/*
The contents of this file are subject to the Mozilla Public License Version 1.1
(the "License"); you may not use this file except in compliance with the License.
You may obtain a copy of the License at http://www.mozilla.org/MPL/
Software distributed under the License is distributed on an "AS IS" basis, WITHOUT
WARRANTY OF ANY KIND, either express or implied. See the License for the specific
language governing rights and limitations under the License.
The Original Code is "CanonicalModel.java". Description:
"Utilities related to state-space models that are in controllable-canonical form.
@author Bryan Tripp"
The Initial Developer of the Original Code is Bryan Tripp & Centre for Theoretical Neuroscience, University of Waterloo. Copyright (C) 2006-2008. All Rights Reserved.
Alternatively, the contents of this file may be used under the terms of the GNU
Public License license (the GPL License), in which case the provisions of GPL
License are applicable instead of those above. If you wish to allow use of your
version of this file only under the terms of the GPL License and not to allow
others to use your version of this file under the MPL, indicate your decision
by deleting the provisions above and replace them with the notice and other
provisions required by the GPL License. If you do not delete the provisions above,
a recipient may use your version of this file under either the MPL or the GPL License.
*/
/*
* Created on 19-Jun-2006
*/
package ca.nengo.dynamics.impl;
import Jama.EigenvalueDecomposition;
import Jama.Matrix;
import ca.nengo.model.Units;
import ca.nengo.util.MU;
/**
* Utilities related to state-space models that are in controllable-canonical form.
*
* @author Bryan Tripp
*/
public class CanonicalModel {
/**
* <p>Realizes a transfer function in the form:</p>
* <p>H(s) = d + (b1*s^(n-1) + b2*s^(n-2) + ... + bn) / (s^n + a1*s^(n-1) + ... + an).</p>
*
* @param numerator Coefficients of the numerator of a transfer function (b1 to bn above)
* @param denominator Coefficients of the denominator of a transfer function (a1 to an above)
* @param passthrough Passthrough value (d above). If your transfer function has numerator and denominator
* of equal degree, divide them, give the result here, and give the remainder as the numerator
* and denominator arguments, so that the new numerator will have degree less than denominator.
* There is no state-space realization for TF with numerator degree > denominator degree.
* @return A controllable-canonical state-space realization of the specified
* transfer function
*/
public static LTISystem getRealization(float[] numerator, float[] denominator, float passthrough) {
if (numerator.length != denominator.length) {
throw new IllegalArgumentException("Should be equal numbers of numerator and denominator coefficients. " +
"(Use zeros instead of shorter lists.) ");
}
float[][] A = new float[denominator.length][];
for (int i = 0; i < A.length; i++) {
A[i] = new float[denominator.length];
if (i == denominator.length-1) {
for (int j = 0; j < denominator.length; j++) {
A[i][j] = -denominator[denominator.length - 1 - j];
}
} else {
A[i][i+1] = 1f;
}
}
float[][] B = new float[denominator.length][];
for (int i = 0; i < B.length; i++) {
B[i] = new float[]{(i == B.length-1) ? 1f : 0f};
}
float[][] C = new float[1][];
C[0] = new float[numerator.length];
for (int i = 0; i < numerator.length; i++) {
C[0][i] = numerator[numerator.length - 1 - i];
}
float[][] D = new float[][]{new float[]{passthrough}};
float[] x0 = new float[numerator.length];
return new LTISystem(A, B, C, D, x0, new Units[]{Units.UNK});
}
/**
* @param system Any SISO linear time-invariant system
* @return True if the system is in controllable-canonical form
*/
public static boolean isControllableCanonical(LTISystem system) {
if (system.getInputDimension() != 1 || system.getOutputDimension() != 1) {
throw new IllegalArgumentException("System must be single-input-single-output");
}
float[][] A = system.getA(0f);
for (int i = 0; i < A.length-1; i++) {
for (int j = 0; j < A.length; j++) {
if (j == i+1) {
if (!isClose(1f, A[i][j])) return false;
} else {
if (!isClose(0f, A[i][j])) return false;
}
}
}
float[][] B = system.getB(0f);
for (int i = 0; i < B.length; i++) {
if (i == B.length-1) {
if (!isClose(1f, B[i][0])) return false;
} else {
if (!isClose(0f, B[i][0])) return false;
}
}
return true;
}
private static boolean isClose(float a, float b) {
float tolerance = .00000001f;
boolean result = true;
if (a > b+tolerance || a < b-tolerance) result = false;
return result;
}
/**
* @param system An LTI system in controllable-canonical form
* @param tau A desired new time constant
* @return A copy of the system, with its slowest time constant changed to the
* new value
*/
public static LTISystem changeTimeConstant(LTISystem system, float tau) {
if (!isControllableCanonical(system)) {
throw new IllegalArgumentException("System must be in controllable-canonical form");
}
EigenvalueDecomposition eig = new Matrix(MU.convert(system.getA(0f))).eig();
double[] eigReal = eig.getRealEigenvalues();
double slowest = eigReal[0];
for (int i = 1; i < eig.getRealEigenvalues().length; i++) {
if (Math.abs(eigReal[i]) < Math.abs(slowest)) slowest = eigReal[i];
}
//may be repeated / complex conjugate so change all slowest real eigenvalues
for (int i = 0; i < eigReal.length; i++) {
if (Math.abs(eigReal[i] - slowest) < .0001*Math.abs(slowest)) eigReal[i] = -1d / (double) tau;
}
//coefficients of new transfer function polynomial ...
double[] eigImag = eig.getImagEigenvalues();
float[][] poly = new float[][]{new float[]{1f, (float) -eigReal[0]}, new float[]{0f, (float) -eigImag[0]}};
for (int i = 1; i < eigReal.length; i++) {
poly = prod(poly, new float[][]{new float[]{1f, (float) -eigReal[i]}, new float[]{0f, (float) -eigImag[i]}});
}
float[][] A = system.getA(0f);
for (int i = 0; i < A[0].length; i++) {
A[A.length-1][i] = - poly[0][poly[0].length-1-i];
}
Units[] units = new Units[system.getOutputDimension()];
for (int i = 0; i < units.length; i++) {
units[i] = system.getOutputUnits(i);
}
//rescale output so that integral of impulse response doesn't change (much; other poles are ignored)
float[][] C = MU.prod(system.getC(0f), -1 / (float) slowest / tau);
return new LTISystem(A, system.getB(0f), C, system.getD(0f), system.getState(), units);
}
//product of complex polynomials
private static float[][] prod(float[][] a, float[][] b) {
if (a[0].length != a[1].length || b[0].length != b[1].length) {
throw new IllegalArgumentException("Number of real and imaginary coefficients must be equal for each polynomial");
}
float[][] result = new float[2][];
result[0] = new float[a[0].length + (b[0].length-1)];
result[1] = new float[a[0].length + (b[0].length-1)];
for (int i = 0; i < a[0].length; i++) {
for (int j = 0; j < b[0].length; j++) {
float[] p = prod(a[0][i], a[1][i], b[0][j], b[1][j]);
int degree = a[0].length + b[0].length - i - j - 2;
result[0][result[0].length - 1 - degree] += p[0];
result[1][result[0].length - 1 - degree] += p[1];
}
}
return result;
}
//product of complex number a0+a1*i and b0+b1*i
private static float[] prod(float a0, float a1, float b0, float b1) {
return new float[]{a0*b0 - a1*b1, a0*b1 + a1*b0};
}
/**
* @param dynamics A linear time-invariant dynamical system
* @return Time constant associated with the system's dominant eigenvalue
*/
public static float getDominantTimeConstant(LTISystem dynamics) {
double[] eig = new Matrix(MU.convert(dynamics.getA(0f))).eig().getRealEigenvalues();
double slowest = eig[0];
for (int i = 1; i < eig.length; i++) {
if (Math.abs(eig[i]) < Math.abs(slowest)) slowest = eig[i];
}
return -1f / (float) slowest;
}
}