3. Reduce the following Boolean expressions to the indicated number of literals a.Ga = a'·c' + a·b•c + a•c' to three literals b. Gb = a'b·(d' +c'·d) + b-(a + a'·c·d) to one literal с. Gc = a-b.c·d + a'·b•d + a·b·c'·d to two literals

A very interesting and useful velocity control system has been designed for a wheelchair control system. A proposed system utilizing velocity sensors mounted in a headgear is shown in Figure 1.The headgear sensor provides an output proportional to the magnitude of the head movement. There is a sensor mounted at 90° intervals so that forward, left, right, or reverse can be commanded. \text { Typical values for the time constants are } \tau_{1}=0.5 \mathrm{~s}, \mathrm{t}_{2}=1 . \mathrm{s}, \text { and } \tau_{3}=0.25 \mathrm{~s} (a) Determine the limiting gain K = K¡K2K3 for a stable system. (b) When the gain K is set equal to one-third (1/3) of the limiting value, determine whetherthe settling time (to within 2% of the final value of the system) is T§< 4 s. (c) Determine the value of gain that results in a system with a settling time of Ts<4 s. Also,obtain the value of the roots of the characteristic equation when the settling time is Ts<4 s.

5. Generate an LTI transfer function for your symbolic representation of F(s) in part 2.problem 2 in both polynomial form and factored form. Start with the F(s) you generated symbolically.

Two carts with negligible rolling friction are connected as shown in Figure 2. An input force u(t) is applied.The masses of the two carts are M, and M, and their displacements are x(t) and q(t), respectively. The carts are connected by a spring k and a damper b. Answer the following questions: By using Newton's Second Law, derive two mathematical equations that describe the motion of thetwo carts. (5 marks)(a) (b)According to your ß value, answer one of the following two questions (10 marks): 0 \leq \beta \leq 13: \text { Determine the transfer function } G_{1}(s)=\frac{Q(s)}{U(s)} 14 \leq \beta \leq 18: \text { Determine the transfer function } G_{2}(s)=\frac{X(s)}{U(s)}

22. State Space(SS) Represent the systems below in state space in phase-variable form. Draw the signal-flow graphs.[Section: 5.7] \text { a. } G(s)=\frac{s+3}{s^{2}+2 s+7}

. For the unity-feedback system in Figure P9.1, with G(s)=\frac{K}{(s+4)(s+6)(s+10)} do the following: a. Design a controller that will yield no more than 25% overshoot and no more than a 2-second settling time for a step input and zero steady-state error for step \text { b. MAILAB Use MATLAB and verify your design. }

2. (19 pts.) (Chapter 3) Given the dc servomotor and load shown, represent the system in state space, where the state variables are the armature current, ia, load displacement, OL, and load angular velocity, wL. Assume that the output is the angular displacement of the armature. Do not neglect armature inductance.

Q1) Given the following block diagram system. a) Reduce the block diagram to a single transfer function. b) Plot the root locus of the system. Explain the plot.

14.Use MATLAB to find the transfer function, G(s)=Y(s)/R(s), for each of the following systems represented in state space: [Section: 3.6] \begin{aligned} &\mathbf{b}\\ &\dot{\mathbf{x}}=\left[\begin{array}{rrrrr} 3 & 1 & 0 & 4 & -2 \\ -3 & 5 & -5 & 2 & -1 \\ 0 & 1 & -1 & 2 & 8 \\ -7 & 6 & -3 & -4 & 0 \\ -6 & 0 & 4 & -3 & 1 \end{array}\right] \mathbf{x}+\left[\begin{array}{l} 2 \\ 7 \\ 8 \\ 5 \\ 4 \end{array}\right] \mathbf{r}\\ &y=\left[\begin{array}{lllll} 1 & -2 & -9 & 7 & 6 \end{array}\right] \mathbf{x} \end{aligned}

The open-loop transfer function of a robot arm drive might be approximated G(s) H(s)=\frac{2 K(s+0.5 a)}{(s+3)\left(s^{2}+4 s+8\right)}, a, K>0 Use the Routh-Hurwitz criterion to show that for sufficiently small a, the system is stable for all(positive) values of K. Let a = 15. Show that the system becomes unstable for sufficiently large K, and find all the roots of the system when it is marginally stable. Mention the frequency of oscillation in Hz. For the values in ii), where the system is marginally stable, plot the system's step response (assume unity gain feedback), and show that there is a sustained oscillation at the frequency in ii) A sample is shown below