The effect of feedback structure on the closed loop dynamics

, (4.4)

where Y_c = ^hY_cp^T Y_cc^TiT. The matrix K ∈ R^rp×(mp+mc) is a constant feedback gain to be computed. The state equations of the controller are

x˙c(t) =Mc·ψ(x(t)). (4.5)

Then, we can partition K and M_c into two blocks as K = [K_p K_c] and M_c = [M_cp M_cc] where Kp ∈ R^rp×mp, Kc ∈ R^rp×mc, Mcp ∈ R^nc×mp and Mcc ∈ R^nc×mc. In that case the equations of the closed loop system are given by

x(t) =˙

"

M_p+B_pK_p B_pK_c

M_cp M_cc

#

·

"

ψ_p(x_p(t)) ψ_c(x_p(t), x_c(t))

#

=M(K, Mc)·ψ(x(t)), (4.6) where the closed loop coefficient matrixM has an affine dependence onK andM_c. The aim of the feedback is choosing a suitable feedback gain K and feedback dynamics Mc such that the closed loop system (4.6)defines a kinetic system with prescribed properties (e.g. complex balance). It is clear from Section 2.2 that this is possible if M(K, M_c) can be factorized as M(K, Mc) = Y ·A_k where A_k ∈ R^{(mp+mc)×(mp+mc)} is a valid Kirchhoff matrix. In the following, let n=n_p+n_c and m=m_p+m_c.

4.3 The effect of feedback structure on the closed loop dy-namics

The results in this section show that involving new monomials (complexes) into the feedback law in (4.3) does not improve the solvability of the feedback problem from the point of view of weak reversibility, deficiency or complex balance. In the following, by additional complexes, we mean complexes corresponding to the monomials of ψ_c in (4.6). Therefore, a closed loop system without additional complexes means a controlled system of the form (4.6), where Kc=0 and Mc=0. Note that this is a technical solution in order to keep the dimension of the state vector and the Kirchhoff matrix of the closed loop system constant in the calculations. In this case, the complexes corresponding to ψc are naturally isolated in the reaction graph.

Lemma 2. Consider the open loop system (4.1) and the feedback law (4.3). Assume that there exists a closed loop system with feedback parameters K, M_c, and it has a realization (Y, A_k) where the jth complex Cj is an additional complex such that [A_k]_j,j 6= 0 (i.e. it is

(a) RealizationAk (b) RealizationA⁰_k

Figure 4.1. Realizations of the open loop system (4.8) with different feedbacks u_p(t) and u⁰_p(t). The realizations 4.1a and 4.1b correspond to the feedbacks u_p(t) andu⁰_p(t), respect-ively.

a source complex in at least one reaction). Then, there exist other feedback parameters K⁰ and M_c⁰ such that the corresponding closed loop system has a realization (Y, A⁰_k), where A⁰_k is given by

[A⁰_k]_·,i= [A_k]_·,i+ [A_k]_j,i

−[A_k]_j,j[A_k]_·,j, ∀i. (4.7) Remark 1 It can be seen from (4.7) that the jth complex is isolated in the realization (Y, A⁰_k), since [A⁰_k]_j,i= 0 and [A⁰_k]_i,j = 0 for all i.

Remark 2 Before the proof, we illustrate the idea behind it by a simple example. Let us consider the following open loop system with a single input

x˙p1(t) =k3xp2(t)−k1xp1(t) +k4xp2(t)−k5xp1(t)²xp2(t) +k6xp1xp2(t)²+up(t) x˙p2(t) =k₁xp1(t) +k₂xp1(t)−k₃xp2(t) +k₅xp1(t)²xp2(t)−k₆xp1xp2(t)²+up(t),

(4.8) wherex_p1,x_p2 are the states,k₁, k₂, . . . , k₆are real parameters andu_p(t) is the input. Let the additional monomial in the feedback be ψc(xp, xc) =xp1xp2. Then, we can give a feedback as follows

up(t) =k₇xp1(t)xp2(t), (4.9) wherek7is a real parameter. With this feedback (4.9) the closed loop system has a realization A_k such that the additional complex X₁+X₂ has outgoing edges in the reaction graph.

Furthermore, we can construct a feedback u⁰_p(t) such that the corresponding realization A⁰_k fulfils (4.7). The constructed feedback has the form

u⁰_p(t) = 0.5k₂x_p1(t) + 0.5k₄x_p2(t). (4.10) The Fig. 4.1 compares the corresponding realizationsAk and A⁰_k.

Proof. It can be seen from (4.7) that the matrix A⁰_k is Kirchhoff, because the sum of its columns is zero and the off-diagonal elements are positive due to [A⁰]_j,i= 0 for all i.

First, we can write (4.7) in the following compact form A⁰_k=A_k+ 1

−[A_k]_j,j[A_k]_·,j·[A_k]_j,·. (4.11) Then, we can construct the feedback gain K⁰ as

K⁰=K+ 1

−[A_k]_j,jK·,j·[A_k]_j,· (4.12) and the matrix M_c⁰ as

M_c⁰ =Mc+ 1

−[A_k]_j,j[Mc]_·,j·[Ak]_j,·. (4.13) Then, the matrix M⁰ is given by

M⁰(K⁰, M_c⁰) =M(K, M_c) + 1

−[A_k]_j,j

"

BpK M_c

#

·,j

[A_k]_j,· (4.14)

which can be written as

M⁰(K⁰, M_c⁰) =Y ·A_k+ 1

−[A_k]_j,jY ·[A_k]_·,j·[A_k]_j,·=Y ·A⁰_k. (4.15) Therefore, M⁰(K⁰, M_c⁰) has the realizationA⁰_k.

Lemma 3. Let A_k and A⁰_k be Kirchhoff matrices where the matrix A⁰_k is constructed by (4.7). Then ker(A_k)⊆ker(A⁰_k).

Proof. Let us take an elementpof ker(A_k), then pj =^X

i6=j

pi

[Ak]_j,i

−[A_k]_j,j. (4.16)

Let us consider the product A⁰_k·p:

m

X

i=1

p_i[A⁰_k]_·,i=^X

i6=j

p_i[A_k]_·,i+^X

i6=j

p_i [A_k]_j,i

−[A_k]_j,j[A_k]_·,j. (4.17) We can now substitute (4.16) into (4.17) to obtain

m

X

i=1

pi[A⁰_k]·,i=^X

i6=j

pi[Ak]·,i+pj[Ak]·,j = 0 (4.18) that means p∈ker(A⁰_k).

Theorem 5. Consider the open loop system (4.1) and the feedback law (4.3). Then the

following statements apply.

(a) If there exists a weakly reversible closed loop system (4.6) with additional complexes, then there exists another weakly reversible closed loop system without additional com-plexes.

(b) Suppose there exists a weakly reversible closed loop system (4.6) with additional com-plexes and deficiency δ. Then there exists another weakly reversible closed loop system without additional complexes, and it has deficiency δ⁰ such that δ⁰≤δ.

(c) Suppose there exists a complex balanced closed loop system (4.6)with additional com-plexes and equilibrium points in the set E. Then there exists another complex balanced closed loop system without additional complexes and it has equilibrium points E⁰ such that E ⊆ E⁰.

Proof. The proofs of the above statements are the following.

(a) If there exists a closed loop system with a weakly reversible realization (Y, A_k), then there exists another one with realization (Y, A⁰_k) where the additional complexes are isolated (Lemma 2) and ker(Ak)⊆ker(A⁰_k) (Lemma 3.). A Kirchhoff matrix is weakly reversible if and only if there exists a positive vector in its kernel. Therefore, matrix A⁰_k is weakly reversible, too.

(b) If there exists a closed loop system with a weakly reversible realization (Y, A_k), then there exists another one with realization (Y, A⁰_k) where the additional complexes are isolated (Lemma 2). Let us denote the corresponding incidence matrices by D and D⁰. The columns of matrix D⁰ are linear combinations of the columns of matrix D by construction. Therefore, im(D⁰) ⊆ im(D). By the definition of deficiency δ = dim(ker(Y)∩im(D)) andδ⁰ = dim(ker(Y)∩im(D⁰)). Soδ⁰ ≤δ.

(c) If there exists a closed loop system with a weakly reversible realization (Y, Ak), then there exists another one with realization (Y, A⁰_k) where the additional complexes are isolated (Lemma 2) and ker(A_k)⊆ker(A⁰_k) (Lemma 3.). According to the assumption, (Y, A_k) is complex balanced, therefore the set of equilibrium points of (4.6) can be described as E ={x|A_k·ψ(x) =0}. Since, ker(A_k) ⊆ker(A⁰_k), then A⁰_k is complex balanced andE ⊆ E⁰.

The kinetic feedback structure The theoretical results described in this section can be summarized in the following simple statements to determine the structure of the kinetic feedback, if one wants to achieve complex balanced closed loop kinetic system, or a weakly reversible zero deficiency closed loop kinetic system.

1. It is not necessary to apply dynamic feedback to achieve complex balance or weak reversibility with zero deficiency for the closed loop system.

2. It is sufficient to use only the monomials of the open loop system (4.1)in the monomial function ψ(x) of the static feedback (4.3).

However, increasing the degrees of freedom of the control design with additional state vari-ables, monomials and the corresponding extra parameters using the general form (4.6) might be advantageous to achieve additional goals, such as improving the time-domain performance of the closed loop system.

In document Nemnegatív polinomiális rendszerek analízise és irányítása (Pldal 36-40)