AN ALGORITHM FOR DESIGN OF A PID CONTROLLER FOR A MULTIVARIABLE SYSTEM USING OUTPUT FEEDBACK

AN ALGORITHM FOR

DESIGN OF A

PID CONTROLLER FOR A

MULTIVARIABLE SYSTEM USING

OUTPUT FEEDBACK

Suma H,

N.M.A.MInstitute of Technology, Nitte, KarKala Karnataka, India-574110

Appukuttan, K. K.

Professor, National Institute of Technology Karnataka, Surathkal, India-575025

Abstract

This paper deals with the design of a proportional-plus-integral-plus-derivative (PID) controller for linear multivariable systems using an output feedback control law. Controller is designed as an equivalent proportional-plus-derivative controller for the augmented system formed due to the integral action. The design equations are formulated in terms of coefficient matrix of the transfer function vector of the equivalent single input (or single output) system. The computation of coefficient matrix of a transfer function vector gives a simple and direct procedure for pole assignment. The design procedure does not require cyclicity as an initial condition. The controller design is illustrated with a numerical example.

Keywords: Multivariable systems, Pole assignment, Eigenvlaues, PID controller

1. Introduction

Many results have been published in the last three decades on pole assignment in a multivariable system using constant gain output feedback [Eric King (2007), Kai Yang and Robert (2006), Kirtisis (2002), Chen et al. (1988), Munro and Hirbod (1979), Seraj (1978), Kimura (1975), Davison and Wang (1975]. The choice of unity rank structure for the output feedback matrix yields an non-linear design equations for a constant gain feedback matrix. Seraji 1978, has suggested a method for pole assignment using output feedback where the feedback matrix is constructed with the sum of the two unity rank matrices (two stage method) and Munro and Hirbod 1979, proposed a full rank design procedure by constructing feedback matrix as sum of minimum m or



unity rank matrices, where m and



are the number of inputs outputs respectively. In both the methods, transfer function matrix is computed and the dyadic form of the feedback matrix yields a solution of linear equations. There is computational advantage, if the transfer function vector is computed in each stage of the design. This paper presents the formulation of the controller design equation using the elements of the transfer function vector of an equivalent single input or single output system. The elements of the transfer function vector are computed from controllable (or observable) matrix of single input (or single output) system and upper triangular Toplitz matrix formed from the coefficient of the characteristic polynomial.

arbitrarily pole assignment as an extension of Hirbod (1979) PI controller design. The PID controller is designed based on the design of an equivalent proportional-plus-derivative (PD) controller for the augmented system. The design technique does not require the cyclicity of the augmented system as an initial condition. The design procedure uses coefficient matrices rather than transfer function matrix which gives direct and simple design procedure.

2. Pole assignment with PID controller

Consider a controllable and observable system described by

x



= Ax+Bu

y = Cx (1) Where x _ℝn _{is the state vector, uℝ}m _{is the input and y ℝ}ℓ _{is the output. For a controllable observable}

multivariable system described in the equation (1), an integral controller is directly acts on the outputs. The additional state variable due to the integral term gives an augmented system. The augmented system matrix are (Hirbod 1979, Seraji 1979):

   

  

          

   

I C C

B B C

A A

xn

xn nx



 

0 0

, 0 ,

0 0

0

0 0

(2)

Let (n+ ℓ) = n’. The augmented system matrices (A0, B0, C0) of dimension n’x n’, n’x m and 2ℓxn’ respectively. The feedback matrix is in the form K1+G(s) where G(s) = (P+Rs), K1 and (P+Rs) are dimensions mx2ℓ. In dyadic form K1 = k1f1T and (P+Rs) = k2(p+rs)T where k1 and k2 are mx1 vectors and p and r are 1x2ℓ vectors.. Here, two stage design suggested by Seraji (1978) is adopted. However, the design procedure presented here can be applied for full rank design procedure [Munro and Hirbod 1979]. In this case, the feedback matrix, K1 is determined to place (m-1) poles in the first stage and P and R are determined for the retention of (m-1) poles assigned in the first stage and additional 4ℓ poles are assigned in the second stage. The feedback matrix K1 + P of dimension mx2ℓ obtained in this case can be partitioned as



K_P| K_I



where Kp and KI are mxℓ proportional and integral components of the PID controller and R gives derivative controller components.

Stage one

In this stage, feedback matrix K1 is determined to assign (m-1) poles. Then, the closed loop system matrix in stage one is:

A01 = A0-B0K1C0 , K1ℝ2ℓxm (3) The closed loop characteristic polynomial for the system with K1, in dyadic form, k1f1T given by Seraji (1978):

H1(s) = H0(s) + f1TC0adj(s-A0)B0k1 (4) where H0(s) = det(s-A0) =



1

0



n*'

T

s

h

, ₀



₀₁

.

.

.

₀n_'



T

h

h

h



,



n



T

n s s

s*_' ' . . . 0 , and k1 and f1T are mx1 and 1x2ℓ vectors respectively. Choosing the vector f1Tarbitrarily, the system reduces to an equivalent single output system, cf = fTC0 and equation (3) can be written as:

H1(s) =



1 0



*n' T

s

h + k1T B0T (cf adj((s-A0))T =



1 0



n*'

T _s

h +k1T



0 N0f



s*n' (5)

where N0f = B0T Vf Dt is the mxn’ coefficient matrix (B0T (cf adj(s-A0))T (derivation is given in the Appendix A), Vf is the observability matrix of equivalent single output system,_







n



T_

f T

f T

f c A c A

c ₀ . . . ₀ ' 1 , and Dt is the

                 _   1 . . . 0 0 0 . . . . . . . . . 1 0 0 . . . 1 0 . . . 1 3 ' , 0 01 2 ' , 0 01 1 ' , 0 02 01 n n n

t h h

h h h h h D

For the distinct (m-1) pole locations, namely, s = (-1), (-



2) , . . . , (-



m1) equation (4) is:





























* ' 1 * ' 1 0 * ' 1 * ' 1 1

1 0 . . . 1 n . . . m n

T n m n of mx

T _{N h}

k   _    _





 

T

T m

k

k

_



















1 1

11

.

.

.

(6)

where



_ℝ(m-1)x(m-1) _, T 1



_ℝ1x(m-1) _and



T_

ℝ1x(m-1) _{. Assuming one element of the vector k}

1, say k1m, the other elements of the vector can be computed as:



₁₁ . . . _{1( 1)}



T 

 

T 1



 _1m₁



m k

k

k  (7)

Let the closed loop system matrix A01 = (A0-B0K1C0) and closed loop characteristic polynomial H1(s) =



1

₁



n*_'

T

s

h

, where T T T _f

N

k

h

h

₁



₀



_{1 0} .

Stage two

In the second stage, PD controller in dyadic form is (P+Rs) = k2(p+rs)T where k2 is the mx1 vectors and pT and r T _{are the 1x 2ℓ vectors. The closed loop characteristic equation in last stage [Rajagopalan 1984, Seraji and} Tarok 1977] is:

H2(s) =

2 0 0

1

1

k

B

rC



[





* ' 1

1

n T

s

h

+ (p+rs)T_W

1(s)k2] (8) The vector k2 is determined for the preservation of the poles placed at s = (-



₁), (-



₂), . . . , (-



m_1)

and vector f2T is determined to assign additional 4ℓ poles. For the retention of the poles assigned in the first stage, H2(s) and H1(s) becomes zero and f2T is a coefficient vector, W1(-



j)k2 = 0 , j = 1, . . ., (m-1). Since W1(-



j) k2

satisfies any row vector of the coefficient matrix C, say c1 ,one can write the design equation as: ₂T ₁₁

 

 _j 0

N

k 

k₂T



0mx₁ N₁₁



.





₁



*n_' . . .



m₁



*n_'



0



. . .



0

1 2

21 

       T m k

k (9)

where N11 = BT V11 Dt, V11 is the observability matrix of an equivalent single output system . Assuming k2m arbitrarily using equation (9), k2 can be determined.

Once k2 is known the system becomes equivalent single input system in the stage two and the coefficient matrix C adj(s-A0) bk is N1k = CUkDt , where Uk =



k



n k

k A b A b

b ₀ . . . ₀ '1 is the controllability matrix.

Hence the design equation (8) become: H2(s) =

2 0 0

1

1

k

B

rC



[





* ' 1

1

n T

s

h

+ (pT _rT_)N

PD

s

*n'1 (10)

where _

      0 0 0 0 1 1 k k PD N N

N . Using the equation(10) additional 4ℓ poles can be assigned. Total number of poles

3. Numerical Example

The example considered is the two shaft turbojet [Muller 1971] which has widely distributed four states, two inputs and two outputs and matrices A, B and C of the model are:

, 100 0 0 10 0 0 0 0 , 0 . 100 0 . 0 0 . 0 0 . 0 0 . 0 0 . 10 0 . 0 0 . 0 1240 52 . 8 957 . 1 002 . 1 4 . 951 498 . 1 0452 . 0 268 . 1                               B

A and _

      0 0 1 0 0 0 0 1 C

The open loop poles are located at s = -1.33, -1.89, -10 and -100. The augmented system will have n’ = 6 states, m = 2 inputs, 2ℓ = 4 outputs and it is controllable and observable.

A PID controller is to be designed to place the six poles of the augmented system at s = -4, -5, -6, -7, -8 and -10.

In the stage one K1 = k1f1T _{be a 2x4 feedback matrix. Choosing k1 = (1 0)}T _{and for the pole location at s =} -4, f1T _{= (1 1 1 -4.17). Hence the feedback matrix is :}

        0 0 0 0 17 . 4 1 1 1 1

K . In the second stage, the pole

assigned at s = -4 in the first stage is preserved by the vector k2 =



1 2.28x103



T_. To locate the additional poles

at s = -5, -6, -7, -8 and -10 the solution of equation (10) gives: p1 = 2.04, p2 = 0.0187, p3 = 2.31, p4 = -5.2, r1 = 0.3625 and r2 = 0. The PID controller components are:

      

 _{3 5}

10 26 . 4 10 65 . 4 018 . 1 04 . 3 x x

KP , 

       _ 011 . 0 10 26 . 5 37 . 9 31 . 3 3 x

KI , 

       _ 0 10 27 . 8 0 63 . 0 4 x R 4. Conclusions

A novel design procedure for pole assignment with a PID controller has been presented. The design equations are formulated using elements of transfer function vector of the equivalent single input (or single output) system and coefficient of the characteristic equation. Computational procedure involves only computation of the controllability and observability matrix of the equivalent single input (or single output) system. PID controller design is carried out by designing an equivalent PD controller for the augmented system formed due to the integral action. The design does not require cyclicity of the augmented system as an initial condition. A numerical example has been illustrated for the design procedure.

REFERENCES

[1] Ang, K.H., G.Chang and Y.Li, (2005): PID control system Analysis, Design and Technology, IEEE Trans. Control system technology, 13 559-579.

[2] Chen-Li-Chen, Tai-Cheng Yang and Neil Munro, (1988): Output feedback pole assignment procedure, Int.J.Control, 48 1503-1518.

[3] Davison, E.J., and S.H.Wang, (1975): On pole assignment in linear multivariable system using output feedback, IEEE Trans. Auto. Control,

20, 516-517.

[4] Eric King-wah Chu, (2007): Pole assignment via the schur form, System and control letters, 56, 303-314.

[5] Hirbod, S.N., (1979): Pole assignment using proportional-plus-integral output feedback control, Int.J.Control, 29, 1035-1046

[6] Kimura, H., (1975): Pole assignment by gain output feedback, IEEE Trans.Auto. Control, 20 509-516.

[7] Kiritsis, K.H., (2002): A necessary condition for pole assignment by constant output feedback, systems and control letters, 45 317-320.

[8] Kaiyang Yang and Robert Orsi, (2006): Generalized pole placement via static output feedback- A methodology based on projections,

Automatica, 42 2143-2150.

[9] Kailath, T., (1980). Linear systems, Prentice Hall, New Delhi.

[10] Munro, N., and S. Novin Hirbod, ( 1979): Pole assignment using full rank output feedback compensators, Int.J.syst. Science , 10 218-226.

[11] Muller, G. S., (1971): Linear model of two-shaft turbojet and its properties, Proc. IEE, 118, 813-815.

[12] Rajagopalan, T., (1984) :Pole assignment with constraints using output feedback, Automatica, 20 423-424.

[13] Seraji, H. and M.Tarokh, (1977) : Design of Proportional-plus-derivative output feedback for pole assignment, Procee IEE, 120 729-732.

[14] Seraji. H. (1978): A new method for pole placement using output feedback, Int. J. Control, .28 147-150.

[15] Seraji, H., (1979): Design of proportional-plus-integral controllers for multivariable systems, Int.J.control, .29 49-63

APPENDIX A

To find the coefficient matrix of C adj (sI-A) bk equal to C U k Dt , consider the resolvent matrix (Kailath 1980) as:

Adj (sI-A) =







1

0 n

k k

A n i

n

k i

k

i s

h 



 



1 1

where h0 = 1 and h1 , h2 , . . . , hn are the coefficients of characteristic polynomial. Hence

Adj (sI –A) bk =





 

1

0

1

(

)

n

k

k k k

s

D

b

A

where D i (s) = [ 1 h 1 . . . h n ] Rn+1 sn* , sn* is defined in equation (4) and

Rn+1 =

     

 

     

 

0 0 0 0 0 0 0

1 0 0 0 0 0 0

. . . . . . .

0 0 0 0 1 0 0

0 0 0 0 0 1 0

Hence, the coefficient matrix





 

1

0 1

) (

n

k

k s

D is equal to the upper triangular Toepliz matrix. Dt and







1

0 n

k k k_b