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University of Pennsylvania ESE 500: Linear Systems Theory Homework II Due: on Wednesday 10/20 at 23:59 on Gradescope INSTRUCTIONS Read the following instructions carefully before beginning to work on the homework. • You must submit your solutions on Gradescope. It must be submitted as a single PDF file, compiled in LATEX or written by hand and scanned into images included in your pdf. It must be readable by our staff to be graded. • Please start a new problem on a fresh page and mark on Gradescope all the pages corre- sponding to each problem. Failure to do so may result in your work not graded completely. • Clearly indicate the name and Penn email of all your collaborators on your submitted solutions. You may discuss the problems but you cannot share solutions and whatever you submit must be your own work. Failure to do so will result in penalties according to Penn’s Code of Conduct policies. • Late days policy reminder: 3 days with no penalty, after which -35% per day. • Regrades are handled via Gradescope within 3 days from publication of grades. After that no regrade request is admissible. When requesting a regrade, the entire submission is regraded. • Please submit only one pdf. If you wrote any code, you have to submit that as well. Errata (Last update 10/8) 1. Problem 8: The output should read y = [ r1 cos θ + (r2 − 1) sin θ −r1 sin θ + (r2 − 1) cos θ ] . Problem 1 (Matrix differentiation - 13 points). Let A(t) be an n × n matrix, which depends on t, i.e. its elements are functions of t: [A(t)]ij = aij(t), for i, j = 1, . . . , n The derivative Ȧ(t) of A(t) with respect to t, is also an n× n matrix and is defined by taking element-wise derivatives: [Ȧ(t)]ij = ȧij(t), for i, j = 1, . . . , n a) (4 points) For n× n differentiable matrices A1(t), A2(t) prove: d dt ( A1(t)A2(t) ) = Ȧ1(t)A2(t) +A1(t)Ȧ2(t) 1 b) (4 points) Using induction, prove that for n × n differentiable matrices A1(t), A2(t), ... , Ak(t), we have: d dt ( A1(t)A2(t)...Ak(t) ) = Ȧ1(t)A2(t)...Ak(t) +A1(t)Ȧ2(t)...Ak(t) + ...+A1(t)A2(t)...Ȧk(t) c) (5 points) The exponential of a n× n matrix A is defined as the following: expA = ∞∑ i=0 Ai i ! where A0 = I. Suppose A(t) and Ȧ(t) commute. Prove: d dt expA(t) = Ȧ(t) expA(t) Problem 2 (Functions of a matrix - 26 points). Let f, g be functions over matrices and A,B ∈ Rn×n. Suppose AB = BA. a) (3 points) Prove f(A)g(B) = g(B)f(A). b) (3 points) Prove f(AT ) = f(A)T . c) (3 points) Let A = QJQ−1 be any matrix decomposition. Prove f(A) = Qf(J)Q−1. d) (4 points) eA+B = eAeB. Hint: if two functions satisfy the same differential equation, then the uniqueness of solution of differential equations says they are equal. e) (4 points) Prove det(eA) = etr(A). Note: you can use known facts about determinant and trace. f) (4 points) Prove that BeAt = eAtB if and only if AB = BA. g) (5 points) For the time invariant linear state equation ddtx(t) = Ax(t) show that given any x0 there exists a constant α(x0) such that det[x(t) Ax(t) . . . An−1x(t)] = α(x0)e tr(A)t Problem 3 (State space representation - 8 points). Consider a system with input u(t) and output y(t) which can be described using the following set of differential equations: z̈1(t) = z1(t) + z2(t) + u̇(t) ż2(t) = ż1(t) + z2(t) + u(t) y(t) = ż1(t) a) (4 points) Define the states of the system such that it can be represented as an 3-dimensional LTI system, i.e., as the following: ẋ(t) = Ax(t) +Bu(t) y(t) = Cx(t) +Du(t) where A,B,C,D are constant matrices. 2 b) (4 points) Consider T defined below, as a new basis for the state space and let x̂(t) be the representation of x(t) with respect to the basis T. T = 10 1 , 01 1 , 11 1 Compute Â, B̂, Ĉ, D̂ in the new representation of the system with respect to T : ˙̂x(t) = Âx̂(t) + B̂u(t) y(t) = Ĉx̂(t) + D̂u(t) Problem 4 (10 points). Using the definition of transition matrix, prove that: a) (2 points) φA(t2, t1)φA(t1, t0) = φA(t2, t0), for all t0, t1, t2. b) (1 point) φA(t, t) = I, for all t. c) (2 points) φ−1A (t, τ) = φA(τ, t), for all t, τ . d) (5 points) d dτ φA(t, τ) = −φA(t, τ)A(τ) Problem 5 (8 points). Find the solution x(t) of the following time variant system ẋ(t) = [ (t+ 1)2 t+ 1 t+ 1 t2 − 1 ] x(t), x(0) = x0 Problem 6 (Periodic System - 13 points). Consider the system ẋ(t) = A(t)x(t) where A(t) is a periodic matrix with period T . That means A(t+ T ) = A(t) for all t ∈ R. a) (2 points) First, consider the state transition matrix Φ(t1, t0) for the system. Define the matrix Ψ(t, 0) = Φ(t+ T, 0). Show the Ψ satisfies: Ψ̇(t, 0) = A(t)Ψ(t, 0) Ψ(0, 0) = Φ(T, 0) b) (2 points) Show that Φ(t+ T, 0) = Φ(t, 0)Φ(T, 0). c) (3 points) Since we know that Φ(T, 0) is invertible, there exists some complex n× n matrix R such that Φ(T, 0) = eTR. Define P (t)−1 := Φ(t, 0)e−tR Show that P (t)−1 is periodic with period T . This implies that P (t) is periodic with period T . Also show that P (T ) = I. d) (4 points) Show that Φ(t, t0) = P (t) −1e(t−t0)RP (t0) Hint: Note that Φ(t, t0) = Φ(t, 0)Φ(0, t0) e) (2 points) Express the system using the coordinate frame z(t) = P (t)x(t). What do you notice about this new system? 3 Problem 7 (Discretization of continuous time LTI systems - 12 points). The dymanics of an aircraft consist of a set of nonlinear coupled differential equations. Under certain assumptions, though, these can be decoupled and reformed in a linear system. Aircraft pitch is governed by the longitudinal dynamics. Let’s assume that the aircraft is in steady-cruise at constant altitude and velocity (thus, the thrust, drag, weight and lift forces balance each other in the x- and y-directions) and that a change in pitch angle will not change the speed of the aircraft under any circumstance (unrealistic but simplifies the problem). Under these assumptions, the longitudinal equations of motion for the aircraft are: ȧ = µΩσ[−(CL + CD)a+ 1 µ− CL q − (CW sin γθ + CL] q̇ = µΩ 2iyy [[CM − η(CL + CD)]a+ [CM + σCM (1− µCL)]q + (ηCW sin γ)δ] θ̇ = Ωq where a is the angle of attack, θ is the pitch angle and q the rate of pitch angle, δ the el- evator deflection angle, µ = ρSc̄4m , ρ the density of air, S the platform’s area of the wing, c̄ the average chord length, m the mass of the aircraft, Ω = 2Uc̄ , U the equilibrium flight speed, CT , CD, CL, CW , CM the coefficients of thrust, drag,lift, weight and pitch moment, γ the flight path angle, σ = 11+µCL , iyy the normalized moment of inertia and η = µσCM . Using the parameters from the data from one of Boeing’s commercial aircraft, the above system becomes: ȧ = −0.313a+ 56.7q + 0.232δ q̇ = −0.0139a− 0.426q + 0.0203δ θ̇ = 56.7q In this problem we will transform this simple continuous time model in discrete time one upon which one may design an autopilot that controls the pitch of an aircraft. a) (2 points) Rewrite the model in state-space space form: ẋ(t) = Ax(t) +Bδ(t), y(t) = Cx(t) Use x(t) := [a, θ, q]T . The elevator deflection angle δ is considered the input of the system and the pitch angle θ of the aircraft is the output. b) (5 points) Let the sampling period be Ts = 0.01s. Compute the discrete-time system. Then, check your derivations using the c2d function of Matlab. Note: You may use MATLAB for simple numerical calculations (e.g., matrix inversion, matrix multiplication), but you should show all the steps in your derivations. 4 c) (3 points) Using δ(t) = 0.2, simulate the continuous time system using ode45 of Matlab for t ∈ [0, 10]. Plot the output y(t) with zero initial conditions, i.e. x(0) = [0, 0, 0]T . d) (2 points) Simulate the discrete time system, for δ(k) = 0.2, k = 0, 1, . . . , N − 1, where N = 10/Ts. In the same plot as before, plot the discretized y(k). Is the response of the discrete-time system close to the one of the continuous time? Repeat the simulation for Ts ∈ {0.1, 0.5, 1, 2, 5, 10} and plot the responses in the previous plot (don’t forget to label what’s what in the plot). For all of the simulations assume zero initial conditions, i.e. x(0) = [0, 0, 0]T . Comment on the graph. Problem 8 (Local linearization around a trajectory: unicycle - 10 points). A single-wheel cart (unicycle) moving on the plane with linear velocity v and angular velocity ω can be modeled by the following system of nonlinear of ODEs: ṙ1 = v cos θ ṙ2 = v sin θ θ̇ = ω, (1) where (r1, r2) denotes the Cartesian coordinates of the wheel and θ its orientation. Regard this as a system with input u = [ v ω ] and output y = [ r1 cos θ + (r2 − 1) sin θ −r1 sin θ + (r2 − 1) cos θ ] . a) (0.5 points) Prove that the map that takes input u(t) into y(t) is nonlinear. b) (2 points) Construct a state-space model for this system with state x = x1x2 x3 := r1 cos θ + (r2 − 1) sin θ−r1 sin θ + (r2 − 1) cos θ θ . c) (0.5 points) Prove that xeq = 0, ueq = 0 is an equilibrium for the state-space system derived in part 2. Is this the only equilibrium point? d) (1.5 points) Compute a local linearization for the state-space system derived in part 2, around the