EE363 Prof. S. Boyd3/17/06–3/18/06 or 3/18/06–3/19/06Final examThis is a 24 hour take-home final exam. Please return it in room 244 of the Packard building24 hours after you pick it up. Please read the following instructions carefully.• You may use any books, notes, or computer programs (e.g., Matlab), but you maynot discuss the exam with anyone until March 20, after everyone has taken the exam.The only exception is that you can ask the TAs or Stephen Boyd for clarification, byemailing to the staff email address.• Please check your email a few times during the exam, just in case we need to send outa clarification or other announcement.• Please address email inquiries to [email protected]. Thisforwards the mail to the professor and the TAs. In particular, please do not use StephenBoyd’s or the TAs’ individual email addresses.• Attach the official exam cover page (available when you pick up or drop off the exam)to your exam, and assemble your solutions to the problems in order, i.e., problem 1,problem 2, . . . , problem 8. Start each solution on a new page.• We are asking you to do only SIX out of the EIGHT problems. You MUSTindicate on the exam cover page which SIX problems you did. If you don’t,we’ll just read your first six problems. (All problems have equal weight.)• Please make a copy of your exam before handing it in. We have never lost one, but itmight occur.• When a problem involves some computation (say, using Matlab), we do not want justthe final answers. We want three things:– A clear discussion and justification of your approach and method, in mathemat-ical terms, i.e., using matrices, matrix multiplication, eigenvalues, singular valuedecomposition, etc. In your mathematical description you cannot refer to Matlaboperators, such as the backslash operator.– A short discussion of how you solved the problem using Matlab (or some equiva-lent), and the source code that produces the result.1– The final numerical result. Be sure to show us your verification that your com-puted solution satisfies whatever properties it is supposed to, at least up to nu-merical precision. For example, if you compute a symmetric matrix P that issupposed to satisfy ATP + P A < 0 and P > 0, you should verify this using thecommands max(eig(A’*P+PA)) and min(eig(P)), and showing us the results.(The first should be negative; the second positive.)• Some problems require you to download and run a Matlab file to generate the dataneeded. These files can be found at the URLhttp://www.stanford.edu/class/ee363/FILENAMEwhere you should substitute the particular filename (given in the problem) for FILENAME.There are no links on the course web page pointing to these files, so you’ll have to typein the whole URL yourself.21. Integral control design via LQR. We consider the LDS ˙x = Ax + Bu, y = Cx, whereA ∈ Rn×n, B ∈ Rn×p, and C ∈ Rm×n. You can think of y(t) as some sort of deviationor error, that we’d like to drive to zero by a good choice of u.We define the signal z to bez(t) =Zt0y(τ) dτ,which is the running integral (componentwise) of the error signal. Now we define thecost functionJ =Z∞0y(τ)Ty(τ) + ρu(τ)Tu(τ) + γz(τ)Tz(τ )dτ,where ρ and γ are given positive constants. This is like an LQR cost function, withthe addition of a term that penalizes the integral of the norm squared of the integralof the error. (Do not say that out loud too quickly.)(a) Show that the input u that minimizes J can be expressed in the formu(t) = KPx(t) + KIz(t),where KP∈ Rp×nand KI∈ Rp×m. The matrix KP∈ Rp×nis called the propor-tional state feedback gain, and the matrix KI∈ Rp×mis called the integral outputgain.Be sure to explain how to find the two matrices KPand KI. You can use anyof the ideas we’ve used in the course: Lyapunov and Sylvester equations, Riccatiequations, LMIs, SDPs, etc.(b) Now consider the specific caseA =0 1 0 00 0 1 00 0 0 11 −3 7 2, B =2 −34 5−1 11 −4, C = [1 0 0 0].with cost parameters ρ = γ = 1. Find KPand KI.Evaluate J (with the optimal u) for x(0) = (1, 0, 0, 0).32. Observer with saturating sensor. We consider a system of the form˙x = Ax + Bu, y = Cx,where A ∈ Rn×n, B ∈ Rn×m, and C ∈ R1×n. To estimate the state x(t), based onobservation of the input signal u and a measurement related to the scalar output signaly, we use a replica of the system, with an extra feedback input e:˙ˆx = Aˆx + Bu + Le, ˆy = C ˆx,where L ∈ Rnis the observer feedback gain. In the usual setup, e(t) = y(t) − ˆy(t),i.e., e is the output prediction error, and L is chosen so that A − LC is stable. Thisensures that the state estimation error ˜x(t) = x(t) − ˆx(t) converges to zero, since˙˜x = (A − LC)˜x.In this case, however, the measurement we have is the saturated output predictionerror,e(t) = sat(y(t) − ˆy(t)).(The signals e, y, and ˆy are all scalar.) In other words, we get the actual predictionerror when it is no more than one in magnitude; otherwise, we get only the sign of theprediction error.The goal is to find an observer feedback gain L that gives quick convergence of thestate estimation error ˜x(t) to zero, for any input u.You can convince yourself (and it is true) that if A is not stable, then no matter howyou choose L, we won’t have ˜x(t) → 0 for all x(0), ˆx(0), and inputs u. So we’regoing to need A to be stable. This means that one possible choice for L is zero, inwhich case the state estimation error satisfies the stable linear dynamics˙˜x = A˜x, andtherefore converges to zero. (This is called an open-loop observer.) By using a nonzeroL, we hope to speed up the estimator convergence, at least when the output errormeasurement isn’t saturated. When the output prediction error isn’t saturated, thestate estimation error dynamics is given by˙˜x = (A − LC)˜x, so we’d like to choose Lto make these dynamics fast, or at least, faster than the open-loop dynamics A.To certify the convergence of ˜x(t) to zero, we also seek a quadratic Lyapunov functionV (˜x) = ˜xTP ˜x with the following properties:• P > 0.•˙V (˜x) < −αV (˜x) for all x 6= ˆx and u.•˙V (˜x) ≤ −βV (˜x) for all x, ˆx, and u, provided |y(t) − ˆy(t)| ≤ 1.Here, β ≥ α > 0 are given parameters that specify required convergence rates for thestate estimation error in the saturated and unsaturated cases,