Abstract
It is well-known that in control theory the stability region of continuoustime system is laid in the left half plane of complex space, while that of discrete-time system is dwelled inside a unit circle. The former fact might be shown by exploiting the Laplace transform and the later by utilizing the corresponding zeta transform. In this paper we revealed the connectivity of both regions by employing Möbius transform. We also used the same transform to derive continuous/discrete-time counterpart of several existing results, including Bode integral and Poisson-Jensen formula. We then demonstrated their unification property by using delta transform. Some numerical examples were provided to verify our results.
INTRODUCTION
In control theory, some results for continuous-time and discrete-time such as regions of stability, expressions for minimum tracking error and energy regulation are derived independently. It is well-known that the stability region of continuous-time system is laid in the left hand side of imaginary axis in complex space, while that of discrete-time system is located inside a unit circle. In this paper, instead of using rigorous derivation we aim to utilize the strength of the Möbius transform to obtain the counterpart of existing results, which cover the Bode integral and Poisson-Jensen formula.
While discrete system is obtained from continuous-time system by sampling, expressions in continuous and discrete domains are not quite clear. This is because the underlying continuous domain descriptions cannot be obtained by setting sampling period to zero in the discrete domain approximations. In optimal tracking error control problem, for instance, contribution of nonminimum phase zeros for continuous and discrete-time systems are provided in completely different ways, see [3, 7] . The delta operator has often been proven entailing many advantages in connecting discrete-time and continuoustime systems, such as control synthesis [4] , control design [5] , estimation [9] and filtering [11] . We then provide the delta domain version of some existing results.
REGION OF STABILITY
The most commonly used definitions of stability are based on the magnitude of the system response in the steady state. A system is said to be (asymptotically) stable if its response to any initial conditions decays to zero asymptotically in the steady state, otherwise is said to be unstable. From the perspective of the forced response of the system for a bounded input, a system is said to be bounded-input and bounded-output (BIBO) stable if its response to any bounded input remains bounded. This section reviews the region of stability for continuous-time and discrete-time systems.
Continuous-time System
Consider a linear continuous-time system given by an initial value problem of n-th order differential equation:
where y(t) and u(t) respectively are the output and input of the system, a i (i = 0, . . . , n) and b j (j = 0, . . . , m) are real coefficients, and a n and b m are nonzeros. Note that in (1), y (i) denotes the i-th derivative of y with respect to time t. System (1) can be expressed in frequency domain, i.e., s-domain, by Laplace transform
where Y (s) := L{y(t)} and U (s) := L{u(t)} respectively denote the Laplace transform of y(t) and u(t). The system possesses the following transfer function
where H(s) represents the transfer function between input U (s) and output Y (s), i.e., Y (s) = H(s)U (s). If it is assumed that H has no zeros and poles in the same location, then we may write (2) as
where z j (j = 1, . . . , m) and p i (i = 1, . . . , n) are zeros and poles of H, respectively, and c is a real constant. As we assume that n ≥ m, (3) can be written in term of partial fraction decomposition as follows,
where
By considering an impulse input function, i.e., U (s) = 1 or equivalently u(t) = δ(t), then the time domain solution for (1) can be obtained by applying the inverse Laplace transform:
It can easily be seen from (5) that system (1) is (asymptotically) stable, i.e., lim t→∞ y(t) = 0, if and only if Re p i < 0 for all i = 1, . . . , n. In other words, the stability region of continuous-time system is laid in the left half plane of complex space.
Discrete-time System
Consider a linear discrete-time system represented by a single n-th order difference equation relating the output y to the input u:
for k = 0, 1, . . . and y(i) = u(j) = 0. Zeta transform provides the z-domain version of system (6), i.e.,
where Y (z) := Z{y(k)} and U (z) := Z{u(k)} respectively denote the zeta transform of y(k) and u(k). Thus we have the following transfer function
Note that if all poles of H(z) lie inside the unit circle {z : |z| < 1}, (7) can be rewritten as a power series
By applying the inverse zeta transform we have
Let |u(i)| ≤ M for all i and C = ∞ i=0 c i . Then,
for each k = 0, 1, . . .. Therefore the linear system (6) is stable. The proof of the converse can be found in [6] . As we completed the proof, it is shown that the stability region of a linear discrete-time system lies inside the unit circle.
MÖBIUS TRANSFORM
It has been revealed in the previous section that the region of stability of continuous-time system is the left half plane of complex space, while that of discrete-time system is the interior of a unit circle. In this section we will utilize the Möbius transform [2] to show the interconnection between these two regions.
In particular, we demonstrate that the region of stability of discrete domain can be asserted by Möbius transforming that of continuous domain. (10) is certainly a bijective mapping of the extended z-domain onto the extended s-domain. When a given point s is the image of some point z under transformation (10), the point z is retrieved by its inverse
Definition 1 (Möbius Transform). Transformation
We can verify that the inverse function (11) is also a bijective mapping by using the definition M −1 (∞) = ∞ for c = 0, and
for c = 0.
Region of Stability Mapping
In Section 2 the regions of stability for continuous-time and discrete-time systems were derived separately by employing the Laplace and zeta transforms, respectively. In this part we will show that the region of stability for discretetime system can be obtained by Möbius transforming that for continuous-time system. To facilitate our analysis, for C the complex space we define the following sets: C − := {s ∈ C : Re s < 0}, C + := {s ∈ C : Re s > 0}, D := {z ∈ C : |z| < 1}, D c := {z ∈ C : |z| ≥ 1} andD c := {z ∈ C : |z| > 1}. For the analysis we consider a special case of Möbius transform where a = c = d = 1 and b = −1, that is
Let s = α + jβ, j = √ −1, is an arbitrary point in C − , that is α < 0. Then we have 2α < −2α ⇔ (α + 1)
The above is nothing but the modulo of complex numbers
By applying (12) to above inequality we have
It is shown that s ∈ C − corresponds to z ∈ D, which reveals the interconnection between region of stability in s-domain and its counterpart in z-domain.
Other Continuous-Discrete Relationships
Now we further exploit the use of special Möbius transform (12) to unveil the counterpart of existing continuous-time or discrete-time results, which covers Bode integral and Poisson-Jensen formula.
Theorem 2 (Bode integral in s-domain). Let g(s)
be an analytic function in C + . Denote that g(jω) = g 1 (ω) + jg 2 (ω) and g(s) = g(s), i.e., g is conjugate symmetric. Then,
Proof. See [10, pp. 53 ].
The discrete-time counterpart of Bode integral is given as follows.
Theorem 3 (Bode integral in z-domain).
Let f (z) be an analytic function in D c . Denote that f (e jθ ) = f 1 (θ) + jf 2 (θ) and f (z) = f (z), i.e., f is conjugate symmetric. Then,
Proof. From (12) we have the following mappings: {s = 0} → {z = 1} and {s = jω} → {z = e jθ }, where ω = tan 1 2 θ. Thus we also have dω = ) we obtain g (0) = 2f (1). Equation (14) is claimed from (13) by substitution. 
Proof. The Poisson-Jensen formula can be found in many standard books on complex analysis. See for instance [1] .
The continuous-time counterpart of Poisson-Jensen formula is provided as follows.
Theorem 5 (Poisson-Jensen formula in s-domain). Let g is analytic in C + and c i (i = 1, . . . , n c ) be the zeros of g in C + , counting their multiplicities. If s ∈ C + and g(s) = 0, then
Proof. Perform transformation (12) over Theorem 4 to prove it.
DELTA TRANSFORM
A book that provides a comprehensive account on delta operator is [8] . The delta operator δ is define as the following forward difference
where q is the forward shift operator commonly used in discrete-time case and T > 0 is the sampling time. For any sequence x(k), k = 1, 2, . . ., delta operator gives
By taking the zeta transform of above equation we obtain
We may say that in z-plane the delta operator will translate a point z ∈ C one unit to the left and then scale it by factor of 1 T
. Later, the variable δ is used as the delta operator variable and is analogous to the Laplace variable s for continuous-time systems and the zeta transform variable z for discrete-time systems. We then obtain the relationship between variable z and variable δ as follows,
For any sequence x(k) we define its delta transform by
Now we ready to present the δ-domain counterparts of previous theorems. For time sampling T we define the following sets: D T := {δ ∈ C : |T δ+1| < 1}, D T . Denote that h(e (jωT −1)/T ) = h 1 (ωT ) + jh 2 (ωT ) and h(δ) = h(δ), i.e., h is conjugate symmetric. Then,
Proof. Consider the relationship h(δ) = f (T δ + 1), where f is defined in Theorem 3. Since f is analytic in D c , then h is analytic in D 
NUMERICAL EXAMPLE
In this section we provide a simple illustrative example to verify our result and to show the unification property between δ-domain and s-domain results.
Verification
To verify the result presented in Theorem 3 we consider a function f defined in z-domain as follows,
Since pole p lies inside the unit circle, indeed f is analytic in D c and it can be written as f (e jθ ) = f 1 (θ) + jf 2 (θ), where
Further, the LHS of (14) is given by
Whereas, for the RHS we have Figure 1 shows that numerical calculation of LHS in (21) and that of RHS in (22) for varying stable pole p are coinciding. 
from which we also have
In the δ-domain, from the relationship h(δ) = f (T δ + 1) we have
and thus
Therefore, the unification property of Theorems 2, 3 and 6 is unveiled by fact that
which show that the LHS of (18) converges to that of (13) as the sampling time T decreases.
CONCLUDING REMARK
It has been shown that Möbius transform can be utilized to derive counterpart result without involving any rigorous derivation. The delta transform, which takes a time sampling into account, can be used to show the unification property between continuous and discrete results. The approach describes in this paper can then be exploited to derive many more counterpart expressions.

{'paper_id': '55261874', 'title': 'MOBIUS AND DELTA TRANSFORMS IN THE UNIFICATION OF CONTINUOUS-DISCRETE SPACES', 's2_url': 'https://api.semanticscholar.org/CorpusID:55261874', 'source': 'S2ORC'}