Description
LAB 3 IMPLEMENTATION ISSUES IN DIGITAL CONTROL
1 Purpose
The purpose of this experiment is to investigate issues that arise in the implementation of
digital controllers. The effects of quantization and finite word-length on signal representation as
well as system parameters are studied. Different forms for the realization of digital controllers and
their numerical accuracy are investigated. The possibility of quantization-induced limit cycles is
demonstrated. Finally, a sampled-data control system is implemented with 16-bit accuracy and
its performance evaluated.
2 Introduction
In our study of digital control design, we have implicitly assumed that computations can
be done with infinite precision. In an actual implementation, the digital control law is realized
through a microcontroller or a digital signal processing chip. These would have finite precision,
usually 8-bit or 16-bit. The analog-to-digital converter (ADC) as well as the digital-to-analog
converter (DAC) also have finite precision, usually 8-bit, 10-bit, or 12-bit. The digital control
law calculation is often done in fixed-point arithmetic, although we do not go into the details of
the implementation.
The finite word-length issue has a significant impact on how to implement a digital controller.
To evaluate the performance of a sampled-data control system, we need to quantize all controller
coefficients as well as signals coming out from the ADC and DAC. In this experiment, we examine
typical situations to illustrate the finite word-length effects. You will also implement the sampleddata control system described in Section 4.6 of the course notes.
3 Preparation
Familiaze yourself with the operation of the quantizer in simulink, which can be found in the
Discontinuities library. In a directory which is in the path for Matlab, add the m-file quant.m,
consisting of the lines:
function [xq]=quant(x,q)
xq=q*round(x/q);
Here q is the quantization value. You will use quant.m to quantize the coefficients of the
control law. Next, complete the following preparation exercises and hand in your answers, on
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a separate sheet, to your T.A. at the beginning of the lab. Every student must hand in
his/her own work. There is space provided below to record your preparation work for the
experiment.
3.1 Control Design Calculations
1. In preparation for the design calculations for the sampled-data control problem described
in Section 4.6, enter the A, B, and D matrices as described on p.123 into Matlab. Take the
sampling interval T = 0.1 sec. Compute [Ad, Bd] = c2d(A, B, T) and partition the matrices
of the resulting discrete time system into the form
x
′ = Adx + Bdu e = Dx
with
x =
x1
x2
, Ad =
A1 A3
0 A2
, Bd =
B1
0
, D =
D1 D2
2. Choose F1 to place the poles of A1 +B1F1 at 0, 0. Repeat the calculation to place the poles
of A1 + B1F1 at 0.8 ± j0.2.
3. Determine the matrices needed to solve the linear algebraic equation (4.8) in the course
notes for the 2 choices of F1. Determine the corresponding solutions for X and F2. Note
that the command to vectorize a matrix A in Matlab is A(:), and the Kronecker product
of A and B is kron(A, B).
4. Determine the gain for the full order observer to place the poles at A + LD at 0, 0, 0, 0, 0.
Repeat to place the poles at 0.1, 0.1 ± j0.3, 0.2 ± j0.2. For future reference, call the output
feedback controller with all poles placed at 0, Controller 1. The second one is Controller
2. Determine their respective transfer functions. Save your data also to a usb stick for use
in the lab.
5. Sketch the simulink diagram that you would use to simulate the sampled-data control
system, incorporating explicitly the quantizer after the zero-order holds. Put also the
sketch in the box below for reference during the experiment.
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