The first general output feedback compensator that can implement state feedback control

THE FIRST GENERAL OUTPUT FEEDBACK COMPENSATOR THAT...

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Copyright © 1999 fF AC 14th Tri~nnial World Congress, Beijing, P.R. China

THE FIRST GENERAL OUTPUT FEEPBACX COMPENSATOR THAT CAN IMPLEMENT STATE FEEDBACK CONTROL Chia-Chi. Tsui. 743 Clove Road r NY l0310 r

USA,

718-727-0670

Abstract: All existing general output feedback compensators (OFC) cannot generate a state feedback control signal Kx(t) where K is a constant and x(t) is the plant system state. This paper presents a simple and systematic design of a new OFC which is applicable for all observable plants with either more outputs than inputs or at least one stable transmission zero(or for most observable plan~s). This new OFC can generate signal Kx(t) where K = KC and where K is a free OFC parameter while C is predetermined by the design of the dynamic part of OFC. This Kx(t) is compatible with static output feedback (SOF) control signal K"Cx (t) ~here Ky is free and C is given. Furthermore, because the rank of C is maxlmized by our design and ranges between the plant system order and the rank of C, this new orc can completely unify the existing result of OFC which estimates x(t) and the existing result of SOF. These two existing results were far from being unified before. Copyright © 19991F.4.C Keyword: Output/state feedback, Separation property, Robustness.

1. INTRODUCTION An output feedback compensator (OFC) has the general structure z(t)

=

Fz(t) + Ly(t) + TEu{t)

(1. a)

u it)

=0

-Kzz it)

(lob)

- Kyy (t)

guarantee that the OFC poles form a part of overall feedback system poles. This property is called "separation property" and the sufficient condition to this property is TA - FT

where z(t)€Rr, while y(t)€Rm and u{t)£RP are the output and input, respectively, of the given controllable and observable open loop plant system (with .x: (t) ERn) :i: (t)

A.x: ( t)

y [t)

Cx (t),

and

TB

+ Bu ( t)

2

= O.

3

Because the observers of state space control theory have the same general structure as (1), the OFC structure is a special case of observer structure with special property (3) (feedback from plant output y(t} only but not from plant input u(t}). The OFC structure is most common in classical control theory, but has been des1gned US1ng state space techniques also, either from pole assignment (Misra et al., 1989; Duan, 1993) point of view or from LTR point of view (Chen et aI, 1991). The references are representative but not exhaustive. However,

the existing design cannot

=

Le.

4 )

The proof can be found in (Tsui, 1996). Hence the existing OFC cannot satisfy (4). Because the above-mentioned existing design of OFC is aimed at the properties of the overall feedback system (such as its poles) only, separation property is the only guarantee of the poles and the stability property of OFC from that of the corresponding overall feedback system, and hence is very important. More important, condi tion (4) together with a stable F, is also well known to be the necessary and sufficient condition for z(t) - Tx(t) for a constant T in (l.a) (Luenberger, 1971). Therefore the existing OFC cannot have z(t)- Tx(t) for a constant 'T and thus cannot generally have u (t) - Kz (t) for a constant K in (l.b). Because :It (t) is the most explicit information about the plant system, the control signal Kx{t), if designed properly, should be able to best

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THE FIRST GENERAL OUTPUT FEEDBACK COMPENSATOR THAT...

improve feedback system performance and robustness. This form of control is the most basic and powerful in state space control theory. The advantage of an OFC over an observer, or the importance of (3), is that if both can satisfy (4) and can generate a state feedback control signal U(t)=-[Kz:K;J[ Z( tl

y(t) A-KCx(t)

r. .

-(Kz :K,.J1 TJ x(tl

le

A -Kx(t)

( 5 )

then the feedback system loop transfer function of (1) and (2), L(s)

=

-K[sI-A)-lB

V

K and

s

{6

if

and only if (3) holds (Doyle, Tsui, 1988J Because (6) is the loop transfer function of the direct state feedback system (with state feedback Kz(t)) and because loop transfer function determines the robustness properties (against plant system model uncertainty and plant control input disturbance) of the corresponding feedback system, an OFC can realize the robustness properties of the corresponding state feedback control while an observer cannot. In fact, this is the main reason that the observer (or Kalman filter) based state space control theory has not been popular in practical design where robustness is often critical. 1978,

Al though the problem of making (6) (called "loop transfer recovery (LTR) " (Doyle et al., 1979, 1981)) was initiated almost two decades ago, all existing systematic LTR design also require ( 1 )

which, tog€thgr with (4), also form the necessary and sufficient condition of the well-known "state observer" (which estimates x et) or Kx(t} for arbitrarily given K in (5) ). Unfortunately, this design (of satisfying (3), (4J, and (7)) is restricted only to the plant system which either has n-m stable transmission zeros or satisfies 1/ .minimurn~phase, 2) .rank(CBJ p, and 3).m ~ p (Kudva et al., 1980). The second set of restrictions is more relaxed than the first because all systems satisfying rank (CB) = p and m = p have n - m transmission zeros while almost all systems with m # p do not have n-m transmission zeros (Davison et al., 1974). Nonetheless, the minimum-phase condi tion is very restrictive. For example if each plant system

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transmission zero is equally likely to be stable or unstable (a very reasonable assumption because the plant system parameters are supposed to be randomly given and because the stable and unstable regions are equally sized), then the probability of minimum-phase among n-m transmission zeros is only (1/2) n-". In addition, rank (CB) =p is also unsatisfied by many practical plant systems such as airborne systems. Hence the existing OFC which generates Kx(t) is invalid for most plant systems. . Another existing and prevailing result is called "asymptotic LTR" (Doyle et. a1., 1979, 1981). This result achieves (6) asymptotically and therefore does not satisfy the OFC definition (3). This result requires asymptotically large observer gain L and is therefore impractical especially in the aspect of robustness. This paper presents a systematic design procedure of a new OFC which satisfies (3) and (4} (not necessarily (7)), with almost arbitrarily assigned (stable) poles. As will be shown in Section 2, this design is valid for all observable plants either with m>p or with at least one stable transmission zero. Because almost all systems w~th ID = P has n - m transmission zeros (Davison et a1., 1914), the probability of having at least one stable transmission zero in such systems is reasonably 1 - (1/2) c,-m .. 1. In other words, the strict conditions of minimum-phase and rank (CB) pare completely eliminated. Hence comparing tne existing OFC' S which satisfy (3) only but not (4), this new OFC has the advantage of generating Kx(t) control signal for a constant K of (5), and of having separation property. In addition, comparing the existing OFC's which satisfy (3), (4) and (7) and which are invalid for most plants, this new OFC is valid for most plant systems. The design procedure is presented in Section 2, an example is presented in Section 3, an additjonal significance on unification, of this result, is presented in Section 4. 2. THE SYSTEMATIC DESIGN

As introduced in Section I, this new OFC (lJ is uniquely required to satisfy (3) and (4) (not necessarily (7) ). The first exact and generaJ solution of (3) and (4), developed based on a new solution of (4) (Tsui,

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ISBN: 008 0432484

THE FIRST GENERAL OUTPUT FEEDBACK COMPENSATOR THAT...

1993), is presented as follows (Tsui, 1992) .

The solution to (4) can be computed based on the form of matrix C as C

[C,:

=

m

0

1.

8 )

n-m

This form can always be computed by orthogonal similarity transformation from any observable plant [A,C). We assume without lose of generality that IC11 F O. We also let

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Moreover, based on the block Hessenberg form of A which can always be computed using orthogonal similarity transformation from any observable (A,C}, these rows can be computed using direct back substitution. Hence the computation of this step is very reliable and effiCient. Once T is determined by choosing the free parameters Cl in (10), then the only undetermined parameter L of (4)

can be uniquely computed from FT)[~mJC1-l

F = r-th order Jordan form matrix ( 9 ) whose eigenvalues li (i=1, ... ,r) are

L =

almost arbitrarily chosen (should be stable of course) . The only requirement on the l's is to match each of the stable plant system transmission zeros. In the case of m>p, even this requirement is not mandatory as long as the existence of the solution of (3) and [4) is concerned and the maximization of rank (C) is not. For complex conjugate l's, the corresponding Jordan block can be a 2x2 real matrix. For multiple of q A's, the co~responding Jordan block is a qxq matrix. The parameter r, which is also the order of our new OFC, can be flexible because (9) uniquely guarantees that the corresponding solution of (3) and (4) are decoupled between Jordan blocks.

Theorem 1: The solution of (8)-{13) satisfies (4}. Proof: The right n-m columns of (4) are satisfied by (8) to (12) and the left m columns of (4) are satisfied by (8) and (13). 0

The corresponding solution T of and (9) is (Tsui, 1993) T

= r

C1

?11

(4) 10 J

l CrD]

where c i (i'=1, .•. , r) are free row vectors, and Dj can be directly computed either from (11

for distinct and real

~"

or from

for the q-dimensional (q>l) Jordan block J j (with AjI j=i+l, ... , i+q) where @ stands for Kronecker product and 1"_111 stands for an identity matrix with dimension n-m. Equation (8) and the observability of (A,C) (rank[(A1I) ':C'J=n Vl) guarantee that all matrices inSide the bracket of (11) and (12) have full column rank. Hence there always exists m or qm linearly independent rows of Di in (11) or (12), respectively. Each row of each D, of (11) and each unknown row of (12) can be computed simultaneously.

(TA -

( 13 )

Noticed that the only condition required for this solution is the observability of (A,C). Condition (3) (TB=O) will be satisfied by the remaining freedom of (4) -- C u i=l, ... ,r. Based on (10), TB=O can be expressed as cdD"BJmKp = 0 Vi and for (11) and c j [Di+l~-':: •• : D.i.+qB! qmx~p = 0 (J-~+l, ... , l+q) for (12).

14 15

Theorem 2(Tsui, 1996): The exact solution (F~L,T) of (3) and (4) can be computed by the above algorithm if the observable plant has either m>p or at least one stable transmission zero. Proof: If m > p, then the respective dimension of the matrices [DiB) and [D 1 + 1 B: .. :D1 +o B] of (14) and (15) guarantee the existence of the nonzero solution c, (i=l, ... r). For m ~ p,if a stable transmission zero (say Zi} is matched by .1. i of F (see (9», then there always exists a vector [ti :1 i ) which satisfies (Chen, 1984) ( 16

Equation (16) shows that t, and I, are the i-th row of matrices T and L respectively, which satisfy (4) and ( 3) •

Because the rows of solution (F,T,L) of (4) and (3) are completely decoupled, its number (which equals the OFC order) can be all flexible and can be as low as 1. 0

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Finally, the OFC order r will be determined such that the resulting matrix [T':C']' of (5) has the maximal row rank (r + m) . If the solution c, of (14)-{15) is not unique, which is true when m>p+l, or if there is remaining freedom of (4) and (3), then this remaining freedom (still expressed in c,) can always be used systematically and generally to maximize the angles between the rows of matrix CA[T':C']' (Kautskyet al., 1985). Maximizing the row angles of a matrix guarantees the maximal row rank of that matrix, even if the first maximization is not fully achieved. Conclusion 1: The entire remaining design freedom of OFC's dynamic part (or of (4)) after the eigenvalue selection of F, is expressed in c (i=l, .. ,r) (or eigenvector assignment freedom). This freedom has been fully used to satisfy (3) and to maximizg the row rank of the corresponding c. Corollary 1: Because of Conclusion 1, and because the eigenvalues of F are similarly selected as in the design of (3), {4) and (7) (Kudva_et al., 1980), the resulting matrix C of our new OFC has maximum rank n i f the solution of (3), (4) and (7) exists ((7) is equivalent of rank(C)

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11 0j1 ,

B,

.~ .. ~I

B2

if 0

°1 ' ~I

0 1

-1 1 1 -1] 1 2 . . . . .J -2 -2 Q , and B. 1 1 1 0

1

0 2

2 3

1

·i . oil

°

-1

Oj 01

1

-1

1 -2 -1 -2 2

1 2 1

1 1

1

1 1

-2 -2 -2 -1 The polynomial matrix fraction description of the corresponding transfer function G(s)=D-l(s)N(s) has a common D(s) and four different N{s) 's for the four plant systems (Chen, 1984):

J'

r (s+1) (s-2): (s-2) J

(s+ 1 )

: {s -1 ) ; 2

l(8+1) (s+ 1): 1 1, [ 2 : (s+2)J (s+3): 1

[ (s+l} (8-2): (s-2)

=

N 3 (s)

(s+l) (s+l)

and N.{s)

= f

l

: :

2 1

l'

(S-I)!.

(8-2) (3+1): (s-2) : (s-l) (s-2) : 2

This example efficiently and broadly demonstrates four plant systems which share a common system matrix pair iA, Cl

The four N(sl 's reveal that only the first and the third system have one common stable transmission zero -I, and only the fourth system has an unstable transmission zero 2. Thus the four corresponding OFC dynamic matrix F can be commonly set according to (9) as

A =

F

=

n).

3. AN EXAMPLE

x x x:l 0 0:0 I

x x x:O 1 0:0 x x x:O 0 1:0 x x x:O 0 0:1 x x x:O 0 0:0

=

diag{-l, -2, -3, -4},

which includes all stable transmission zeros (-1) of the four plant systems. All other eigenvalues of F are randomly selected.

x x x:D 0 0:0

~·~·~;o·o·6;6·J C

°

0:0 OO:OJ 1 [ x 1 0:0 Q 0:0 x x 1:0 0 0:0

Because (A,C,F} are common for the four plant systems, the D, matrices can be commonly computed according to (11); 0:-1 0 0: 0 -1 0: 0 1 0; 0 -1: 0 0 1 : 0: 4 0 0:-2 0 D2 1 0: 0 -2 0: 0 0 1: 0 0 -2: 0 0;-3 0 0; 9 0 D3 = 0 -3 0: 0 1 0: 0 0 -3: 0 0 1 : 16 0 0;-4 0 and D. = 0: 0 1 0 -4 0 0 -4: 0 0 D1

where "x" are arbitrary. Thus this example is also very general. The above matrix pair (A,C) is in observable canonical form which is a special case of observable Hessenberg form (Chen, 1984). All four plants satisfy the sufficient condition m > p of Theorem 2, and are distinguished by their respective B matrices:

1

{ 00

gj ,

I

g) ,

I

g]

l

!

1

0; 0: 1:

~9 .

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THE FIRST GENERAL OUTPUT FEEDBACK COMPENSATOR THAT ...

Now (14) can be solved and the T matrices for the four plant systems become respective ly, according ( I D) , T, =

=

[1

5 J D,

0

I

[1 4/5 16/5]0; [1 5/ 6 2 5/6 1D3 [ 1 6/7 36/7)D, 0 -5 -1 4 -8/5 - 3 2/5 -2 9 -15/6 -75/6 -3 16 -24/7 -144/7 -4

11

T2 =

I

[1 [0

[1

o

[0 -1

4 -2 9 -3 o -4 T,

r [1[0 . [1

J -J]D, 1 -1) D2

1 1 1

and

T,

=

I

OlD, 2 J D: 0 1 -1 2 -2 1 -1 o - 3 1 0)0 -8 0 1 ....

o1 o o

-2JD, 41 D? 5J D3

[1 -1 2 4 0 -8 9 o -15 16 o -24

o

o 5 11 4 / 5 15/5 ~ 516 25/6 11 6/7 36/7

I

f(

I

01

be i mp l emented by t he third and fourt h OFC . Th is i s p r edi ctable be ca use t h e 3 rd plan t ha s r a nk (C B)~ 1 < 2~ p and the 4-th plant has an nonm i nimum-phase zero (2). Although arbitrary state feedback cont r ol cannot be implemented by the OFC's of the last t wo plant systems, arbitrary eigenva lue assignment fo r the corres£o~di n g feedback system matrix A-BKjCL- i s still possible because rank(Ci) +p =6+2=8 >7=n (1=3,4 ) (Kimura, 1975). Thi s is imp o ssib l e for the ordina r y s t at i c outpu t f eedback (SOF) cont r ol be c ause r a n k (C)+p=3+ 2= 5 f 7=n . Th is overall qu alify (arbi t r a r y pole assignment by state feedback control and t he full realization of the robustness properties of this c on trol) cannot b e achieved genera l ly and systematically b y othe r existing designs for the third and fourth plan t systems.

6] Dl

1 -2

0

-2 -3 -4

0

0 0

4

:. 6

0 1 1 1

4 .COMPLET E UNIFICATION

1[2 - 1 llD~l [1 -1/5 615 J D2 I

. [1

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0

[11/7 ..., 1 -1 "4 2 /5 - 12 /5 9 0 -6 1 6 -4/7 -80/7

2} DJ

2017lD.J 1 -2 -1 -2 -1/5 61 5 -3 2 0 -4 1/7 20/7 1 J

t!

It can be easily verified that the above four matri c es satisfy (4) (the right n-m=4 columns) and (3 ) (T i Bi =O, i=l, . . ,4). Because t h e left m (=3) columns of (4) can alwa ys be satisfied by matr ix L as shown in (13), the exac t solution (F,Ti/L, ) (i=l, .. ,4) of (4) and (3) are derived, for the four plant syst e ms and for the dynamic part of their corresponding OFC's. Thus Theorem 2 is conformed for all four plant cases. These four OFC's can genera te a sta~~ feedback control signal Kix(t)=K jC, x(t) of ( 5 ) and can guarantee that the corresponding feedback system lQo~ transfer functions equa l -K i C (sI-A) ·'Bi' i=1, ... ,4. Because rank(C 1= [l i ' :C'1') = 7=n for i=l and 2, the first two OFe's can generate arbitrary state feedback control of (5). This result confo r ms with Corollary 1 because the first two plant systems satisfy the three rest r ictions of (Kudva et al., 1980). On the other hand, rank (Ci =[T i ' :C'1' ):6<7=n (1 = 3 ,4). Hence only constrai~~ s t ate feedback control K,-=KiC i ( i=3,4 ) can

As shown i n the f irst two sec t i ons, o ur ne w OFC is the first g en era l OFC whi c h can generate a state f e edback control signal Kx( t )= KCx(t ) of (5) where K is free and the row rank of C is maximized. This row rank of which is flexible and maximized, can reach its maximum possible v alue n whenever possib l e (Corollary 1) . Therefore our new OFC can completely and uniquely unify two basic existing control structures which were far from being unified before-the OFC which impl e ments arbitrarily given state feedback control ( rank( C ) = ma x imal n in (5») and SOF contro l (rank (C = C) = minimal m). Spe c i f ically, the OFC with rank~C) = n has the hiqhest order (n-m), implements the strongest (arbitrary) state feedback control, but is most restricted (to almo st all plant systems), while SOF has the lowest order (0), imp l ements the weakest (mos t con strained ) state f eedback cont ro l, but has no restrictions on the plant systems, and our OFC completely fill s t he gap between these two e x i s ting resu l ts which can now be considered as t wo extreme cases of our OFC.

er

The unique common ground of this unification is that both the state feedback control s i gnal Kx (t) (K is constant, constra i n ed or arbitrary) and its loop transfer function are realized. We re c all Section 1 that state feedback control is the best form of c ontrol and that only OFC can full y realize the robustness properti e s of this control.

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This unification can also invalidate the two consistent criticisms of our result because these two criticisms are derived from the above two existing extreme results -- the existing state feedback control cannot have its critical robustness properties realized and the existing SOF control is too constrained. 5. CONCLUSION The cr i tical advantage of OFC over observer is that it can fully realize the robustness properties of the corresponding state feedback control. Information wise, state feedback control is perhaps the most powerful and general form of control. One of the key advantages of this control is separation property. This paper proposed a new OFC which uniquely can implement state feedback control Kx(tj of (5) generally. Comparing the existing OFC's which can generate Kx(t) with arbitrarily given K but is invalid for most plant systems, this new OFC is valid for most plant sY3tems and unifies that existing result as its special and extreme case (the other special and extreme case of our new OFC is static output feedback). These significant developments are also simple and basic from the theoretical point of view. This is because the real challenge has not been to understand why, but to know how to actually design such an OFC generally and systematically (or how to solve matrix equations (4)-(3) systematically and generally). This particular challenge has been successfully met only now. Another key advantage of this design is its simplicity. In fact, the 7th order example of this paper is computed by hand. This feature uniquely makes this design tractable -- adjustable based on the finally computed design results. Experience shows that only the control theory with a tractable design procedure can really become practical and useful. This result also initiated a new approach to the strong stabilization problem -- stabilizing the feedback system by a stable OFC (Vidyasagar, 1985). This is because state feedback control is uniquely powe r ful in stabilizing the system while our new OFC can uniquely generate a state feedback control generally.

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REFERENCES Chen, B.M., A. Saberi and P. Sannuti (1991) "A new stable compensator design for exact and approximate loop transfer recovery," Automatica, 27, 257-280. Chen, C.T. (1984) Linear Sys t em: Theory and Design. Holt, Rinehart and Winston. Davison, E.J. and S.H. Wang ( 1974) "Properties and calculation of transmission zeros of linear multivariable systems," Automatica, 10, 643-658. Doyle, J. ( 1978) "Guaranteed margins for LQG regulators," IEEE T-AC-23, 8, 756-757. Doyle J. and G. Stein (19 7 9) "Robustness with observers," IEEE T-AC-24, 6, 607-611. , (1981) "Mult i variable feedback --aesign; concepts for classical! modern analysis," IEEE T-AC-26, I, 4-16 . Duan, G.R. (1993) "Robust eigenstructure assignment via dynamic compensators," Autornatica, 29, 469-474. Kautsky, Nichols, and Van Doo ren (1985) "Robust pole assignment in linear state feedback," In t. J. Control, 41, 5, 1129-1155. Kiml.lra, H. (1975) "Pole aSSignment by gain output feedback,~ IEEE TAC-20, 509-516. Kudva, Viswanadham, and Ramakrishna (1980) "Observers for linear systems with unknown inputs," IEEE T-AC-25, 1, 113-1115. Luenberger, D.G. (1971) "Introduction to observers," IEEE T-AC-16,596603. Misra, P. and R.V. Patel (l989) "Numeri cal a l gorithms for eigenvalue assignment by constant and dynamic output feedback," IEEE T-AC-34, 6, 579-588. Tsui, C.C., (1988) "On robust observer compensator design," Automatica, 24, 692-697. , (1992) "Unified output feedback --aesign and loop transfer recovery," Proc. 11-th American Control Conf., 3, 3113-311B. , (1993) "On the solution to --matriX equation TA-FT=LC and its applications," SIAM J. Matrix Analysis, 14, 1, 33-44. , (1996) Robust Control System --oesign, Advanced State Space Techniques, Marcel Dekker, NY. Vidyasagar, M. (1985) Control System Synthesis: A Factorization Approach, Cambridge MA: MIT Press. Wang, S.H., E.J. Davison & P. Dorato (1975) "Observing the states of systems with unmeasurable disturbances," IEEE T-AC-20, 716717.

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