I cannot understand the purpose of your question.

The purpose is to make a program in Matlab which will be able to show the relationship between the Controllability Indices and the inputs of the system.

You can have a look at the following papers.

1. Liu, Yang-Yu, Jean-Jacques Slotine, and Albert-László Barabási. "Controllability of complex networks." Nature 473.7346 (2011): 167-173.

2. Pasqualetti, Fabio, Sandro Zampieri, and Francesco Bullo. "Controllability metrics, limitations and algorithms for complex networks." IEEE Transactions on Control of Network Systems 1.1 (2014): 40-52.

3. Summers, Tyler H., Fabrizio L. Cortesi, and John Lygeros. "On submodularity and controllability in complex dynamical networks." IEEE Transactions on Control of Network Systems 3.1 (2016): 91-101

Given a system (A, B), there are different algorithms to compute different sets of controllability indices. In general, a controllability index only tells how many states are attached (controlled) by a control input.

However, together with matrix A in special forms such as controllable-Hessenberg form or controllable-canonical form, it is possible to tell what input controls what set of system states. These special forms of (A, B) can be computed by direct similarity transformations on A and B. The computational algorithms, discussions, and examples can be found in my book "Robust Control System Design -- Advanced state space techniques", 2nd Edition 2004, Taylor & Francis.

Compare to other sets of controllability indices, the indices of my book are more even between each other. Compare the two different forms in my book, the Hessenberg form can be computed by much more reliable orthogonal similarity transformation, while the canonical form has the unique advantage of direct parametric relation with the polynomial-matrix-fraction-description form of the transfer function model of the same system. See my book for many details.

The fastest and the simplest way is the usage of the  "ctrbf" command in MATLAB in order to find the controllable and uncontrollable parts of the system:

[ABAR,BBAR,CBAR,T,K] = ctrbf(A,B,C) returns a decomposition

into the controllable/uncontrollable subspaces.

This command essentially simplifies further analysis. Estimating the H2-norms of the transfer functions  between each input and each state ("ss2tf") you can receive the numerical estimation of all links between inputs and states.

I wish to add two more points to this question.

1) The property raised in this question -- knowing what input controls what set of system states, an additional knowledge of the internal physical structure of the system, can be very useful in subsequent design. For example in robust eigenstructure assignment, it is possible to partially decouple this assignment into separate blocks each is corresponding to one input, and the dimension of each block equals the controllability index of that input (see my book). To compute the solution of the key matrix equation of this assignment, simple back-substitution operation can be used if based on the controllable Hessenberg form.

2) Regarding the separation of controllable and uncontrollable parts of the system. My impression is that the algorithm used in the MATLAB software, is perhaps most numerically reliable in determining the overall dimensions of the controllable/ uncontrollable parts, although from the result of this algorithm, it is impossible to tell what input controls what set of system states, and it is not possible to compute eigenstructure assignment by back-substitution. The result of this algorithm is very similar to the controllable Hessenberg form. The only difference is that in the common main operations of making a matrix (say B) into the form of [B1': 0' ]' (B1 has full row rank), SVD method (not Householder's method) is used in this algorithm. While SVD method is most numerically reliable in determining the overall number of independent/dependent columns of B, its resulting B1 does not tell what column is independent/dependent, and does allow back-substitution based on it.

Could you point me in which chapter of your book i could find the relatng theory for the 1st point that you gave?

Chapter 8 (Subsection 2.2) for decoupling eigenstructure assignment.

Chapter 5 (Section 2) for more back-substitution (in dual case).

Hello,

I want to demonstrate the physical interpretation of Controllabiliy and Observability indices as well as the Canonical forms in the controller and observer design.

Moreover i want to demonstrate the potential applications of the above.

I have tried seaching many different soures without any luck.

If you knew or you could suggest me a source, i would be grateful.

I can suggest some numerical and graphical examples in my book: "Robust Control System Design" (the 2nd edition):

On (block) observable canonical forms and observability indices:

Example 1.6 and Figure 1.8 on pages 17-18.

Example 1.7 on page 19.

Exercises 1.3 to 1.6 on pages 24 to 26.

Example 5.1 to 5.2 on pages 109 to 110.

Example 5.3 on pages 112-114.

On observer design based on block observable Hessenberg/canonical forms:

Example 5.5 on pages 118-119.

Example 6.1 on pages 138, 140-141.

Example 6.2 on page 145.

Example 6.3 on page 147.

Example 7.3 on pages 166-167.

Thank you very much for your answer. It really helped me a lot!!

I have another question that maybe someone could explain me.

I have for example a 10x40 matrix that is nothing more than the controllability matrix of a state space system which was chosen to be uncontrollable.

The matrix A has rank=10 and lets say that the controllability matrix has rank=5.

This means the system is uncontrollable and it has 5 uncontrollable states. Moreover, the controllability matrix has at maximum 5 linearly independent columns.

When i chech the linear independency with Matlab, i find out that the controllability matrix has 10 linearly independent columns! and 5 linearly independent rows. Is this normal or possible?

Thank you very much for your help!

I am not sure how did you actually use that Matlab subroutine, and I think your way of counting the # of linearly independent rows/columns is wrong (should be the same). 

I have described in detail another algorithm of determining the observability indices and separating the unobservable part -- a dual of your problem. The working matrix that I use, in your case. would be 10 x (10+p). This algorithm is definitely superior than the one working on a 10 x 40 matrix, both technically and numerically.

You can find that algorithm in my book (pp.107-112.). A numerical example of computing the controllable Hessenberg form can be found on pages 210-211 of the same book. In computing the Hessenberg form, the determination of controllablity indices and the separation of uncontrollable part (if any) can be achieved.

Thank you very much for your quick answe

With matlab i was just checking each column with each left for the linear independency according to the rank.

I will check the pages that you told me.

The way you propose is using the Hessenberg. What i wanted is having the Controllability matrix C=[B AB ....] and when its rank is less than the rank of matrix A to make the seperation.

Lets suppose that the rankA=n and rankC=n1 and n1n1.

Is the way that i am thinking wrong?

I have not used the matrix [B AB ...] for three decades. I think you can determine the controllability indices and determine controllable/uncontrollable by working on that matrix, but NOT the separation of these two parts.

The algorithm is to upper echelonize matrix [B AB ...], so you can find linearly independent columns one by one from left to right. You will stop incrementing the controllability index of an input if the corresponding column of that input becomes linearly dependent on the columns at its left side.

To answer the second part of your question. You need to find the TOTAL rank of the 10 x 40 matrix, system is controllable or uncontrollable if that rank equals n or is < n. The n1 rank that you mentioned seems to be the rank of C (or B), but not of the 10 x 40 matrix.

The matrix C that i talked about is nothing more that the Controllability matrix [B AB...].

With the matlab command rank i find the total rank of this 10 x 40 matrix. But when i check for linear independent columns, then these are more than the rank of the matrix.

The amount of the linearly independent rows is equal with the rank of this matrix. And this has confused me.  The linear independence i checked it as i explained previously using the ranks. By using this method do i need to upper echelonize the matrix [B AB...] first?

For a same matrix say [B AB ...], the # 's of linearly independent rows and columns should be the same.