duan-guangren

Parametric control systems design with applications in missile control
Guang-Ren Duan1,Hai-Hua Yu1,2,Feng Tan1
1.Center for Control Theory and Guidance Technology,Harbin Institute of Technology,Harbin150001,China
E-mail:g.r.duan@
2.Department of Automation,Heilongjiang University,Harbin150080
E-mail:yuhaihua@
Abstract:This paper considers parametric control of high-order descriptor linear systems via proportional plus deriva-tive state feedback.By employing general parametric solutions to a type of so-called high-order Sylvester matrix equa-tions,complete parametric control approaches for high-order linear systems are presented.The proposed approaches give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices,and produce all the design degrees of freedom.Furthermore,important special cases are particular treated,and an application of the proposed parametric design approaches to two missile control problems are also investigated.The simulation results show that the method is superior to the traditional one in sense of either global stability or system performance.
Key Words:High-order linear systems,parametric approaches,Sylvester matrix equations,BTT missiles
1Introduction
Parametric control systems design approaches possess the advantage of providing all the degree of freedom in the sys-tem design and are very effective in dealing with multiple objective design.This paper is concerned with the para-metric control of the following high-order dynamical linear system
A m x(m)+A m−1x(m−1)+···+A1˙x+A0x=Bu,(1) where x∈R n and u∈R r are the state vector and the control vector,respectively;A i∈R n×n,i=0,1,2,...,m, and B∈R n×r are the system coefficient matrices,which satisfy the following assumptions.
Assumption A1.rank(A m)=n0,0=n0≤n, Assumption A2.rank(B)=r.
The above system(1)clearly contains the followingfirst-order system
E˙x=Ax+Bu(2) and the following second-order dynamical linear system
M¨x+D˙x+Kx=Bu,(3) as special cases.Let s denote the differential operator,then the system(1)can be represented in the following operator form:
A(s)x(s)=Bu(s),
where x(s)and u(s)are the Laplace transforms of x(t) and u(t),respectively,and
A(s)=s m A m+···+s2A2+sA1+A0.(4)
This work has been partially supported by a grant from Major Pro-gram of National Natural Science Foundation of China under Grant 60710002,Program for Changjiang Scholars and Innovative Research Team in University.
Like the cases offirst-and second-order systems,we call (1)an m-th order descriptor linear system when n0<n.It is called a m-th order normal linear system when n0=n.
Furthermore,det A(s)is called the characteristic polyno-mial of the system,and the zeros of det A(s)are called the poles of the system.In this paper we are presenting a unified approach for the control of the abovefirst-,second-and high-order systems.The presentation is given for the general m-th order system(1).When A(s)is specified as A(s)=Ms2+Ds+K and A(s)=−Es+A,cor-responding results for the second-andfirst-order systems can be obtained.
As is well known,thefirst-order system(2)has fundamen-tal importance in control systems theory.As a special case of(1),the second-order linear system(3)has found ap-plications in manyfields,such as vibration and structural analysis,spacecraft control and robotics control,and hence have attracted much attention([1,2,3,4,5,6,7,8,9, 10,11]).However,concerning the control of second-order linear systems,most of the results are focused on stabiliza-tion([2,3])and pole assignment([4,5,6,7]).Regarding eigenstructure assignment in second-order linear systems, there have been only a few results([8,9,10,11]).Refer-ence[8]considers eigenstructure assignment in a special class of second-order linear systems using inverse eigen-value methods.Reference[9]proposes an algorithm for eigenstructure assignment in second-order linear systems, with the system coefficient matrices satisfying certain sym-metric positive definiteness condition.This algorithm is attractive because it utilizes only the original system data.
Very recently,eigenstructure assignment in second-order linear systems using a proportional plus derivative feed-back controller is considered in[10].Simple general and complete parametric expressions in direct closed forms for both the closed-loop eigenvector matrix and the feedback 1
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gains are established.As in [9],the approach utilizes di-rectly the original system data,and involves manipulations on only n −dimensional matrices.However,the approach has the disadvantage that it requires the controllability of the matrix pair (A 1,B ),which is not satisfied in some ap-plications.
Control of the high-order descriptor linear system (1)can be realized by converting the system into the following cor-responding extended first-order state-space descriptor sys-tem model
E e ˙z =A e z +B e u,(5)where
z T
=
x
T
˙
x T
x
(m −1) T
,
E e =Blockdiag (I n ,...,I n ,A m ),
(6)
A e =⎡
⎢⎢⎢
⎣
0I n
...I n
−A 0−A 1...−A m −1
⎤
⎥⎥⎥⎦,B e =⎡⎢⎢⎢⎣0...0B ⎤
⎥⎥⎥⎦.(7)
We point out that such a conversion is not preferable in most practical analysis and design problems because of the following several reasons:
1.it gives additional computation load and makes the treatment more complicated;
2.it is often not numerically reliable,and the error given in this very first conversion step will be carried to the final analysis or design results and makes the final re-sults inaccurate;
3.it destroys the physical meanings of the original sys-tem parameters.
In this paper we consider the problem of generalized eigen-structure assignment in the high-order descriptor linear sys-tem (1)via proportional plus derivative coordinate con-trol.The intension is to provide simple direct methods which utilizes only the original system coefficients A i ,i =0,1,2,...,m and B.Based on a complete parametric so-lution to a type of high-order generalized Sylvester ma-trix equations,a complete parametric solution is provided.Very simple,complete parametric expressions for both the closed-loop eigenvector matrices and the feedback gains are established.These expressions contain a matrix pa-rameter which represents the design degrees of freedom,which can be properly further chosen to produce a closed-loop system with some desired system specifications.Fur-thermore,complete parametric approaches are proposed for three important special cases.The first one employs the right factorization of the system,happens to be a nat-ural generalization of the parametric method in [12,13]proposed for eigenstructure assignment in first-order state-space descriptor linear systems.With this method,besides the group of parameter vectors,the closed-loop eigenvalues may also be treated as part of the design degrees of freedom since they appear directly in the expressions of the eigen-vector matrix and the feedback gains,and hence are not necessarily chosen a priori .The second one depends only on a series of singular value decompositions,hence is nu-merically very simple and reliable.
Finally,in order to realized the BTT missile autopilot,we propose a parametric method for the gain-scheduling con-troller design of a linear time-varying system based on the parametric design approaches.The simulation results show that the method is superior to the traditional one in sense of either global stability or system performance.
The paper is composed of six sections.Section 2gives the formulation of the generalized eigenstructure assignment problem for high-order descriptor linear systems.Solutions to the general parametric control problem are presented in Sections 3.Three important special cases are treated in Section 4.Section 5considers the modes decoupling con-trol for un-rotating missile.Section 6proposes a paramet-ric gain-scheduling controller design method for the BTT missile autopilot.
2Problem statement
For the high-order dynamical system (1),by choosing the following control law
u =K 0x +K 1˙x +···+K m −1x (m −1),K i ∈R r ×n ,(8)we obtain the closed-loop system as follows:
A m x (m )+A c m −1x (m −1)
+···+A c 0x =0,(9)
with
A c i =A i −BK i ,i =0,1,···,m −1,
which can be written in the first-order state-space form
E ec ˙z
=A ec z,(10)
where
E ec =Blockdiag (I n ,...,I n ,A m ),
(11)A ec =⎡⎢⎢⎢⎣0I n ...I n
−A c 0−A c 1...−A c m −1
⎤
⎥⎥⎥⎦,
(12)
Following the pole assignment theory for first-order de-scriptor linear systems,under the R-and I-controllability of system (1),the maximum number of finite eigenvalues can be assigned to the closed-loop system (10)-(12)is
n e =n (m −1)+n 0.
(13)
Therefore,the desired Jordan matrix J which can be as-signed to the matrix pair (E ec ,A ec )is of order n e .In this paper,we consider a generalized eigenstructure assignment problem:instead of assigning a Jordan matrix J to the ma-trix pair (E ec ,A ec ),we assign an arbitrary square matrix F ∈R n e ×n e to the matrix pair (E ec ,A ec ),that is,find a
matrix V f
ec
,of full column rank,satisfying A ec V f ec =E ec V f
ec F.
In this case,we say F is a core matrix of the matrix pair (E ec ,A ec ),while V f
ec is the generalized finite eigenvec-tor matrix of the matrix pair (E ec ,A ec )associated with the core matrix F.
22009Chinese Control and Decision Conference (CCDC 2009)
Lemma1.Let E ec and A ec be given by(11)and(12), and F∈C n e×n e be arbitrarily given.Then the following statements hold.
1.The core matrix of the matrix pair(E ec,A ec)is F if
and only if there exists a matrix V∈C n×n e satisfying
A m V F m+A c m−1V F m−1+···+A c0V=0,(14)
and in this case the corresponding generalizedfinite eigenvector matrix of the matrix pair(E ec,A ec)is given by
V f ec=
V T(V F)T···(V F m−1)T
T
.
2.The infinite eigenvector matrix of the matrix pair
(E ec,A ec)is given by
V∞ec=
0(V∞)T
T
,
where V∞∈R n×(n−n0)is determined by
A m V∞=0,rank(V∞)=n−n0.(15)
3.The entire eigenvector matrix of the matrix pair
(E ec,A ec)is
V ec=⎡
⎢⎢
⎢⎣
V0
..
.
..
.
V F m−20
V F m−1V∞
⎤
⎥⎥
⎥⎦.
For the proof of the Lemma1,one can refer to the Lemma 1in[14].
With the above understanding,the problem of general-ized eigenstructure assignment in the high-order descriptor dynamical system(1)via the proportional plus derivative feedback law(8)can be stated as follows.
Problem GESA(Generalized eigenstructure assignment) Given the system(1)satisfying Assumptions A1and A2, and an arbitrary matrix F∈C n e×n e,find a general parametric form for the matrices K i∈R r×n,i= 0,1,2,...,m−1,and V∈C n×n e such that the matrix
equation(14)and the condition
det V ec=0(16) are satisfied.
3Solution to Problem GESA
Letting
W=K m−1V F m−1+···+K1V F+K0V
=
K0K1···K m−1
⎡
⎢⎢
⎢⎣
V
V F
..
.
V F m−1
⎤
⎥⎥
⎥⎦
(17)
then(14)becomes the following high-order Sylvester ma-
trix equation
A m V F m+···+A1V F+A0V=BW.(18)
Introducing the following auxiliary equation
W∞=K m−1V∞,(19)
then combining(19)with(17),gives
W W∞
=
K0K1···K m−1
V ec.(20)
Therefore,when condition(16)holds,the feedback gain
matrices can be obtained as
K0K1···K m−1
=
W W∞
V−1
ec.(21)
By specifying the general solution to the high-order
Sylvester matrix equation(18)for this special case,we can
immediately obtain the corresponding solution to the gen-
eralized eigenstructure assignment problem.
3.1Solution to the Sylvester matrix equation(18)
Definition1.Let F∈C p×p be an arbitrary matrix,then
1.a pair of polynomial matrices N(s)∈R n×r[s]and
D(s)∈R r×r[s]is said to be F-right coprime if
rank
N(s)
D(s)
=r,∀s∈σ(F);(22)
2.a pair of polynomial matrices H(s)∈R m×n[s]and
L(s)∈R m×m[s]is said to be F-left coprime if
rank
H(s)L(s)
=m,∀s∈σ(F).
According to the above definition,A(s)and B are F-left
coprime if
rank
A(s)B
=n,∀s∈σ(F).(23)
Regarding the existence of solutions and the degree of free-
dom in the solution(V,W)to the high-order Sylvester ma-
trix equation(18),we have the following result([15]).
Lemma2.Let A(s)∈R n×n[s]be given in(4),F∈
C n e×n e,B∈R n×r with rank(B)=r,A(s)and B be
F-left coprime,and N(s)and D(s)be given by
⎧
⎪⎪⎨
⎪⎪⎩
N(s)=
ω
i=0
N i s i,N i∈R n×r
D(s)=
ω
i=0
D i s i,D i∈R r×r
.(24)
and satisfy
A(s)N(s)−BD(s)=0.(25)
1.The matrices V∈C n×n e,W∈C r×n e given by
V=N0Z+N1ZF+···+NωZFω
W=D0Z+D1ZF+···+DωZFω
(26)
satisfy the high-order Sylvester matrix equation(18)
for arbitrary matrix Z∈C r×n e;
2009Chinese Control and Decision Conference(CCDC2009)3
2.All the matrices V∈C n×n e,W∈C r×n e satisfy-
ing the matrix equation(18)can be parameterized as in(26)if and only if N(s)and D(s)are F-right co-prime,i.e.,condition(22)holds.
Remark1.In the case that(A(s),B)is R-controllable, A(s)and B are F-left coprime for arbitrary F.Therefore, in this case the F matrix can be arbitrarily chosen.
The solution(26)can be immediately written out as soon as the two polynomial matrices N(s)and D(s)determined by(25)are obtained.In many applications,we have B= I n.In this case we can obviously take
N(s)=I,D(s)=A(s).
When the polynomial det A(s)is not identically zero,we can take
N(s)=adj(A(s))B,D(s)=det(A(s))I r.
For more general case,general numerical algorithms, e.g.,[16]and[17],for solving the right factorization A−1(s)B=N(s)D−1(s)can also be readily applied to find the polynomial matrices N(s)and D(s).
3.2The complete parametric solution
Theorem1.Let A(s)∈R n×n[s]be given in(4),F∈C n e×n e,B∈R n×r with rank(B)=r,and A(s)and B be F-left coprime.Further,let N(s)and D(s)be given by(24)and satisfy(25),and be also F-coprime.Then
1.Problem GESA has a solution if and only if there exist
a matrix Z∈C r×n e,satisfying
det V ca=0
where V ca=⎡
⎢⎢
⎢⎣
N0Z+···+NωZFω
(N0Z+···+NωZFω)F
..
.
(N0Z+···+NωZFω)F m−1
..
.
V∞
⎤
⎥⎥
⎥⎦;
2.when the above condition is met,all the solutions to
the Problem GESA are given by
V=N0Z+N1ZF+···+NωZFω, and
K0K1···K m−1
=(D0Z+D1ZF+···+DωZFω)V−1
ca, where Z∈C r×n e is an arbitrary parameter ma-trix making the matrix V ca nonsingular,and W∞∈R r×(n−n0)is an arbitrary parameter matrix. Remark2.The above theorem gives a complete paramet-ric solution to the Problem GESA.The free parameter ma-trices Z and W∞represent the degrees of freedom in the eigenstructure assignment design,and can be sought to meet certain desired system performances.Furthermore,it
should be noted that the core matrix F appears explicitly in
the general solution,hence does not need to be prescribed.
It may be sought together with the parameter matrices Z
and W∞to meet certain closed-loop specifications.This is
in fact a very important feature of the approach. Remark3.When rank(A m)=n0<n,both the open-loop system(1)and the closed-loop system(9)are sin-
gular ones.For this case,closed-loop regularity has to
be addressed.Note that the control of system(1)via the
feedback control law(8)is equivalent to the state feedback
control in thefirst-order descriptor linear system(5)-(7).
Therefore,under the controllability condition of system(1),
thefirst-order descriptor linear system(5)-(7)is regulariz-
able via state feedback([18]),and hence for“almost all”
controllers in the form of(8),the corresponding closed-
loop system(10)-(12)is regular.
4Special cases
In this section,let us consider the important case of
F=diag
s1,s2,···,s n e
.(27) In accordance with the structure of the matrix F,we denote
V=
v1v2···v q
,(28) W=
w1w2···w q
.(29) By specifying the general solution to the high-order Sylvester matrix equations for this special case,we can im-mediately obtain the corresponding solution to the general-ized eigenstructure assignment problem.
4.1Case of undetermined s i,i=1,2,···,q Following from the above deduction and the results in Sec-tion3,we can now obtain the following corollary regarding the solution to the Problem GESA.
Corollary 1.Let system(1)be R-controllable,and N(s)∈R n×r[s]and D(s)∈R r×r[s]be a pair of polynomial matrices satisfying the right factorization(25). Then
1.Problem GESA has a solution if and only if there exist
a group of parameters f i∈C r,i=1,2,...,n e,satis-
fying
Constraint C0:f i=¯f j if s i=¯s j.
Constraint C a:det V ca=0with
V ca=
⎡
⎢⎢
⎢⎣
N(s1)f1···N(s n e)f n e
s1N(s1)f1···s n e N(s n e)f n e
..
.
..
.
..
.
s m−1
1N(s1)f1···s m−1
n e N(s n e)f n e
..
.
V∞
⎤
⎥⎥
⎥⎦;
2.when the above condition is met,all the solutions to
the Problem GESA are given by
V=
N(s1)f1···N(s n e)f n e
42009Chinese Control and Decision Conference(CCDC2009)
and
K0K1···K m−1
=
D(s1)f1···D(s n e)f n e W∞
V−1
ca,
where f i∈C r,i=1,2,...,n e,are arbitrary pa-rameter vectors satisfying Constraints C0and C a, W∞∈R r×(n−n0)is an arbitrary parameter matrix.
4.2Case of prescribed s i,i=1,2,···,q
Let
Πi=
A(s i)−B
,i=1,2,···,q.(30)
The following algorithm produces two sets of constant ma-trices N i and D i,i=1,2,···,q,to be used in the repre-sentation of the solution to the matrix equation(18). Algorithm ND(Solving N i and D i,i=1,2,···,q) Step1Through applying SVD to the matrixΠi,i= 1,2,···,q,obtain two sets of unitary matrices P i∈R n×n and Q i∈R(n+r)×(n+r),i=1,2,···,q,satis-
fying
P iΠi Q i=
diag(σ1,σ2,···,σn i)0
00
,(31) i=1,2,···,q,
whereσk>0,k=1,2,···,n i,are the singular val-ues ofΠi,and
n i=rank
A(s i)B
,i=1,2,···,q.(32)
Step2Obtain the matrices N i∈R n×(n+r−n i)and D i∈R r×(n+r−n i),i=1,2,···,q,by partitioning the ma-trix Q i as follows:
Q i=
∗N i
∗D i
,i=1,2,···,q.(33)
Corresponding to the above Algorithm and Theorem1,we have the following result for solution to Problem GESA. Corollary2.Let n i,i=1,2,...,n e,be given by(32), and N i∈R r×(n+r−n i)and D i∈R n×(n+r−n i),i=
1,2,...,n e,be given by Algorithm ND.Then
1.Problem GESA has a solution if and only if there exist
a group of parameters f i∈C n+r−n i,i=1,2,...,n e,
satisfying constraints C0and
Constraint C b:det V cb=0with
V cb=⎡
⎢⎢
⎢⎣
N1f1···N n e f n e
s1N1f1···s n e N n e f n e
..
.
..
.
..
.
s m−1
1N1f1···s m−1
n e N n e f n e
..
.
V∞
⎤
⎥⎥
⎥⎦;
(34)
2.when the above condition is met,all the solutions to
the Problem GESA are given by
V=
N1f1N2f2···N n e f n e
,(35)
and
K0K1···K m−1
=
D1f1···D n e f n e W∞
V−1
cb,
(36)
where f i∈C n+r−n i,i=1,2,...,n e,are arbitrary
parameter vectors satisfying Constraints C0and C b,
W∞∈R r×(n−n0)is an arbitrary parameter matrix.
4.3Case of normalfirst-order linear system
When m=1,and A m=I,the system(1)becomes a
normalfirst-order linear system in the following form
˙x=Ax+Bu.(37)
In this case,following from the results in Section3,we can
immediately obtain the following corollary regarding the
solution to the Problem GESA.
Corollary3.Let system(37)be controllable,and N(s)∈
R n×r[s]and D(s)∈R r×r[s]be a pair of right coprime
polynomial matrices satisfying
(sI−A)N(s)−BD(s)=0.(38)
1.Problem GESA has a solution if and only if there exist
a group of parameters f i∈C r,i=1,2,...,n e,satis-
fying Constraint C0and
Constraint C v:det V=0with
V=
N(s1)f1N(s2)f2···N(s n)f n
.
2.When the above condition is met,all the solutions to
the Problem GESA are given by
K=
D(s1)f1···D(s n)f n
V−1,(39)
where f i∈C r,i=1,2,...,n,are arbitrary parame-
ter vectors satisfying Constraints C0and C v.
In the rest of this section,let us make some remarks on the
above results.
Remark4.The above three corollaries give complete
parametric solutions to the Problem GESA.The free pa-
rameter matrix W∞and the free parameter vectors f i,i=
1,2,...,n e,represent the degrees of freedom in the eigen-
structure assignment design,and can be sought to meet cer-
tain desired system performances.It should be noted that
Constraint C0is not a restriction at all,it only gives a way
of selecting these design parameter vectors.
Remark5.It follows from well-known pole assignment re-
sult that Problem GESA has a solution when the system(1)
is R-and I-controllable and the closed-loop eigenvalues
s i,i=1,2,...,n e,are restricted to be distinct.In this
case,there exist parameter vectors f i,i=1,2,...,n e,sat-
isfying Constraints C a,C b or C v.As a matter of fact,it can
be reasoned that,in this case,“almost all”parameter vec-
tors f i,i=1,2,...,n e,satisfy Constraints C a,C b or C v.
Therefore,in such applications Constraints C a,C b or C v
can often be neglected.
2009Chinese Control and Decision Conference(CCDC2009)5
Remark 6.The solution given in Corollary 2utilizes only a series of singular value decompositions,and hence is numerically very simple and reliable.As for the solution given in Corollary 1,it has the advantage that the closed-loop eigenvalues s i ,i =1,2,...,n e ,can be set undeter-mined and used as a part of extra design degrees of freedom to be sought with f i ,i =1,2,...,n e ,by certain optimiza-tion procedures.Furthermore,it happens that the solution given in Corollary 1is a natural generalization of the para-metric solution in [12,13,18]proposed for eigenstructure assignment in first-order descriptor state-space systems.
5
Modes decoupling for un-rotating missile
5.1
The problem
Consider the yaw/roll channel attitude stabilization system of an un-rotating missile.Suppose β,ωy and ωx are re-spectively the sideslip angle,the yaw rate and the rolling angular velocity.δx,y and δxc,yc stand for the actual and input actuator deflections.Then,by choosing the state vec-tor
x =[βωy ωx δx δy ]T and the control vector
u =[δxc
δyc ]T
,
the state space representation of the system can be given as:
˙x =Ax +Bu with
A =⎡
⎢⎢⎢⎢
⎣
1V m c β
y qS 1
01J y m β
y qSL 12V J y m nr qSL
201J x m β
x qSL
012V J x m lp qSL 2
0000
00
00
01J y m δy y qSL 1J x m δx
x qSL 1J x m δy x qSL −1τx 00−1
τy
⎤
⎥⎥
⎥⎥⎥⎦,B
=
⎡
⎢⎢⎢⎢⎣
00000
01τx
00
1τy
⎤⎥⎥⎥⎥⎦
,where V ,m and J x,y are the velocity,the mass and the rotary inertia of the missile,
c βy ,m βy ,m nr ,m δy y ,m βx ,m lp ,m δx
x ,m δy x
are some aero-dynamic coefficients and τx ,τy are the actuator time constants.
Consider a certain characteristic point,the parameter ma-trices are given as ([20])
A =⎡
⎢⎢
⎢⎢
⎣−0.51000−62−0.160030100−50405000−4000000−20
⎤⎥⎥⎥⎥⎦
,B
=
⎡
⎢⎢⎢⎢⎣
000000400020
⎤⎥⎥⎥⎥⎦
.The objective is to find a state feedback controller for the system at the characteristic point,such that the following two requirements are met simultaneously:
1.The closed-loop eigenvalues are located in the regions as:
s 1,2=a +bj,−10≤a ≤−5,2≤b ≤6,−90≤s 3≤−65,−65≤s 4≤−50,−40≤
s 5≤−25;
2.The first two states are decoupled with the third eigen-value,the third state is decoupled with the first two eigenvalues.5.2The solution
Proposition 1.The above second requirement is satisfied if and only if the eigenvector matrix of the closed-loop system possesses the following structure:
V =⎡
⎢⎢⎢⎢
⎣∗∗0∗∗∗∗0∗∗00∗∗∗∗∗∗∗∗∗∗∗∗∗⎤⎥⎥
⎥⎥⎦,(40)where ∗represents any complex numbers.
Based on the above Proposition,we can solve the problem based on the parametric control systems design approaches.We can obtain,according to the factorization (38),
N (s )=⎡
⎢⎢
⎢⎢⎣10
0.5+s 001−[122+p (s )]/240(50+s )/40[62+p (s )]/6000
⎤⎥⎥⎥⎥
⎦
,D (s )=
−(40+s )[122+p (s )]/960(20+s )[62+p (s )]/600
(40+s )(50+s )/160
,
with
p (s )
=
(0.5+s )(0.16+3),
then the controller can be given as
K =
D (s 1)f 1···D (s 5)f 5 V −1,(41)
where
V =
N (s 1)f 1
···
N (s 5)f 5
.
(42)
62009Chinese Control and Decision Conference (CCDC 2009)
Denoting
f 1,2=
f 11±f 21j f 12±f 22j
,f i =
f i 1f i 2
,i =3,4,5,
(43)
it can be easily shown,according to (42),that the vectors matrix V has the form of (40)if and only if,
f 12=f 22=f 31=0.
(44)
Therefore,the solution to the modes decoupling control is given by (41)-(43),where the parameters s i ,f ij ,i =1,···,5,j =1,2,satisfying Constraint C0,(44)and
det N (s 1)f 1···N (s 5)f 5
=0.If we choose specially the parameters as s 1,2=−6±4j,s 3=−70,s 4=−60,s 5=−30,f 11=f 32=1,f 21=0.5455,
f 41
=
f 52=0,f 42=−4,f 51=0.03184,
then the controller is obtained as
K =
−3.6844−1.5266−1.2500−10−5.9703
1.9463−0.573700−1.0671 .The modes decoupling control problem of un-rotating mis-sile is solved based on the parametric control system design
method.Considering the solution,we can get the following conclusions.
•The method convert simply the modes decoupling problem into the linear equations problem;
•The polynomial matrices N (s )and D (s )are accurate (without rounding error),hence the result is more pre-cise;
•There are still some freedom degrees in the solution,which can be utilized to meet some other requirements in the system design.
6
Application on autopilot design of a BTT mis-sile
In this section,we propose a new scheduling method for linear time-varying systems based on the parametric ap-proaches of linear invariant systems and apply the approach to the design of a BTT missile autopilot.6.1
Gain-scheduled controller design of time-varying systems
Consider a linear time-varying system represented by
˙x =A (t )x +B (t )u,t ∈[t 0,t e ],
(45)
where x ∈R n and u ∈R r are respectively the state vec-tor and the input vector,A (t )and B (t )are system coef-ficient matrices of appropriate dimensions which are con-tinuously time-varying.The problem is to find a controller
for the system (45)which guarantees both the global sta-bility and desired performance of the resulted closed-loop system.In the following,state feedback controllers are designed for the local subsystems,and new justified tech-nique for scheduling is presented to construct the global
controller.Suppose t i ,i =1,2,···,r 0,are the character-istic time points selected in the operating range,then the local subsystems are
Σi :˙x =A i x +B i u,i =1,2,···,r 0,
(46)
where A i =A (t i ),B i =B (t i ).Linear system theory can be used to derive local controllers for each subsystem Σi .State feedback controllers in the form
u i =K i x,i =1,2,···,r 0.
(47)
are employed to stabilize the subsystems.In order to con-struct the global controller,certain interpolation method should be adopted to link K i ,i =1,2,···,r 0together.The problem here is how to construct the global state feed-back controller K (t )such that the resulted closed-loop sys-tem
˙x =A c (t )x,A c (t )=A (t )+B (t )K (t ),t ≥t 0
(48)
can be stabilized and at the same time desired performance is guaranteed.K (t )is a time-varying controller,which can be obtained by combining the local controllers using cer-tain gain-scheduled method.Suppose system (45)is con-trollable for all t ≥t 0.According to the proposed para-metric approaches we can design the local controllers K i ,i =1,2,···,r 0,as
K = D (s 1)f 1D (s 2)f 2···D (s n )f n
V −1,V =
N (s 1)f 1N (s 2)f 2···N (s n )f n ,where f j ∈C r ,j =1,2,···,n are arbitrary vectors
which satisfy Constraints C0and C v ,N (s )and D (s )are right coprime polynomial matrices satisfying (38).Then we can derive the local controller gain matrices K i ,i =1,2,···,r 0,by properly assign the closed-loop eigenval-ues s i j and the free vectors f i
j ,j =1,2,···,n .The eigenvalues can be determined by desired system perfor-mance such as overshoot and so on,and additional require-ments can also be fulfilled by extra freedoms proposed by f i j .In order to maintain the desired performance of the closed-loop system between adjacent characteristic points,the eigenvalues of the closed-loop system for all fixed time points should stay in a small neighborhood of the desired area.The local controllers of two adjacent characteristic points t i and t i +1are respectively noted by K i and K i +1
which are determined by s i,i +1j
,f i,i +1
j ,j =1,2,···,n .Denote s j (t ),j =1,2,···,n ,the point-wise eigenvalues of the closed-loop system.Since the performance require-ments on the whole operating range are usually similar,we can simply choose the same set of desired eigenvalues be-tween characteristic points,that is
s j (t )=s i j =s i +1
j
=s j ,j =1,2,···,n,t ∈(t i ;t i +1),(49)
where Re (s j )<0.Then the controller between charac-teristic points t i and t i +1can be obtained by the following scheduling procedure:
K (t )
=K (s j (t ),f j (t );j =1,2,···,n )
=
D (s 1)f 1(t )···D (s n )f n (t )
N (s 1)f 1(t )···N (s n )f n (t ) −1,
(50)2009Chinese Control and Decision Conference (CCDC 2009)7。