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Computation of K

: Design methods

Dr. Abdul Qayyum Khan

Room No. SE-302Department of Electrical Engineering,Pakistan Institute of Engineering and Applied Sciences,

P.O. Nilore Islamabd Pakistan

Email: [email protected]

http://www.pieas.edu.pk/aqayyum/

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Lecture Outline

Discussed so far: how controllability is necessary and sufficientcondition for design of state feedback controller

Today; we shall discussComputation methods for state feedback gain matrix K

Design of K using the transformation methodDesign of K using the Direct substitution methodDesign of K using Ackermann formula


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Design of K using the method of Transformation matrix T

I

Consider the system

ẋ = Ax + Bu (1)

where x ∈ ℜn

, A ∈ ℜn×n

matrix and B ∈ ℜn×1

. Consider the control of the form u = −Kx . Then

ẋ = (A − BK ) x (2)

Let µ1, µ2, · · ·µn be the desired eigenvalues.

K must be computed such away that the closed loop eigenvalues shouldbe µ1, µ2, · · · , µn


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II

The following algorithm is used to compute K .

Algorithm 1

Step I. Check the controllability, if the system is controllable, then go tostep II. Otherwise exit.

Step II. From the characteristics polynomial of A, i.e.;

|sI − A| = s n + a1s n−1 + a2s

n−2 + · · · + an−1s + an (3)

determine the values of the a′s


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III

Step III. Determine the transformation matrix T that transform the

system (1) into CCF. If the system is in CCF, then T = I .Otherwise it is given by

T = MW (4)

where

M =

B ... AB

... · · ·... An−1B

(5)

W =

an−1 an−2 · · · a1 1an−2 an−3 · · · 1 0

... ...

... ...

...a1 1 0 0 0

1 0 0 0 0

(6)


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IV

This transformation is not only necessary to find CCF, it is also used tocompute K

Step IV. Form a characteristics polynomial using the desired eigenvalues

µ1, µ2, · · · , µn as

(s − µ1) (s − µ2) · · · (s − µn)

= s n + α1s n−1 + α2s

n−2 + · · · + αn−1s + αn

Step V. Compute K as

K = αn − an

... αn−1 − an−1... · · ·

... α1 − a1

T −1 (7)


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V

Example 1: Consider

ẋ =

0 1 00 0 1

−1 −5 −6

x +

00

1

u (8)

Find, if possible, a state feedback control of the form u = −Kx whichplaces the closed loop poles at s 12 = −2 ± j 4, s 3 = −10. Use the methodof transformation.

Solution:

Following the steps given in algorithm 1, we have


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VI

Step I. Controllability:

M =

B AB A2B

=

0 0 10 −1 −6

1 −6 −31

(9)

Here the Rank (M ) = 3, since |M | = −1.

Step II. Form the characteristic polynomial of A

|sI − A| =

s −1 00 s −11 5 s + 6

= s 3 + 6s 2 + 5s + 1 (10)

Here a1 = 6, a2 = 5, a3 = 1

Step III. Since the system is already in CCF, the transformation matrix

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VII

Step IV. Form the characteristic polynomial using the desired poles; i.e.;

(s + 2 + j 4) (s + 2 − j 4) (s + 10) (11)

= s 3 + 14s 2 + 60s + 200 (12)

Here α1 = 14, α2 = 60, α3 = 200

Step V. Form K as

K = α3 − a3 α2 − a2 α1 − a1 = 199 55 8 (13)


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Design of K using the direct substitution method I

This method is relatively straight forward and easy

Very suitable for the systems with n ≤ 3

For n > 3, the computations are tidy

Consider the case n = 3. The K matrix is then

K = k 1 k 2 k 3 Also note that

|sI − A + BK | = s 3 + δ 1s 2 + δ 2s + δ 3 (14)

is a polynomial whose coefficient contains k 1, k 2 and k 3.

Form a characteristic polynomial of the desired poles µ1, µ2, µ3; i.e;

(s − µ1) (s − µ2) (s − µ3) = s 3 + α1s

2 + α2s + α3 (15)


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Design of K using the direct substitution method II

Comparing the coefficient in the similar power of s in (14) and (15),

Calculate the value of K Example 2

Consider the same example as discussed under the section of the methodof transformation.

Solution:

Here in this example, we have

K =

k 1 k 2 k 3

(16)


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Design of K using the direct substitution method III

Then

|sI − A + BK | = s −1 00 s −1

1 + k 1 5 + k 2 s + 6 + k 3

= s 3 + (6 + k 3s 2 + (5 + k 2)

(17)

The desired polynomial is

(s + 2 + j 4) (s + 2 − j 4) (s + 10) (18)

= s 3 + 14s 2 + 60s + 200 (19)

Comparing the coefficients, the K matrix is

K =

k 1 k 2 k 3

=

200 − 1 60 − 5 14 − 6

=

199 55 8


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Design of K using the formula of Ackermann I

Consider again (14)

ẋ = (A − BK ) x (20)

= Ãx (21)

and the desired eigenvalues are µ1, µ2, · · · µn. The required characteristics

equation is

|sI − A + BK | = |sI − Ã|

= (s − µ1) (s − µ2) · · · (s − µn)

= s n

+ α1s n−1

+ · · · + αn−1s + αn (22)

Recall Calay-Hamilton Theorem which is

φ(Ã) = Ãn + α1Ãn−1 + α2Ã

n−2 + · · · + αn−1Ã + αnI (23)


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Design of K using the formula of Ackermann II

(23) is used to derive Ackermann’s formula

Consider the case for n = 3. Notice from (23) that

I = I

Ã = A − BK

Ã2 = (A − BK )2 = A2 − ABK − BK Ã

Ã3

= (A − BK )3

= A3

− A2

BK − ABK Ã − BK Ã2


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Design of K using the formula of Ackermann III

multiply these equation by α1,α2,α3 and adding them as follows

φ(Ã) = α3I + α2Ã + α1Ã2 + Ã3

= α3I + α2(A − BK ) + α1(A2 − ABK − BK Ã)

+ A3 − A2BK − ABK Ã − BK Ã2

= α3I + α2A + α1A2 + A3

φ(A)

−α2BK

− α1ABK − α1BK Ã − A2BK − ABK Ã − BK Ã2

φ(Ã) = φ(A) − α2BK − α1ABK − α1BK Ã − A2BK − ABK Ã − BK Ã2


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Design of K using the formula of Ackermann IV

As φ(Ã) = 0 from CH-Theorem. Here note that φ(A) = 0, since it is notthe characteristic polynomial of A. The above equation can be re-writtenas

φ(A) = B (α2K + α1K Ã + K Ã2) + AB (α1K + K Ã) + A

2BK

=

B ... AB

... A2B

α2K + α1K Ã + K Ã2

α1K + K ÃK

φ(A) = M α2K + α1K Ã + K Ã

2

α1K + K ÃK


D i f K i h f l f A k V

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Design of K using the formula of Ackermann V

As φ(Ã) = 0 from CH-Theorem. Here note that φ(A) = 0, since it is notthe characteristic polynomial of A. The above equation can be re-writtenas

φ(A) = B (α2K + α1K Ã + K Ã2) + AB (α1K + K Ã) + A

2BK

=

B ... AB

... A2B

α2K + α1K Ã + K Ã2

α1K + K ÃK

φ(A) = M Controllability Matrix

α2K + α1K Ã + K Ã

2

α1K + K ÃK


D i f K i h f l f A k VI

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Design of K using the formula of Ackermann VI

Since the system is completely state controllable, M −1 exist. Therefore

M −1φ(A) =

α2K + α1K Ã + K Ã2α1K + K Ã

K

or

α2K + α1K

Ã + K Ã2

α1K + K ÃK

= M −1φ(A)

Pre-multiplying both sides by 0 0 1 , we getK =

0 0 1

M −1φ(A)

=

0 0 1

B ... AB

... A2B

φ(A)


D i f K i h f l f A k VII

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Design of K using the formula of Ackermann VII

Generalizing this equation for n-dimensional system, we have

K = 0 0 · · · 1 M −1φ(A)=

0 0 · · · 1

B

... AB ... · · · An−1B

φ(A)

This equation is referred to as Ackermann’s formula for the determination

of state feedback gain matrix.

Example 3:

Consider the same example as discussed before.

Solution


D i f K i th f l f A k VIII

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Design of K using the formula of Ackermann VIII

Here the controllability matrix M is

M =

B AB A2B

= 0 0 10 1 −6

1 −6 31

From example 1 and 2, we have α1 = 14, α2 = 60, α3 = 200. Therefore,

φ(A) = α3I + α2A + α1A2 + A3

= 200I + 60A + 14A2 + A3 =

199 55 8−8 159 7

−7 −43 117

The matrix K can be computed as

K =

0 0 1

M −1φ(A)

=

199 55 8


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Comments and Remarks

For SISO systems, K is unique for the desired poles from the client

For MIMO systems, K is not unique.

The desired closed loop poles is a compromise between rapidity of theresponse of the error and sensitivity to disturbance and noises, If these are at the dispense of designer.

For 2nd order system, the response characteristics can be precisely

correlated with the location of the desired closed loop poles.For higher order system, this is not easy job.

MATLAB function: acker and place ⇒ These function compute K byplacing the closed loop poles at the desired location; i.e.;Given matrices A and B and a vector of desired closed loop poles p ,

K can be computed as

K = acker (A,B , p )

K = place (A,B , p )



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For SISO systems, K is unique for the desired poles from the clientFor MIMO systems, K is not unique.

The desired closed loop poles is a compromise between rapidity of theresponse of the error and sensitivity to disturbance and noises, If these are at the dispense of designer.For 2nd order system, the response characteristics can be preciselycorrelated with the location of the desired closed loop poles.For higher order system, this is not easy job.MATLAB function: acker and place ⇒ These function compute K byplacing the closed loop poles at the desired location; i.e.;Given matrices A and B and a vector of desired closed loop poles p ,

K can be computed asK = acker(A,B,p)

⇑Function ’Acker’ is used for single input systems

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