Grid-based decoupling of parameter-varying subsystems

sequence. The K_u^⋆(ϱ) lower rank approximation of the blend matrix is computed by (5.12).

Then the K_u,i^⋆ reduced vertex matrices are given based on (5.6) by solving a system of linear equations. These K_u,i^⋆ matrices are used in the next iteration cycle, by adapting Lemma 21 to the polytopic model formulation.

Remark 13. It is important to identify some metrics, to decide which singular value function is the least significant on the diagonal ofS(ϱ). In practice anyl_p norm defined in Appendix A.1 may be a suitable measure.

Remark 14. Due to the parameter variation, the least significant singular value functions on the diagonal ofS(ϱ) may cross each other. At every time it occurs, the parameter space is divided into segments, and separate blend vector functions are computed, leading to piecewise LPV designs. The continuity of the U(ϱ), S(ϱ), V(ϱ) matrix functions assure that, these piecewise blend vector functions are connected at the boundaries of their domain.

Once the rank oneKu(ϱ) blend matrix function is achieved, theku(ϱ) blend vector function is computed as the parameter-varying singular vector, corresponding to the only non-zero singular value in S(ϱ).

The LMI conditions for the parameter-varying output blend design are similar. Define the blended, parameter dependent input and direct feedthrough matrices as ¯B(ϱ) =B(ϱ)k_u(ϱ) and D(ϱ) =¯ D(ϱ)k_u(ϱ), respectively. With these, the synthesis LMIs for the output blend are written

as

Ac(ϱ) B¯c(ϱ)

I 0

T

Ξ

⋆ +

Cc(ϱ) D(ϱ)¯

0 I

T

−Ky(ϱ) 0 0 β²I

⋆

≺0, (5.13)

and

A^T_d(ϱ)X_d+X_dA(ϱ)_d+C_d^T(ϱ)K_y(ϱ)C_d(ϱ) X_dB¯_d(ϱ) +C_d^T(ϱ)K_y(ϱ) ¯D_d(ϱ)

⋆ D¯^T_d(ϱ)K_y(ϱ) ¯D_d(ϱ)−γ²I

⪯0. (5.14) Then the straightforward modification of Proposition 8 provides the k_y(ϱ) output blend vector function.

Theorem 5.: The minimum gain of a linear parameter-varying system over a fi-nite frequency range. Consider the LPV system given by (2.55). Let Π =

−I 0 0 β²I

∈ R^{(nx+ny)×(nx+ny)} with β ≥0, and select Φ according to the Ω frequency range from Table 2.1. Then the system has a finite minimum gain, i.e., ||G(ϱ)||^[Ω]₋ ≥ β, if there exists X(ϱ) and Y ≻0 such that

A(ϱ) B(ϱ)

I 0

T

Φ₁₁Y X(ϱ) + Φ₁₂Y X(ϱ) + Φ₂₁Y X(ϱ) + Φ˙ ₂₂Y

⋆ +

C(ϱ) D(ϱ)

0 I

T

Π

⋆

≺0, (5.15) holds for all ϱ∈ F_P^V and u∈ Lⁿ_2Ω^u, with X(T) = 0.

Proof. The multiplication of the inequality in (5.15) by

x^T u^T

from the left and by its transpose from the right yields:

d

dt(x^TXx) +β²u^Tu−y^Ty+ Φ₁₁x˙^TYx˙+ Φ₁₂x˙^TY x+ Φ₂₁x^TYx˙+ Φ₂₂x^TY x <0. (5.16) Integrating along the state trajectory from t = 0 to t = T, and considering that x ∈ L2e

one gets:

−x(0)^TXx(0) +β² ZT 0

u^Tudt− ZT 0

y^Tydt

+ ZT

0

Φ11x˙^TYx˙+ Φ12x˙^TY x+ Φ21x^TYx˙+ Φ22x^TY x

dt <0.

(5.17)

This can be rewritten as

−x(0)^TXx(0) +β² ZT 0

u^Tudt− ZT

0

y^Tydt+

+ tr

Y RT 0

Φ₁₁x˙^Tx˙+ Φ₁₂x˙^Tx+ Φ₂₁x^Tx˙+ Φ₂₂x^Tx dt

<0.

(5.18)

Since Y ≻0 andu∈ Lⁿ_2Ω^u, the tr[·] term is non-negative, and the inequality reduces to

−x(0)^TY x(0)<

ZT 0

y^Tydt−β² ZT

0

u^Tudt, (5.19)

which by definition of the minimum gain in (2.19) completes the proof.

Grid-based blend calculation

The aim of the subsection is to find an input blend vector function ku(ϱ), which maximizes the excitation of the selected LPV subsystem, while minimizes the impact on the one to be decoupled. Similarly to the LTI case, this is achieved in two consecutive steps. First an optimal parameter dependent input blend is found, and applied to the system, next a corresponding output blend function is calculated. The necessary synthesis LMIs are based on (5.15) and (2.59), i.e., the minimum and maximum sensitivity conditions for parameter-varying systems.

Gc(%)

Gd(%) ku(%)

B_c(%) R A_c(%)

C_c(%) k_y0

+

+ ¯yc

B_d(%) R A_d(%)

C_d(%) k_y0

+

+ ¯y_d

¯u

Figure 5.1: Parameter-varying input blend calculation

In order to assure linearity in the synthesis inequalities for the input blend design, the concept of duality for LPV systems needs to be introduced. This is summarized based on [130].

Definition 10.: The dual LPV system [130]. IfG(ϱ) =

"

A(ϱ) B(ϱ) C(ϱ) D(ϱ)

#

is a (primal) LPV system then G^T(ϱ) =

"

A^T(ϱ) C^T(ϱ) B^T(ϱ) D^T(ϱ)

#

is the corresponding dual system.

At the same time, it is emphasized again, that theH−index can only be calculated for tall or square systems [71]. For this reason the same discussion applies as for the polytopic LPV design technique in Section 5.2, and an initial k_y0 output blend is used for the input blend design. The problem layout is shown in Figure 5.1.

If one writes the LMI constraints of (5.15) and (2.59) for the dual system, and then expresses the formulas in terms of the original representation, one gets

A^T_c(ϱ) C_c^T(ϱ)

I 0

T

Ξ

⋆ +

B^T_c(ϱ) D^T(ϱ)

0 I

T

Π

⋆

≺0, (5.20)

X_d(ϱ)A^T_d(ϱ) +A_d(ϱ)X_d(ϱ) +B_d(ϱ)K_u(ϱ)B^T_d(ϱ) + ˙X(ϱ) ⋆

k_y0^T C_d(ϱ)X_d(ϱ) +k^T_y0D(ϱ)Ku(ϱ)B^T_d(ϱ) k^T_y0D(ϱ)Ku(ϱ)D^T(ϱ)ky0−γ²I

≺0, (5.21) where Π =

−K_u(ϱ) 0 0 β²I

. The newly introduced parameter dependent matrix function K_u(ϱ) = k_u(ϱ)k_u(ϱ)^T, is the dyadic product of the parameter dependent input blend vectors, and K_u(ϱ)∈R^nu×nu ∀ϱ∈ F_P^V. Furthermore, it has rank 1 for allϱ∈ F_P^V, which has to be taken into consideration in the solution. The input blend calculation is summarized in Proposition 9.

Proposition 9.: The parameter-varying input blend calculation. The optimalku(ϱ) input blend for the system given in the form of (2.55) can be calculated as the left parameter dependent singular vector corresponding to the largest singular value of the K_u(ϱ) blend matrix, whereKu(ϱ) satisfies the following optimization problem

minimize

X_d(ϱ), Ku(ϱ), Xc(ϱ), Y, β², γ² −β²+γ² subject to (5.20), (5.21),

Y ≻0, X_d(ϱ)≻0,

Ku(ϱ) =Ku(ϱ)^T, 0⪯Ku(ϱ)⪯I, rank (Ku(ϱ)) = 1, ∀ϱ∈ F_P^V,

(5.22)

with I being the identity matrix.

Similarly for the LTI case in Proposition 1, (5.22) is a multi-objective optimization problem with two competing objectives which are merged into a single value by using scalarization. The solution follows a similar track as for Proposition 1, but one has to consider the continuous parameter dependency of the K_u(ϱ) matrix function through the ASVD algorithm, discussed in the previous section.

The LMIs (5.20) and (5.21) form an infinite number of constraints over the admissible set of the ϱ scheduling parameter. Therefore for achieving a finite dimensional convex problem, they are evaluated over a finite grid. More precisely, the parameter variation set is discretized and the corresponding LTI dynamics are obtained. Then, the LMI constraints are written for the finite set of systems, taking into account the bounds of the change in the scheduling parameter.

Following the discussion in Section 2.5, these rewritten LMI constraints for X(ϱ) =

nbf

X

i=1

fi(ϱ)Xi and Ku(ϱ) =

nXbfK

i=1

gi(ϱ)Ku,i, have the form

Ac(ϱ_k) C_c^T(ϱ_k)

I 0

T

Ξ

⋆ +

B_c^T(ϱ_k) D^T(ϱ_k)

0 I

T

Π

⋆

≺0, (5.23)



 Υ ⋆

k_y0^TCd(ϱk)

nP_bf

i=1

fi(ϱk)Xd,i+

nP_bfK

i=1

gi(ϱk)k^T_y0D(ϱk)Ku,iB_d^T(ϱk) D^T(ϱk)D(ϱk)−γ²I



≺0, (5.24)

with

Ξ =







Φ₁₁Y

_nbf P

i=1

f_i(ϱ_k)X_c,i

+ Φ₁₂Y _nbf

P

i=1

f_i(ϱ_k)X_c,i

+ Φ₂₁Y Φ₂₂Y +

nϱ

P

j=1±

ν_j

nPbf

i=1

∂fi

∂ϱiX_i





, Π =



−

nP_bfK

i=1

g_i(ϱ_k)K_u,i 0

0 β²I



,

Υ =

nbf

X

i=1

f_i(ϱ_k) (X_d,iA_d(ϱ_k) +⋆) +

nXbfK

i=1

g_i(ϱ_k)B_d(ϱ_k)K_u,iB_d^T(ϱ_k) +

nϱ

X

j=1

± ν_j

nbf

X

i=1

∂f_i

∂ϱi

X_d,i

! , corresponding to (5.20) and (5.21), respectively. They are evaluated for all ϱ_k in the gridded parameter space. Then Algorithm 4 summarizes the parameter-varying input blend calculation.

The output blend will maximize the information of the controlled subsystem to the single output, while it suppresses the effects of the undesired dynamics. Define the blended, parameter dependent input and direct feedthrough matrices as ¯B(ϱ) =B(ϱ)k_u(ϱ) and ¯D(ϱ) =D(ϱ)k_u(ϱ), respectively. The problem layout for the output blend calculation process is shown in Figure 5.2. The synthesis inequalities (similarly to (5.20) and (5.21)) with applied k_u(ϱ) input blend function are given as

Ac(ϱ) B¯c(ϱ)

I 0

T

Ξ

⋆ +

Cc(ϱ) D(ϱ)¯

0 I

T

Π

⋆

≺0, (5.26)

X_d(ϱ)A_d(ϱ) +A^T_d(ϱ)X_d(ϱ) +C_d^T(ϱ)K_y(ϱ)C_d(ϱ) + ˙X(ϱ) ⋆

B¯_d^T(ϱ)X_d(ϱ) + ¯D_d(ϱ)^TK_y(ϱ)C_d(ϱ) D¯^T_d(ϱ)K_y(ϱ)D_d(ϱ)k_u−γ²I

≺0, (5.27) where Π =

−K_y(ϱ) 0 0 β²I

. The next proposition summarizes the parameter-varying output blend calculation problem.

Algorithm 4 Parameter-varying input blend calculation

1: The subsystemsGc(ϱ) and Gd(ϱ) are given in the form as shown in Figure 5.1.

2: Find a K_u0(ϱ) initial solution for the optimization problem minimize

Xd(ϱ), Ku(ϱ), Xc(ϱ), Y, β², γ² −β²+γ² s.t.:(5.23), (5.24), X_d(ϱ)≻0, Y ≻0, Ku(ϱ) =Ku(ϱ)^T,0⪯Ku(ϱ)⪯I, ∀ϱ∈ F_P^V.

(5.25)

Note that the rank constraint is removed. The resulting β and γ are held fix during the following iterations. Set the counter variable to k= 1.

3: Alternating projection. Once reached, this point is iterated till convergence is achieved by a suitable selected error metric, such as the relative change in the solution. The previously obtained values of β and γ are kept constant during the iteration, which consists of two steps.

a: By an ASVD algorithm project K_u(ϱ) to an n_u−k dimensional subset with Lemma 20 to obtain K_u^⋆(ϱ).

b: Project the achieved reduced rank K_u^⋆(ϱ) to the LMI constraint set by the following optimization problem

minimize

X_d(ϱ), Ku(ϱ), Xc(ϱ), Y, S(ϱ)trace (S(ϱ)) s.t.:(5.23), (5.24), 0⪯K_u(ϱ)⪯I, Y ≻0,

S(ϱ) K_u(ϱ)−K_u^⋆(ϱ) K_u(ϱ)−K_u^⋆(ϱ) I

⪰0 for ∀ϱ∈ F_P^V.

4: Setk=k+ 1 and return to step 3, until rank 1 is achieved, then go to step 5.

5: By an ASVD algorithm projectK_u(ϱ) to an n_u−kdimensional subset with Lemma 20. The result isK_u^⋆(ϱ).

6: Calculate k_u(ϱ) as the parameter dependent singular vector, corresponding to the largest singular value in the ASVD of K_u^⋆(ϱ).

Proposition 10.: The parameter-varying output blend calculation. The optimal k_y(ϱ) output blend for the system shown in Figure 5.2 can be calculated as the left parameter dependent singular vector function corresponding to the largest singular value of theKy(ϱ) blend matrix, where K_y(ϱ) satisfies the following optimization problem

minimize

X_d(ϱ), Ky(ϱ), Xc(ϱ), Y, β², γ² −β²+γ² subject to (5.26), (5.27),

Y ≻0, X_d(ϱ)≻0,

K_y(ϱ) =K_y(ϱ)^T, 0⪯K_y(ϱ)⪯I, rank (K_y(ϱ)) = 1, ∀ϱ∈ F_P^V,

(5.28)

with I being the identity matrix with appropriate dimensions.

The calculation of the output blend vector function can be carried out in the same way as in Algorithm 4, hence it is omitted here. In the remaining of the chapter, the two parameter-varying approaches are evaluated, based on a simple example.

Gc(%)

Gd(%)

B¯_c(%) R A_c(%)

Cc(%) k_y(%)

+

+ ¯y_c

B¯_d(%) R A_d(%)

C_d(%) k_y(%)

+

+ ¯y_d

u¯

Figure 5.2: Problem layout for parameter-varying output blend calculation

In document Optimal Decoupling of Dynamical Systems (Pldal 90-95)