Stability Analysis of Discrete System
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This document discusses different models for representing digital control systems and filters in MATLAB, including transfer functions, zero-pole-gain, state space, partial fraction expansion, second-order sections, lattice structure, and convolution matrix representations. It provides examples of converting between these different representations, such as using the poly and roots functions to convert between polynomial and zero-pole-gain forms. The toolbox functions residuez, tf2latc, and convmtx are also demonstrated for partial fraction expansion, converting to lattice structure, and generating a convolution matrix respectively.
![The poly and roots functions convert between polynomial and zero-pole-gain
representations.
For example, a simple IIR filter is
b = [2 3 4]; a = [1 3 3 1];
The zeros and poles of this filter are
q = roots(b) p = roots(a)
% Gain factor k = b(1)/a(1)
Returning to the original polynomials,
bb = k*poly(q)
aa = poly(p)](https://image.slidesharecdn.com/stability3rdlecture-181014183753/85/Stability-Analysis-of-Discrete-System-38-320.jpg)
![Partial fraction expansion arises in signal processing as one method of finding the
inverse Z-transform of a transfer function. For example, the partial fraction expansion of
certain systemb = [-4 8];
a = [1 6 8];
[r,p,k] = residuez(b,a)
To find the inverse Z-transform of H(z), find the sum of the inverse Z-transforms of the
two addends of H(z), giving the causal impulse response:
h(n)=−12(−4)^n +8(−2)^n n=0,1,2,…](https://image.slidesharecdn.com/stability3rdlecture-181014183753/85/Stability-Analysis-of-Discrete-System-39-320.jpg)
![To verify this in the MATLAB environment, type
imp = [1 0 0 0 0];
resptf = filter(b,a,imp)
respres = filter(r(1),[1 -p(1)],imp)+...
filter(r(2),[1 -p(2)],imp)
The toolbox function tf2latc accepts an FIR or IIR filter in polynomial form and returns
the corresponding reflection coefficients. An example FIR filter in polynomial form is
b = [1.0000 0.6149 0.9899 0.0000 0.0031 -0.0082];
This filter's lattice (reflection coefficient) representation is
k = tf2latc(b)](https://image.slidesharecdn.com/stability3rdlecture-181014183753/85/Stability-Analysis-of-Discrete-System-40-320.jpg)
![he convolution matrix for a vector b, representing the numerator coefficients for a
digital filter, is
b = [1 2 3];
x = randn(3,1);
C = convmtx(b',3);
Two equivalent ways to convolve b with x are as follows.
y1 = C*x;
y2 = conv(b,x);](https://image.slidesharecdn.com/stability3rdlecture-181014183753/85/Stability-Analysis-of-Discrete-System-41-320.jpg)
Stability Analysis of Discrete System
- 1. Assist. Prof. Dr. Khalaf S Gaeid Electrical Engineering Department Tikrit University 3rd lecture Digital Control systems Stability Analysis of Discrete Systems with Examples
- 37. MATLAB DISCRETE-TIME SYSTEM MODELS The discrete-time system models are representational schemes for digital filters. The MATLAB technical computing environment supports several discrete-time system models, which are described in the following sections: 1. Transfer Function 2. Zero-Pole-Gain 3.State Space 4. Partial Fraction Expansion (Residue Form) 5.Second-Order Sections (SOS) 6. Lattice Structure 7.Convolution Matrix
- 38. The poly and roots functions convert between polynomial and zero-pole-gain representations. For example, a simple IIR filter is b = [2 3 4]; a = [1 3 3 1]; The zeros and poles of this filter are q = roots(b) p = roots(a) % Gain factor k = b(1)/a(1) Returning to the original polynomials, bb = k*poly(q) aa = poly(p)
- 39. Partial fraction expansion arises in signal processing as one method of finding the inverse Z-transform of a transfer function. For example, the partial fraction expansion of certain systemb = [-4 8]; a = [1 6 8]; [r,p,k] = residuez(b,a) To find the inverse Z-transform of H(z), find the sum of the inverse Z-transforms of the two addends of H(z), giving the causal impulse response: h(n)=−12(−4)^n +8(−2)^n n=0,1,2,…
- 40. To verify this in the MATLAB environment, type imp = [1 0 0 0 0]; resptf = filter(b,a,imp) respres = filter(r(1),[1 -p(1)],imp)+... filter(r(2),[1 -p(2)],imp) The toolbox function tf2latc accepts an FIR or IIR filter in polynomial form and returns the corresponding reflection coefficients. An example FIR filter in polynomial form is b = [1.0000 0.6149 0.9899 0.0000 0.0031 -0.0082]; This filter's lattice (reflection coefficient) representation is k = tf2latc(b)
- 41. he convolution matrix for a vector b, representing the numerator coefficients for a digital filter, is b = [1 2 3]; x = randn(3,1); C = convmtx(b',3); Two equivalent ways to convolve b with x are as follows. y1 = C*x; y2 = conv(b,x);
- 42. Thanks