unique and some solutions can be such that E5 = (u(0))2 + (u(1))2 + (u(2))2 + (u(3))2 + (u(4))2 < E4 in (c) above); or

(d) (ii) (for ECE6501) work out Part (i) above, ﬁnd the control input u(k),k = 0, 1,2,3,4, which can control the system state vector from x(0) = [1,0,0,0]T to x(5) = [0,0,0,1]T, such that E5 = (u(0))2 + (u(1))2 + (u(2))2 + (u(3))2 + (u(4))2 is minimized (and compare it with E4 in Part (c)).

Problem 3 (20%): Consider x(k + 1) =Ax(k) +Bu(k), with A =

B =

−4.2 −3.5 , 1 .

(a) Check the stability ofx(k + 1) =Ax(k), and with x(0) = [1,0,0]T, using Matlab, calculate kx(1)k2 , kx(10)k2 and kx(20)k2 (is kx(k)k2 growing? why?);

(b) ﬁnd the gain vector K = [k1,k2,k3] such that the eigenvalues ofA − BK are: 0.5, 0.3 and 0.2;

(c) with K in the feedback control law u(k) = −Kx(k)+ r(k) (where r(k) is an external signal not to be concerned for our study), derive the closed-loop system equation, and with x(0) = [1,0,0]T and r(k) = 0, using Matlab, calculate kx(1)k2 , kx(10)k2 and kx(20)k2 (compare it to Part (a) above); and

(d) (i) (for ECE4501) compare the controllability of the open-loop system x(k + 1) = Ax(k) +Bu(k) and the closed-loop system; or

(d) (ii) (for ECE6501) work out Part (i) above, and further verify

B AKB AB B AB A2 , AK = A − BK,

for anyA ∈R3×3,B ∈R3,K ∈R1×3, and answer the question: does state feedback control u(k) = −Kx(k)+ r(k) change system controllability? why?

Problem 4 (40%): Consider the system: x(k + 1) =Ax(k)+Bu(k), withA =

−0.3 ,

B = 25 , with the measured output y(k) = x1(k) and applied input u(k), whose parameters 0.01875,

We have studied such an algorithm (see the lecture notes), and learned some important concepts: the input-output system model, the input-output system model with delayed signals, the parametrized input- output system model, the estimation error, the parameter estimate vector, the parameter error vector.

(a) For the given system, derive its input-output system model with delayed signals; (b) Derive its parametrized input-output system model and specify θ∗;

(c) design the speciﬁc (not just a copy of the general form) iterative (adaptive) learning algorithm to generate the estimate θ(k) of θ∗, with your choice of the design parameters a and c;

(d) conduct a simulation study of your adaptive algorithm, with y(k) = u(k) = 0, k < 0, and θ(0) = 0.95θ∗, for the following cases:

(i) for u(k) = 1, k ≥ 0, calculate e(k) and θ(k) for k = 1,2,...,100, and plot kθ(k) − θ∗ k2 and e(k) for k = 0,1,2,...,100;

(ii) for u(k) = cos(0.2k), k ≥ 0, calculate e(k) and θ(k) for k = 1,2,...,100, and plot kθ(k) − θ∗ k2 and e(k) for k = 0, 1,2,..., 100; and

(iii) for u(k) = cos(0.3k) + 1.3 cos(1. 1k), k ≥ 0, calculate e(k) and θ(k) for k = 1,2,...,100, and plot kθ(k) − θ∗ k2 and e(k) for k = 0, 1,2,..., 100; and

(e) based on your simulation results, discuss the performance of the adaptive algorithm. (f) re-do Parts (i)-(iii) with θ(0) = 1.05θ∗ (optional).

In practice, the adaptive parameter estimation algorithm runs iteratively in real time, with the system signals u andy being collected in real time, as time k goes, to generate the improved estimates θ(k) of the unknown system parameter θ∗, as k goes.

For simulation, a Matlab program can be written to simulate such a process, to solve this project problem, to obtain θ(k) for k = 1,2,..., while y(k) is calculated from y(k) = θ∗Tϕ(k − 1) to simulate the response of the real-life system which takes the input u and produces the outputy (here, θ∗ is used only to build a system to generate (calculate)y(k), and θ∗ is unknown to the algorithm generating θ(k)).