Robust Stability in Discrete Control Systems via Linear Controllers with Single and Delayed Time

In this work, a discrete feedback of single and a delayed time is introduced in a LTI control discrete system, yielding a monoparametric family of LTI systems. A polynomial approach technique to compute the maximal robust stability interval of the monoparametric system with single and delayed time controller is developed by using the zero exclusion principle and the boundary crossing theorem. Illustrative examples are given to show the technique.


Introduction
Discrete LTI control systems have been object of study and design in the last years due to the great quantity of applications as in digital devices as well in mechanical plants and others.Particularly, the discrete-time control systems of interest are the so-called SISO (single input-single output), which are LTI (linear time invariant [1][2][3][4]).It is well known that the internal evolution of the system is determined by its poles (also referred to as the eigenvalues of its associated matrix in state space representation) which can be computed by finding the roots (zeroes or solutions) of its characteristic polynomial.One of the most worked problems concerning the dynamics of a system is the stabilization problem.Namely, a LTI discrete system is BIBO stable (bounded input-bounded output) if and only if its transfer function has no poles with modulus greater than or equal to one.That is, the poles of the system must lie inside the open unit complex disk.This class of stability for discrete systems is also called Schur stability ( [5][6][7][8]).In this sense, a linear discrete-time invariant system is said to be stable if and only if its characteristic polynomial is a Schur polynomial.A polynomial is said to be stable if its roots lie in the stability zone (in our case of study, the stability zone is the open unit complex disk for Schur stability).There exists a lot of criteria to determinate the Schur stability of a polynomial, as the Schur-Cohn and Jury's criterion [9][10][11][12].In [13,14], modified Schur-Cohn and Jury's criterion are given.With respect to polynomial families in topological sense, sufficient conditions to determine an interval of Schur polynomials are given in [15][16][17]; techniques to locate the roots inside the unit circle of a polytope of polynomials are developed; moreover, a robust Hurwitz stability criterion for polynomial intervals is proposed in [18] based on Kharitonov rectangles and in [19] set values are given for polynomials Schur stable, while a method to study the Schur stability of a segment of polynomials was reported recently in [20].Concerning rays of polynomials, the case of finding the maximal Hurwitz stability (roots with negative real part) interval was solved by Bialas in 1985 [21].An improved technique to determinate the Hurwitz stability of the ray was given in [8], and these results are used in [22] in order to generate chaos.Good references where the stability of families of polynomials can be consulted are in the books [5][6][7].
A related problem is the inclusion of discrete controllers with single delays; for instance, in [23] a technique to determinate an interval of Schur stability for sample-data systems with a delayed control is given and in [24] Schur stability in linear discrete-time systems with single-delay using the method of Lyapunov functions is studied.In the following work, we shall provide a technique that ensure the Schur stability of a ray of polynomials, which is the characteristic family of polynomials of a linear discrete-time system in closed-loop with a feedback linear control in both single and delayed time.
The rest of the paper is organized as follows: In Section 3, the description of the system is given as well as the problems statement.In Section 4, preliminary tool to solve the proposed problems is introduced.The main results are provided in Section 5 which is divided into three subsections: In Section 5.1, generalizations of the boundary theorems in its discrete version are shown; in Section 5.2, the obtained generalizations are applied on a technique to find the maximal robust Schur stability interval of the uncertainty present on a polynomial ray, and, therefore, robust stability of a discrete LTI system in closed-loop with a linear controller.An illustrative example that shows the technique is also given.Finally, in Section 5.3 the technique is extended to discrete control systems with a controller in single time-delay and applied to an example.

Problems Statement
Consider the discrete LTI control system where   ∈ R  is the states variable vector,  ∈ R × and  ∈ R  are given in controllable canonical form (also called Jerk realization) as and   ∈ R is the linear actuator of the system.The openloop characteristic polynomial is   () =   +  1  −1 + ⋅ ⋅ ⋅ +  −1  +   , where  is the complex -transform variable.
If we suppose that   () is a Schur polynomial, then we can to perturb the parameter  of the controller (for both of the Cases 1 and 2) to keep stability.Then, the objective is to compute the maximal perturbation of , that is, the minimum  < 0 and the maximum  > 0, to obtain  − min and  + max , respectively, such that system (1) has poles inside the unit complex disk D for all  in the maximal stability interval  = ( − min ,  + max ).In other words, the system is robustly Schur stable in the maximal stability interval .Any other subset [, ] ⊂  is just an interval of robust stability.The technique will be based on the boundary theorems: the zero exclusion principle (ZEP) and the boundary crossing theorem (BCT).

Preliminaries
4.1.The Boundary Theorems Generalized.Let us consider the complex plane C and let S ⊂ C be any given open set.We know that S, its boundary S, and the complement U = C − S form a disjoint partition of the complex plane.In the following we give the tool used in [8] to find the maximal Hurwitz stability interval for a ray of real polynomials.The discrete versions are given in the following section of main results.
The classical boundary theorems consider the following hypothesis.
Mathematical Problems in Engineering 3 Hypothesis 1.Consider the polynomial family (, ) has at least one root in R.
Similarly, under the same hypothesis and with the same stability zone S = C − , the generalization of the zero exclusion principle is given as follows.
Theorem 2 (generalization of ZEP).Consider the polynomial family (, ) under Hypothesis 1. Suppose that for some  ∈ (, ) the family has  1 roots in C − and  −  1 roots in C + .Then the entire family keeps  1 roots in C − and  −  1 roots in C + if and only if An important result that is used to prove the previous generalized theorems is the continuous dependence of roots (CDR), which establishes the preservation of continuity between roots and coefficients of a polynomial family.

Main Results
We have divided this section in three parts.In the Section 5.1 we present adaptation of the before generalizations theorems to discrete case.In Section 5.2, boundary theorems based technique for calculating the maximal Schur stability interval for a discrete family of systems using linear feedback with single time is developed.And in Section 5.3 the technique is extended to a single time-delay controller.

The Discrete Generalized Boundary Theorems.
It is not too hard to see that the generalizations of the boundary theorems can be easily adaptable in the case of S = D instead of C − .However, it is necessary to present the discrete version of these generalizations in order to apply them in the aforementioned technique.
Remark 4. The polynomials to consider come from a real discrete control system whose characteristic polynomial comes from the -transform applied to the discrete system.Thus, all of the polynomials to deal with in the coming development have real coefficients and complex variable  instead of .Proof.Since (, ) satisfies Hypothesis 1, then by Lemma 3 there exist  continuous function roots of (, ), say  1 (), . . .,   () ∈ C. Without loss of generality we can suppose that, for  = 1, . . .,  1 , we have that   () ∈ D and, for  =  1 + 1, . . ., , we have that   () ∈ C − D. Since  1 ̸ =  1 , by continuity at least one root traveled through the complex plane collapsing into D for some  ∈ (, ], resulting in the items claimed. Similarly, the generalized zero exclusion principle can be adapted to the discrete case in the following way.=  1 .Then, from Theorem 5, there exists a  ∈ (, ] such that (  0 , ) = 0 for some 0 ≤  0 ≤ 2, which is a contradiction.

The Roots Criterion for Discrete-Time Linear Systems.
Next, we shall develop the technique to compute the maximal Schur stability interval.Let   () =   +  1  −1 + ⋅ ⋅ ⋅ +   be an -fixed degree polynomial and   () =  1   + ⋅ ⋅ ⋅ +    +  +1 be a polynomial such that  > .We desire to find the values of  around  = 0 such that the ray (, ) =   () +   () has  roots inside D. Since   () is a Schur polynomial, then   ( −1 ) has  roots outside D. Thus, the discrete generalized zero exclusion principle will be applied to the family P(, ) =   ( −1 )(, ), which has  roots inside D and  roots outside D for  = 0.The idea is to extend (, ) to the maximum interval for  where the zero exclusion principle is satisfied and, therefore, we obtain robust stability.Firstly, let us carry P(, )| =  in a suitable form.
By evaluating   () and   () in  =   , 0 ≤  ≤ 2 we get where Thus, the family (, )| =  can be written as Now, since   ( −1 )| =  =   ( − ) = ()−(), the family P(, ) evaluated in the unit circle becomes on and this is the polynomial family form to be applied the zero exclusion principle in order to compute the maximal stability interval.
(iii)  is an even function.
Proof.Before anything, it is not hard to see that and consequently,  is an odd function.
Finally, proof of (iii): and, therefore,  is an even function.
Item (ii) shows that  = 0,  are always zeroes of  and if  0 ∈ () then its conjugate − 0 is also a zero of .In the incoming development it will be shown a condition where (0) ̸ = 0 in item (iii).Let () = { ∈ C | () = 0} denote the set of zeroes for a function ().In our case of studio we have the variable 0 ≤  ≤ 2; thus the set () is the set of zeroes of  modulo 2.With the functions (), (), and () defined above, let us define the sets If there are no elements in () such that (  ) > 0, then we will define  + = {0 + }.Similarly, if there are no elements in () such that (  ) < 0, then we will define  − = {0 − }.
That is, all of its poles lie inside the unit circle for all  ∈ (−0.1875, 0.625).Recall that the characteristic equation   () is Schur stable.In Figure 1, one can see the Nyquist diagram intersecting the point of -1+0i for the extreme values of the maximal stability interval obtaining instability in the system, while, in Figure 2, stability for the system arises for values of the gain within the interval of robustness; moreover, in Figure 3 the point -1+0i is encircled by the Nyquist diagram for a value not in the computed interval.Finally, robust stability is depicted by the set values in Figure 4, since origin is excluded and there is a Schur stable polynomial member of the family (, ) for  = 0.

The Roots Criterion for Discrete Systems with Single Time-Delay Control.
Consider the discrete-time system with a single time-delay  with ,  ∈ R × , whose stability is in charge of the roots of its characteristic equation given by () = det( +1 −  −) = 0 [24].
Based on the aforementioned reference, we shall consider the control system where for which the linear delayed control  − = −   − , with   = (  , . . .,  1 ), endows the system (27) with a single timedelay as required in system (26).
In the same essence as the technique in single time case, by evaluating in the unit complex circle we get where , , , and  are the functions given in (7).The product to consider in this case is P  (, ) = p ( −1 )  (, ).Thus, P  (, ) in  =   can be written as , , and  are the functions defined in (10).Note that if  = 0, we recover the single time case from the previous section.Write then the family P  (  , ) in the form for () defined in (10).Analogous to the sets  − and  + from ( 14), let us define the sets Thence, the following result is held.

Conclusions
In this work, a linear feedback control has been designed to obtain robust stability in discrete-time control systems and although the stability regions of both analogue and discrete systems are different, we were able to adapt a technique which is given for analogue systems to find the maximal Schur stability interval for a ray of polynomials emerging from the closed-loop discrete control system.A pair of examples have been developed to illustrate the functionality of the technique.Finally, the obtained technique has been analogously used for discrete control systems with a single time-delay controller, which comes to be an easy algebraic stability test.

N
: set of natural numbers R: set of real numbers R  : set of −tuples of real numbers R × : set of real square matrices of size  ×  C: set of complex numbers C − : set of complex numbers with negative real part C + : set of complex numbers with positive real part R: imaginary pure complex numbers-complex axis D: open unit complex disk-complex numbers with modulus less than 1 : boundary of the set  : closure of the set   − : difference between sets-set  without elements in common with  (): set of zeroes (roots) of the function (polynomial)  det(): determinant of a square matrix .

Theorem 5 (
discrete BCT generalized).Consider the polynomial family (, ) on the complex variable  under Hypothesis 1. Suppose that (, ) has  1 roots in D and  −  1 roots in C − D; and (, ) has  1 roots in D and  −  1 roots in C − D.Then, if  1 ̸ =  1 there exist at least one  in (, ] such that (i) (, ) has at least  1 roots in D, (ii) (, ) has at least  −  1 roots in C − D, (iii) (, ) has at least one root in D.

Theorem 6 (
generalization of ZEP).Consider the polynomial family (, ) under Hypothesis 1. Suppose that for some  ∈ (, ) the family has  1 roots in D and  −  1 roots in C − D.Then, the entire family keeps  1 roots in D and  −  1 roots in C − D if and only if  (  , ) ̸ = 0 ∀ ∈ [, ] and ∀ ∈ [0, 2] .(5) Proof.(⇒) If all of the elements of the family have  1 roots in D and  −  1 roots in C − D, then it is clear that there are no roots in the unit circumference D.Therefore, (  , ) ̸ = 0, for all 0 ≤  ≤ 2 and for all  ∈ [, ].(⇐) Suppose that (  , ) ̸ = 0 for all 0 ≤  ≤ 2 and for all  ∈ [, ].Let us proceed by contradiction.Suppose that there is  0 ∈ (, ] such that the polynomial (,  0 ) has  1 roots in D and − 1 roots in C − D with  1 ̸

Figure 1 :
Figure 1: Nyquist diagram of the closed-loop system for  − min (left image) and  + max (right image) extremes of the maximal stability interval.
discrete system (38), with the delayed time feedback −2 (, ) = − [1.5 1 ( − 2) + 0.8 2 ( − 2)] , (48)has all of its roots inside the unit complex disk (and then, stable) ∈  = (−0.00000232,0.06654) .(49)Since the extremes of the maximal stability interval  are so close to each other, a Nyquist diagram becomes sensible under the choice of gains.Consequently, it is hard to see that the Nyquist plot passes throught the point -1+0i.However, since the closed-loop characteristic equation 4   () is Schur stable, in Figure5stability is depicted by Nyquist diagram (-1+0i point not enclosed) where values of the gain  belong to  and in Figure6instability (-1+0i point enclosed) is depicted for gains not in .