]> 2.9 Discussion of results

2.9 Discussion of results

The direct spectral approach to the semigroup generation problem is effective if it is possible to establish whether a system of eigenfuctions of the linear closed–loop operator forms a Riesz basis in the state space. To get some idea about the applicability of different Riesz–basis criteria, we consider a generalization of (2.1) with a finite–dimensional system considered instead of the proportional controller:
v̇(t) =Fv(t) + x(1,t)g,t 0 x(θ,t) t =2x(θ,t) θ2 , t 0,0 θ 1 x(θ,t) θ θ=1 =0, t 0 hT v + qx(1,t) =x(0,t), t 0 (2.56)

where F L(n), g, h n, and q , with appropriate initial conditions.

The space H = n L2(0, 1) and the operator 𝒜 given by

𝒜v x = Fv + x(1)g x D(𝒜) = v x H : x H2(0, 1),x(1) = 0,hv + qx(1) = x(0) ,

are naturally connected with the system (2.56). The system (2.56) is a particular case of a class of hybrid feedback system depicted in Figure 2.7


Figure 2.7: The hybrid feedback control system


x =x, D() ={x H2(0, 1) : x(1) = 0} Γ1x =x(1), Γ2x =x(0) .

An eigenvalue problem for 𝒜 takes the form

(λI F)v + x(1)g =0 x =λx hv + qx(1) =x(0) x(1) =0 ,v n,x H2(0, 1) .

Premultiplying the first equation by h adj(λI F) and the second equation by det(λI F), and adding the resulting equations, we come to "the reduced eigenproblem":

x =λx x(1) =0 x(1)[h adj(λI F)g + q det(λI F)] =x(0) det(λI F) (2.57)

The polynomials of the spectral parameter λ, appearing in the boundary conditions, are easily identified as the numerator and denominator of the transfer function of the finite–dimensional part. Similar equations can be derived for the conventional PID controller.

Mikhaylov [65] and Kesel’man [54], referring the results due to J.Schwartz, have obtained a Riesz–basis criterion for such operators in the case of boundary conditions not containing the spectral parameter. Their criterion reduces to checking the so–called strict regularity of boundary conditions. In what follows we present briefly this theory.

In the Hilbert space L2(0, 1) equipped in the scalar product we consider a differential operator with coefficients pi Wni,1(0, 1), i = 1, 2,,n and the normalized boundary conditions,

(𝒜x)(θ) =x(n)(θ) + p 1(θ)x(n1)(θ) + + p n(θ)x(θ) D(𝒜) = x Hn(0, 1) : j=0kν ανjx(j)(0) + β νjx(j)(1) = 0 (2.58)

where n 1 k1 k2 kn 0, kν+2 < kν, ανkν + βνkν 0, ν = 1, 2,,n.

Definition 2.9.1. Let

ωk := exp (2k 1)πj n ,k (2.59)

and assume αν = ανkν, βν = βvkν, ν = 1, 2,,n for simplicity of notation. The boundary conditions in (2.58) are called regular if

n = 2μ 1,Θ00,Θ10

where Θ0, Θ1 are defined by the identity

det α1ω1k1α 1ωμ1k1 (α 1 + β1s)ωμk1 β 1ωμ+1k1β 1ωnk1 α2ω1k2α 2ωμ1k2 (α 2 + β2s)ωμk2 β 2ωμ+1k2β 2ωnk2 αnω1knα nωμ1kn(α n + βns)ωμknβ nωμ+1knβ nωnkn = Θ0+Θ1s (2.60)


n = 2μ,Θ12 + Θ 120

with Θ1, Θ0, Θ1 defined now by the identity

det α1ω1k1 α 1ωμ1k1 (α 1 + β1s)ωμk1 α1s + β1 s ωμ+1k1 β 1ωμ+2k1 β 1ωnk1 α2ω1k2 α 2ωμ1k2 (α 2 + β2s)ωμk2 α2s + β2 s ωμ+1k2 β 2ωμ+2k2 β 2ωnk2 αnω1kn α nωμ1kn(α n + βns)ωμknαns + βn s ωμ+1knβ nωμ+2kn β nωnkn = Θ1 s + Θ0 + Θ1s (2.61)

The boundary conditions in (2.58) are strictly regular if they are regular and additionally

Θ024Θ 1Θ1ifn = 2μ (2.62)

Theorem 2.9.1 (Schwartz–Mikhaylov–Kesel’man). Let 𝒜 be the operator of the form (2.58) with the strictly regular boundary conditions. Then only the finitely many of eigenvalues are nonsimple, and there exists a system of generalized eigenvectors which forms a Riesz basis in L2(0, 1).

Corollary 2.9.1. Consider the differential operator

𝒜x = x,D(𝒜) = x H2(0, 1) : a1b1a0b0 c1d1c0d0 x(0) x(1) x(0) x(1) = 0 0 (2.63)

Denote by Aik the determinant of a matrix constructed of the i–th and k–th columns of the matrix of boundary conditions, i,k = 1, 2, 3, 4. The operator (2.63) has the form (2.58) with n = 2 and μ = 1 in the following three cases:

k1 = k2 = 1. Then: α1 = a1, β1 = b1, a1 + b1 0, α2 = c1, β2 = d1, c1 + d1 0.
k1 = 1, k2 = 0. Then: α1 = a1, β1 = b1, a1 + b1 0, α2 = c0, β2 = d0, c0 + d0 0 and c1 = d1 = 0.
k1 = k2 = 0. Then: α1 = a0, β1 = b0, a0 + b0 0, α2 = c0, β2 = d0, c0 + d0 0 and a1 = b1 = c1 = d1 = 0.

By Definition 2.9.1 in the case 1 the boundary conditions in (2.63) are regular or equivalently strictly regular if

A120 (2.64)

In the case 2 the boundary conditions are regular if

c1 = d1 = 0,A14 + A230 (2.65)

and strictly regular if additionally

A13 + A24 ± (A14 + A23) (2.66)

In the case 3 the boundary condition are regular or equivalently strictly regular if

a1 = b1 = c1 = d1 = 0,A340 (2.67)

Moreover, (2.65) can be expressed in terms of Aik as

A12 = 0,A14 + A230 (2.68)

while (2.67) can be expressed as

A12 = A13 = A14 = A23 = A24 = 0,A340

or equivalently in the form

A12 = 0,A14 + A23 = 0,A340,A13 + A24 = 0,A13 = A24 (2.69)

Exercise 2.9.1. Derive Corollary 2.9.1 from Theorem 2.9.1.

The above theory applies to the hybrid feedback systems with a plant describable by an n–th order ordinary differential operator with a spatial domain equal to the interval [0, 1]. The control and observation should act at the ends of the interval [0, 1], i.e., they are of the boundary type. Finally, the feedback controller must be purely proportional (the case of a purely static control without any dynamics).

Exercise 2.9.2. Derive Proposition 2.5.1 from Corollary 2.9.1 proving that the boundary conditions in (2.4) are regular for any K and strictly regular iff K1.

Remark 2.9.1. For the operator

𝒜x = x,D(𝒜) = {x H2(0, 1) : x(0) x(1) + x(1) = 0,x(0) = 0}

we have: n = 2, k1 = 1, α1 = 1, β1 = 1 (hence α1 + β1 0), k2 = 0, α2 = 1, β2 = 0 (hence α2 + β2 0). From (2.59) we get ω1 = j, ω2 = j and by (2.61)

Θ1 s +Θ0+Θ1s = det (1 s)j 1 1 sj 1 1 .

This yields Θ0 = 2j, Θ1 = Θ1 = j, Θ12 + Θ 120 and Θ02 4Θ 1Θ1 = 0. Therefore (2.62) is not satisfied and the boundary conditions are regular but not strictly regular. The spectrum σ(A) of the operator A consists of two series of eigenvalues. The first one is {4k2π2} k, while the second one tends asymptotically as k to the first one. There is no system of eigenvectors which forms a basis because the angle between eigenspaces corresponding to the two series of eigenvalues, tends to 0 as k [54], [85]. This example shows that (2.62) is essential for the validity of the assertion of Theorem 2.9.1. Any system of eigenvectors does not generally constitutes a Riesz basis, however, a Riesz basis of subspaces can be constructed from an appropriately chosen sequence of subspaces [77], [5].

Shkalikov [78] gives an extensive discussion of the Riesz–basis problem of a system of eigenvectors or subspaces in the case of a spectral parameter λ polynomially entering the boundary conditions. Some criteria are established for basisness in various spaces, particularly related to (2.57). The results directly concern the operator (2.57) or its realizations in those spaces. Application of Shkalikov’s theory requires, however, a separate detail presentation [40414243].

If the series (2.28) admits an explicit summation, then it is possible to obtain a compact formula expressing the performance index in terms of the system parameters. From the example discussed, one may conclude that an exact knowledge of the spectrum simplifies these attempts. In the opposite case, the series (2.28) should be approximated by its truncations, with eigenvalues calculated numerically.

If L2(0, 1) is the state space, and the assumptions of Theorem 2.4.1 hold, then the information about the general form of a solution to the Lyapunov equation is sufficient to characterize the kernel of this solution by an integral equation. For computational purposes, this equation can be converted, with the aid of the inflation technique, into a matrix Lyapunov equation. Further investigations should be focused on a maximal elimination of the need of exact knowledge of a spectrum.