The estimation problem for dynamic systems with several time scales is considered. The method of asymptotic decomposition is used to reduce dimension and to simplify the structure of the observers.
The existence of the processes with essentially di®erent velocities is a feature of the complex dynamic systems. The mathematical models of such systems are the singularly perturbed di®erential systems, which contain one or several small parameters at some derivatives. The problems of numerical analysis and control of such systems are very di±cult due to high dimension and multirate components.
For analyzing of the dynamic systems' behavior and making control law we must measure the state vector (phase vector). In real problems the measuring of the state vector is di±cult due to technical or economic reasons. Furthermore the measurement devices have a complex structure and can modify the dynamic of the control object.
It makes relevant the application of indirect state estimation methods. The most widespread approach to the estimation problem is the creation of the system which is called \observer". Any solution of such system tends to the solution of initial system. We will consider two types of observers: full order observer and lower order observer (Luenberger observer). We analyzed the features of structure of observers for linear dynamic systems with slow and fast variables. We created the algorithm of the construction of the observers for such systems, which is based on the method of asymptotic decomposition.
Let us consider a linear dynamic model
x_ (t) = A(t)x(t) + B(t)u(t);
where state vector x(t) 2 Rn, input vector u(t) 2 Rm; output vector y 2 Rl;
y(t) = C(t)x(t);
(
C is ( l £ n )-matrix of measurements.
The so called "full order observer" is m_(t) = A(t)m(t) + B(t)u(t) + V (t)(y(t) ¡ C(t)m(t)); where ( m £ l )-matrix V (t) is chosen to guarantee the asymptotic convergence of the state estimation error ¢(t) = x(t) ¡ m(t): For autonomous systems any eigenvalue of matrix (A ¡ V C) must satisfy the inequality Re ¸j (A ¡ V C) < 0: Let rank(C) = l; ((n ¡ l) £ n){ matrix W be such that (n £ n){matrix Q = µ C
W
¶ is nonsingular. Write matrix Q¡1 in the form Q¡1 = (R D), where R is (n £ l){ matrix, D is (n £ (n ¡ l)){matrix.
The so called \lower order observer" (Luenberger observer) dynamic is
m(t) = D®(t) + (R + DV )y(t);
where
®_ = (W ¡ V C)[AD® + Bu + A(DV + R)y]; ®(0) = ®0:
The state estimation error ¢ satis¯es the equation
¢_ (t) = D(W ¡ V C)A¢(t):
For autonomous systems any eigenvalue of matrix (W ¡ V C)AD must satisfy
the inequality Re ¸j ((W ¡ V C)AD) < 0:
Asymptotic decomposition of linear singularly perturbed
systems
Let us consider a system
x_ 1 = A11x1 + A12x2;
"x_ 2 = A21x1 + A22x2
with output vector
y = C1x1 + C2x2;
(
Using a coordinate transformation x2 = z + Lx1; = ¡ ³A(202)´¡1 A(201); = ¡ ³A(202)´¡1 hA(211) + A(212)L(0) ¡ L(0)A(101) ¡ L(0)A(102)L(0) ¡ L_ (0)i ; = = + The output vector y takes the form y = C~1v + C~2z; were C~1 = C1 + C2L;
C~2 = "C1P + C2 + "C2LP:
L = L(t; ") = L(0)(t) + "L(
A1v;
"z_ = A2z;
where A1 = A1(t; ") = A11 + A12L; A2 = A2(t; ") = A22 ¡ "LA12:
It can be proved that matrix functions L; P may be constructed with any degree
of accuracy as asymptotic series in small parameter "
(
: V = µ V1 ¶
V2
: Then the observer takes the form µ m_v ¶ m_z = µ A1 ¡~V1C~1 ¡V2C1 ¡V1C~2 A2"¡1 ¡ V2C~2 ¶ µ mv ¶ mz + µ V1 ¶
V2 y: Since all eigenvalues of the matrix A(202)(t) satisfy inequalities Re ¸j < 0, we can take V2 = 0: The system (11) takes the block-triangular form µ m_v " m_z ¶ = µ A1 ¡ V1C~1 0 ¡V1C~2 ¶ µ mv ¶ + A2 mz µ V1 ¶ 0 y:
We can choose the block V1 from the condition of asymptotic stability of the slow subsystem, which for an autonomous system takes the form Re ¸j (A1 ¡ V1C~1) < 0: For the estimation of state vector of the initial system we have m1 = mv + "P mz;
m2 = mz + Lm1:
Let us construct the full order observer for slow subsystem of the block diagonal
system (
It can be proved that (12) (13) (14) Consider the model of a longitudinal motion of an aircraft, see Figure 1, [10] As an estimation of fast variable we can use any solution of the system For estimation of state vector of the initial system we have The similar reasoning can be used for construction of the Luenberger observer. µ C1 ¶ ~ For the slow subsystem (13) we choose matrix W such that matrix Q = W is nonsingular. Let Q¡1 = (R D): The estimations of state vector take the form m1 =
D® + (R + DV1)y;
m2 = mz; where ®_ = (W ¡ V1C~1)(A1D® + A1(DV1 + R)y); ®(0) = ®0; " m_z = A2mz:
vÄ = d1® ¡ d2± µ_ = d3®; T ±_ + ± = Krmu:
x2 = ±: The system (14) takes the form
A11x1 + A12x2 ¡x2; x_1 "x_2 where = =
0 ¡d2 1 A12 = @ 0 A ;
0 A21 = ¡ 0 0 0 ¢ ;
A22 = ¡1: Let the outputs be º and µ, then y = C µ x1 ¶ x2 ;
C = where P = P (") is the matrix function, which satis¯es the equation ¡P = "A11P + A12; v_ = A11v;
"z_ = ¡z:
we can transform the system (15) to the block diagonal form
The matrix function P (") may be constructed with any degree of accuracy as
asymptotic series in small parameter "
P = P (") = P (0) + "P (
1 0.8 0.6 0.4 0.2 0 2 4 t 6 8
10 2 4 t 6 8 10 0 2 4 t 6 8 10
Fig. 4. ±; m± Fig. 5. µ; mµ For slow subsystem we notice that the second row of matrix C~1 is zero. We choose matrix W such that matrix Q = µ CW¹1 ¶ ; where C¹1 = ¡ 0 1 0 ¢ ; is nonsingular. For example, W = µ 10 00 01 ¶ : Write matrix Q¡1 in the form Q¡1 = (R D); where
0 0 1 0 1 0 1 R = @ 1 A ; D = @ 0 0 A :
0 0 1 The estimation of the vector v takes the form mv = D® + (R + DV1)y; where ®_ = (W ¡ V1C¹1)(A11D® + A11(DV1 + R)y); ®(0) = ®0: Let V1 = µ ba ¶ : We have (W ¡ V1C¹1)A11D = µ ¡a ¡b d1 ¡d3 ¶
: For example, let us put a = 0; b > 0: As an estimation of the fast variable we can use any solution of the system For the estimation of the state vector of the initial system we have The Figures 6{9 demonstrate the dynamic of the state vector and its estimation. 2 0 –2 –4 –6 –8
1 0.8 0.6 0.4 0.2 0 2 4 t 6 8
10 The asymptotic decomposition method helped us to reduce the observation problems for the dynamic systems with slow and fast variables. This approach can be also used for solving the observation problems in a stochastic case.
The research has been supported by the Ministry of Education and Science of the Russian Federation (the Project 204).