EE363
Prof. S. Boyd
EE363 homework 1
1. LQR for a triple accumulator. We consider the system xt+1 = Axt + But , yt = Cxt , with 1 1 0 0 C= 0 0 1 . B = 0 , A = 1 1 0 , 0 0 1 1
This system has transfer function H(z) = (z - 1)-3 , and is called a triple accu
EE363
Winter 2008-09
Lecture 7
Estimation
Gaussian random vectors
minimum mean-square estimation (MMSE)
MMSE with linear measurements
relation to least-squares, pseudo-inverse
71
Gaussian random vectors
random vector x Rn is Gaussian if it has density
EE363
Winter 2008-09
Lecture 4
Continuous time linear quadratic regulator
continuous-time LQR problem
dynamic programming solution
Hamiltonian system and two point boundary value problem
infinite horizon LQR
direct solution of ARE via Hamiltonian
41
EE363
Winter 2008-09
Lecture 10
Linear Quadratic Stochastic Control with
Partial State Observation
partially observed linear-quadratic stochastic control problem
estimation-control separation principle
solution via dynamic programming
101
Linear stocha
EE363
Winter 2008-09
Lecture 14
Lyapunov theory with inputs and outputs
systems with inputs and outputs
reachability bounding
bounds on RMS gain
bounded-real lemma
feedback synthesis via control-Lyapunov functions
141
Systems with inputs
we now consi
EE363
Winter 2008-09
Lecture 15
Linear matrix inequalities and the S-procedure
Linear matrix inequalities
Semidefinite programming
S-procedure for quadratic forms and quadratic functions
151
Linear matrix inequalities
suppose F0, . . . , Fn are symmetr
EE363
Winter 2008-09
Lecture 16
Analysis of systems with sector nonlinearities
Sector nonlinearities
Lure system
Analysis via quadratic Lyapunov functions
Extension to multiple nonlinearities
161
Sector nonlinearities
a function : R R is said to be in
EE363
Prof. S. Boyd
EE363 homework 5
1. One-step ahead prediction of an autoregressive time series. We consider the following autoregressive (AR) system pt+1 = pt + pt-1 + pt-2 + wt , yt = pt + vt .
Here p is the (scalar) time series we are interested in,
EE363
Prof. S. Boyd
EE363 homework 4
1. Estimating an unknown constant from repeated measurements. We wish to estimate x N (0, 1) from measurements yi = x + vi , i = 1, . . . , N, where vi are IID N (0, 2), uncorrelated with x. Find an explicit expression
EE363
Prof. S. Boyd
EE363 homework 3
1. Solution of a two-point boundary value problem. We consider a linear dynamical system x = Ax, with x(t) Rn . There is an n-dimensional subspace of solutions of this equation, so to single out one of the trajectories
EE363
Prof. S. Boyd
EE363 homework 2
1. Derivative of matrix inverse. Suppose that X : R Rnn , and that X(t) is invertible. Show that d d X(t)-1 = -X(t)-1 X(t) X(t)-1 . dt dt Hint: differentiate X(t)X(t)-1 = I with respect to t. 2. Infinite horizon LQR fo
EE363
Winter 2008-09
Lecture 17
Perron-Frobenius Theory
Positive and nonnegative matrices and vectors
Perron-Frobenius theorems
Markov chains
Economic growth
Population dynamics
Max-min and min-max characterization
Power control
Linear Lyapunov fu