Digital Control System

Q1 What is a linear control system. In state space method what does the matrices A, B, C, D stand for?

Ans. A linear control system is that which satisfies the rules of homogeneity and superposition.
If y1(t)= x1(t);
Y2(t)= x2(t);
For a linear system ay1(t)+by2(t) = ax1(t) +bx2(t);
The state space equations are written as
X’ = Ax + Bu
Y = Cx + Du
Where x= state variable
A = State matrix
B = Input matrix
C = Output matrix
D = Disturbance Matrix

Q2 What is the sampling time which has been selected for the nonlinear control system & how?

Ans. Sampling time is the discrete time interval for which data is obtained for the system. It depends on the dynamics of the system, the required accuracy & hardware constraints.
The sampling time is generally selected according to the thumbrule
1/30Fc < T < 1/5Fc
Where Fc = system bandwidth
Here System Bandwidth < 5 Hz (3.88Hz) as Rise Time * BW = 0 .35
Sampling Time = 100Hz (10msec) (ST should be between 25Hz to 150 Hz)

1. Sampling times lower than 5 Fc are not selected because it may result in aliasing errors. (According to the Sampling Theorem the sampling frequency must be more than 2 times the highest frequency component in the signal.)

2. The sampling period must be compatible with the update rate specifications of the ADC & DAC used in system design.

3. The shorter the sampling period the more closely the discrete time system resembles a continuous time system. In the limit as the sampling period approaches zero, the responses of the discrete time & continuous time & continuous time systems become indistinguishable.

4. The sampling period must be long enough so that sufficient time is available for the execution of the controller algorithm & I/O operations during each discrete time step. This criterion decides the higher limit of the time step.

5. The time required for the control system to complete one loop of the algorithm is the sampling time T. It depends on the time required for the computer to calculate the control algorithm & the time required by the interfaces to convert data. The step size for the simulation must be less than the smallest local time constant of the model simulated. If the step size chosen is higher then the simulation will exhibit numerical instability.

Q3 What is the method for selecting a good controller sampling frequency?

Ans. 1. Develop a linear plant model & design a continuous time controller. Plot the step response & frequency response of the closed loop control system.
2. Choose a very short sampling period that provides a good discrete time system approximation to the continuous time system performance.

3. Discretize the controller algorithm using c2d command & an appropriate discretization method.

4. Plot the step response & frequency response of closed loop system using the discrete controller in place of the continuous time controller.
5. Increase the sampling period & repeat till the step & frequency response of the system vary unacceptably from the continuous time response.

Q4a. What is quantization error? Explain

It is the error due to the computer’s finite word size.


Q4b. What are the errors which can arise due to numerical solution of differential equations?

Ans. Many numerical methods are interative, that is they involve repeating the same calculation many times. In terms of error analysis, two types of error emerge, local and global errors. The local error is the error introduced during one operation of the iterative process. The global error is the accumulative error over many iterations. Note that the global error is not simply the sum of the local errors due to the nonlinear nature of many problems although often it is assumed to be so because of the difficulties in measuring the global error.

There are at least two sources of errors in numerical calculations:
1. Rounding Errors
2. Truncation Errors
Rounding errors originate from the fact that computers can only represent numbers using a fixed and limited number of significant figures. Thus, numbers such as or cannot be represented exactly in computer memory. The discrepancy introduced by this limitation is call round-off error. Even simple addition can result in round-off error.
Truncation errors in numerical analysis arise when approximations are used to estimate some quantity. Often a Taylor series is used to approximate a solution which is then truncated.
Errors enter the numerical solution of the initial value problem from two sources:
² discretization error,
² roundoff error.
Discretization error is a property of the differential equation and the numerical method. If all the arithmetic could be performed with infinite precision, discretization error would be the only error present. Roundoff error is a property of the computer hardware and the program. It is usually far less important than the discretization error, except when we try to achieve very high accuracy. Discretization error can be assessed from two points of view, local and global.
Local discretization error is the error that would be made in one step if the previous
values were exact and if there were no roundoff error. Let un(t) be the solution of
the differential equation determined not by the original initial condition at t0 but
by the value of the computed solution at tn. That is, un(t) is the function of t
defined by
u_n = f(t; un);
un(tn) = yn:
The local discretization error dn is the difference between this theoretical solution
and the computed solution (ignoring roundoff) determined by the same data at tn:
dn = yn+1 ¡ un(tn+1):
Global discretization error is the difference between the computed solution,
still ignoring roundoff, and the true solution determined by the original initial condition at t0, that is,
en = yn ¡ y(tn):
The distinction between local and global discretization error can be easily seen
in the special case where f(t; y) does not depend on y.

Q4c. How is the step size of the interval selected?

Ans. The accuracy of the solution will depend on how small we make the step size, h. If you look at the figure below you will see that the Euler method projects the slope forward and uses it to estimate the next solution point. However, because the exact solution may be nonlinear, the projection will inevitably yield some error, called the truncation error. In panel B), one can see that the smaller we make h, the smaller the truncation error. However, a small h means it will take longer to traverse the solution and if h is too small we can begin to run into round-off errors.


Q5. How was the feedback selected? Why is feedback used?

Ans. Feedback was selected based on the linear system transfer function root locus plot.
Feedback is used to
a) Modify the transient response of the system
b) Reduce the effects of disturbance on system response
c) Reduce steady state tracking errors
d) Decrease the sensitivity of the system to plant variations
e) Disadvantages of feedback are increased system complexity, reduction of closed loop gain, introduction of instabilities in the system

Q6. What are the different ways in which differential equations can be solved?
Ans Second order Differential equations with linear constant coefficients can be solved if they can be put in quadratic form by solving for the roots of the quadratic equation. First order differential equations can be solved by separation of the variables & then integration of both sides.
Numerical methods of solution are – Runge Kutta’s , Euler’s method, Rosenbrock’s method, etc.


Q7. Explain the Runge Kutta 4 method of solution of differential equations.

Ans.) Consider the differential eqn
Y’ = f(x,y)
With initial condition y(xo) = y0
If y(x) is the exact solution of the equation then the Taylor’s series for y(x) around x= xo is given by
Y(x) = yo + (x-xo)yo’ + (x-xo)2/2yo’ + .............
4th order Runge Kutta method
K1 = h f(xo,yo)
K2 = h f(xo+h/2, yo+k1/2)
K3 = h f(xo+h/2, yo+k2/2)
K4 = h f(xo+h,yo+k3)
Y1 = y0 + 1/6*(k1 + 2k2 + 2k3 + k4)
The RK4 method takes a series of slopes from which it constructs a weighted average.

Q8. What are stiff problems?

Ans. One class of problem, terms stiff problems, is characterized by widely separated time scales. For example one part of a model might be dominated by very fast reactions, while another by slow reactions. In such situations, a numerical integrator must ensure that the step size is small enough to capture the fast dynamics. This results in extremely slow integration times and often failure to integrate the solutions at all. As a result, numerical analysts have developed sophisticated methods to deal with stiff problems.

Q9. What are error estimates? How are they obtained for MATLAB ode solvers?

Ans Error estimates are errors in the solution of the differential equations.
Modern numerical methods automatically determine the step sizes
hn = tn+1 - tn
so that the estimated error in the numerical solution is controlled by a specified tolerance.
For example, ode45 obtains its error estimate by comparing a fourth-order and a fifth-order formula.
An estimate of the error that would occur with this step is provided by yet another
linear combination of the slopes:
en+1 = hXki=1±isi:
If this error is less than the specified tolerance, then the step is successful and yn+1 is accepted. If not, the step is a failure and yn+1 is rejected. In either case, the error estimate is used to compute the step size h for the next step.

Q10. What is a phase plane plot?
Ans. It is the plot of the free response of a 2nd order nonlinear system where a pair of state variables generally displacement and velocity are taken for plotting the system response.


Q11. What is limit cycle oscillation ?
'Limit cycle oscillations' can be thought of as in between decaying oscillations and flutter. After an initial displacement the oscillations grow for a short period. The oscillations then settle down to a constant magnitude, instead of continuing to grow.
Limit-cycle oscillations arise from physical systems that show large deviations from equilibrium;
In a limit-cycle oscillator, the amplitude tends to be more or less constant but the frequency can vary greatly.