BIEN 462: Engineering Principles in Physiological Systems

Fall 2015 Assignment #1 – Basic principles of signals and systems analysis

Thursday, September 17, 2015 Deadline: Thursday, October 1, 2015

Reading: - Lectures 1-5. - Oppenheim and Willsky: chapters 1, 2, 4, 6, 9 or Lathi 1, 2, 4, 7 (only material related to continuous time systems covered in the lectures). 1. Consider a continuous-time system described by the following input-output relation:

1

( ) ( )t

t

y t x dτ α τ+

= −∫

where α is a real constant . (α) Determine whether this system is: (i) static (ii) invertible (iii) BIBO stable (iv) time-invariant (v) linear (β) For shich valus of α is this system causal?

2. Determine and sketch the response of the LTI system with impulse response h(t) to the input x(t), where x(t) and h(t) are given below, using the convolution integral.

3. Consider the following continuous-time linear time invariant (LTI) system:

One  period  of  sin(πt)  

where the causal subsystems F, G, H are described by the following ordinary linear differential equations (ODEs):

F: ( ) ( ) ( )dz t z t v tdt

+ =

G: ( )dz twdt

= −

H:( ) 2 ( ) ( )dy t y t z tdt

+ =

(i) Find the total system transfer function (from u to y). Draw the pole-zero diagram and

the region of convergence. Is the system stable? (ii) Determine and plot the system frequency response. (iii) Find the response of the systen to the input signal u(t)=sin(2t)

4. Consider the LTI system that is described by the following ordinary differential equation (ODE):

3 2

3 2

( ) ( ) ( )6 11 6 ( ) ( )d y t d y t dy t y t x tdt dt dt

+ + + =

Using the (one-sided) Laplace transform, determine the zero-input response yzi(t) and the zero-state response yzs(t) for t>0- when the input is x(t) = e−4tu(t) for the following initial conditions:

y(0-)=1, dy(0− )

dt= −1, d 2 y(0− )

dt2 = 1

5. Consider the simplified model of the human respiratory system shown below. The resistance R and compliance C represent the overall mechanical resistive and compliant properties of the respiratory system. Therefore, R corresponds to the combination of resistance to airflow in the airways, lung-tissue resistance and chest-wall resistance. C represents the combined compliance of lung tissue, chest wall and airways. Finally, the induction L represents fluid inertance in the airways. Also, PA is the alveolar pressure and Pa0 is the airway opening pressure. Also, consider Pa0 as the input and PA as the output of this simplified model.

(i) Show that the transfer function between Pa0 and PA is given by:

H (s) = 1LCs2 + RCs+1

(ii) Now, consider that we want to attenuate large fluctuations of the airway pressure

PA, which could be damaging to the lungs (e.g. in a clinical setting) by measuring esophageal pressure (which has been shown to closely reflect alveolar pressure fluctuations) and feeding back and subtracting a scaled version of this signal from the input as shown below:

Derive the transfer function of the above closed-loop system. (iii) Assume that R=0, L=0.01 (cmH2O)s2L-1, C=0.1 L (cmH2O)-1 and that the feedback gain k=1. Using Matlab, plot the impulse response, step response and frequency response (gain and phase) of both the open-loop and closed-loop systems described in (i) and (ii) respectively (you can use the commands impulse, step, freqs). Comment on your results and relate to the poles of the system. What is the effect of closing the loop? (iv) Repeat (iii) for R=0.5 (cmH2O)sL-1 and the rest of the parameters as above and comment on your results. What are the poles of both systems in this case? (v) Repeat (iii) for R=1 (cmH2O)sL-1 and the rest of the parameters as above and comment on your results, relating once more to the pole values of both open-loop and closed-loop systems.

6. In many applications, unstable subsystems are used. However, negative feedback can be used to obtain a closed-loop system that is stable. Consider the closed-loop system shown in Fig 1 where the subsyetem with transfer function Η(s) is causal and unstable and the subsystem with transfer function G(s) is stable.

Figure 1

Even if Η(s) is unstable, the closed loop system ΗCL(s) may be stable if G(s) is selected such that all poles of HCL(s) have a negative real part (i.e. they lie in the left half s-plane). Assume that the unstable subsystem H(s) is described by:

𝑑𝑦(𝑡)𝑑𝑡

− 3𝑦 𝑡 = 2𝑥(𝑡)  

and that the feedback is of the form G(s)=K.

(a) Determine the closed-loop transfer function ΗCL(s) as a function of Κ.

(b) Determine analytically the location of the pole of the closed-loop system as a function of K. For which values of K is the system stable?

(c) Use the Matlab command rlocus to plot the pole location as a function of K. Specifically, this command calculates the roots of the equation 1+ΚΗ(s)=0 as a function of Κ, which are the closed-loop system poles. If the vectors b and a contain the polynomial coefficients of the enumerator and denominator of HCL(s) then rlocus(b,a) plots the roors for Κ≥0. The roots for K≤0 may be plotted using rlocus(-b,a). These diagrams do not provide arithmetic values of K which correspond to specific poles. In order to get these precise values you can use the command syntax [r,k]=rlocus(b,a). (Note you can also use the syntax[r,k]=rlocus(sys,K) where sys=tf(b,a)).

Use rlocus to verify the results of (b). Plot separately the roots for Κ≤0 and Κ≥0. Use [r,k]=rlocus(b,a) to verify the values of Κ for which the closed-loop system is stable.

(d) Let the open-loop system be:

4 3 2

4 3 2

40 600 4000 10000( )123 50 2500

s s s sH ss s s s+ + + +

=− + + +

Use rlocus to plot the location of the poles for G(s)=K (separately for Κ≥0 and Κ≤0)

(e) Use the same command to specify a value of Κ such that all the closed-loop poles pk satisfy: -15<Re{pk}<0 and -5<Im{pk}<5.