Chapter 3 Canonical Form and Irreducible Realization of Linear Time-invariant Systems

§3-1Canonical form of systems 1. Canonical forms of single variable systems The characteristic polynomial is The controllable and observable matrices are

Steps: Find a equivalence transformation for a given , such that with canonical form. where Two methods to compute the equivalence transformations for canonical form: Compute P firstly. Compute P1 firstly.

1) Realization of controllable canonical form Theorem 3-1 Let the system (3-1) be controllable. Then, we can transform it into the following controllable canonical form by an equivalence transformation.

The first method for computing controllable canonical form: computing the transfer matrix P firstly. Compute the controllability matrix Compute and write its last row as h. c) Construct the transform matrix

d). where Cayley-Hamilton theorem

Note that we have

Question Is the matrix P nonsingular? In order to prove that P is nonsingular, we should prove that

Consider that 1)Mutipling the above equation by b, and noting that we have 2) Multiplying Equation (*) by Ab, and noting Equation (3-4) and , we have By the same token, we have

The second method for computing controllable canonical form: compute the transfer matrix P1 firstly. 1). Let the base vectors be Note that

it is clear that and

With the same method, we have At last, from and Cayley-Hamilton theorem, we have

Hence where we have used the following equations:

2) 3)

Argumentation 1) By using the uniqueness of the transformation, we have

2) Uniqueness of the transformation Proposition Let (A, b) be controllable. If there exist two nonsingular matrices P1 and P2, such that Then, we have Proof In fact, we have Q.E.D

Example Consider that following state-variable equation Transform the equation into controllable canonical form. First of all, we have to examine the controllability of the system. If it is controllable, we can transform the state-variable equation into controllable canonical form.

hence, the system is controllable. Now, we construct the matrix P.

The matrix is i.e.

2) Observable canonical form Theorem 3-2 Let the system (3-1) be observable. With an equivalence transformation, we can transform it into an observable canonical form as follows

The first method for computing observable canonical form: compute the transfer matrix P1 firstly. 1) Compute the observability matrix 2) Compute , and denote its last column as h; 3) Construct the transformation matrix

The second method for computing observable canonical form: computing the transfer matrix P firstly.

2. The canonical forms for multivariable systems 1) Luenberger controllable canonical form Consider the system Theorem 3-3 Let the system (3-15) be controllable. Then, there exists an equivalence transformation which transforms the system (3-15) into controllable canonical form as follows (3-16) where

where are matrices, respectively.

2) Design steps We assume that B=[b1 b2, … bp ] is full column rank; Construct the controllability matrix: Select n linear independent vectors of U, and rearrange them as follows It is clear that

Remark If Ab2 is linearly dependent with , i.e. then, all cannot be elected for . which means that can be linearly expressed by the forgoing vectors. 3) Let

4) Compute P1. Let hi represent the rows. row the nth row.

5) Construct the transform matrix Consider the nonsingular transformation we have

P2 is nonsingular: We have to prove that all the column vectors such that From

In particular, we have b) It is clear that the set of vectors of without is a basis of the zero-space of . Hence, can be expressed as the linear combination of the above vectors. That is because (take p=2 for example)

which shows that the zero-space of can be formed by the vectors above.

Generally, we have and c) Express as follows Multiplying the two sides of the equation by , and noting the Equation and , we have

Then, multiplying the above equation by , we have By the same token, we can prove that . Q.E.D

About .

Take p=2 for example: Hence, we have

If we first study how the set of base vectors is chosen. If is one of the vectors ,then ; 2) If is not one of the vectors, it can be expressed as the linear combination of Hence, we still have

In generally, if the matrix (P1)1 is given as Then, we have

P.82 Example 3-2 Consider the dynamical system (A, B, C), where Find the controllable canonical form. Compute the controllability matrix

It is clear that the first four linear independent columns are the first, second, third and fifth columns, respectively. Hence, 1=3,  2=1, h1=[2 1 0 0 ], h2=[0 0 1 0 ], and then we have

from which, we can figure out the controllable canonical form.

2) Observability canonical forms for multi-output systems Omitted.

Question Can we write the Luenberger observability canonical form? Hint: