Control Systems and Number Theory

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Introduction and Preliminaries
It turns out there is great similarity in control theory and number theory in their treatment of the signals in time domain t and frequency domain ω, s σ jω which is conducted by the Laplace transform in the case of control theory while, in the theory of zeta-functions, this role is played by the Mellin transform, both of which convert the signals in time domain to those in the right half-plane.For integral transforms, compare Section 11.
Section 5 introduces the Hardy space H p which consists of functions analytic in RHP-right half-plane σ > 0.

State Space Representation and the Visualization Principle
Let x x t ∈ R n , u u t ∈ R r , and y y t ∈ R m be the state function, input function, and output function, respectively.We write ẋ for d/dt x.The system of differential equations DEs The state state x is not visible while the input and output are so, and the state may be thought of as an interface between the past and the present information since it contains all the information contained in the system from the past.The x being invisible, 2.1 would read y Du, 2.2 which appears in many places in the literature in disguised form.All the subsequent systems, for example, 3.1 , are variations of 2.2 .And whenever we would like to obtain the state equation, we are to restore the state x to make a recourse to 2.1 , which we would call the visualization principle.In the case of feedback system, it is often the case that 2.2 is given in the form of 3.8 .It is quite remarkable that this controller S works for the matrix variable in the symplectic geometry compare Section 4 .
Using the matrix exponential function e At , the first equation in 2.1 can be solved in the same way as for the scalar case: called an autonomous system, is said to be asymptotically stable if for all initial values, x t approaches a limit as t → ∞.
Since the solution of 2.4 is given by x e At x 0 , 2.5 the system is asymptotically stable if and only if e At −→ 0 as t −→ ∞.

2.6
A linear system is said to be stable if 2.6 holds, which is the case if all the eigenvalues of A have negative real parts.Compare Section 5 in this regard.It also amounts to saying that the step response of the system approaches a limit as time elapses, where step response means a response y t t 0 e A t−τ u τ dτ, 2.7 with the unit step function u u t as the input function, which is 0 for t < 0 and 1 for t ≥ 0.

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Up here, the things are happening in the time domain.We now move to a frequency domain.For this purpose, we refer to the Laplace transform to be discussed in Section 11.It has the effect of shifting from the time domain to frequency domain and vice versa.For more details, see, for example, 1 .Taking the Laplace transform of 2. where where I indicates the identity matrix, which is sometimes denoted by I n to show its size.
In general, supposing that the initial values of all the signals in a system are 0, we call the ratio of output/input of the signal, the transfer function, and denote it by G s , Φ s , and so forth.We may suppose so because, if the system is in equilibrium, then we may take the values of parameters at that moment as standard and may suppose the initial values to be 0. Equation 2.10 is called the state space representation form, realization, description, characterization of the transfer function G s of the system 2.1 and is written as

2.11
According to the visualization principle above, we have the embedding principle.Given a state space representation of a transfer function G s , it is to be embedded in the state equation 2.1 .

Example 2.2. If
12 then it follows from 2.10 that

2.13
The principle above will establish the most important cascade connection concatenation rule 1, 2.13 , page 15 .Given two state space representations

2.15
Proof of 2.15 .We have the input/output relation 2.10

2.18
Eliminating u, we conclude that

2.19
Hence Indeed, we have 2.17 , and for 2.18 , we have

2.22
Hence for 2.20 , we have
As an example, combining 2.15 and 2.21 we deduce

2.24
Example 2.4.For 2.1 , we consider the inversion U s G −1 s Y s .Solving the second equality in 2.1 for u we obtain u −D −1 Cx D −1 y.

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Substituting this in the first equality in 2.1 , we obtain then we are to embed it in the linear system 2.30

Chain-Scattering Representation
Following 1, pages 7 and 67 , we first give the definition of a chain-scattering representation of a system.Suppose We must impose the nonconstant condition |Θ| / 0. Then Θ ∈ GL m r R .If S is obtained from S under the action of Θ , S Θ S , then its composition J with 3.14 yields JS ΦΦ ΘΘ S , that is, which is referred to as the cascade connection or the cascade structure of Θ and Θ .Thus the chain-scattering representation of a system allows us to treat the feedback connection as a cascade connection.
Suppose a closed-loop system is given with z a 1 ∈ R m , y a 2 ∈ R q , w b 1 ∈ R r , and u b 2 ∈ R p and Φ given by 3.2 .
H ∞ -Control Problem.Find a controller K such that the closed-loop system is internally stable and the transfer function Φ satisfies for a positive constant γ.For the meaning of the norm, compare Section 5.

Siegel Upper Space
Let * denote the conjugate transpose of a square matrix: S * t S, and let the imaginary part of S defined by Im S 1/2j S − S * .Let H n be the Siegel upper half-space consisting of all the matrices S recall 3.8 whose imaginary parts are positive definite Im S > 0-imaginary parts of all eigen values are positive and satisfies S t S: International Journal of Mathematics and Mathematical Sciences 9 and let Sp n, R denote the symplectic group of order n: The action of Sp n, R on H n is defined by 3.14 which we restate as SL n R and the theory of modular forms of one variable is well known.Siegel modular forms are a generalization of the one variable case into several variables.As in the case of the sushmna principle in 2 , there is a need to rotate the upper half-space into the right half-space RHS, which is a counter part of the right-half plane RHP.In the case of Siegel modular forms, the matrices are constant, while in control theory, they are analytic functions mostly rational functions analytic in RHP .A general theory would be useful for controlling theory.See Section 7 for physically realizable cases.There are many research problems lying in this direction.

Norm of the Function Spaces
The norm x x 1 . . .
x n ∈ C n is defined to be the Euclidean norm or by the sup norm or anything that satisfies the axioms of the norm.They introduce the same topology on C n .The definition of the norm of a matrix should be given in a similar way by viewing its elements as an n 2 -dimensional vector, that is, embedding it in or otherwise.The sup norm is a limit of the p-norm as p → ∞.For a a 1 , . . ., a n , lim

5.4
Suppose For p > 1, the Bernoulli inequality gives Hence the right-hand side of 5.5 tends to |a 1 |.The proof of 5.4 can be readily generalized to give The p-norm in 5.6 is defined by where f t is any Euclidean norm.Note that the functions are not ordinary functions but classes of functions which are regarded as the same if they differ only at measure 0 set.L p is a Banach space i.e., a complete metric space , and in particular L 2 is a Hilbert space.The 2-norm • 2 is induced from the inner product where * refers to the transposed complex conjugation.The Parseval identity holds true if and only if the system is complete.However, the restriction that f t → 0 as t → ∞ excludes signals of infinite duration such as unit step signals or periodic ones from L p .To circumvent the inconvenience, the notion of averaged norm, or similar, is important and the power norm has been introduced:

5.9
Remark 5.1.In mathematics and in particular in analytic number theory, studying the mean square in the form of a sum or an integral is quite common.Especially, this idea is applied to finding out the true order of magnitude of the error term on average.Such an average result will give a hint on the order of the error term itself.
Example 5.2.Let ζ s denote the Riemann zeta-function defined for σ > 1 s σ it , in the first instance, where it is analytic and then continued meromorphically over the whole complex plane with a simple pole at s 1.It is essential that it does not vanish on the line σ 1 for the prime number theorem PNT to hold.The plausible best bound for the error term for the PNT is equivalent to the celebrated Riemann hypothesis RH to the effect that the Riemann zeta-function does not vanish on the critical line σ 1/2.Since the values on the critical line are expected to be small, the averaged norm would imply the weak Lindel öf hypothesis LH in the form for every ε > 0. It is apparent that the RH implies the LH.
The Hardy space H p cf. e.g., 1, page 39 is well known.It consists of all f s which are analytic in RHP-right half-plane σ > 0 such that f jω ∈ L p , in particular, H ∞ with sup norm.Thus H ∞ -control problem is about those rational functions which are analytic in RHP, a fortiori stable, with regard to the sup norm.Thus the above-mentioned meanvalue problem for the Riemann zeta-function is related to the H 2k -control problem with finite Dirichlet series main ingredients in the approximate functional equation .Since the H ∞control problem asks for all individual values, it flows afar from the H 2k -control problem and goes up to the LH or the RH.

(Unity) Feedback System
The synthesis problem of a controller of the unity feedback system, depicted in Figure 1, refers to the sensitivity reduction problem, which asks for the estimation of the sensitivity function S S s multiplied by an appropriate frequency weighting function W W s : is a transfer function from r to e, where C K is a compensator and P is a plant.The problem consists in reducing the magnitude of S over a specified frequency range Ω, which amounts to finding a compensator C stabilizing the closed-loop system such that for a positive constant γ.
To accommodate this in the H ∞ control problem 3.

J-Lossless Factorization and Dualization
In this section we mostly follow Helton 4-6 who uses the unit ball in place of RHP.They shift to each other under the complex exponential map.For conventional control theory, the unit ball is to be replaced by the critical line σ 0 .In practice what appears is the algebra of functions 5, page 2 , Table 1 R functions defined on the unit ball having the rational continuation to the whole space , 7.1 or still larger algebra ψ consisting of those functions which have pseudo meromorphic continuations 5, footnote 6, page 27 .The occurrence of the gamma function 5,

FOPID
"FO" means "Fractional order" and "PID" refers to "Proportional, Integral, Differential," whence "Proportional" means just constant times the input function e t , "Integral" means the fractional order integration I λ t D −λ t of e t λ > 0 , and "Differential" the fractional order differentiation D δ t of e t δ > 0 .The FO PI λ D δ controller control signal in the time domain is one of the most refined feed-forward compensators defined by The derivation of 8.3 from 8.1 depends on the following.The general fractional calculus operator a D α t is symbolically stated as where a and t are the lower and upper limits of integration and α is the order of calculus.More precisely, the definition of the fractional differintegral is given by the Riemann-Liouville expression where {α} α − α indicates the fractional part of α, with α the integral part of α.Thus we are also led to the Riemann-Liouville fractional integral transform: For applications, compare Section 13.When α ∈ N, 8.5 reads the αth derivative of f.We will see that the definition 8.5 is a natural outcome of the general formula for the difference operator of order α ∈ N with difference y ≥ 0:

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If f has the α-th derivative f α , then 8.9 The special case of 8.9 with t ν a, a y → x ϕ t f α t reads whose far-right hand side is RL ϕ .Let F s be the Laplace transform of the input function f t .Then 8.11

Fourier, Mellin, and (Two-Sided) Laplace Transforms
We state the Mellin, two-sided Laplace, and the Fourier transforms.If Under the change of variable x e −t , the Mellin transform and the two-sided Laplace transform shift each other: where we write ϕ t f e −t .The ordinary Laplace transform one-sided Laplace transform is obtained by multiplying the integrand by the unit step function u u t cf. the passage immediately after 2.7 : the Fourier transform of g.
We explain Plancherel's theorem for functions in L 2 R .Let where lim is a short-hand for "limit in the mean."The Parseval identity reads If we apply 9.7 to a causal function f, then it leads to 1, 3.19 Hence we see that 1, 3.19 is indeed the Parseval identity for the Fourier or Plancherel transform for f ∈ L 2 R .

Electrical Circuits
The electric current i i t flowing an electrical circuit which consists of four ingredients, electromotive-force e e t , resistance R, coil L, and condenser C, satisfies 10.1

Newton's Equation of Motion (cf. [7])
One has where M is the inertance of mass, R is the viscous resistance of the dashpot, and K is the spring stiffness.
Introducing the new parameters

Partial Fraction Expansion and Examples
As long as the input function is a sinusoidal function, Example 11.2 will suffice to compute its Laplace transform.To go back to the time domain from the frequency domain, we need to solve the DE and, for most purposes, the following partial fraction expansion will give the answer almost automatically.
The following theorem, which is well known, provides us with the partial fraction expansion.
Theorem 12.1.If the denominator C z of the rational function S z P z /C z is given by where where the coefficients are given by 12.3 Proof.By 12.1 , for each i, 1 ≤ i ≤ q, we may write International Journal of Mathematics and Mathematical Sciences and S i z has no pole at z β i .We write where H i z ∈ C z has no pole at z β i .By successively differentiating and setting z β i , we obtain 12.3 .Now, the rational function has no pole, so that it must be a polynomial.But, since lim z → ∞ F z 0 where we use the assumption deg P < deg C , it follows that F z must be zero.Now we will give examples of 2.2 for the second-order systems which do not appear anywhere else save for 2 .
Example 12.2.We find the output signal current y y t described by the DE y y y e − 1/2 t sin where the initial values are assumed to be 0: y 0 0, y 0 0.
Proof.Let Y s L y s be the Laplace transform of y t .Then we have 12.8 and we may obtain the partial fraction expansion 1 where ρ e 2πi/3 −1 √ 3i /2 is the first primitive cube root of 1. Hence where the initial values are assumed to be 0: y 0 0, y 0 0.
We have

The Product of Zeta-Functions: ΓΓ-Type
In this section, we illustrate the use of fractional integrals by proving a slight generalization of the result of Chandrasekharan and Narasimhan 8 involving the ΓΓ-type functional International Journal of Mathematics and Mathematical Sciences equation, which is the first instance beyond Hecke theory of the functional equation with a single gamma factor.First we state the basic settings.

Statement of the Situation
Let {λ k }, {μ k } be increasing sequences of positive numbers tending to ∞, and let {α k }, {β k } be complex sequences.We form the Dirichlet series and suppose that they have finite abscissas of absolute convergence σ ϕ , σ ψ , respectively.We suppose the existence of the meromorphic function χ satisfying the functional equation of ΓΓ-type of the form with r a real number and having a finite number of poles We introduce the processing gamma factor and suppose that for any real numbers In the w-plane we take two deformed Bromwich paths

H-1
In the special case, where A j B j 1, the H-function reduces to G-functions and denoted by G with other parameters remaining the same.We also define the χ-function X z, s by which is for χ 1 one of H-functions.Hereafter we always assume that z > 0, which may be extended to Re z > 0.

13.27
where S stands for the Lommel function.
Equation 13.26 is 12, 63 , page 194 and 13.27 is 12, page 196 .We only need 13.27 and 13.27 is for treating the J-Bessel function.
Arguing in the same way as in 8 , we may prove the following.

2 International
Journal of Mathematics and Mathematical Sciences ẋ Ax Bu, y Cx Du 2.1 is called a state equation for a linear system, where A ∈ M n,n R , B, C, D are given constant matrices.

Definition 2 . 1 .
A linear system with the input u o

Figure 2 .
We have E R − Y , so that Y PR PK R − Y .Solving in Y , we deduce that I PK −1 PKR.We take into account the disturbance d, and we obtain since U CE C R − Y Y PU PD PC R − Y PD, 6.5 whence Y PC R − Y PD.It follows that Y I PC −1 PCR I PC −1 PD.In the case where d 0, PC being the open-loop transfer function, we have SR is the tracking error for the input R. Hence 6.1 holds true.

γ 1 13. 5 such 1 Γ 1 Γj m 1 13. 7 lie
that they squeeze a compact set S with boundary C for which s k ∈ S 1 ≤ k ≤ L and all the poles of Γ b j − B j s B j w m j a j − A j s A j w p j n 1 Γ b j B j s − B j w q j m 1 13.6 lie to the left of L 2 s and those of Γ a j A j s − A j w n j a j − A j s A j w p j n 1 Γ b j B j s − B j w q to the right of L 1 s .Then we define the H-function by 0

Suppose that a 2 is fed back to b 2 by b 2 Sa 2 ,
and b 2 ∈ R p are related by S , where in the last equality we mean the action of Θ on the variable S.
With action, we may introduce the orbit decomposition of H n and whence the fundamental domain.We note that, in the special case of n 1, we have H 1 H and Sp 1, R 4.1.For a controller S living in the Siegel upper space, its rotation Z −jS lies in the right half-space RHS, that is, stable having positive real parts.For the controller Z, the feedback connection −jb 2 Z −ja 2 4.4 is accommodated in the cascade connection of the chain-scattering representation Θ 3.15 , which is then viewed as the action 3.15 of Θ ∈ Sp n, R on S ∈ H n : ΘΘ S Θ Θ S ; or HM Θ; HM Θ ; S HM ΘΘ ; S , 4.5 where Θ is subject to the condition t Θ UΘ U, 4.6 with U O I n −I n O .An FOPID controller (in Section 6), being a unity feedback connection, is also accommodated in this framework.Remark 4.2.
1 , we choose the matrix elements P ij of P in such a way that the closed-loop transfer function Φ in 3.11 coincides with WS.First we are to choose P 22 −P .Then we would choose P 12 P 21 WP .Then Φ becomes P 11 WP C I PC −1 P 11 − W W I PC −1 .Hence choosing P 11 W, we have Φ WS.
Example 6.1.First we treat the case of general feedback scheme.Denoting the Laplace transforms by the corresponding capital letters, we haveY PR PU, U KE, 6.4whence Y PR PKE.Now if it so happens that e r − y and P is replaced by PK, that is, in the case of unity FD, we derive 6.1 directly from

Table 1 :
Correspondence between control systems and zeta-functions.
Then the only mapping Θ ∈ RU m, n acting on BH ∞ must satisfy the J-lossless property.Let Θ denote an m n × m n matrix.ThenΘ * J mn Θ ≤ J mn , 7.3which is interpreted to be the power preservation of the system in the chain-scattering representation 3.6 1, page 82 .We now briefly refer to the dual chain-scattering representation of the plant P in 3.2 .
is the input function, e is the deviation, and K p , K i , K d are constant parameters which are to be specified K p : the position feedback gain, K d : the velocity feedback gain .DE 8.1 translates into the state equation To solve 10.1 , we use the Laplace transform which has been defined by 9.3 and we state its definition independently.The integral converges absolutely in Re s > a and represents an analytic function there.Re α in the first instance.The right-hand side of 11.2 gives a meromorphic continuation of the left-hand side to the punctured domain C \ {α}.Furthermore, 11.2 with α replaced by iα reads By definition 11.2 clearly holds true.Since the right-hand side is analytic in C \ {α}, the consistency theorem establishes the last assertion.Once 11.2 is established, we have Definition 11.1.Suppose y t O e at , t → ∞ for an a ∈ R. The Laplace transform Y s L y s of y y t is defined by In the same vein as with Example 12.2, we may find the solution of the DE