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Control problem for linear systems with inputderivatives controlLIYI DAI aa Institute of Systems Science, Academia Sinica , Beijing, 100080, People's Republic of ChinaPublished online: 30 May 2007.

To cite this article: LIYI DAI (1988) Control problem for linear systems with input derivatives control, International Journal ofSystems Science, 19:8, 1645-1653, DOI: 10.1080/00207728808964065

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INT. J. SYSTEMS SCI., 1988, VOL. 19, No.8, 1645-1653

Control problem for linear systems with input derivatives control

LIYI DAIt

Descriptor variable system theory is applied to one kind of linear system with inputderivatives control, focusing on the problems of controllability and compensation.Necessary and sufficient conditions for the existence and design methods of itsdynamic compensator are given. Examples are also considered to illustrate theresults.

1. IntroductionIn the analysis and synthesis of systems it is common to consider the problems of

controllability and compensation. For the linear system

x(t) = Ax(t) + Bu(t), y(t) = Cx(t)

the classical linear system theory has given rise to many design methods. Thesemethods make it possible to choose control schemes to meet our various demands.However, real systems are always complicated and have various forms. For thisreason, some system models may not appear in the above form. In the sequel, thelinear system theory may not be suitable for or may be even impossible to apply tothese systems. For example, the above-mentioned results cannot be applied directly toa system described by

x(t) = Ax(t) + Bou(t) + e, u(t) + ... + B 1u(l)(t), x(to) = xo}(1.1)

y(t) = Cx(t)

where x(t) E lRn is the state, u(t) E IR' is the feasible control input, y(t) E IRm is themeasure output. Here it is assumed that u(t) is sufficiently piecewise continuousdifferentiable, i.e. u(t) E n~, which represents the set of 1 times piecewise continuousdifferentiable functions. The main feature of such systems is that their input includesnot only the function u(t), as for classical linear systems, but also its derivatives. Thiskind of system is well-known (Rosenbrock 1970, Al-Nasr 1984, Porter and Bradshaw1972 a, b), and problems relating to its state controllability and pole placement havebeen studied from various points of view (Al-Nasr 1984, Porter and Bradshaw 1972 a, b)for B2 = ... = B, = 0, producing some interesting results. In extending these results togeneral systems some difficulties may be met if the classical method is involved. Thispaper studies the problem of its state controllability and compensation from the pointof view of the newly-developed descriptor variable systems approach.

Received 8 April 1987.t Institute of Systems Science, Academia Sinica, Beijing 100080, People's Republic of

China.

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1646 L. Dai

Lemma 1.1 (Dai and Wang 1987)

The descriptor variable system

Ex(t) = Ax(t) + BU(t)}

y(t) = Cx(t)(1.2)

(1.3)

has a dynamic compensator of the following form:

xc(t) = Acxc(t) + BcY(t)}

u(t) = Fcxc(t) + Fy(t)

if and only if the system (1.2) is sta bilizable and detectable (Dai and Wang 1987,Campbell 1982).

2. State controllabilityThe system (1.1) is called state controllable in the sense of Porter (1972 a) which

means that for any initial condition Xo and state WE !Ria, there always exist a timeT ~ to and a feasible control u(t) E n~ such that x( to) = Xo and x( T) = w.

To study the state controllability of the system (1.1) we now change the statedescription to a more convenient form.

Consider the matrix polynomial

P(s) = B 1 + B 2 S + ... + BI S'- 1

Then the rational matrix

~pG)=~BI+GYB2+... +G)'B1

is a strictly proper rational matrix with only zero poles. Regarding it as the transfermatrix of a linear system, from the linear system theory we know that it has a minimalrealization (N, M, H), where N is nilpotent, such that

M(SI-N)-IH=~pG) (2.1)

The comparison of corresponding matrix coefficients yields

MH=B1, MNH=B 2 , ••• , MN'-1H=B"

N'=O

Let (N, M, H) be determined in the above way. We now construct a descriptorvariable system

with an output

Nv(t) = v(t) + HU(t)}

y(t) = Mv(t)

- y(t) = B1 u( t) + B2 u(t) + ... + BI u(l- 1>( t)

(2.2)

(2.3)

For the convenience of our discussion, the impulse terms in the state responses ofthe system (2.2) at the initial point are not considered here. This simplification does noharm to our analysis of state controllability and compensation because these featuresonly reflect the asymptotic properties of the system, while the impulse terms only have

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Linear systems with input derivatives control 1647

an effect at the initial point. This omission greatJy simplifies our procedure and it isthus reasonable.

Combining (1.1) with (2.3) it is easy to show that the system (1.1) can be changedinto

x(t) + Mv(t) = Ax(t) + Bou(t)

Nv(t) = vet) + Hu(t)

y(t) = [C OJ [~;n

Let

y= -M-AMN- ... -A'-IMN'-I, X= -AY

Then X and Y satisfy

Y+M +XN=O, X +AY=O

Under the transform

Q = [01 X], p= [1 Y], [X(t)] = p[WI(t)]I 0 1 vet) w2 (t)

(2.4)

(2.5)

(2.6)

a direct computation shows that the system (2.4) is a restricted system equivalent(RSE) to

WI (t) = AWl (t) + (B o + AB I + ... + A' B,)u(t)

Nw2 (t) = w2 (t) + Hu(t)

y(t) = CWI (t) + CYw2 (t)

x(t) = WI (t) + Yw2 (t)

The subsystems (2.5) and (2.6) are called the slow and fast subsystems of the descriptorvariable system (2.4), respectively. It can be seen from (2.5) and (2.6) that the statecontrollability of the system (1.1) is now changed into the controllability of the outputx( t) for the system (2.5) and (2.6).

On the 'other hand, we have

x(t) = WI (t) + Yw2 (t)

= exp (A(t - to»W I(to) + ft exp (A(t -t»)(Bo + AB t + ... + A' B,)u(r) d:to

+(-M-AMN- ... -A'-lMN'-l)

x ( - Hu(t) - N Hu(t) - ... - N'-l Hu(l-l)(t»

= exp (A(t - to))w 1 (to) + ff exp (A(t -t»(Bo + AB t + ... + A' B,)u(t) dtfo

+ (B 1 + AB2 t ... + A'-l B,)u(t) + (B 2 + AB3 + ... + A'-2 B,)u(t)

+ ... + B,uO-I)(t), u(t) E n~ (2.7)

with

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1648 L. Dai

Theorem 2.1The system (1.1) is state controllable if and only if

1

(A IB) + L 9l(Bj + ABj + I + ... + A'-iB,) = lRII

i= 1

where

B = Bo + AB I + ... + A'Bz, (A IB) = 9l(B) + A9l(B) + .., + A'9l(B)

and 9l(B) represents the range space of B.

Proof

(2.8)

I

For any X2 E L gf(Bj + ABi + 1 + ... + A'-iB,), there exist some constantsl:=z 1

(Xl' C(2' ... , (x, such that,

X 2 = L (B j + ABi + 1 + ... + A'-iB')(Xji= 1

For any t 1 > to let

ul(t) =lXl +1X2(t-td + ... +IX,(t-td'-t (l~ I)!

Then

(2.9)

Let

It is easy to verify that

u~) ( t 1) = U~) (0) = 0, i = 0, 1, ... , I - 1

Let

p= It I exp (A(t - r))Bu l (r) ·dt E (A IB)to

It has been proved by Yip and Sincovec (1981) that for any Xl E (AlB) and t 1 > tothere exists a u3(t) E n~ such that

ft l

exp (A(t - ~))Bu2(r) dt = Xl - PE (A IB) (2.10)to

For the U1(t) and U2(t) determined in the above, let u(t) = U 1(t) + u2(t). From (2.9) and(2.10) it is easy to verify that

II exp (AU - r»Bu(r) dt + (B l + AB2 + ... + A'-l B,)u(tll

+ ... + B1u(l-1)(td = Xl + X 2

that is,

I

x(t) E 9l( w1(to)) + (A IB) + L 9t( B i + ... + A1-i BI) (2.11)i= 1

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Linear systems with input derivatives control 1649

(3.1)

(3.2)

From the definition of state controllability and (2.11) we can conclude that thesystem (1.1) is state controllable if and only if the condition (2.8) holds. 0

In particular, Theorem 2.1 leads to the following corollaries.

Corollary 2.1

Let 1= 1; then the system (1.1) is state controllable if and only if

Bl[Bo + AB 1 , A(Bo+ AB1 ) , ••• , An-l(Bo+ ABd, Bl ]

= <AI Bo + AB1 ) + at(B 1) = lRn

This is just the condition given by Porter (1972 a).

Corollary 2.2

The system (1.1) is output controllable if and only if

I

C (A IB) + L C9t(Bj + AB i + 1 + ... + Al-

i B,) = lRm (2.12)i= 1

3. Normal dynamic compensatorsWe now tum to consider the existence and the design methods of dynamic

compensators for the system (1.1). The compensator stated here is restricted to theform (1.3) and in this sense makes the closed-loop system formed by (1.1) and (1.3)stable. This is a kind of output-dynamical feedback problem.

Theorem 3.1The system (1.1) has a dynamic compensator of the form (1.3) if and only if (A, Bo

+ AB 1 + ... + A' B,) is stabilizable and (A, C) is detectable.

Proof(a) Necessity. Let (1.3) be a compensator for the system (1.1). From the definition of

dynamic compensators we know that the dosed-loop system is

x(t) = Ax(t) + Bou(t) + Bl u(t) + ... + B,u(l)(t)

xc(t) = Acxc(t) + Bcy(t)

y(t) = Cx(t)

u(t) = Fcxc(t) + Fy(t)

i.e.

[X(t)] = [A + BoFC BoFe] [X(t)] + [B 1 FC

xc(t) Bee Ac xe(t) 0

+ [BIFC BIFc][X(I)(t)]o 0 x~l)( t)

which is a linear system described by m.~trix polynomial operators. The system (3.2) is

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1650

stable for

L. Dai

if and only if its characteristical polynomial

(3.3)s1-(A+BoFC)-B1FCs- ... -B,FCs' -BoFc- ... -B,Fcs'

-BeC s1- A c

is a stable polynomial, i.e. all its roots lie in the open-left-half complex plane (Kailath1980). Noting the fact that (Ns -I) is invertible, we obtain by direct verification that

sl- (A + BoFC) -BoFc Ms

-BeC s1-Ae 0

-HFC -HFc Ns-I

s1- (A + BoFC) - Ms(Ns _1)-1 HFC -Ms(Ns _1)-1 HFe - BoFc= INs-II

-BeC sl - Ac

=(_l)h~, h = dim (N) = dim (v(t)) (3.4)

From (3.1) and by also noting the fact that ~ is a stable polynomial, we knowimmediately that the system (1.3) is a normal dynamic compensator for the descriptorvariable system (2.5) and (2.6). Thus the system (2.5) and (2.6) is stabilizable anddetectable (Dai and Wang 1986). This is consequently equivalent to the condition thatits slow subsystem is stabilizable and detectable (Dai and Wang 1987, Campbell1982). This is precisely the given condition.

(b) Sufficiency. Assume that (A, B, C) is stabilizable and detectable. It is easy toverify that the system (2.5) and (2.6) is stabilizable and detectable (Dai and Wang1987, Campbell 1982). Under this condition, from Lemma 1.1 we know that thesystem (2.5) and (2.6) has a compensator of the form

Xe(t) = Acxc(t) + BCY(t)}

u(t) = Fcxc(t) + Fy(t)(3.5)

such that the closed-loop system is stable. Inverting the procedure in the proof of (a)we know that (3.5) is also a compensator for the system (1.1). Thus our proof iscomplete. 0

Remark 1

If the triple (A, B, C) is controllable and observable, then Theorem 3.1 shows thatthe system (1.1) has a compensator of the form (1.3) and its closed-loop system is

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Linear systems with input derivatives control 1651

stable with poles that can be placed at arbitrary preassigned positions on the complexplane. In this case the system (1.1) is state controllable. However, it can be seen fromTheorems 2.1 and 3.1 that the state controllability and observability of (A, C) do notnecessarily imply that there exists a compensator for the system (1.1). The inversestatement is also true.

Remark 2It can also be seen from the proof of Theorem 3.1 that to construct a dynamic

compensator (1.3) for the system (1.1) is equivalent to constructing a compensator ofthe same form for the system (2.4), and this compensator is precisely the compensatorfor the system (1.1). So Theorem 3.1 tells us not only when the compensator exists butalso how to construct it (the construction of the compensator for descriptor variablesystems has been given by Dai and Wang 1987).

Remark 3If the descriptor variable system (2.4) is stabilizable and detectable with additional

conditions of impulse controllability and impulse observability (Cobb 1984), it can beproved (Dai and Wang 1987) that there must exist a compensator of the form (1.3)such that the closed-loop system has n + "c + rank N finite stable poles, where nc =dim (xc(t». In this case, two appropriate vector variables xl(t), X2(t) can be chosensuch that the closed-loop system is RSE to the following system:

Xl(t) = AX l (t) + 111 r(t) )

0= x 2 (t) + B2 r(t)

y(t) = C1 Xl (t) + C2 X 2 (t)

(3.6)

where r(t) is the new input. Thus the closed-loop system does not include the effect ofderivatives of the input function.

4. ExamplesExample 1

Consider "the scalar system

x(t) = -x(t), y(t) =x(t)

This is not state controllable, but has a compensator of the form (1.3).

Example 2Consider another scalar system

x(t) = x(t) + u(t) - u(t), y(t) = x(t)

It is obviously state controllable, but we note the fact that Bo+ AB 1 = 0, (A, B) is notstabilizable. Hence it does not have a compensator of the form (1.3).

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Example 3Let the system be described by

L. Dai

In this case

X(t) = [~ 'l-+G}(t) +G}(l)+ G}(l) }y(t) = [1 O]x(t)

1=2, n=2, m=r=l, A=G aBo=[~J

Bl=[~J. B2=[~J. C=[l OJ

(4.1)

Using the previously given method we can construct the following descriptor variablesystem realization of P(s) = B1 + B2s (Kailath 1980):

[~ ~] ~t) = u(t)+ [~}(t)

}itt)= [~ ~}(t)Therefore

N=[~ ~J. H=[~J. M=[~ ~JThe corresponding extended descriptor variable system is

0 0 o 0 0

0 0 2 [X(t)] = 0 o 0 [x(t)J + 1u(t)

0 0 0 v(t) 0 0 1 0 v(t) 0(4.2)

0 0 0 0 0 0 o 1

y(t) = [1 o 0 [x(t)]0] .v(t)

For this system (A, Bo + AB 1 + A 2B2 ) is controllable and (A, C) is observable. Thus it

is not only state controllable but also has a compensator of the form (1.3). Followingthe design algorithm given by Dai and Wang (1987) we can construct the following asits compensator:

-210

34

--3 3

.-2 -1xc{t) = xc(t) + y(t) (4.3)

-113

-44 2

--3 3 2

-2 0 0

u(t) = [1 0 0 -l]xc{t) + ( -l)y(t)

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Linear systems with input derivatives control 1653

Remark 3 shows that it is a compensator for the system (4.2). Furthermore, it can beverified that the closed-loop system is stable with the stable poles { -1, -1, - 1, -1,--I, - 1, -I}.

5. Conclusion and remarksBy applying the descriptor variable system theory to the synthesis of systems, this

paper successfully solves the problem of state controllability and compensation for thesystem (1.1). The practical design method is given by Theorem 3.1, together with theresults in Dai and Wang (1987) which are omitted here. Furthermore, all the results inthis paper can be extended to the discrete systems case. Interested readers may like to

try it.

ACKNOWLEDGMENT

This work was supported by the Science Foundation of Academia Sinica.

REFERENCES

AL-NASR, N., 1984, Int. J. Systems Sci., 10, 1031.CAMPBELL, S. L., 1982, Singular Systems of Differential Equations, Vol. II (New York: Pitman).COBB, D., 1984, 1.£.£.£. Trans. autom. Control, 29, 1076.DAI, L., and WANG, C., 1987, Acta math. Scientia, 7, 273.KAILATH, T., 1980, Linear Systems (New Jersey: Prentice-Hall).PORTER, B., and BRADSHAW, A., 1972 a, Int. J. Control, 1, 101; 1972 b, Ibid., 1, 109.ROSENBROCK, H. H., 1970, State Space and Multiuariable Theory (London: Nelson), p. 216.YIP, E. L., and SINCOVEC, R. E, 1981, I.E.E.E. Trans. autom. Control, 26, 702.

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