Transformation Step: Block Diagonal Matrix

, (4.78)

with matrix A^h₁₁^ki(t)∈ ^{R(k−1)×(k−1), vectors ah}₁₂^ki(t), a^h₂₁^ki(t) ∈^{R(k−1)×1, and the} scalar a^h₂₂^ki(t). Superscript h^ki indicates affiliation to the k–dimensional system (4.74). The first transformation (4.75) converts system (4.74) into an upper block triangular form, comparable to the transformation (4.65) in connection with ma-trix (4.68), where

y˙_k(t) =

"

A^h₁₁^ki(t) +a^h₁₂^ki(t)[p_k(t)]^T a^h₁₂^ki(t) 0^T λ_k(t)

#

y_k(t) (4.79)

iffp_k(t)is any solution of the system of Riccati differential equations p˙_k(t) =−p_k(t) [a^h₁₂^ki(t)]^Tp_k(t)−[A^h₁₁^ki(t)]^Tp_k(t)

+p_k(t)a^h₂₂^ki(t) +a^h₂₁^ki(t), (4.80) whileλ_k(t)is called dynamic eigenvalue and is calculated with

λ_k(t) = a^h₂₂^ki(t)−[p_k(t)]^Ta^h₁₂^ki(t). (4.81) At the end of the first transformation step, it is important to emphasize that any solution of the Riccati equation (4.80) transfers equation (4.74) into the upper block triangular matrix of (4.79), since dynamic eigenvalues and -vectors are not unique.

2. Transformation Step: Block Diagonal Matrix

The second coordinate transformation converts equation (4.79) into a block diag-onal form. This is obtained via the relation that is equivalent to equation (4.69) but specified again fork =n, . . . , 2, so that

y_k(t) = Q_k(t)z_k(t). (4.82) The transformation matrixQ_k(t) ∈^Rk×kis comparable to matrix (4.71) but spec-ified for the transformation algorithm as follows: The dimensions of the identity matricesI_aandI_b are further ona=k−^{1 andb} =1, respectively. Consequently,

matrixL(t)is a column vector of dimension(k−¹)×1. Again, for clarity, I spec-ify and redefine−^L(t) = q_k(t). Hence,

Q_k(t) =

"

I_k−1 q_k(t) 0^T 1

#

, (4.83)

where

q_k(t) =









q_k,1(t) q_k,2(t)

...

q_k,k−1(t)







, (4.84)

and the scalar functions of time q_k,i(t) for i =1, . . . , k−1. The submatrix 0^T ∈ ^R1×(k−1) is a zero (row) vector. This transformation ensures that equation (4.79) goes into the following block diagonal form, comparable to the transforma-tion (4.70) in connectransforma-tion with matrix (4.73), where

˙

z_k(t) =

"

A^h₁₁^ki(t) +a^h₁₂^ki(t) [p_k(t)]^T 0 0^T λ_k(t)

#

z_k(t) (4.85)

iffq_k(t)is any particular solution of the system of differential equations

˙

q_k(t) = A^h₁₁^ki(t) +a^h₁₂^ki(t) [p_k(t)]^T−^λk(t)I_k−1

q_k(t) +a^h₁₂^ki(t). (4.86) The combination of the two consecutive transformations results in

x_k(t) =P_k(t)Q_k(t)z_k(t),

which is called Riccati transform. The second transformation step completes the first round of transformations.

Second Transformation Round

In order to decouple the whole system (4.74), these two transformation steps must be applied(n−¹)times in total. As a result, each round decouples an additional row in equation (4.85). The next transformation round starts with an order re-duction of system (4.74) by one (i.e.k := k - 1), leading to the starting system of the

second transformation round with

x˙_k(t) =A_k(t)x_k(t), for k=n−^{1, (4.87)} where the updated system matrix in (4.87) is

A_k =A^h₁₁^k+1i(t) +a^h₁₂^k+1i(t) [p_k+1(t)]^T. (4.88)

Dynamic Eigenvalues and the State Transition Matrix

After completing(n−¹)transformation rounds, the dynamic eigenvalueλ₁(t)is still pending. It can be calculated via the connection

λ₁(t) = trace[An(t)]−

n

∑

i=2

λ_i(t), (4.89)

since the Riccati transform possesses the property of preserving the trace.¹¹¹ Dy-namic eigenvalues “are invariant under any algebraic transformation.”¹¹²

The results of the two transformation steps, which are performed(n−¹)times, are(n−¹)P-matrices and(n−¹)Q-matrices. Van der Kloet and Neerhoff (2001, 2004) combine these matrices to form one transformation matrix T(t) ∈^Rn×n. The transformation matrixT(t)diagonalizes system (4.44), as shown in equation (4.58), whereT(t)is calculated with

T(t) =

(n−¹)

∏

i=1

S_n−i+1(t), (4.90)

and

S_k(t) =

"

P_k(t)Q_k(t) 0 0^T I_n−k

#

, (4.91)

fork =n, n−1, . . . , 2. The submatrix0is of dimensionk×(n−^k). As a result,

111 See van der Kloet, Neerhoff, and de Anda (2001), p. 79, and Zhu and Johnson (1991), pp.

204–206.

112 Wu (1980), p. 825.

matrixT(t)transforms equation (4.44) into the diagonal system

z˙(t) =







λ₁(t) · · · ⁰ ... ... ...

0 · · · ^λn(t)





z(t),

which is comparable to equation (4.59), via the relation (4.60), that is Λ(t) = [T(t)]⁻¹A(t)T(t)−[T(t)]⁻¹T^˙(t).

The state transition matrixΦ(t,t0)to system (4.44) results with Φ(t,t₀) = T(t)e^{Rtt0Λ(τ)dτ} [T(t₀)]⁻¹,

for the initial time t₀. Any particular solution to p_k(t) and q_k(t) is feasible, and mode-vectors are not unique.

Lyapunov Transformation

Finally, it must be ensured that the stability properties do not change due to the successive Riccati transforms: MatricesP_k(t) and Q_k(t) must be Lyapunov ma-trices. This condition implies that the chosen solutions to the systems of Ric-cati differential equationsp_k(t), ˙p_k(t) and the chosen solutions to the systems of Sylvester differential equationsq_k(t), ˙q_k(t)have to be bounded for allt.

For the first transformation step, the solutionsp_k(t)to the matrix Riccati equation can have the property of blowing up on a finite interval. This phenomenon is called finite escape time and it arises when there are singularities.¹¹³ Since any solution to equation (4.80) is feasible, one can try to find a bounded solution. This approach works for the system (4.22), so I do not go into details about non-blow-up conditions here. Detailed information can be found in Abou–Kandil, Freiling, Ionescu, and Jank (2003) or Freiling, Jank, and Sarychev (2000).

For the second transformation step, the solution to the systems of differential equationsq_k(t)can be unbounded, i.e. unstable. It is well known that the stability analysis of LTI systems is characterized by its eigenvalues:

“A linear homogeneous system with constant coefficients [like equation (4.51)] is

113 See Abou–Kandil, Freiling, Ionescu, and Jank (2003), p. 91.

1) stable if and only if all the eigenvalues of the coefficient matrix have non-positive real parts, and simple elementary divisors correspond to the eigen-values with zero real part,

2) asymptotically stable if and only if all the eigenvalues of the coefficient ma-trix have negative real parts.”¹¹⁴ [Emphasis deleted.]

The stability analysis of LTV systems is, in general, not as easy as it is for LTI sys-tems. Eigenvalues are misleading in the LTV case and the application of dynamic eigenvalues is not clear yet.¹¹⁵ In general, the characteristics of the state transi-tion matrix are examined. But, in most cases, the state transitransi-tion matrix must be calculated numerically, since an analytic solution is not available. The transfor-mation algorithm provided by van der Kloet and Neerhoff (2001, 2004) and van der Kloet, Neerhoff, and de Anda (2001) shows that—in principle¹¹⁶—an analytic solution can be derived. The transformation algorithm can be applied again to find a solution ofq_k(t)in equation (4.86). The stability ofq_k(t)can be verified by the corresponding state transition matrix:¹¹⁷

The LTV homogeneous system (4.44) is

1. uniformly stable iff there exists a constantD₁ >_{0 so that}

k^Φ(t,t₀)k ≤ ^D1, for 0≤^t0 ≤^t<∞, (4.92) wherek·kis the matrix norm.¹¹⁸ Uniform stability implies that all solutions to the homogeneous system remain bounded fort≥^t0.

2. uniformly asymptotically stable iff two constants D₂ > _{0, α} > 0 (indepen-dent oft₀) exist so that

k^Φ(t,t₀)k ≤ ^D2e^{−α(t−t0)}, for 0 ≤^t0 ≤^t<∞. (4.93)

114 Adrianova (1995), p. 84.

115 Van der Kloet, Neerhoff, and Waning (2007) suggest Lyapunov characteristic exponents, cal-culated by the mean value of the dynamic eigenvalues, to determine the stability of a LTV system. But the counterexample of de Anda (2012) shows that some more research has to be done on this topic.

116 An analytic solution for an LTV system is possible if analytic solutions to the Riccati equa-tions (4.80) and differential equaequa-tions (4.86) can be derived. This is feasible for 2ײ sys-tems, but it can become challenging for higher order systems. Kolas (2008, ch. 7) develops a computer algorithm for calculating dynamic eigenvalues. Additionally, it can handle some singularities.

117 See Zadeh and Desoer (1963), ch. 7 and Adrianova (1995), ch. IV.

118 See the definition for matrix norm in Adrianova (1995), ch. I §2.

Uniform asymptotic stability implies that all solutions to the homogeneous system tend to zero ast→^∞. This is equal to

tlim→^∞k^Φ(t,t₀)k =0, ∀^t0.

Hence, uniform asymptotic stability includes uniform stability.

The corresponding inhomogeneous LTV system (4.43) is bounded, i.e. stable, 1. iff the corresponding homogeneous system is stable, and

2. iffR_t

t0Φ(t,v)b(v)dvis bounded. Hence, it must be ensured that there exists a numberM₁such that

k^b(t)k< _{M1, (4.94)} i.e.b(t)is bounded∀^t >_t0. Additionally, it must be ensured that there are positive constantsM₂and βsuch that

k^Φ(t,t₀)k_II ≤ ^M2e^{−β(t−t0)}, for 0≤^t0 ≤^t<∞.¹¹⁹ (4.95)

119 Subscript II denotes that the normk^A(t)kII=max

j n

∑

i=1

aijis used. See Adrianova (1995), p.

3 for the definition of matrix norms.

4B Appendix: Solution of the System of Differential

Im Dokument Liquidity Shocks in Over-the-Counter Markets (Seite 99-105)