Exercise13

  • Due No Due Date
  • Points None

13.1 Model Reduction and Model Reduction Error

Find balanced state space realizations  for the following systems LaTeX: G\left(s\right)\:=\:C\left(sI-A\right)^{-1}B and truncate them to find suitable approximations LaTeX: \widehat{G} that make  LaTeX: |G(i\omega) -\widehat{G}(i\omega)|small for all LaTeX: \omega (hint: balreal+modred). Plot Bode diagrams for LaTeX: G and LaTeX: \widehat{G} to illustrate the results for different model orders.

a) LaTeX: A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -0.1 & 0 \\ 0 & 0 & -0.101 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 1 & 1 \end{pmatrix}

b) LaTeX: A = \begin{pmatrix} -1 & 0 & 0 \\ 0 & -0.1 & 0 \\ 0 & 0 & -0.101 \end{pmatrix}, \quad B = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \quad C = \begin{pmatrix} 1 & 1 & 1 \end{pmatrix}

c) LaTeX: G(s) = \dfrac{(s+2)(s+4)(s+6)(s+8)}{(s+1)(s+3)(s+5)(s+7)}

d) It is possible to show that the truncation error lies between the following bounds

LaTeX: \sigma_{r+1} \leq |G(i\omega)-\widehat{G}(i\omega)| \leq 2 (\sigma_{r+1} + \ldots \sigma_{n})      (the "twice the tail-bound")

The bound is valid if LaTeX: A is stable, and you are keeping the first LaTeX: r states and have LaTeX: \sigma_{r} > \sigma_{r+1} .

Compare these bounds with the result you got in c).

Hint: Either use a Bode  plot of LaTeX: G-\widehat{G}, or use the function hinfnorm(H) which calculates LaTeX: \sup_\omega|H(i\omega)|.

13.2 Balanced Realization Transformation

Show that the coordinate transformation LaTeX: x=T\bar x changes matrices the following way

a) LaTeX: \bar A = T^{-1} A T, LaTeX: \quad \bar B = T^{-1}B, LaTeX: \quad \bar C = C T, \quad \bar D = D (state-space matrices)

b) LaTeX: \bar P = T^{-1}PT^{-T}, \quad \bar Q = T^TQT (reachability and observability Gramians)

c) Show that the following procedure transforms the system to a form where LaTeX: \bar P = \bar Q = \Sigma

  • Compute Gramians LaTeX: P and LaTeX: Q for the system given by LaTeX: (A,B,C,D)
  • Compute a matrix LaTeX: R so that LaTeX: P = RR^T (this is called Cholesky factorization)
  • Compute a SVD of LaTeX: R^TQR and write it on the form LaTeX: R^TQR = U\Sigma^2 U^T
  • Use the coordinate transformation LaTeX: x=T\bar x with LaTeX: T = RU\Sigma^{-1/2}

13.3 Grey box Identification of a Heated Rod

The file greyrod.m on Lecture 12 does Grey Box Identification of the parameters kappa and htf. A compartment model of the system is produced by heatd.m.

Try different model orders 1,2,5 and 100 and compare the result of the grey box identification. Study both the Bode diagrams, and the estimated values of kappa and htf and comment on the results.

 (You will need greyrod.m and heatd.m )

13.4 State Space Identification with measurable states

If the full state vector LaTeX: x is known, then the state space model LaTeX: (A,B,C,D) can be found by standard least squares regression in

LaTeX: \begin{bmatrix} x_2 & \cdots & x_{N+1} \\ y_1 & \cdots & y_{N} \end{bmatrix} = \begin{bmatrix} A &B \\ C & D \end{bmatrix} \begin{bmatrix} x_1 & \cdots & x_{N} \\ u_1 & \cdots & u_{N} \end{bmatrix} + \textrm{noise}

a) Study the file ex13_4.m and explain how the code works

b) Suggest a practical method to reduce the number of nonzero elements in LaTeX: (A,B,C,D). Such a method can be useful if you suspect that many elements in theses matrices are zero. (You don't need to implement it).

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Solutions:

Matlab files with solutions

ex 13_1: See the code ex13_1.m

ex 13_2:  See this solution exercise-13solutions.pdf 

ex 13_3: See the code ex13_3.m (you will need greyrod.m and heatd.m )

ex 13_4: 

a) The code in the exercise implements solutions to the normal equations for a linear matrix equation of the form

Measurement-matrix = Theta-matrix * Regressor-matrix

in the code the  Measurement-matrix is called "lhs", for "left hand side", and Regressor-matrix is called "rhs".

We have talked about such a generalization of linear regression to the matrix case at an earlier exercise, and concluded then that the corresponding normal equations has similar structure as before. In the code the solution is obtained both from such a normal equation, and from matlab's internal implementation of solution to over and underdetermined linear equations (which often have better numerics)

b) One can use the "lasso" method from an early lecture in the course, adding an L_1 penalty on the sum of all coefficients in the A,B,C and D matrices. The simplest way to implement this would probably be to vectorize the matrix equation, i.e. rewrite the given linear equations system in the problem into the traditional form LaTeX: y = \Phi \theta , where LaTeX: \theta would be a parameter vector with all the matrix elements of A,B,C and D.

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