ECE-520: Discrete-Time Control Systems Homework 3 Due: Tuesday December18 in class 1) Prove or disprove the following claims: if are linearly independent vectors, then so are ,,anduv w a) ,,uu vu v w+++b) 2,2,4uvwuvw+− −− vc) ,,uvvwwu−− −d) ,,uvwuvwuvw−++ −+ −+− Note: You must do this for arbitrary vectors. Do Not assume are specific vectors. ,,anduv w 2) For the following matrix 1021110 2A⎡⎤=⎢⎥⎣⎦ a) Find a set of vectors that form a basis for the null space ofA. b) Is the vector []22 22Tn =−in the null space ofA? That is, can you represent this vector as a linear combination of your basis vectors? c) Is the vector []12Ta =in the range (column) space ofA? 3) For the following matrix 101001221010A⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦ a) Find a set of vectors that form a basis for the null space ofA. b) Is the vector []2621Tn =−−in the null space ofA? That is, can you represent this vector as a linear combination of your basis vectors? c) Is the vector[]123Ta = in the range (column) space of A? 14) For the following matrix 101201112135A⎡⎤⎢⎥=⎢⎥⎢⎥⎣⎦ a) Find the rank of A (the number of linearly independent rows or columns). b) Determine two vectors that span the null space of A. c) Determine two vectors that span the row space of A. d) Show that any vector in the row space of A is orthogonal to any vector in the null space of A. e) Determine two vectors that span the column space of A. 5) Suppose we want to minimize a function while satisfying a constraint. For example, find the point in the plane closest to the origin. We want to write this as a minimization problem with a constraint, such as 5xy+= 22minimize (distance from origin)subject to x+y-5 = 0 (constraint)xy+ We do this with Lagrange multipliers (λ) and form the minimization problem 22minimize ( , , ) ( 5)Lxy x y x yλλ=++ +− To solve the problem we now set 0LLLxyλ∂∂∂===∂∂∂. Show that the optimal point is 52xy==. 6) Consider the discrete-time state variable system (1) () ()xkGxkHuk+=+ with initial state (0) 0x =. a) Show that after three steps(0 we have the system of equations ,1,2k = ) 2(0)(3) (1)(2)uxGHGHHuu⎡⎤⎢⎥⎡⎤=⎣⎦⎢⎥⎢⎥⎣⎦ 2b) Assume we want to go from the origin to the final state fxin three time steps with a penalty on the amount of input (i.e., the signal energy) We can formulate this problem as minimize (minimize engergy)subject to (constraint: must reach final state)TfuRuxQu− Ris a symmetric weighting matrix, indicating how much to penalize the input energy at each time step. What are andQu? c) We can again solve this problem using Lagrange multipliers. The form of the Lagrange multiplier is chosen so the L function makes mathematical sense. For example, for this problem, the function to be minimized is a scalar, so the Lagrange multiplier must be chosen to make the problem a scalar problem. Specifically, we form (, ) ( )TTfLu u Ru Qu xλλ=+ − Assuming 1R− exists but does not exist, show that the optimal control signal is 1Q−11()TT1fuRQQRQ x−−−= d) How would your answer change if (0) 0x≠? e) For , determine the matrix and then determine vector(s) that span the null space of Q 12 1and 11 0GH⎡⎤ ⎡=⎢⎥ ⎢⎣⎦ ⎣⎤=⎥⎦Q f) For[]12Tfx =, find a control vector uthat takes the system from the origin to fxthat minimizes the energy (is a minimum norm solution). Assume and RI=. g) Show that your control vector u to part e is orthogonal to the vector(s) that span the null space ofQ, hence u has no components in the null space ofQ. 7) Prelab: In this (and the next) lab you will be using Matlab’s sisotool to simulate and implement discrete-time PI and PID controllers for your one degree of freedom systems. This prelab presents a brief review of Matlab’s sisotool (the 6.5.1 version) and some of the things you will need to know to apply this to our problem. The file DT_PID.mdl is a Simulink model that implements a discrete-time PID controller. It is somewhat unusual in that the plant is represented in state-variable form, but this is the usual form we will using in this class. The Simulink model looks like the following: 3The file DT_PID_driver.m is the Matlab file that runs this code. We will be utilizing Matlab’s sisotool for determining the pole placement and the values of the gains. Before we go on, we need to remember the following two things about discrete-time systems: • For stability, all poles of the system must be within the unit circle. However, zeros can be outside of the unit circle. • The closer to the origin your dominant poles are, the faster your system will respond. However, the control effort will generally be larger. The basic transfer function form of the components of a discrete-time PID controller are as follows: Proportional (P) term : () ()pCz KEz=Integral (I) term: 1()11iiKKzCzzz−==−− Derivative (D) term : 1(1)() (1 )ddKzCz K zz−−=−= 4PI Controller: To construct a PI controller, we add the P and I controllers together to get the overall transfer function: ()()11piipKKzKKzCz Kzzp+−=+ =−− In sisotool this will be represented as 2()(()(1) (1)Kz az Kz aCzzz z)++==−− In order to get the coefficients we need out of the sisotool format we equate coefficients to get: ,pipKKa K K K=−=− PID Controller: To construct a PID controller, we add the P, I, and D controllers together to get the overall transfer function: 22 2(1) (1) ( ) ( 2)(1)()1(1) (1)pid pid pdidpKzz Kz K z K K K z K K z KKz K zCz Kz z zz zz−+ − − + + +− − +−=+ + = =−− −d In sisotool this will be represented as 2()()(1)Kz az bCzzz++=− In order to get the coefficients we need out of the sisotool format we equate coefficients to get: ,2,dp dipdKKb K Ka K K K K K==−− =−− For the PID controller, we can have either two complex conjugate zeros or two real zeros. Sisotool (Breif) Example A) Run the Matlab program DT_PID_driver.m. This program is set up to read the data file bobs_1dof_205.mat, which is a continuous time state variable model for a one degree of freedom torsional system, and implement a P controller with gain 0.0116. It will put the value of the transfer function for your system, , in your workspace. ()pGz • Type sisotool in the command window • Click close when the help window comes up • Click on view → open loop bode to turn off the bode plot.