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

Authors: B. L. Hernández-Galván; J. A. López-Rentería; B. Aguirre-Hernández; G. Fernández-Anaya
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

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.


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## Body

## 1. 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 [1–4]). 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 calledSchur stability ([5–8]). In this sense, a linear discrete-time invariant system is said to be stable if and only if its characteristic polynomial is aSchur 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 [9–12]. 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 [15–17]; 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]. Concerningrays 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 [5–7].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 Section3, 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.


## 2. Basic Notations

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


## 3. Problems Statement

Consider the discrete LTI control system(1)xm+1=Axm+bum,where xm∈Rn is the states variable vector, A∈Rn×n and b∈Rn are given in controllable canonical form (also called Jerk realization) as (2)A=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1-an-an-1-an-2⋯-a1,b=00⋮1,and um∈R is the linear actuator of the system. The open-loop characteristic polynomial is pA(z)=zn+a1zn-1+⋯+an-1z+an, where z is the complex z-transform variable.Case 1.
For system (1) with feedback control um=-kcTxm, with cT=(cn,…,c1), the closed-loop characteristic polynomial is the ray of polynomials P(z,k)=pA(z)+kpc(z), where pc(z)=c1zn-1+c2zn-2+⋯+cn-1z+cn.Case 2.
For system (1) with the feedback control um-τ=-kcTxm-τ, where τ∈N and cT=(cn,…,c1), the closed-loop characteristic polynomial is the ray of polynomials Pτ(z,k)=p~A(z)+kp~c(z) (further details about this characteristic polynomial will be given in Section 5.3).If we suppose thatpA(z) is a Schur polynomial, then we can to perturb the parameter k of the controller (for both of the Cases 1 and 2) to keep stability. Then, the objective is to compute the maximal perturbation of k, that is, the minimum k<0 and the maximum k>0, to obtain kmin- and kmax+, respectively, such that system (1) has poles inside the unit complex disk D for all k in the maximal stability interval K=(kmin-,kmax+). In other words, the system is robustly Schur stable in the maximal stability interval K. Any other subset [a,b]⊂K 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).


## 4. Preliminaries

### 4.1. The Boundary Theorems Generalized

Let us consider the complex planeC 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 ofreal polynomials. The discrete versions are given in the following section of main results.The classical boundary theorems consider the following hypothesis.Hypothesis 1.
Consider the polynomial family(3)Pλ,k=a0kλn+a1kλn-1+⋯+an-1kλ+ank,in the variable λ∈C and uncertainty parameter k∈R, satisfying the following:(i)
having fixed degreen,(ii)
with continuous coefficients with respect tok on a fixed interval I=[a,b].The generalizations of the boundary theorems used in [8] also use the same hypothesis for the roots criterion by taking S=C- which is the Hurwitz stability zone. In the following we shall give a version of them.Theorem 1 (generalization of BCT).
Under Hypothesis1, suppose that P(λ,a) has n1 roots in C- and n-n1 roots in C+, and P(λ,b) has m1 roots in C- and n-m1 roots in C+. If n1≠m1, then there exists at least one ρ in (a,b] such that(i)
P(λ,ρ) has at least n1 roots in C-∪iR,(ii)
P(λ,ρ) has at least n-n1 roots in C+∪iR,(iii)
P(λ,ρ) has at least one root in iR.Similarly, under the same hypothesis and with the same stability zoneS=C-, the generalization of the zero exclusion principle is given as follows.Theorem 2 (generalization of ZEP).
Consider the polynomial familyP(λ,k) under Hypothesis 1. Suppose that for some ρ∈(a,b) the family has n1 roots in C- and n-n1 roots in C+. Then the entire family keeps n1 roots in C- and n-n1 roots in C+ if and only if (4)Piω,k≠0∀k∈a,band∀ω∈R.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.Lemma 3 (CDR, [5]).
Consider the monoparametric family of polynomialsP(λ,k)=∑j=0najkλn-j under Hypothesis 1. Then the roots of P(λ,k) vary continuously with respect to k∈[a,b]. That is, there exist continuous mappings λj:[a,b]→C for j=1,2,,…,n such that λ1(k),…,λn(k) are the roots of P(λ,k).


## 4.1. The Boundary Theorems Generalized

Let us consider the complex planeC 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 ofreal polynomials. The discrete versions are given in the following section of main results.The classical boundary theorems consider the following hypothesis.Hypothesis 1.
Consider the polynomial family(3)Pλ,k=a0kλn+a1kλn-1+⋯+an-1kλ+ank,in the variable λ∈C and uncertainty parameter k∈R, satisfying the following:(i)
having fixed degreen,(ii)
with continuous coefficients with respect tok on a fixed interval I=[a,b].The generalizations of the boundary theorems used in [8] also use the same hypothesis for the roots criterion by taking S=C- which is the Hurwitz stability zone. In the following we shall give a version of them.Theorem 1 (generalization of BCT).
Under Hypothesis1, suppose that P(λ,a) has n1 roots in C- and n-n1 roots in C+, and P(λ,b) has m1 roots in C- and n-m1 roots in C+. If n1≠m1, then there exists at least one ρ in (a,b] such that(i)
P(λ,ρ) has at least n1 roots in C-∪iR,(ii)
P(λ,ρ) has at least n-n1 roots in C+∪iR,(iii)
P(λ,ρ) has at least one root in iR.Similarly, under the same hypothesis and with the same stability zoneS=C-, the generalization of the zero exclusion principle is given as follows.Theorem 2 (generalization of ZEP).
Consider the polynomial familyP(λ,k) under Hypothesis 1. Suppose that for some ρ∈(a,b) the family has n1 roots in C- and n-n1 roots in C+. Then the entire family keeps n1 roots in C- and n-n1 roots in C+ if and only if (4)Piω,k≠0∀k∈a,band∀ω∈R.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.Lemma 3 (CDR, [5]).
Consider the monoparametric family of polynomialsP(λ,k)=∑j=0najkλn-j under Hypothesis 1. Then the roots of P(λ,k) vary continuously with respect to k∈[a,b]. That is, there exist continuous mappings λj:[a,b]→C for j=1,2,,…,n such that λ1(k),…,λn(k) are the roots of P(λ,k).


## 5. Main Results

We have divided this section in three parts. In the Section5.1 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.


### 5.1. 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 ofS=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 thez-transform applied to the discrete system. Thus, all of the polynomials to deal with in the coming development have real coefficients and complex variable z instead of λ.Theorem 5 (discrete BCT generalized).
Consider the polynomial familyP(z,k) on the complex variable z under Hypothesis 1. Suppose that P(z,a) has n1 roots in D and n-n1 roots in C-D¯; and P(z,b) has m1 roots in D and n-m1 roots in C-D¯. Then, if n1≠m1 there exist at least one ρ in (a,b] such that (i)
P(z,ρ) has at least n1 roots in D¯,(ii)
P(z,ρ) has at least n-n1 roots in C-D¯,(iii)
P(z,ρ) has at least one root in ∂D.Proof.
SinceP(z,k) satisfies Hypothesis 1, then by Lemma 3 there exist n continuous function roots of P(z,k), say z1(k),…,zn(k)∈C. Without loss of generality we can suppose that, for j=1,…,n1, we have that zj(a)∈D and, for j=n1+1,…,n, we have that zj(a)∈C-D¯. Since n1≠m1, by continuity at least one root traveled through the complex plane collapsing into ∂D for some ρ∈(a,b], resulting in the items claimed.Similarly, the generalized zero exclusion principle can be adapted to the discrete case in the following way.Theorem 6 (generalization of ZEP).
Consider the polynomial familyP(z,k) under Hypothesis 1. Suppose that for some ρ∈(a,b) the family has n1 roots in D and n-n1 roots in C-D¯. Then, the entire family keeps n1 roots in D and n-n1 roots in C-D¯ if and only if (5)Peiθ,k≠0∀k∈a,band∀θ∈0,2π.Proof.
( ⇒ ) If all of the elements of the family have n1 roots in D and n-n1 roots in C-D¯, then it is clear that there are no roots in the unit circumference ∂D. Therefore, P(eiθ,k)≠0, for all 0≤θ≤2π and for all k∈[a,b]. (⇐) Suppose that P(eiθ,k)≠0 for all 0≤θ≤2π and for all k∈[a,b]. Let us proceed by contradiction. Suppose that there is k0∈(a,b] such that the polynomial P(z,k0) has m1 roots in D and n-m1 roots in C-D¯ with n1≠m1. Then, from Theorem 5, there exists a ρ∈(a,b] such that P(eiθ0,ρ)=0 for some 0≤θ0≤2π, which is a contradiction.


### 5.2. The Roots Criterion for Discrete-Time Linear Systems

Next, we shall develop the technique to compute the maximal Schur stability interval. LetpA(z)=zn+a1zn-1+⋯+an be an n-fixed degree polynomial and pc(z)=c1zm+⋯+cmz+cm+1 be a polynomial such that n>m. We desire to find the values of k around k=0 such that the ray P(z,k)=pA(z)+kpc(z) has n roots inside D. Since pA(z) is a Schur polynomial, then pA(z-1) has n roots outside D. Thus, the discrete generalized zero exclusion principle will be applied to the family P(z,k)=pA(z-1)P(z,k), which has n roots inside D and n roots outside D for k=0. The idea is to extend (a,b) to the maximum interval for k where the zero exclusion principle is satisfied and, therefore, we obtain robust stability. Firstly, let us carry P(z,k)z=eiθ in a suitable form.By evaluatingpA(z) and pc(z) in z=eiθ,0≤θ≤2π we get (6)pAeiθ=Rθ+iQθpceiθ=rθ+iqθwhere(7)Rθ=cos⁡nθ+∑j=1najcos⁡n-jθ,Qθ=sin⁡nθ+∑j=1n-1ajsin⁡n-jθ,rθ=∑j=0mcj+1cos⁡m-jθ,qθ=∑j=0m-1cj+1sinm-jθ.Thus, the family P(z,k)z=eiθ can be written as(8)Peiθ,k=pAeiθ+kpceiθ=R+krθ+iQ+kqθNow, since pA(z-1)z=eiθ=pA(e-iθ)=R(θ)-iQ(θ), the family P(z,k) evaluated in the unit circle becomes on(9)Peiθ,k=pAe-iθPeiθ,k=pA-eiθpAeiθ+kpAe-iθpceiθ=R2+Q2θ+kRr+Qqθ+ikRq-Qrθ.Next, define the functions(10)Fθ=Rr+Qqθ;Gθ=R2+Q2θ;Hθ=Rq-Qrθ.Thence, we can rewrite P(eiθ,k) as(11)Peiθ,k=Gθ+kFθ+ikHθ,and this is the polynomial family form to be applied the zero exclusion principle in order to compute the maximal stability interval.Remark 7.
Some properties ofF,G, and H are as follows:(i)
G(θ)≠0, for all 0≤θ≤2π.(ii)
H is an odd function and H(0)=0, H(π)=0.(iii)
F is an even function.Proof.
Before anything, it is not hard to see thatQ(0)=q(0)=0, Q(π)=q(π)=0, Q(-θ)=-Q(θ), q(-θ)=-q(θ), R(-θ)=R(θ), and r(-θ)=r(θ).
Proof of (i):G(θ)=[R2+Q2](θ)=|pA(eiθ)|2≠0, since pA(eiθ)≠0, for all 0≤θ≤2π, since pA(z) is Schur.
Proof of (ii):H(0)=Q(0)r(0)-R(0)q(0)=0 and similarly H(π)=0. Now, if θ∈[0,2π], then (12)H-θ=Q-θr-θ-R-θq-θ=-Qθrθ+Rθqθ=-Hθ,and consequently, H is an odd function.
Finally, proof of (iii):(13)F-θ=R-θr-θ+Q-θq-θ=Rθrθ+Qθqθ=Fθ,and, therefore, F is an even function.Item (ii) shows thatθ=0, π are always zeroes of H and if θ0∈σ(H) then its conjugate -θ0 is also a zero of H. In the incoming development it will be shown a condition where F(0)≠0 in item (iii).Letσ(f)={ζ∈C∣f(ζ)=0} denote the set of zeroes for a function f(ζ). In our case of studio we have the variable 0≤θ≤2π; thus the set σ(H) is the set of zeroes of H modulo 2π. With the functions F(θ), G(θ), and H(θ) defined above, let us define the sets(14)K+=Fθl∣θl∈σH,Fθl>0)K-=Fθl∣θl∈σH,Fθl<0)If there are no elements in σ(H) such that F(θl)>0, then we will define K+={0+}. Similarly, if there are no elements in σ(H) such that F(θl)<0, then we will define K-={0-}.Finally, the method to compute the maximal robust Schur stability interval is derived in the following result.Theorem 8.
LetpA(z)=zn+a1zn-1+⋯+an be a real Schur polynomial and consider any real polynomial pc(z)=c1zm+⋯+cmz+cm+1, with n>m. Then, the maximal Schur stability interval for the monoparametric polynomial family P(z,k)=pA(z)+kpc(z) is (kmin-,kmax+), where(15)kmin-=max⁡-GθlFθl∣Fθl∈K+kmax+=min⁡-GθlFθl∣Fθl∈K-.Proof.
The proof will be based on the discrete generalized zero exclusion principle applied toP(z,k)=pA(z-1)P(z,k); since k=0, the polynomial P(z,0)=pA(z-1)pA(z) has n roots in D and n roots in C-D¯. That is, we have a member of the family P(z,k) with n roots inside the open unit disk and n inside its open complement. Thus, it is necessary to find the minimum kmin- and the maximum kmax+ such that P(eiθ,k)≠0, for all 0<θ<2π and for all k∈(kmin-,kmax+) and keep the quantity of roots in each zone. If k satisfies P(eiθ,k)=0, then k∉(kmin-,kmax+) and therefore we seek the maximum k<0 and the minimum k>0 such that P(eiθ,k)=0; that is, (16)Gθ+kFθ+ikHθ=0.for some 0≤θ≤2π and k≠0. Consequently, the following system must be solved:(17)Gθ+kFθ=0kHθ=0.For each θl∈σ(H), the before system has solution by taking k=-G(θl)/F(θl). Now, if z∈∂D, then z-1=z¯, and then G(θ)=pAeiθ2>0. Consequently, -G(θ)/F(θ) is positive if F(θ)<0 and -G(θ)/F(θ) is negative if F(θ)>0. Notice that all the values of k satisfying system (19) are determined by the zeroes of H(θ). Therefore, k=-G(θl)/F(θl) is positive if F(θl)∈K- and it is negative if F(θl)∈K+, θl∈σ(H). Thence, the maximum k<0 and minimum k>0 satisfying G(θl)+kF(θl)+ikH(θl)=0 for θl∈σ(H) are, respectively, given by(18)kmin-=max⁡-GθlFθl∣Fθl∈K+,kmax+=min⁡-GθlFθl∣Fθl∈K-.Finally, if K-=0- (K+=0+, resp.), then kmax+=limF→0-⁡(-G(θ)/F)=+∞ (kmin-=limF→0+⁡(-G(θ)/F)=-∞, resp.), and this ends the proof.Remark 9.
If we suppose thatG(θ)+kF(θ)+ikH(θ)=0, zeroes θl∈σ(H) must not be considered such that F(θl)=0; if so, then 0=G(θl)+kF(θl)+ikH(θl)=G(θl), no matter the values of k, which is a contradiction since G(θ)=pAeiθ2≠0 for all θ∈[0,2π], by the fact that pA(z) is a Schur polynomial.Remark 10.
Note that(kmin-,kmax+) extends any interval (a,b) for the uncertainty k to keep stability of the ray P(z,k) and such that the family P(z,k) satisfies the zero exclusion principle in the closure [a,b].Example 11.
Consider the discrete-time system(19)xm+1=0100010140xm+001um,having as characteristic polynomial to pA(z)=z3-(1/4)z, which is trivially Schur stable. If we need to design the feedback um=-k(1,2,1)xm, then we obtain the polynomial pc(z)=z2+2z+1 and the polynomial family P(z,k)=pA(z)+kpc(z). We will now compute the maximal stability interval with our proposed method. Evaluate the polynomials pA(z) and pc(z) in the unit circle to obtain(20)pAeiθ=e3iθ-14eiθ=cos⁡3θ-14cos⁡θ+isin⁡3θ-14sin⁡θ=Rθ+iQθ,pceiθ=e2iθ+2eiθ+1=cos⁡2θ+2cos⁡θ+1+isin⁡2θ+2sin⁡θ=rθ+iqθThus,(21)Fθ=cos⁡3θ+2cos⁡2θ+12cos⁡θ-12Gθ=1716-12cos⁡2θHθ=-sin⁡3θ-2sin⁡2θ-sin⁡θNote that just zeroes of Hmod⁡(2π) must be considered. Now, it is not hard to see that H(θ) can be written as (22)Hθ=-4sin⁡θcos⁡θcos⁡θ+1,and, thence, its zeroes set is σ(H)=0,π/2,π,3π/2. One can see that F(π)=0; therefore θ=π will not be considered in K- and K+. Next, (23)F0=3;Fπ2=-2.5;F3π2=-2.5;G0=0.5625;Gπ2=1.5625;G3π2=1.5625.Therefore,(24)kmin-=-G0F0=-0.1875(25)kmax+=-Gπ/2Fπ/2=0.625.Finally, system (19) is stable for all k∈(-0.1875,0.625). That is, all of its poles lie inside the unit circle for all k∈(-0.1875,0.625).Recall that the characteristic equationpA(z) 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 P(z,k) for k=0.Figure 1
Nyquist diagram of the closed-loop system forkmin- (left image) and kmax+ (right image) extremes of the maximal stability interval.Figure 2
Nyquist diagram fork=-0.1 (left image) and k=0.1 (right image).Figure 3
Nyquist diagram fork=1.Figure 4
Set value of the familyP(z,k)=z3+kz2+2kz+(k-1/4) with k∈(-0.1875,0.625); right image with zoom around the origin.


### 5.3. The Roots Criterion for Discrete Systems with Single Time-Delay Control

Consider the discrete-time system with a single time-delay(26)xm+1=Axm+Bxm-τ,τ∈N,with A,B∈Rn×n, whose stability is in charge of the roots of its characteristic equation given by p(z)=det(zτ+1I-zτA-B)=0 [24].Based on the aforementioned reference, we shall consider the control system(27)xm+1=Axm+bum-τ,τ∈N,where(28)A=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1-an-an-1-an-2⋯-a1,b=00⋮1,for which the linear delayed control um-τ=-kcTxm-τ, with cT=(cn,…,c1), endows the system (27) with a single time-delay as required in system (26).Lemma 12.
The closed-loop characteristic equation of system (27) is given by the ray Pτ(z,k)=znτpA(z)+kznpc(z), where pA(z)=det⁡(zI-A) and pc(z)=c1zn-1+⋯+cn-1z+cn.Proof.
If we closed-loop system (27) with the control um-τ=-kcTxm-τ the following arises:(29)xm+1=Axm-kbcTxm-τ,and by taking B(k)=-kbcT, we get that the characteristic equation is (30)det⁡zτ+1I-zτA-Bk=det⁡zτzI-A-z-τBk=det⁡zτzI-A+kz-τbcT=znτdet⁡zI-A-kz-τbcT=znτpAz+kz-τpcz=znτpAz+kznpcz,as we claim.In the same essence as the technique in single time case, by evaluating in the unit complex circle we get(31)p~Aeiθ=einτθpAeiθ=einτθR+iQθp~ceiθ=einθpceiθ=einθr+iqθwhere R,Q,r, and q are the functions given in (7). The product to consider in this case is Pτ(z,k)=p~A(z-1)Pτ(z,k). Thus, Pτ(z,k) in z=eiθ can be written as (32)Pτeiθ,k=p~Ae-iθp~Aeiθ+p~Ae-iθp~ceiθ=e-inτθpAe-iθeinτθpAeiθ+e-inτθpAe-iθeinθpceiθ=R2+Q2θ+kcos⁡τθRr+Qqθ+sin⁡τθRq-Qrθ+ikcos⁡τθRq-Qrθ-sin⁡τθRr+Qqθ.Define(33)Fτθ=cos⁡τθRr+Qqθ+sin⁡τθRq-Qrθ=cos⁡τθFθ+sin⁡τθHθ,Hτθ=cos⁡τθRq-Qrθ-sin⁡τθRr+Qqθ=cos⁡τθHθ-sin⁡τθFθ,Gτθ=GθF, G, and H are the functions defined in (10). Note that if τ=0, we recover the single time case from the previous section. Write then the family Pτ(eiθ,k) in the form (34)Pτeiθ,k=Gτθ+kFτθ+ikHτθ,for G(θ) defined in (10). Analogous to the sets K- and K+ from (14), let us define the sets(35)Kτ+=Fτθl∣θl∈σHτ,Fτθl>0)Kτ-=Fτθl∣θl∈σHτ,Fτθl<0).Thence, the following result is held.Theorem 13.
LetpA(z)=zn+a1zn-1+⋯+an be a real Schur polynomial and consider any real polynomial pc(z)=c1zm+⋯+cmz+cm+1, with n>m. Then, the maximal Schur stability interval for the monoparametric polynomial Pτ(z,k)=znτpA(z)+kznpc(z) is (kmin-,kmax+), where(36)kmin-=max⁡-GτθlFτθl∣Fτθl∈Kτ+kmax+=min⁡-GτθlFτθl∣Fτθl∈Kτ-.Proof.
It is analogous to the proof of Theorem8.Remark 14.
As in Remark9, we must not consider zeroes θl∈σ(Hτ) such that Fτ(θl)=0, since (37)Gτθl+kFτθl+ikHτθl=Gτθl=Gθl=0,which is a contradiction due to the fact that G(θ)=pAeiθ2≠0 for all θ∈[0,2π], by the fact that pA(z) is a Schur polynomial.Example 15 (Phillips and Nagle (1995) [4]).
Consider the digital system described by the difference equation(38)ym+2-1.7ym+1+0.72ym=um-τ.Then, the change of variables x1(m)=y[m] and x2(m)=x1(m+1)=y[m+1] transforms the difference equation (38) in to the states variable system(39)x1m+1=x2mx2m+1=-0.72x1m+1.7x2m+um-τWe may express these equations in state variable representation as (40)xm+1=01-0.721.7xm+01um-τ,where xmT=(x1(m),x2(m)) and um-τ=u[m-τ]. The open-loop characteristic polynomial of the system is pA(z)=z2-1.7z+0.72 with roots z1=0.9 and z2=0.8, which indicates that the open-loop system is Schur stable. Let us design the feedback(41)um-τx,k=-k1.5x1m-τ+0.8x2m-τto obtain the polynomial pc(z)=0.8z+1.5 and the closed-loop characteristic polynomial family(42)Pτz,k=z2τpAz+kz2pcz=z2τz2-1.7z+0.72+kz20.8z+1.5Thence, by evaluating z=eiθ,θ∈[0,2π], the following complex functions arise: (43)pAeiθ=ei2θ-1.7eiθ+0.72=cos⁡2θ-1.7cos⁡θ+0.72+isin⁡2θ-1.7sin⁡θ=Rθ+iQθ;pceiθ=0.8eiθ+1.5=0.8cos⁡θ+1.5+i0.8sin⁡θ=rθ+iqθ,and, consequently,(44)Fθ=Rr+Qqθ=3cos2⁡θ-1.174cos⁡θ-1.78;Hθ=Rq-Qrθ=sin⁡θ-3cos⁡θ+2.326;Gθ=R2+Q2θ=2.88cos2⁡θ-5.848cos⁡θ+2.9684.Therefore, if τ=2 the directly involved functions Fτ, Gτ, and Hτ are given by (45)Fτθ=cos⁡τθFθ+sin⁡τθHθ=9cos4⁡θ+168.17cos3θ-12.56cos2⁡θ+5.826cos⁡θ+1.78Hτθ=cos⁡τθHθ-sin⁡τθFθ=sin⁡θ-12cos3⁡θ+7cos2⁡θ+6.56cos⁡θ-2.326Gτθ=Gθ.The change of variable x=cos⁡θ in the function -12cos3⁡θ+7cos2⁡θ+6.56cos⁡θ-2.326=0 arises at the roots x1=0.94479, x2=0.30693, and x3=-0.6684. Consequently, θ1=cos-1⁡(x1), θ2=cos-1⁡(x2), and θ3=cos-1⁡(x3) are real zeroes of Hτ. Moreover, it is not hard to see that θ3=0 and θ4=π are also roots of Hτ. Thence, (46)Fτ0=172.046,Gτ0=0.0004;Fτπ=-175.776,Gτπ=11.6964;Fτθ1=145.07198,Gτθ1=3.01403;Fτθ2=7.32785,Gτθ2=4.44474;Fτθ3=-56.14728,Gτθ3=11.16388.Therefore, (47)kmin-=max⁡-G0F0,-Gθ1Fθ1,-Gθ2Fθ2=max⁡-0.00000232,-0.02077,-0.60655=-0.00000232.kmax+=min⁡-GπFπ,-Gθ3Fθ3=min⁡0.06654,0.19883=0.06654.Consequently, the discrete system (38), with the delayed time feedback(48)um-2x,k=-k1.5x1m-2+0.8x2m-2,has all of its roots inside the unit complex disk (and then, stable) (49)k∈K=-0.00000232,0.06654.Since the extremes of the maximal stability interval K 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 z4pA(z) is Schur stable, in Figure 5 stability is depicted by Nyquist diagram (-1+0i point not enclosed) where values of the gain k belong to K and in Figure 6 instability (-1+0i point enclosed) is depicted for gains not in K. In Figure 7, one can see that the origin is excluded from the set values for k∈(-0.00000232,0.06654) and since Pτ=2(z,0) is Schur stable, then the whole polynomial family Pτ=2(z,k) is robustly stable.Figure 5
Nyquist diagram fork=0.01 (left image) and k=0.04 (right image).Figure 6
Nyquist diagram fork=-0.60655 (left image) and k=0.19883 (right image).Figure 7
Set value of the polynomial familyPτ=2(z,k)=z4(z2-1.7z+0.72)+kz2(0.8z+1.5) with k∈(-0.00000232,0.06654); right image with zoom around the origin.


## 5.1. 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 ofS=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 thez-transform applied to the discrete system. Thus, all of the polynomials to deal with in the coming development have real coefficients and complex variable z instead of λ.Theorem 5 (discrete BCT generalized).
Consider the polynomial familyP(z,k) on the complex variable z under Hypothesis 1. Suppose that P(z,a) has n1 roots in D and n-n1 roots in C-D¯; and P(z,b) has m1 roots in D and n-m1 roots in C-D¯. Then, if n1≠m1 there exist at least one ρ in (a,b] such that (i)
P(z,ρ) has at least n1 roots in D¯,(ii)
P(z,ρ) has at least n-n1 roots in C-D¯,(iii)
P(z,ρ) has at least one root in ∂D.Proof.
SinceP(z,k) satisfies Hypothesis 1, then by Lemma 3 there exist n continuous function roots of P(z,k), say z1(k),…,zn(k)∈C. Without loss of generality we can suppose that, for j=1,…,n1, we have that zj(a)∈D and, for j=n1+1,…,n, we have that zj(a)∈C-D¯. Since n1≠m1, by continuity at least one root traveled through the complex plane collapsing into ∂D for some ρ∈(a,b], resulting in the items claimed.Similarly, the generalized zero exclusion principle can be adapted to the discrete case in the following way.Theorem 6 (generalization of ZEP).
Consider the polynomial familyP(z,k) under Hypothesis 1. Suppose that for some ρ∈(a,b) the family has n1 roots in D and n-n1 roots in C-D¯. Then, the entire family keeps n1 roots in D and n-n1 roots in C-D¯ if and only if (5)Peiθ,k≠0∀k∈a,band∀θ∈0,2π.Proof.
( ⇒ ) If all of the elements of the family have n1 roots in D and n-n1 roots in C-D¯, then it is clear that there are no roots in the unit circumference ∂D. Therefore, P(eiθ,k)≠0, for all 0≤θ≤2π and for all k∈[a,b]. (⇐) Suppose that P(eiθ,k)≠0 for all 0≤θ≤2π and for all k∈[a,b]. Let us proceed by contradiction. Suppose that there is k0∈(a,b] such that the polynomial P(z,k0) has m1 roots in D and n-m1 roots in C-D¯ with n1≠m1. Then, from Theorem 5, there exists a ρ∈(a,b] such that P(eiθ0,ρ)=0 for some 0≤θ0≤2π, which is a contradiction.


## 5.2. The Roots Criterion for Discrete-Time Linear Systems

Next, we shall develop the technique to compute the maximal Schur stability interval. LetpA(z)=zn+a1zn-1+⋯+an be an n-fixed degree polynomial and pc(z)=c1zm+⋯+cmz+cm+1 be a polynomial such that n>m. We desire to find the values of k around k=0 such that the ray P(z,k)=pA(z)+kpc(z) has n roots inside D. Since pA(z) is a Schur polynomial, then pA(z-1) has n roots outside D. Thus, the discrete generalized zero exclusion principle will be applied to the family P(z,k)=pA(z-1)P(z,k), which has n roots inside D and n roots outside D for k=0. The idea is to extend (a,b) to the maximum interval for k where the zero exclusion principle is satisfied and, therefore, we obtain robust stability. Firstly, let us carry P(z,k)z=eiθ in a suitable form.By evaluatingpA(z) and pc(z) in z=eiθ,0≤θ≤2π we get (6)pAeiθ=Rθ+iQθpceiθ=rθ+iqθwhere(7)Rθ=cos⁡nθ+∑j=1najcos⁡n-jθ,Qθ=sin⁡nθ+∑j=1n-1ajsin⁡n-jθ,rθ=∑j=0mcj+1cos⁡m-jθ,qθ=∑j=0m-1cj+1sinm-jθ.Thus, the family P(z,k)z=eiθ can be written as(8)Peiθ,k=pAeiθ+kpceiθ=R+krθ+iQ+kqθNow, since pA(z-1)z=eiθ=pA(e-iθ)=R(θ)-iQ(θ), the family P(z,k) evaluated in the unit circle becomes on(9)Peiθ,k=pAe-iθPeiθ,k=pA-eiθpAeiθ+kpAe-iθpceiθ=R2+Q2θ+kRr+Qqθ+ikRq-Qrθ.Next, define the functions(10)Fθ=Rr+Qqθ;Gθ=R2+Q2θ;Hθ=Rq-Qrθ.Thence, we can rewrite P(eiθ,k) as(11)Peiθ,k=Gθ+kFθ+ikHθ,and this is the polynomial family form to be applied the zero exclusion principle in order to compute the maximal stability interval.Remark 7.
Some properties ofF,G, and H are as follows:(i)
G(θ)≠0, for all 0≤θ≤2π.(ii)
H is an odd function and H(0)=0, H(π)=0.(iii)
F is an even function.Proof.
Before anything, it is not hard to see thatQ(0)=q(0)=0, Q(π)=q(π)=0, Q(-θ)=-Q(θ), q(-θ)=-q(θ), R(-θ)=R(θ), and r(-θ)=r(θ).
Proof of (i):G(θ)=[R2+Q2](θ)=|pA(eiθ)|2≠0, since pA(eiθ)≠0, for all 0≤θ≤2π, since pA(z) is Schur.
Proof of (ii):H(0)=Q(0)r(0)-R(0)q(0)=0 and similarly H(π)=0. Now, if θ∈[0,2π], then (12)H-θ=Q-θr-θ-R-θq-θ=-Qθrθ+Rθqθ=-Hθ,and consequently, H is an odd function.
Finally, proof of (iii):(13)F-θ=R-θr-θ+Q-θq-θ=Rθrθ+Qθqθ=Fθ,and, therefore, F is an even function.Item (ii) shows thatθ=0, π are always zeroes of H and if θ0∈σ(H) then its conjugate -θ0 is also a zero of H. In the incoming development it will be shown a condition where F(0)≠0 in item (iii).Letσ(f)={ζ∈C∣f(ζ)=0} denote the set of zeroes for a function f(ζ). In our case of studio we have the variable 0≤θ≤2π; thus the set σ(H) is the set of zeroes of H modulo 2π. With the functions F(θ), G(θ), and H(θ) defined above, let us define the sets(14)K+=Fθl∣θl∈σH,Fθl>0)K-=Fθl∣θl∈σH,Fθl<0)If there are no elements in σ(H) such that F(θl)>0, then we will define K+={0+}. Similarly, if there are no elements in σ(H) such that F(θl)<0, then we will define K-={0-}.Finally, the method to compute the maximal robust Schur stability interval is derived in the following result.Theorem 8.
LetpA(z)=zn+a1zn-1+⋯+an be a real Schur polynomial and consider any real polynomial pc(z)=c1zm+⋯+cmz+cm+1, with n>m. Then, the maximal Schur stability interval for the monoparametric polynomial family P(z,k)=pA(z)+kpc(z) is (kmin-,kmax+), where(15)kmin-=max⁡-GθlFθl∣Fθl∈K+kmax+=min⁡-GθlFθl∣Fθl∈K-.Proof.
The proof will be based on the discrete generalized zero exclusion principle applied toP(z,k)=pA(z-1)P(z,k); since k=0, the polynomial P(z,0)=pA(z-1)pA(z) has n roots in D and n roots in C-D¯. That is, we have a member of the family P(z,k) with n roots inside the open unit disk and n inside its open complement. Thus, it is necessary to find the minimum kmin- and the maximum kmax+ such that P(eiθ,k)≠0, for all 0<θ<2π and for all k∈(kmin-,kmax+) and keep the quantity of roots in each zone. If k satisfies P(eiθ,k)=0, then k∉(kmin-,kmax+) and therefore we seek the maximum k<0 and the minimum k>0 such that P(eiθ,k)=0; that is, (16)Gθ+kFθ+ikHθ=0.for some 0≤θ≤2π and k≠0. Consequently, the following system must be solved:(17)Gθ+kFθ=0kHθ=0.For each θl∈σ(H), the before system has solution by taking k=-G(θl)/F(θl). Now, if z∈∂D, then z-1=z¯, and then G(θ)=pAeiθ2>0. Consequently, -G(θ)/F(θ) is positive if F(θ)<0 and -G(θ)/F(θ) is negative if F(θ)>0. Notice that all the values of k satisfying system (19) are determined by the zeroes of H(θ). Therefore, k=-G(θl)/F(θl) is positive if F(θl)∈K- and it is negative if F(θl)∈K+, θl∈σ(H). Thence, the maximum k<0 and minimum k>0 satisfying G(θl)+kF(θl)+ikH(θl)=0 for θl∈σ(H) are, respectively, given by(18)kmin-=max⁡-GθlFθl∣Fθl∈K+,kmax+=min⁡-GθlFθl∣Fθl∈K-.Finally, if K-=0- (K+=0+, resp.), then kmax+=limF→0-⁡(-G(θ)/F)=+∞ (kmin-=limF→0+⁡(-G(θ)/F)=-∞, resp.), and this ends the proof.Remark 9.
If we suppose thatG(θ)+kF(θ)+ikH(θ)=0, zeroes θl∈σ(H) must not be considered such that F(θl)=0; if so, then 0=G(θl)+kF(θl)+ikH(θl)=G(θl), no matter the values of k, which is a contradiction since G(θ)=pAeiθ2≠0 for all θ∈[0,2π], by the fact that pA(z) is a Schur polynomial.Remark 10.
Note that(kmin-,kmax+) extends any interval (a,b) for the uncertainty k to keep stability of the ray P(z,k) and such that the family P(z,k) satisfies the zero exclusion principle in the closure [a,b].Example 11.
Consider the discrete-time system(19)xm+1=0100010140xm+001um,having as characteristic polynomial to pA(z)=z3-(1/4)z, which is trivially Schur stable. If we need to design the feedback um=-k(1,2,1)xm, then we obtain the polynomial pc(z)=z2+2z+1 and the polynomial family P(z,k)=pA(z)+kpc(z). We will now compute the maximal stability interval with our proposed method. Evaluate the polynomials pA(z) and pc(z) in the unit circle to obtain(20)pAeiθ=e3iθ-14eiθ=cos⁡3θ-14cos⁡θ+isin⁡3θ-14sin⁡θ=Rθ+iQθ,pceiθ=e2iθ+2eiθ+1=cos⁡2θ+2cos⁡θ+1+isin⁡2θ+2sin⁡θ=rθ+iqθThus,(21)Fθ=cos⁡3θ+2cos⁡2θ+12cos⁡θ-12Gθ=1716-12cos⁡2θHθ=-sin⁡3θ-2sin⁡2θ-sin⁡θNote that just zeroes of Hmod⁡(2π) must be considered. Now, it is not hard to see that H(θ) can be written as (22)Hθ=-4sin⁡θcos⁡θcos⁡θ+1,and, thence, its zeroes set is σ(H)=0,π/2,π,3π/2. One can see that F(π)=0; therefore θ=π will not be considered in K- and K+. Next, (23)F0=3;Fπ2=-2.5;F3π2=-2.5;G0=0.5625;Gπ2=1.5625;G3π2=1.5625.Therefore,(24)kmin-=-G0F0=-0.1875(25)kmax+=-Gπ/2Fπ/2=0.625.Finally, system (19) is stable for all k∈(-0.1875,0.625). That is, all of its poles lie inside the unit circle for all k∈(-0.1875,0.625).Recall that the characteristic equationpA(z) 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 P(z,k) for k=0.Figure 1
Nyquist diagram of the closed-loop system forkmin- (left image) and kmax+ (right image) extremes of the maximal stability interval.Figure 2
Nyquist diagram fork=-0.1 (left image) and k=0.1 (right image).Figure 3
Nyquist diagram fork=1.Figure 4
Set value of the familyP(z,k)=z3+kz2+2kz+(k-1/4) with k∈(-0.1875,0.625); right image with zoom around the origin.


## 5.3. The Roots Criterion for Discrete Systems with Single Time-Delay Control

Consider the discrete-time system with a single time-delay(26)xm+1=Axm+Bxm-τ,τ∈N,with A,B∈Rn×n, whose stability is in charge of the roots of its characteristic equation given by p(z)=det(zτ+1I-zτA-B)=0 [24].Based on the aforementioned reference, we shall consider the control system(27)xm+1=Axm+bum-τ,τ∈N,where(28)A=010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1-an-an-1-an-2⋯-a1,b=00⋮1,for which the linear delayed control um-τ=-kcTxm-τ, with cT=(cn,…,c1), endows the system (27) with a single time-delay as required in system (26).Lemma 12.
The closed-loop characteristic equation of system (27) is given by the ray Pτ(z,k)=znτpA(z)+kznpc(z), where pA(z)=det⁡(zI-A) and pc(z)=c1zn-1+⋯+cn-1z+cn.Proof.
If we closed-loop system (27) with the control um-τ=-kcTxm-τ the following arises:(29)xm+1=Axm-kbcTxm-τ,and by taking B(k)=-kbcT, we get that the characteristic equation is (30)det⁡zτ+1I-zτA-Bk=det⁡zτzI-A-z-τBk=det⁡zτzI-A+kz-τbcT=znτdet⁡zI-A-kz-τbcT=znτpAz+kz-τpcz=znτpAz+kznpcz,as we claim.In the same essence as the technique in single time case, by evaluating in the unit complex circle we get(31)p~Aeiθ=einτθpAeiθ=einτθR+iQθp~ceiθ=einθpceiθ=einθr+iqθwhere R,Q,r, and q are the functions given in (7). The product to consider in this case is Pτ(z,k)=p~A(z-1)Pτ(z,k). Thus, Pτ(z,k) in z=eiθ can be written as (32)Pτeiθ,k=p~Ae-iθp~Aeiθ+p~Ae-iθp~ceiθ=e-inτθpAe-iθeinτθpAeiθ+e-inτθpAe-iθeinθpceiθ=R2+Q2θ+kcos⁡τθRr+Qqθ+sin⁡τθRq-Qrθ+ikcos⁡τθRq-Qrθ-sin⁡τθRr+Qqθ.Define(33)Fτθ=cos⁡τθRr+Qqθ+sin⁡τθRq-Qrθ=cos⁡τθFθ+sin⁡τθHθ,Hτθ=cos⁡τθRq-Qrθ-sin⁡τθRr+Qqθ=cos⁡τθHθ-sin⁡τθFθ,Gτθ=GθF, G, and H are the functions defined in (10). Note that if τ=0, we recover the single time case from the previous section. Write then the family Pτ(eiθ,k) in the form (34)Pτeiθ,k=Gτθ+kFτθ+ikHτθ,for G(θ) defined in (10). Analogous to the sets K- and K+ from (14), let us define the sets(35)Kτ+=Fτθl∣θl∈σHτ,Fτθl>0)Kτ-=Fτθl∣θl∈σHτ,Fτθl<0).Thence, the following result is held.Theorem 13.
LetpA(z)=zn+a1zn-1+⋯+an be a real Schur polynomial and consider any real polynomial pc(z)=c1zm+⋯+cmz+cm+1, with n>m. Then, the maximal Schur stability interval for the monoparametric polynomial Pτ(z,k)=znτpA(z)+kznpc(z) is (kmin-,kmax+), where(36)kmin-=max⁡-GτθlFτθl∣Fτθl∈Kτ+kmax+=min⁡-GτθlFτθl∣Fτθl∈Kτ-.Proof.
It is analogous to the proof of Theorem8.Remark 14.
As in Remark9, we must not consider zeroes θl∈σ(Hτ) such that Fτ(θl)=0, since (37)Gτθl+kFτθl+ikHτθl=Gτθl=Gθl=0,which is a contradiction due to the fact that G(θ)=pAeiθ2≠0 for all θ∈[0,2π], by the fact that pA(z) is a Schur polynomial.Example 15 (Phillips and Nagle (1995) [4]).
Consider the digital system described by the difference equation(38)ym+2-1.7ym+1+0.72ym=um-τ.Then, the change of variables x1(m)=y[m] and x2(m)=x1(m+1)=y[m+1] transforms the difference equation (38) in to the states variable system(39)x1m+1=x2mx2m+1=-0.72x1m+1.7x2m+um-τWe may express these equations in state variable representation as (40)xm+1=01-0.721.7xm+01um-τ,where xmT=(x1(m),x2(m)) and um-τ=u[m-τ]. The open-loop characteristic polynomial of the system is pA(z)=z2-1.7z+0.72 with roots z1=0.9 and z2=0.8, which indicates that the open-loop system is Schur stable. Let us design the feedback(41)um-τx,k=-k1.5x1m-τ+0.8x2m-τto obtain the polynomial pc(z)=0.8z+1.5 and the closed-loop characteristic polynomial family(42)Pτz,k=z2τpAz+kz2pcz=z2τz2-1.7z+0.72+kz20.8z+1.5Thence, by evaluating z=eiθ,θ∈[0,2π], the following complex functions arise: (43)pAeiθ=ei2θ-1.7eiθ+0.72=cos⁡2θ-1.7cos⁡θ+0.72+isin⁡2θ-1.7sin⁡θ=Rθ+iQθ;pceiθ=0.8eiθ+1.5=0.8cos⁡θ+1.5+i0.8sin⁡θ=rθ+iqθ,and, consequently,(44)Fθ=Rr+Qqθ=3cos2⁡θ-1.174cos⁡θ-1.78;Hθ=Rq-Qrθ=sin⁡θ-3cos⁡θ+2.326;Gθ=R2+Q2θ=2.88cos2⁡θ-5.848cos⁡θ+2.9684.Therefore, if τ=2 the directly involved functions Fτ, Gτ, and Hτ are given by (45)Fτθ=cos⁡τθFθ+sin⁡τθHθ=9cos4⁡θ+168.17cos3θ-12.56cos2⁡θ+5.826cos⁡θ+1.78Hτθ=cos⁡τθHθ-sin⁡τθFθ=sin⁡θ-12cos3⁡θ+7cos2⁡θ+6.56cos⁡θ-2.326Gτθ=Gθ.The change of variable x=cos⁡θ in the function -12cos3⁡θ+7cos2⁡θ+6.56cos⁡θ-2.326=0 arises at the roots x1=0.94479, x2=0.30693, and x3=-0.6684. Consequently, θ1=cos-1⁡(x1), θ2=cos-1⁡(x2), and θ3=cos-1⁡(x3) are real zeroes of Hτ. Moreover, it is not hard to see that θ3=0 and θ4=π are also roots of Hτ. Thence, (46)Fτ0=172.046,Gτ0=0.0004;Fτπ=-175.776,Gτπ=11.6964;Fτθ1=145.07198,Gτθ1=3.01403;Fτθ2=7.32785,Gτθ2=4.44474;Fτθ3=-56.14728,Gτθ3=11.16388.Therefore, (47)kmin-=max⁡-G0F0,-Gθ1Fθ1,-Gθ2Fθ2=max⁡-0.00000232,-0.02077,-0.60655=-0.00000232.kmax+=min⁡-GπFπ,-Gθ3Fθ3=min⁡0.06654,0.19883=0.06654.Consequently, the discrete system (38), with the delayed time feedback(48)um-2x,k=-k1.5x1m-2+0.8x2m-2,has all of its roots inside the unit complex disk (and then, stable) (49)k∈K=-0.00000232,0.06654.Since the extremes of the maximal stability interval K 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 z4pA(z) is Schur stable, in Figure 5 stability is depicted by Nyquist diagram (-1+0i point not enclosed) where values of the gain k belong to K and in Figure 6 instability (-1+0i point enclosed) is depicted for gains not in K. In Figure 7, one can see that the origin is excluded from the set values for k∈(-0.00000232,0.06654) and since Pτ=2(z,0) is Schur stable, then the whole polynomial family Pτ=2(z,k) is robustly stable.Figure 5
Nyquist diagram fork=0.01 (left image) and k=0.04 (right image).Figure 6
Nyquist diagram fork=-0.60655 (left image) and k=0.19883 (right image).Figure 7
Set value of the polynomial familyPτ=2(z,k)=z4(z2-1.7z+0.72)+kz2(0.8z+1.5) with k∈(-0.00000232,0.06654); right image with zoom around the origin.


## 6. 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.



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*Source: 3674628-2018-06-27.xml*