Question 1: In the following problem we analyze control of a ball and beam system.  The parameters of the         system include the inertia of the beam J, the mass and  radius of the ball m and r, gravity g. The variables of the system include the angle of the beam e , and the linear   position of the ball p, while the applied effort to the         system is T . Then the nonlinear equations of motion are:

(mp2  + J) + 2mpṗ ė + mgp cos e = T

below.  Solve for Ẋ 3  and Ẋ 4 .

X"  = p

1

 The operating point e =  e%  = 0 and p = p% corresponds to the beam being at rest horizontally, while the ball is at

f2  ≈?

f2 (e%, p% ) =?

D&!,p!   =?

6e ≡?

Ẋ 4  ≈?

Question 2: Assume m = 1,  M!  = 1, M!  = 2, g = 10, p0 = 0.5 for the linearized ball and beam

system. Determine the state space representation for this system in terms of matrices A, B, C, and

D in normal measurement space with ball position p, velocity p-dot, small angular deviation 6e , and

X.3  = X4

Question 3: Given the state space system in represented below in measurement space, Use          Matlab and matrix manipulations to solve for the state space representation in phase variable form. Compare the characteristic equations of the new and original representations.

J   K̇̇ẋẋxx = L         − 0(1)0(0)ML    M + L0 M u

y = [1 0 0 0] L    M +0*u

Question 4: A state space representation in phase variable form is given below. Show that all the   states are controllable using the controllability matrix.  Design feedback gains Kx=[K1 K2 K3 K4] so that the closed loop characteristic equation is (s) = s4   +  14s3   +  71s2   +  154s  +  120 . Show all work.

Question 5: Assume you have a set of gains Kx=[121 155 72 15] designed in phase variable         space. Convert these gains into measurement variables Kz. Compare the characteristic equations of the closed loop systems for Ax-Bx*Kx and Az-Bz*Kz.

uremt 0(ce) 0(1)0(0)M

Phase variable space

A)  = L             M

B*  = L0 M

B)  = L  M

C*  = [1 0 0 0]

C)  = [1 0 0 0]

Question 6: The differential equation below represents the transfer function for the linearized ball and beam with 6e as output and p as input.  Assume a controller C(s) and a closed loop transfer function T(s)=C(s)*G(s)/(1+C(s)*G(s)). Determine the steady state response p(t = ∞ ) to a unit    impulse as a reference input r(s). Hint: Pick your own controller and then use the final value

theorem to find the value of the state as time goes to infinity .

 = − 10 . 6e

p(s)          10

G(s) =               =   2

Question 7: After the ball and beam system state space control has been implemented, a closed  loop system is formed G2(s) shown below. It is proposed to place the resulting system within in    another unity feedback loop so that the open loop plant is K*G2(s). The root locus and Nyquist     plots for the new open loop system are shown. Write down the Nyquist stability criterion, and note the number of unstable open loop poles, and desired number of counter-clockwise encirclements. Using the information from the plots shown below, make a quick estimate of the upper and lower  limits of gain K for stability.                                                                                                                KC(s)G2 (s) =

Question 8: The Bode plots of the complete system and individual factors of T(s) are shown. A)      Compute expressions for the magnitude of each individual Bode factor, and determine the total DC gain. B) On the magnitude and phase plot below, label each item that was determine in (A).

T(s) = 

Bode  Diagram

0

-50

- 100

- 150

0

-90

- 180

-270

-360

10

Frequency    (rad/s)

Question 9: The continuous state space observer equations of the ball and beam system are given below.  Two cases are shown for measurement output: A) ball position, and B) beam angle.  Pick   one of the cases and determine the discrete observer matrices Ad, Bd, Cd, and Ld, using the         explicit approximation of the derivative operator “s” . Assume the sample time is T. Do not solve for the observer gains.  Just express your answers in terms of the matrix L or L1, L2, L3, and L4.

Explicit:  ≈  (z − 1)

A = L         − 0(1)0(0)M

L = J    K

Case A) C*  = [1 0 0 0]

Case B) C*  = [0 0 1 0]

B = L0 M

D = 0

Question 10: For this problem, we consider the ball and beam system, using the ball position p(s) is as output and (clockwise) beam angle e(s) is considered input. The continuous system transfer function G3 (s) and proposed controller C(s) are given below. The equivalent sampled transfer         function G3 (z) from the z-transform table (see back page), as well as the discretized approximation of the controller C(z) using the explicit approximation of the Laplace operator “s” are also provided.

Evaluate stability for K=10 and a sampling time of T=2.0 seconds. Hint: Use Matlab to find the roots of the closed loop characteristic equation.

p(s)        10

G3 (s)  =            =   2