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**Multivariable Control Systems**

Ali Karimpour Assistant Professor Ferdowsi University of Mashhad <<<1.1>>> ###Control System Design### {{{Control, Design}}}

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**Topics to be covered include:**

Chapter 5 Controllability, Observability and Realization Topics to be covered include: Controllability of Linear Dynamical Equations Observability of Linear Dynamical Equations Canonical Decomposition of a Linear Time-invariant Dynamical Equation Realization of Proper Rational Transfer Function Matrices Irreducible Realizations Irreducible realization of proper rational transfer functions Irreducible Realization of Proper Rational Transfer Function Vectors Irreducible Realization of Proper Rational Matrices

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**Controllability and Observability of Linear Dynamical Equations**

Definition 5-1 Definition 5-2

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**Controllability and Observability of Linear Dynamical Equations**

Theorem 5-1

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**Controllability and Observability of Linear Dynamical Equations**

Theorem 5-2

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**Controllability and Observability of Linear Dynamical Equations**

Theorem 5-2(continue)

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**Canonical Decomposition of a Linear Time-invariant**

Dynamical Equation Theorem 5-3 The controllability and observability of a linear time-invariant dynamical equation are invariant under any equivalence transformation. Proof: Let we first consider controllability Similarly we can consider observability

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**Canonical Decomposition of a Linear Time-invariant**

Dynamical Equation Theorem 5-4 Consider the n-dimensional linear time –invariant dynamical equation If the controllability matrix of the dynamical equation has rank n1 (where n1<n ), then there exists an equivalence transformation which transform the dynamical equation to and the n1-dimensional sub-equation is controllable and has the same transfer function matrix as the first system.

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**Canonical Decomposition of a Linear Time-invariant**

Dynamical Equation Theorem 5-4 (Continue) Furthermore P=[q1 q2 … qn1 … qn]-1 where q1, q2, …, qn1 be any n1 linearly independent column of S (controllability matrix) and the last n-n1 column of P are entirely arbitrary so long as the matrix [q1 q2 … qn1 … qn] is nonsingular. Proof: See “Linear system theory and design” Chi-Tsong Chen G(s) Hence, we derive the reduced order controllable equation.

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**Canonical Decomposition of a Linear Time-invariant**

Dynamical Equation Theorem 5-5 Consider the n-dimensional linear time –invariant dynamical equation If the observability matrix of the dynamical equation has rank n2 (where n2<n ), then there exists an equivalence transformation which transform the dynamical equation to and the n2-dimensional sub-equation is observable and has the same transfer function matrix as the first system.

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**Canonical Decomposition of a Linear Time-invariant**

Dynamical Equation Theorem 5-5 (Continue) Furthermore the first n2 row of P are any n2 linearly independent rows of V (observability matrix) and the last and the last n-n2 row of P is entirely arbitrary so long as the matrix P is nonsingular. Proof: See “Linear system theory and design” Chi-Tsong Chen G(s) Hence, we derive the reduced order observable equation.

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**Canonical Decomposition of a Linear Time-invariant**

Dynamical Equation Theorem 5-6 (Canonical decomposition theorem) Consider the n-dimensional linear time –invariant dynamical equation There exists an equivalence transformation which transform the dynamical equation to and the reduced dimensional sub-equation is observable and controllable and has the same transfer function matrix as the first system.

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**Canonical Decomposition of a Linear Time-invariant**

Dynamical Equation Definition 5-3 A linear time-invariant dynamical equation is said to be reducible if and only if there exist a linear time-invariant dynamical equation of lesser dimension that has the same transfer function matrix. Otherwise, the equation is irreducible. Theorem 5-7 A linear time invariant dynamical equation is irreducible if and only if it is controllable and observable. Theorem 5-8

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**Realization of Proper Rational Transfer Function Matrices**

Dynamical equation (state-space) description The input-output description (transfer function matrix) This transformation is unique The input-output description (transfer function matrix) Dynamical equation (state-space) description Realization This transformation is not unique Is it possible at all to obtain the state-space description from the transfer function matrix of a system? 2. If yes, how do we obtain the state space description from the transfer function matrix?

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**Realization of Proper Rational Transfer Function Matrices**

Theorem 5-9 A transfer function matrix G(s) is realizable by a finite dimensional linear time invariant dynamical equation if and only if G(s) is a proper rational matrix. Proof: See “Linear system theory and design” Chi-Tsong Chen

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**Irreducible realizations**

Definition 5-4 Theorem 5-10

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**Irreducible realizations**

Before considering the general case (irreducible realization of proper rational matrices) we start the following parts: 1. Irreducible realization of Proper Rational Transfer Functions 2. Irreducible Realization of Proper Rational Transfer Function Vectors 3. Irreducible Realization of Proper Rational Matrices

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**Irreducible realization of proper rational transfer functions**

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**Irreducible realization of proper rational transfer functions**

There are different forms of realization Observable canonical form realization Controllable canonical form realization Realization from the Hankel matrix

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**Observable canonical form realization of proper rational transfer functions**

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**Observable canonical form realization of proper rational transfer functions**

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**Observable canonical form realization of proper rational transfer functions**

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**Observable canonical form realization of proper rational transfer functions**

The derived dynamical equation is observable. Exersise 1: Why? The derived dynamical equation controllable as well if numerator and denominator of g(s) are coprime. Exersise 2: Why?

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**Controllable canonical form realization of proper rational transfer functions**

Let us introduce a new variable We may define the state variable as: Clearly

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**Controllable canonical form realization of proper rational transfer functions**

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**Controllable canonical form realization of proper rational transfer functions**

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**Controllable canonical form realization of proper rational transfer functions**

The derived dynamical equation is controllable . Exersise 3: Why? The derived dynamical equation observable as well if numerator and denominator of g(s) are coprime. Exersise 4: Why?

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**Controllable and observable canonical form realization of **

proper rational transfer functions Example 5-2 Derive controllable and observable canonical realization for following system. Observable canonical form realization is: Controllable canonical form realization is: It is not controllable. Why? It is not observable. Why?

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**Irreducible realization of proper rational transfer functions**

Example 5-3 Derive irreducible realization for following transfer function. Observable canonical form realization is: Controllable canonical form realization is: It is controllable too. Why? It is observable too. Why?

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**Irreducible realization of proper rational transfer functions**

Realization from the Hankel matrix The coefficients h(i) will be called Markov parameters.

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**Irreducible realization of proper rational transfer functions**

Realization from the Hankel matrix Theorem Consider the proper transfer function g(s) as then g(s) has degree m if and only if

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**Irreducible realization of proper rational transfer functions**

Realization from the Hankel matrix Now consider the dynamical equation Let the first σ rows be linearly independent and the (σ+1) th row of H(n+1,n) be linearly dependent on its previous rows. So

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**Irreducible realization of proper rational transfer functions**

Realization from the Hankel matrix We claim that the σ-dimensional dynamical equation (I) is a controllable and observable (irreducible realization). Exercise 5: Show that (I) is a controllable and observable (irreducible realization) of

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**Irreducible realization of proper rational transfer functions**

Example 5-4 Derive irreducible realization for following transfer function. We can show that the rank of H(4,3) is 2. So Hence an irreducible realization of g(s) is

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**Realization of Proper Rational Transfer Function Vectors**

Consider the rational function vector

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**We see that the transfer function from**

Realization of Proper Rational Transfer Function Vectors This is a controllable form realization of G(s). We see that the transfer function from u to yi is equal to

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**Realization of Proper Rational Transfer Function Vectors**

Example 5-5 Derive a realization for following transfer function vector. Hence a minimal dimensional realization of G(s) is given by

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**Realization of Proper Rational Matrices**

There are many approaches to find irreducible realizations for proper rational matrices. One approach is to first find a reducible realization and then apply the reduction procedure to reduce it to an irreducible one. Method I, Method II, Method III and Method IV In the second approach irreducible realization will yield directly.

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**Realization of Proper Rational Matrices**

Method I: Given a proper rational matrix G(s), if we first find an irreducible realization for every element gij(s) of G(s) as Clearly this equation is generally not controllable and not observable. To reduce this realization to irreducible one requires the application of the reduction procedure twice (theorems 5-4 and 5-5).

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**Realization of Proper Rational Matrices**

Proof:

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**Realization of Proper Rational Matrices**

Method II: Given a proper rational matrix G(s), if we find the controllable canonical- form realization for the ith column, Gi(s), of G(s) say, This realization is always controllable. It is however generally not observable. Proof:

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**Realization of Proper Rational Matrices**

Method III: Let a proper rational matrix G(s), where consider the monic least common denominator of G(s) as Then we can write G(s) as Then the following dynamic equation is a realization of G(s). Exercise 6: Show that the above dynamical equation is a controllable realization of G(s)

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**Realization of Proper Rational Matrices**

Method IV: It is possible to obtain observable realization of a proper G(s). Let Consider the monic least common denominator of G(s) as Then after deriving H(i) one can simply show Exercise 7: Proof equation (I) Let {A, B, C and E} be a realization of G(s) then we have

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**Realization of Proper Rational Matrices**

Then {A, B, C and D} be a realization of G(s) if and only if Now we claim that the following dynamical equation is a realization of G(s). We can readily verify that

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**Realization of Proper Rational Matrices**

Now we shall discuss in the following a method which will yield directly irreducible realizations. This method is based on the Hankel matrices. Let G(s) be Consider the monic least common denominator of G(s) as Define

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**Realization of Proper Rational Matrices**

We also define the two following Hankel matrices It can be readily verified that

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**Realization of Proper Rational Matrices**

It can be readily verified that Let be as the form Note that the left-upper-corner of M iT = TN i is H(i+1) so: It can be readily verified that But we want Irreducible Realization of Proper Rational Matrices

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**Irreducible Realization of Proper Rational Matrices**

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**Irreducible Realization of Proper Rational Matrices**

Example 5-6 Derive an irreducible realization for the following proper rational function. Least common denominator of G(s), is Non-zero singular values of T are 10.23, 5.79, 0.90 and 0.23. So, r = 4.

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**Irreducible Realization of Proper Rational Matrices**

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**Irreducible Realization of Proper Rational Matrices**

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