Mimo System

978-1-4244-8542-0/10/$26.00 2010 IEEE

Comparison of State Feedback Controller Design Methods for MIMO Systems

Prachi Barsaiyan and Shubhi Purwar

Abstract The objective of this paper is to compare the time

specification performance among four state feedback tracking controller design methods and determine which control method delivers better performance for linear MIMO systems. The four controller design methods are Pole Placement method, Linear Quadratic Regulator (LQR), an Eigenvalue Assignment method for shaping the output response and a method based on Moores Eigenstructure Assignment for controlling the linearized model of MIMO systems. In the Pole Placement method, some components overshoot, the convergence is slow and does not provide robust solution. LQR assures stability robustness with full state feedback but does not provide the flexibility of assigning eigenvalues and eigenvectors in placing closed loop structure. Eigenvalue Assignment is used to regulate the output response of minimum phase linear multivariable square plants. A method based on Moores Eigenstructure Assignment provides nonovershooting and fast response.

Index Termss Pole Placement method, LQR, Eigenvalue

Assignment, Eigenstructure Assignment.

I. INTRODUCTION The state feedback controller design methods have been

considered in many papers. In past decades, considerable efforts have been done to obtain state feedback gain.

Arbitrary assignment of closed loop poles to the system discussed in [1]. Pole placement by state feedback, has considered in [2]. Robust pole assignment presented in [3], [4]. Linear Quadratic Regulator (LQR) method considered in many papers [5]-[8], these papers are based on the selection of state and control input weight matrix, to minimize the cost of performance index. Controller design methods for shaping of the output response in a class of Linear multivariable systems, considered in many papers [9], [10]. Flexibility offered by state feedback in multivariable systems beyond closed loop eigenvalue assignment, proposed a design scheme based on eigenvalue- eigenvector assignment [11]. Papers on undershoot, overshoot, minimum and nonminimum phase systems considered in [12],[13]. The purpose of this paper is to study the different state feedback controller design methods with step reference trajectory control for linear MIMO systems. In this paper, investigations on performance comparison among modern control Pole Placement method, LQR method, an Eigenvalue Assignment method for shaping the output response and a method based on Moores Eigenstructure Assignment method,

Authors are with the Department of Electrical Engineering, MNNIT,

Allahabad,India (e-mail: [email protected],[email protected]).

are considered. Pole Placement design allows all closed loop poles to be placed in desirable locations but does not provide robust controller. LQR is an optimal control design method that results in a system with guaranteed robustness. The state and control input weighting matrices, for the LQR cost function, must be selected by the designer in an attempt to achieve the desired system performance. Third method considered, is a Eigenstructure Assignment method of designing a state feedback controller to regulate as desired the output response of minimum phase linear multivariable square plants with CB having full rank (B, input matrix and C, output matrix). This method provides the advantage of allowing great flexibility in shaping closed-loop system responses by allowing specification of closed-loop eigenvalues and corresponding eigenvectors, but has the disadvantage that stability-robustness is not guaranteed. On the other hand, LQR assures stability-robustness with full state feedback but does not provide the flexibility of assigning eigenvalues and eigenvectors in placing closed-loop structure.

II. CONTROLLER DESIGN METHODS In this section, four state feedback controller design

methods Pole Placement method, Linear Quadratic Regulator (LQR), an Eigenvalue Assignment method for shaping the output response and a method based on Moores Eigenstructure Assignment, are discussed.

A. Pole Placement or Pole Assignment Method The Pole Placement design allows all closed loop poles to

be placed in desirable locations. The necessary and sufficient condition for the arbitrary placement of closed loop eigenvalues in the complex plane (with the restriction that complex eigenvalues occur in complex conjugate pairs) is that the system

( ) ( ) ( )( ) ( ) ( )

x t Ax t Bu ty t Cx t Du t

= +

= +

(1)

is controllable[1], where x(t) Rn is the state, u(t) Rm is the input, and y(t) Rp is the output. If all n state variables x1, x2,.., xn can be accurately measured at all times, it is possible to implement a linear control law of the form

1 1 2 2( ) ( ) ( ) .... ( ) ( )n nu t k x t k x t k x t Kx t= = (2)

where K=[k1, k2,.., kn] is a constant state feedback gain matrix [4]. If this state feedback control law is connected to the system, the closed loop system is described by the state differential equation

( ) ( ) ( )x t A BK x t= (3) The feedback gain matrix K should be such that the closed

loop system is robust, in the sense that its poles are as

2

insensitive to perturbations as possible. Let xi and yi be the right and left eigenvectors of the closed loop system matrix A-BK, corresponding to eigenvalue i, that is

( )

( )i i i

T Ti i

A BK x xy A BK

=

=

(4)

If closed loop system A-BK has n linearly independent eigenvectors, then it is diagonalizable and the sensitivity of the eigenvalue i, to the perturbations in the components of A, B and K, depends upon the magnitude of the condition number ci, where

2 21 1i ii Ti i i

y xc

s y x= = (5)

For real i, the sensitivity si is the cosine of the angle between the right and left eigenvectors corresponding to i. A bound on the sensitivities of the eigenvalues is given by

12 2i

max c X X (6)

It provides the condition number of the matrix of eigenvectors X=[x1, x2,, xn].

B. Linear Quadratic Regulator (LQR) Linear Quadratic Regulator (LQR) design is a widely used

method for controller design based on finding the state feedback gain K that minimizes a quadratic cost function

( ( ) ( ) ( ) ( )0

T TJ x t Qx t u t Ru t dt

= + (7)

subject to the linear system in (1) and the state feedback control law in (2) minimizes the cost function. State feedback control law K is given by

1 TK R B P= (8) where P is symmetric, positive definite constant matrix,

found by solving the continuous time algebraic Riccati equation

1 0T TA P PA PBR B P Q+ + = (9)

where Q Rn, symmetric, positive definite (or positive semidefinite) constant matrix, can be used as a performance measure. The simplest form of Q is the diagonal matrix

1

2

0 .... 00 .... 0: : : :0 0 0 n

qq

Q

q

= (10)

The ith {i = 1,2..,n} entry of Q represents the weight that designer places the constraint on the state variable. For larger value of qi, relative to the other values of q, more control effort is spent to regulate xi. R Rm is, symmetric, positive definite, constant matrix. By giving sufficient weight to control terms, the amplitudes of control signals which minimize overall performance index, may be kept within practical bounds although at the expense of increased error in xi[4]. Limitations:

The pair (A, B) is stabilizable. R > 0 and Q 0.

C. Eigenvalue Assignment Method In this method, a state feedback controller is designed to

regulate the output response of minimum phase linear multivariable square plants as desired with CB having full rank (B, input matrix and C, output matrix). This is a simple method for computing a state feedback for a certain class of linear multivariable systems, such that a subset of the closed-loop poles is assigned to the location of finite zeros of the system, while the rest of the poles are arbitrarily assigned accordingly, the output response is shaped as desired [9].

Consider the linear time-invariant system (1) and apply the following state feedback to it

( ) 1 1 ( )K CB CA CB JC = + (11) where J Rm. Then (n - m) closed loop poles are located at

the position of the invariant zeros of the system, while the remaining m poles are assigned to the eigenvalues of the matrix J and the output response could be freely shaped by proper choice of J. The feedback presented in (2) could be decomposed to a state feedback (-(CB)-1CA) and an output feedback ((CB)-1J ). The role of state feedback is to assign m poles to the origin and (n - m) poles to the locations of the invariant zeros to obtain (n - m) fixed modes [5]. The role of output feedback is to reassign the (m) poles, located at the origin, to the arbitrary positions of the eigenvalues of J, while keeping the rest at the location of invariant zeros. Choosing

( )jJ diag = , 0, 0, 1,2,...,j j m > < = (12) Then control law (2), provides the output response y(t)=eJty(0) for arbitrary initial condition x0.

D. Eigenstructure Assignment Method An Eigenstructure Assignment method, based on Moores algorithm is used to achieve a nonovershooting step response. A method is given for designing a linear time invariant state feedback controller to asymptotically track a constant step reference with zero overshoot. Moores Algorithm: Let A and B be as in (1). For

C denote S=[A-I B] and a compatibly partitioned

matrix, N

KM

=

whose columns constitute a basis for

ker (S). Let L = {1, 2,.., n} be a self-conjugate set of distinct eigenvalues, and let V = {v1, v2,, vn} be a set of vectors in complex plane. Then, there exists a matrix K of real numbers such that (A+BK)i = vii, if and only if the following conditions are satisfied for all i {i = 1,2,..,n}:

The vectors vi are linearly independent; vi=vi* whenever i = i* ; vi im(N i );

If K exists, then it is unique. Let S = {s1,..,sn} be a set of n (not necessarily distinct) vectors in Rp. Assume that, for each i {i = 1,2,...,n} , the matrix

0iii i

vA I Bw SC D

=

(13)

3

has solution sets V={v1,v2,,vn} and W={w1,w2,..,wn}. Then, provided V is linearly independent, a unique real feedback matrix K exists such that, for all i n,

( )( )

i i i

i i

A BK v vC DK v S

+ =+ =

(14)

where control law 1K WV = (15) If a system has at least n p distinct stable invariant zeros,

Z = {z1 ,z2 ,.., zn]} be freely chosen from among the distinct stable invariant zeros of the system. Then, we choose i for

{1, ..., }i n p= , these modes are stable as the zi all lie in left half of complex plane. For the continuous time case, for

{ 1, ..., }i n p n= + , the i may be freely chosen to be any real distinct stable modes not coincident with invariant zeros of the system. If { , ..., }1e ep be the canonical basis of R

p, and

Si = {1,2,,n} such that

0 {1, ..., }11

:

for i n pe for i n p

Sie for i np

= +

=

=

(16)

When the invariant zeros {1,2,..,n-p} are distinct, the null

spaces of the system A I Bip C D

= for all

{1, ..., }i n p are one-dimensional subspaces of Rn+p, as

such, vector ker ( )vi p iwi=

satisfies (14). For all { 1, ..., }i n p n + , the solution is given by

viwi =

1( )p i

0

( )ei n p .

For any given step reference, the matrix K found above can

be used to obtain a state feedback to the step reference r without overshoot, from all initial conditions.

III. DESIGN OF TRACKING CONTROLLER If a system is right invertible, stabilizable and has no

invariant zeros at the origin, the following method for designing a tracking controller for a step reference signal is standard. Choose a feedback gain matrix K such that A+BK is stable. Then two vectors xss Rn desired equilibrium state vector and uss Rn desired equilibrium control input vector, exist that satisfy[13]

0 ss ssss ss

Ax Bur Cx Du

= +

= + (17)

For any r Rp. Applying the state feedback control law

( )ss ssu K x x u= + (18)

to (1) and employing the change of variable = x - xss, then the closed loop homogeneous system is

( )( )

A BKy C DK r

= +

= + +

0(0) ssx x = (19)

Since is A+BK stable, x converges to xss and y converges to r as t goes to infinity. Then a system has a nonovershooting response for r from the initial condition x0 Rn , if the output y of (1) arising from x0 yields a tracking error = r - y that satisfies 0, when t . A system has a globally nonovershooting response for r, if the output y is nonovershooting for all initial conditions x0.

IV. RESULT AND DISCUSSION In this section, the simulation results using four different

state feedback controller design methods, which is performed on the linearized model of a jet transport aircraft, are presented. Comparative assessment of control methods to the system performance also is discussed in this section. Considering a linear time invariant system

x Ax Buy Cx Du

= +

= +

(20)

where

,

-0.0558 -0.9968 0.0802 0.04150.5980 -0.1150 -0.0318 0-3.0500 0.3880 -0.4650 0

0 0.0805 1 0

0.0073 0-0.4750 0.00770.1530 0.1430

0 0.03

0 1 0 0 0 00 0 0 1 0 0

A

B

C D

=

=

= =

Above system is a state space model of a linearized jet aircraft model during cruise flight at MACH = 0.8 and H = 40,000 ft. This system is a minimum phase square system. The states are sideslip (); x1, yaw rate (r); x2 and roll rate (); x3, bank angle (); x4. Inputs are rudder deflection (u1) and aileron deflection (u2) in degree and outputs are yaw rate (y1) in degree/sec and bank angle (y2) in degree. Step reference r=[2,-2]. For this model, xss = [-18.5837, 2, 15.3059, -2] and uss = [-25.7406, -51.5564]. Initial condition x0=[1, 1, 1, 0]. The following design requirements, to examine the performance of all state feedback controller design methods are:

The system overshoot of yaw rate should be less than 15/sec and overshoot of bank angle should be within 9.

Rudder deflection should be less than 80 and aileron deflection should be within 35.

The settling time of system should not be more than 7.5sec.

E. Using Pole Placement Method Closed loop poles are considered at the locations P = [-

0.8,-0.9,-1,-3] that gives feedback gain matrix K from equation (2) is

3.5721 5.6877 1.8127 1 .67130.6108 0.8158 1.8383 1.3699

K

=

For the large magnitudes of the closed loop poles, system responses fast but the input signals to the plant are large. As

4

the magnitude of the control signals increase, the possibility of the system entering in the non linear regions of the operation also increases.

F. Using LQR Method Matrices Q = diag[9,6,1,2] and R = diag[1,1] are

considered, then feedback gain matrix K from equation (8) is

2.5027 3.8707 0.0621 0.00871.2205 0.5556 1.2060 1.3973

K

=

For larger value of q2 (10) relative to the q4, more control effort is spent to regulate x2 than x4 and error magnitude is also large. For the large value of R, magnitudes of control efforts are less but system response is slow and error magnitudes are also large.

G. Using Eigenvalue Assignment Method In this method, (n - m) two closed loop poles are located at

the position of invariant zeroes such as -3.4163 and -0.1139, remaining two poles -8 and -9 are assigned to the eigenvalues of matrix J(12). Simulation is done for closed loop poles [-3.4163, -0.1139, -8, -9] and = 0.5. The feedback gain matrix K using (11) is

1.2589 8.1746 0.1210 0.2432 0 0.2683 3.3333 15.0000

K

=

H. Using Eigenstructure Assignment Method Closed loop poles are considered at the locations [-3.4163,

-0.1139, -2, -3], where -3.4163 and -0.1139 are stable invariant zeroes of the system. Solve (25) for the vector S = {[0 0]T, [0 0]T, [0 0]T, [1 0]T, [0 1]T}, gives the linearly independent closed loop eigenvectors. These eigenvectors should satisfy (14). After applying Moores algorithm, feedback gain matrix using (15) is

1.2615 3.9643 0.1178 0.09390.0000 0.2684 3.3332 9.9964

K

=

Control efforts increase with large negative closed loop poles, but output response is fast.

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

Time (sec)

Yaw

rate

(deg

ree/

sec)

Pole placemant LQREigenvalue assignmentEigenstructure assignment

Yaw rate characteristics

Fig. 1 Yaw Rate Characteristics

0 1 2 3 4 5 6 7 8 9 10-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Time (sec)

Ban

k an

gle

(deg

ree)

Pole placementLQREigenvalue assignmentEigenstructure assignment

Bank angle characteristics

Fig. 2 Bank Angle characteristics

0 1 2 3 4 5 6 7 8 9 10-80

-70

-60

-50

-40

-30

-20

-10

0

Time (sec)

Rud

der d

efle

ctio

n (d

egre

e)


Rudder deflection characteristics

Fig. 3 Rudder Deflection Characteristics

0 1 2 3 4 5 6 7 8 9 10-60

-50

-40

-30

-20

-10

0

10

Time (sec)

Aile

ron

defle

ctio

n (d

egre

e)


Aileron deflection characteristics

Fig. 4 Aileron Deflection Characteristics

0 1 2 3 4 5 6 7 8 9 10-12

-10

-8

-6

-4

-2

0

2

Time (sec)

Erro

r1 (r

ms)


Fig. 5 Error 1 Characteristics

5

0 1 2 3 4 5 6 7 8 9 10-2

-1

0

1

2

3

4

5

6

7

Time (sec)

Erro

r2 (r

ms)


Simulations for different methods have done with best available data. Using Pole Placement and LQR, output components overshoot and convergence is slow as shown in table.. An Eigenvalue Assignment method and Eigenstructure Assignment method provide nonovershoot, fast output responses and low values of root mean square errors.

Fig. 6 Error 2 Characteristics

TABLE I SUMMARY OF PERFORMANCE CHARACTERISTICS USING FOUR METHOD

V. CONCLUSION In this paper, state feedback controller is designed using four methods such as Pole Placement method, LQR, an Eigenvalue Assignment method for shaping the output response and an Eigenstructure Assignment method, based on Moores algorithm. Based on the results and the analysis, a conclusion has been made that all four methods are capable of designing the state feedback controller for a linearized model of jet transport aircraft MIMO system. All methods have been compared successfully. The responses of each method have plotted in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6 and summarized in Table. Simulation results show that last two methods Eigenvalue Assignment and an Eigenstructure method provide fast and nonovershoot output responses than pole placement and LQR methods. But Eigenvalue Assignment method is applicable for minimum phase linear square systems. Further improvements need to be done for all of the controller methods for reducing the control efforts.

VI. REFERENCES [1] W. M. Wonham, On Pole Assignment in Multiinput Controllable

Linear Systems, IEEE Transections on Automatic Control, vol. AC-12, pp. 660-665, Dec. 1967.

[2] J. Kautsky, N. K. Nichols & P. Van Dooren, Robust Pole Assignment in Linear State Feedback, International Journal of Control, vol. 41, pp. 1129-1155, 1985.

[3] S. Srinathkumar, Robust eigenvalue/ eigenvector selection in linear state feedback systems, Proceedings of the 27th Conference on decision and Control, pp. 1303-1307, Austin, Texas Dec. 1988.

[4] M. Gopal, Digital Control and State Variable Methods, Conventional and Neuro- Fuzzy Control Systems , 2nd Edition, Tata McGraw Hill, 1997.

[5] P. Tomas Larsson, A. Galip Ulsoy, Scaling the Speed of Response using LQR Design, Proceedings of the 37th IEEE Conference on Decision & Control, pp. 1171-1176, Dec. 1998.

[6] John R. Broussard, A Quadratic Weight Selection Algorithm, IEEE Transactions on Automatic Control, vol. AC-27, no. 4, pp. 945-947, Aug. 1982.

[7] Jae Weon Choi, Young Bong Seo, LQR Design with Eigenstructure Assignment Capability, IEEE Transactions on Aerospace and Electronic System, vol. 35, no. 2, pp. 700-708, Apr. 1999.

[8] Douik Ali, Liouane Hend, Messaoud Hassani, Optimized Eigenstructure Assignment by ant System and LQR Approaches, International Journal of Computer Science and Applications, vol. 5, no. 4, pp. 45 - 56, 2008.

[9] A. N. K. Nasir, M. A. Ahmad, M.F. Rahmat, Performance Comparison Between LQR and PID Controller for an Inverted Pendulum system, International Conference on Power Control and Optimization, Chiang May, Thailand,pp. 18-20, July 2008.

[10] O. A. Sebakhy, M. I. Elsingaby, I. F. Elarabawy, Shaping of the Output Response in a Class of Linear Multivariable Systems, IEEE Transactions on Automatic Control, vol 33, no. 5, pp. 457-458, May 1988.

[11] Hidenori Kimura, New Approach to the Perfect Regulation and the Bounded Peaking in Linear Multivariable Control Systems, IEEE Transection on Automatic Control, vol. AC-26, No. 1, pp. 253-270, Feb. 1981.

[12] B. C. Moore, On the Flexibility Offered by State Feedback in Multivariable Systems Beyond Closed Loop Eigenvalue Assignment, IEEE Transactions on Automatic Control, vol. 21, no. 5,pp. 689-692, 1976.

[13] Robert Schamid, Lorenzo Ntogramatzidis, A Unified Method for the Design of Nonovershooting Linear Multivariable State Feedback Tracking Controllers, Automatica, vol. AC-21, no. 5, pp. 312-321, 2009.

[14] B. Porter and J. J. D'azzo, Algorithm for Closed loop Eigenstructure Assignment by State Feedback in Multivariable Linear Systems, International Journal of Control, vol. 27, no.6, pp 943-947, 1978

[15] S. Askarpour, T. J. Owens, Integrated Approach to Eigenstructure Assignment by Output Feedback Control, IEEE Proceedings of IEEE Conference on Control Theory, vol 144, no. 5, pp 435-438, Sep. 1997

Methods Settling time (sec)

Overshoot Control effort Error (rms) y1 (degree/sec)

y2 (degree)

u1 (degree)

u2 (degree) e1 e2

Pole Placement 7.5 9.27 -3.1

[-78.8 -4.8]

[-16.85 -3.25] 2.689 0.5798

LQR 7.0 [13.341.42] [-8.57 -1.76] [-79.5 -4.81]

[-11.15 -53.67] 3.83 2.66

Eigenvalue Assignment 1 0 0

[-8.02 -1.22]

[-33.6 3.17] 0.114 0.2154

Eigenstructure Assignment 1.3 0 0

[-16.92 -0.61]

[-63.6 5.74] 0.082 0.1558

6

[16] G. P. Liu and R. J. Patton, Paremetric State Feedback Design of Multivariable Control System Using Eigenstructure Assignment, Proceedings of 32nd IEEE Conference on Decision and Control, pp. 835-836, Taxas, Dec. 1993.

[17] G. Klein, B.C. Moore, Eigenvalue Generalized Eigenvector Assignment with State Feedback, IEEE Transections on Automatic Control, vol 22, no. 1, pp. 140-141, 1977.

[18] Wilson, Robert F.; Cloutier, James R., Optimal Eigenstructure Achievement with Robustness Guarantees, American Control Conference, pp. 952-957, 1990.

[19] Jie Chen, Li Qiu, and Onur Toker, Limitations on Maximal Tracking Accuracy, IEEE Transections on Automatic Control, vol. 45, no. 2, pp. 326-331, Feb. 2000.

[20] James Stewart and Daniel E. Davison, On Overshoot and Nonminimum Phase Zeros, IEEE Transections on Automatic Control, vol. 51, no. 8, pp. 1378-1382, Aug. 2006.

/ColorImageDict > /JPEG2000ColorACSImageDict > /JPEG2000ColorImageDict > /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 200 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 2.00333 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict > /GrayImageDict > /JPEG2000GrayACSImageDict > /JPEG2000GrayImageDict > /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 400 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 600 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.00167 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict > /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile (None) /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False

/CreateJDFFile false /Description > /Namespace [ (Adobe) (Common) (1.0) ] /OtherNamespaces [ > /FormElements false /GenerateStructure false /IncludeBookmarks false /IncludeHyperlinks false /IncludeInteractive false /IncludeLayers false /IncludeProfiles true /MultimediaHandling /UseObjectSettings /Namespace [ (Adobe) (CreativeSuite) (2.0) ] /PDFXOutputIntentProfileSelector /NA /PreserveEditing false /UntaggedCMYKHandling /UseDocumentProfile /UntaggedRGBHandling /UseDocumentProfile /UseDocumentBleed false >> ]>> setdistillerparams> setpagedevice

Mimo System

Documents

Transcript of Mimo System