Assignability of Certain Lyapunov Invariants for Linear Discrete-Time Systems⁎

17th IFAC Workshop on Control Applications of Optimization 17th IFAC Workshop on Control Applications of Optimization Yekaterinburg, Russia, 2018 of Optimization 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, October 15-19, 2018 Available online at www.sciencedirect.com 17th IFAC Workshop onOctober Control 15-19, Applications Yekaterinburg, Russia, 2018 of Optimization Yekaterinburg, Russia, October 15-19, 2018

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IFAC PapersOnLine 51-32 (2018) 40–45

Assignability of Certain Lyapunov Assignability of Certain Lyapunov Assignability of Certain Lyapunov Invariants for Linear Discrete-Time Assignability of Certain Lyapunov Invariants for Linear Discrete-Time Invariants forSystems Linear Discrete-Time  Invariants forSystems Linear Discrete-Time Systems  Systems Svetlana N. Popova ∗∗

Svetlana N. Popova ∗∗ Svetlana N. Popova ∗ ∗ Svetlana N. Popova ∗ Udmurt State University, Izhevsk, Russia ∗ Udmurt State University, Izhevsk, Russia ∗ Udmurt University, Izhevsk, Russia (e-mail:State [email protected]) ∗ (e-mail: [email protected]) Udmurt University, Izhevsk, Russia (e-mail:State [email protected]) (e-mail: [email protected]) Abstract: Abstract: We We prove prove that that the the property property of of uniform uniform complete complete controllability controllability is is sufficient sufficient for for the simultaneous proportional assignability of the Lyapunov spectrum and Lyapunov Abstract: We prove that thelocal property of uniform complete controllability is the sufficient for the simultaneous proportional local assignability of the Lyapunov spectrum and the Lyapunov Abstract: We prove thatlinear thelocal property of uniform complete controllability is the sufficient for irregularity coefficient for control systems with discrete time. the simultaneous proportional assignability of the Lyapunov spectrum and Lyapunov irregularity coefficient for linearlocal control systems with discrete time.spectrum and the Lyapunov the simultaneous proportional assignability of the Lyapunov irregularity coefficient for linear control systems with discrete time. © 2018, IFACcoefficient (International Federation of Automatic Hostingtime. by Elsevier Ltd. All rights reserved. irregularity for linear control systems Control) with discrete Keywords: Keywords: linear linear discrete discrete time-varying time-varying systems, systems, controllability, controllability, Lyapunov Lyapunov exponents, exponents, pole pole Keywords: linear discrete time-varying systems, controllability, Lyapunov exponents, pole assignment problem. assignment problem. Keywords: linear discrete time-varying systems, controllability, Lyapunov exponents, pole assignment problem. assignment problem. 1. zero. 1. INTRODUCTION INTRODUCTION zero. Therefore, Therefore, by by assigning assigning the the Lyapunov Lyapunov spectrum spectrum of of 1. INTRODUCTION aa linear system, one can control its stability. In addition, zero. Therefore, by assigning the Lyapunov spectrum of linear system, one can control its stability. In addition, 1. INTRODUCTION zero. Therefore, byLyapunov assigning theitsLyapunov spectrum of assigning other invariants of this system, one aby linear system, one can control stability. In addition, One of the main methods of designing the control strategy by assigning other Lyapunov invariants of this In system, one One of the main methods of designing the control strategy a linear system, one can control its stability. addition, by assigning other Lyapunov invariants of this system, one influence on stability of systems with One of thesystems main methods of designing the control strategy for linear linear with time-invariant time-invariant coefficients is pole can canassigning influenceother on the the stability invariants of all all nonlinear nonlinear systems for with coefficients is Lyapunov of this system,with one Onelinear of thesystems main methods of(2013). designing the control strategy aa linear approximation given by this linear system. for systems with time-invariant coefficients is pole pole by can influence on the stability of all nonlinear systems with placement method Sontag It is known (see Dickinlinear approximation given by this linear system. placement method Sontag (2013). It is (see influence on the stability nonlinear systems with for linear systems with time-invariant coefficients is pole acan placement method Sontag (2013). It is known known (see DickinDickinlinear approximation given of byall this linear system. son (1974), (Elaydi, 1999, p. 458)) that the controllability son (1974),method (Elaydi,Sontag 1999, p. 458)) that the controllability These an argument linearconsiderations approximationare given by this linear system.in placement (2013). is to known (see that Dickinconsiderations are an important important argument in favor favor son (1974), (Elaydi, 1999,isp. 458)) It that the of the stationary system equivalent thecontrollability fact for aThese of considering various problems of control over Lyapunov of the stationary system is equivalent to the fact that for These considerations are an important argument favor son (1974), (Elaydi, 1999, p. 458)) that the controllability of considering various problems of control over Lyapunov of theset stationary system is equivalent to the fact that for These considerations are an important argument in each of complex numbers with cardinality equal to the in favor invariants of linear system, not limited to control over each set of complex numbers with cardinality equal to the of considering various problems of control over Lyapunov of theset stationary is equivalent to the fact that for of linear system, not limited to control over the the each of of complex numbers with equal to the dimension the system state vector andcardinality symmetric relative to invariants of considering various problems of control over Lyapunov Lyapunov spectrum. dimension of the state vector and symmetric relative to invariants of linear system, not limited to control over the eachreal set axis, of of complex numbers equal to the spectrum. dimension the state vectorwith andcardinality symmetric relative to Lyapunov the there is a stationary feedback, such that the invariants linear system, not limited to control over the Lyapunov of spectrum. the real axis, there is a stationary feedback, such that the dimension the state vector symmetric relative to In the real there is a stationary feedback, that the this paper, we poles of axis, theofclosed-loop closed-loop system and form this set. set.such Lyapunov spectrum. In this paper, we solve solve the the problem problem of of simultaneous simultaneous assignassignpoles of the system form this the real there is a stationary feedback, thisofpaper, we solve thenamely problemthe of simultaneous assignpoles of axis, the closed-loop system form this set.such that the In ment two invariants, Lyapunov spectrum ment of two invariants, namely the Lyapunov spectrum This problem for time-varying systems is much more comthis we solve thenamely problem of simultaneous assignpoles problem of the closed-loop systemsystems form this set. more com- In ment ofpaper, two invariants, the Lyapunov spectrum This for time-varying is much and the Lyapunov irregularity coefficient. The problem is the irregularity The problem is This time-varying is much plex problem and less less for studied. In the the systems literature most more of the thecomre- and ment of Lyapunov two invariants, namelycoefficient. theit Lyapunov spectrum considered in the local formulation: is required to assign plex and studied. In literature most of reand the Lyapunov irregularity coefficient. The problem is This problem for time-varying systems is much more in the local formulation: it is required to assign plex and lesscontinuous-time studied. In the systems literature most of thecomre- considered sults are for (see Makarov and and the Lyapunov irregularity coefficient. The problem is these invariants into a small neighborhood of the invariants sults are for continuous-time systems (see Makarov and considered in the local formulation: it is required to assign plex and lesscontinuous-time studied. In the systems literature most ofare thevarire- these invariants into a small neighborhood of the invariants sults are(2012)). for (see Makarov and Popova For time-varying systems, there considered in the local formulation: it is required to assign these invariants into a small neighborhood of the invariants the In we condition Popova For systems, there are varisultsconcepts are(2012)). for of continuous-time systems (see Makarov and of of the original original system. system. In addition, addition, we set set the invariants condition Popova (2012)). For time-varying time-varying systems, there are various controllability (uniform, complete, uniform these intofeedback a small neighborhood of the of theainvariants original system. In addition, weproportional set condition that norm of the control be to ous concepts of controllability (uniform, complete, uniform the Popova (2012)). For time-varying systems, there are varithat a norm of the feedback control be proportional to the the ous concepts of controllability (uniform, complete, uniform complete, output, etc., see Halanay and Ionescu (1994); of thea norm original system. In addition, weproportional set the condition required shifts of the invariants. complete, output, etc., see Halanay and Ionescu (1994); that of the feedback control be to the ous concepts of controllability (uniform, complete, uniform required shifts of the invariants. complete, output, etc., see Halanay and Ionescu (1994); Klamka (1991); (1991); Kwakernaak Kwakernaak and and Sivan Sivan (1972)). (1972)). Also, Also, for for that a norm of the feedback control be proportional to the Klamka required shifts of the invariants. complete,(1991); output, etc.,weseehave Halanay and(1972)). IonescuAlso, (1994); Klamka Kwakernaak and Sivan The is as time-varying systems, no proper replacement for required shifts of the invariants. The paper paper is organized organized as follows. follows. In In Section Section 2, 2, we we inintime-varying systems, we have no proper replacement for Klamka (1991); Kwakernaak and Sivan (1972)). Also, time-varying systems, we have no proper replacement for The troduce the basic notation. Section 3 contains the formal the concept of poles, but their role, to a certain extent, paper is organized as follows. In Section 2, we introduce the basic notation. Section 3 contains the formal the concept of poles, but their role, to a certain extent, paper is uniform organized as Section follows. Sectionthe 2, we intime-varying systems, we have no proper for The the concept poles, but their role, toAdrianova a replacement certain (1995); extent, definition complete controllability and some is taken by of the Lyapunov exponents troduce theof notation. 3Incontains definition ofbasic uniform complete controllability andformal some is taken by the Lyapunov exponents Adrianova (1995); troduce the basic notation. Section 3 contains the formal the concept of poles, but their role, to a certain extent, definition of uniform complete controllability and some is taken by the Lyapunov exponents Adrianova (1995); auxiliary about this of which Barreira and and Pesin Pesin (2002). (2002). auxiliary results results aboutcomplete this type typecontrollability of controllability controllability which Barreira of the uniform and some is taken and by the Lyapunov Barreira Pesin (2002). exponents Adrianova (1995); definition are used in rest the paper. Section 4, we inauxiliary results aboutof this type of In controllability which are used in the rest of the paper. In Section 4, we inThe problems of assignability of the Lyapunov spectrum auxiliary results aboutof this type of equivalence controllability which Barreira and Pesin (2002). troduce the concepts dynamical and LyaThe problems of assignability of the Lyapunov spectrum are used in the rest the paper. In Section 4, we introduce the concepts of dynamical equivalence and LyaThe problems of assignability of the Lyapunov spectrum for discrete discrete time-varying time-varying systems systems were were considered considered in in BabiBabi- are used in concepts the rest of the paper. equivalence Inthe Section 4, weLyainpunov invariants. Section 5 contains main results of for troduce the of dynamical and Thediscrete problems of assignability of were the Lyapunov invariants. Section 5 containsequivalence the main results of for time-varying systems consideredspectrum in Babi- punov arz et al. (2017a,b,c). troduce the concepts of dynamical and Lyapunov invariants. Section 5 contains the main results of arz et al. (2017a,b,c). the paper. Here we give the definition of the simultaneous for discrete time-varying systems were considered in Babi- the paper. Here we give the definition of the simultaneous arz et al. (2017a,b,c). punov invariants. Section 5 contains the main results of proportional local assignability of the Lyapunov spectrum the paper. Here we give the definition of the simultaneous Characteristics of linear systems systems that that are preserved preserved under under proportional local assignability of the Lyapunov spectrum arz et al. (2017a,b,c). Characteristics of the paper. Here weassignability give the definition of the simultaneous and the Lyapunov irregularity coefficient, obtain sufficient Characteristics of linear linear systems that are are preserved under proportional local of the Lyapunov spectrum the action of Lyapunov transformations are called the irregularity coefficient, obtain spectrum sufficient the action of Lyapunov transformations are called the and the Lyapunov local assignability of the Lyapunov Characteristics of linear systems that the are Lyapunov preserved under conditions for such assignability, and illustrate possible the action invariants. of Lyapunov transformations are called the proportional and the Lyapunov irregularity coefficient, obtain sufficient Lyapunov These include specconditions for such assignability, and illustrate possible Lyapunov invariants. These include the Lyapunov specand the Lyapunov irregularity coefficient, obtain sufficient conditions for such assignability, and illustrate possible the action of Lyapunov transformations are calledspecthe applications of our results by an example. Lyapunov invariants. These include the Lyapunov trum, stability and asymptotic stability properties, the applications of our results by an example. trum, stability and asymptotic stability properties, the conditions for such assignability, and illustrate possible Lyapunov invariants. These include the Lyapunov spectrum, stability and asymptotic stability properties, the applications of our results by an example. Lyapunov irregularity irregularity coefficient, coefficient, aa property property of of the regregLyapunov trum, stability and asymptotic properties, the applications of our results by an example. Lyapunov irregularity coefficient,stability a property of the the regularity, and many others. ularity, and many others. 2. Lyapunov coefficient, a property of the regularity, andirregularity many others. 2. BASIC BASIC NOTATION NOTATION Most ofand themany Lyapunov invariants, which which have have ever ever atatularity,of others.invariants, 2. BASIC NOTATION Most the Lyapunov Most of the the attention Lyapunovofinvariants, which have everwith attracted researchers, is connected 2. BASIC NOTATIONspace with a fixed s tracted the attention of researchers, is connected with Mostproblem of the the attention Lyapunov which have ever at- Let Let R Rsss be be the the s-dimensional s-dimensional Euclidean Euclidean space with a fixed tracted ofinvariants, researchers, is stability connected the of stability. The Lyapunov is with natLet R be the s-dimensional Euclidean space with a fixed the problem of stability. The Lyapunov stability is natorthonormal basis ee1 ,, .. .. .. ,, ees and the norm  tracted the attention of researchers, is stability connected orthonormal basis and the standard standard norm  ·· . . the problem of stability. The Lyapunov is with nats urally associated with the Lyapunov spectrum, since for 1 s s×t Let Rs×tbewe theshall s-dimensional Euclidean space with a fixed urally associated with the Lyapunov spectrum, since for By R denote the space of all real matrices orthonormal basis e , . . . , e and the standard norm  · of . 1 s 1 s space of all real matrices the problem of stability. The Lyapunov stability is naturally associated with the Lyapunov spectrum, since for By R we shall denote the of the asymptotic asymptotic stability stability of of aa linear linear system system it it is is sufficient sufficient orthonormal s×t s×t basisthe e1 , spectral . . . , the es and theofi.e. standard norm  · of . the the size s × t with norm, with the operator By R we shall denote space all real matrices urally associated with the spectrum, since for the size spectral norm,ofi.e.allwith the operator the asymptotic stability of Lyapunov a linear system it is sufficient s×ts × t with the s×t that its Lyapunov spectrum be located to the left of s By R we shall denote the space real matrices of that its Lyapunov spectrum be located to the left of the size s × t with the spectral norm, i.e. with the operator generated in by Euclidean norms in R the asymptotic stability of a linear system to it is norm generated intheR Rs×t inoperator Rsss and and that its Lyapunov spectrum be located thesufficient left of norm s×t t size s×sby Euclidean s×t the s × t with spectral norm, i.e. norms with the  This work is financially supported by the Russian Foundation for R II ∈ R is the identity matrix. For any norm generated in R Euclidean norms in Rs and t , respectively; s×sby    that its Lyapunov spectrum be located to the left of R , respectively; ∈ R is the identity matrix. For any s×t This work is financially supported by the Russian Foundation for t s×s  s×t t s×s  norm generated in RR ⊂ by Euclidean norms in For R and  sequence F = F (n) R , we define R , respectively; I ∈ is the identity matrix. any Basic projects no. 16–01–00346–A and 18–51–41005–Uzb. ThisResearch, work is financially supported by the Russian Foundation for s×t   n∈N sequence F = F (n) ⊂ R , we define Basic Research, projects no. 16–01–00346–A and 18–51–41005–Uzb. t s×s s×t identity matrix. For any  This work is financially supported by the Russian Foundation for s×t n∈N ⊂isRthe R , respectively; ∈R sequence F = F I(n) , we define Basic Research, projects no. 16–01–00346–A and 18–51–41005–Uzb. n∈N n∈N s×t sequence F = FLtd. (n)All ⊂R , we define Basic Research, projects no. 16–01–00346–A and 18–51–41005–Uzb. 2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier rights reserved. n∈N

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F ∞ = sup F (n). n∈N   A bounded sequence L(n) n∈N ⊂ Rs×s of invertible  −1  matrices, such that L (n) n∈N is bounded, will be called a Lyapunov sequence. For a positive number δ, we denote by Rδ the set of all Lyapunov sequences L satisfying the estimate L − I∞ < δ.

zero, output, etc., see Halanay and Ionescu (1994); Klamka (1991); Kwakernaak and Sivan (1972) and other). For our purposes, the most suitable is the concept of uniform complete controllability. Definition 1. (Halanay and Ionescu (1994)). System (1) is called uniformly completely controllable if there exist γ > 0 and K ∈ N, such that W (k0 , k0 + K) ≥ γI for all k0 ∈ N, where n−1 .  W (k, n) = ΦA (k, j + 1)B(j)B T (j)ΦTA (k, j + 1)

By Rs≤ we denote the set of all nondecreasing sequences of s real numbers. For a fixed sequence µ = (µ1 , . . . , µs ) ∈ Rs≤ and any δ > 0 let us denote by Oδ (µ) the set of all sequences ν = (ν1 , . . . , νs ) ∈ Rs≤ , such that

j=k

max |νj − µj | < δ.

is the Kalman controllability matrix.

j=1,...,s

For any bounded sequence U = (U (n))n∈N ⊂ Rt×s , we consider a linear feedback control u(n) = U (n)x(n), n ∈ N for system (1). We identify this control u with the sequence U and call this sequence U a feedback control for system (1). Definition 2. (Babiarz et al. (2017c)). A sequence   U = U (n) n∈N ⊂ Rt×s

We consider a discrete linear time-varying system x(n + 1) = A(n)x(n) + B(n)u(n), n ∈ N, (1) s×s is a Lyapunov sequence, where A = (A(n))n∈N ⊂ R B = (B(n))n∈N ⊂ Rs×t is a bounded sequence, and u = (u(n))n∈N ⊂ Rt is a control sequence. Suppose that zero control is chosen in system (1). Then, we get the free system x(n + 1) = A(n)x(n), n ∈ N, (2) corresponding to (1). Denote by ΦA (n, m), n, m ∈ N, the transition matrix of system (2). Let 1 λ[x] = lim sup ln x(n) n n→∞ be the Lyapunov exponent of an arbitrary nontrivial solution x of system (2),   λ(A) = λ1 (A), λ2 (A), . . . , λs (A) ∈ Rs≤ be the Lyapunov spectrum of this system, that is, the set of the Lyapunov exponents of all nontrivial solutions of system (2), taking into account their multiplicities (see (Ga˘ıshun, 2001, p. 57), Babiarz et al. (2017c) for details).

is said to be an admissible feedback control for system (1)  if A(n) + B(n)U (n) n∈N is a Lyapunov sequence.

Let U = (U (n))n∈N be any admissible feedback control for system (1). Let us consider the closed-loop system x(n + 1) = (A(n) + B(n)U (n)) x(n), n ∈ N. (3)

Below we need the following theorem (Babiarz et al. (2017b)). Theorem 3. If system (1) is uniformly completely controllable,  then there exist δ > 0 and l > 0, such that for each R(n) n∈N ∈ Rδ there exists an admissible for system (1)   feedback control U = U (n) n∈N , such that

Recall that the fundamental basis of system (2) is a collection of s linearly independent solutions of this system. A fundamental basis of (2) is said to be normal (Ga˘ıshun, 2001, p. 53) if it realizes the Lyapunov spectrum of this system. It is known (Ga˘ıshun, 2001, p. 55) that a fundamental basis x1 , . . . , xs is normal if and only if it has the incompressibility property, that is, for every nontrivial linear combinations of this solutions the equality s     cj xj = max λ[xj ] : cj = 0 λ j=1

U ∞ ≤ lR − I∞ and system (3) is dynamically equivalent to the system z(n + 1) = A(n)R(n)z(n), n ∈ N.

The concept of dynamical equivalence is described in the next section. 4. LYAPUNOV INVARIANTS

j=1,...,s

In our further considerations we shall use the concept of dynamically equivalent systems (see (Halanay and Ionescu, 1994, p. 15), (Ga˘ıshun, 2001, p. 100), Gohberg et al. (1996)).   Definition 4. Let L(n) n∈N ⊂ Rs×s be a Lyapunov sequence. A linear transformation y = L(n)x, n ∈ N, (4) of the space Rs is called a Lyapunov transformation. Definition 5. We say that system (2) is dynamically equivalent to the system (5) y(n + 1) = C(n)y(n), n ∈ N, y ∈ Rs , if there exists a Lyapunov transformation (4) which connects these systems, i.e. for every solution x(n) of system (2) the function y(n) = L(n)x(n) is a solution of

holds.

We call upper and lower mean values of the sequence  φ = φ(n) n∈N ⊂ R the numbers φ = lim sup n→∞

n−1 1 ln |φ(k)|, n k=1

φ = lim inf n→∞

41

n−1 1 ln |φ(k)|, n k=1

respectively. If φ = φ we say that the sequence φ has the exact mean value. 3. UNIFORM COMPLETE CONTROLLABILITY There is a number definitions of controllability of system (1) (global, local, complete, uniform, to zero, from 41

IFAC CAO 2018 42 Yekaterinburg, Russia, October 15-19, 2018 Svetlana N. Popova / IFAC PapersOnLine 51-32 (2018) 40–45

n−1    1    ln det L(k + 1) − lndet L(k) n→∞ n k=1    1   ln det L(n) − lndet L(1) = σL (A). = σL (A) − lim n→∞ n Here the last limit is 0, since L is a Lyapunov sequence with the two-sided estimate 0 < l1 ≤ | det L(n)| ≤ l2 < ∞, n ∈ N. Corollary 9. The regularity of linear systems is a Lyapunov invariant.

system (5) and for every solution y(n) of system (5) the function x(n) = L−1 (n)y(n) is a solution of system (2).

= σL (A) − lim

Let us note that if the Lyapunov transformation (4) establishes the dynamic equivalence between systems (2) and (5), then y(n + 1) = L(n + 1)x(n + 1) = L(n + 1)A(n)x(n) = L(n + 1)A(n)L−1 (n)y(n), n ∈ N, hence, C(n) = L(n + 1)A(n)L−1 (n),

n ∈ N.

(6) 5. MAIN RESULT

Thus, systems (2) and (5) are dynamically equivalent if  and only if there exists a Lyapunov sequence L(n) n∈N ⊂ Rs×s , such that equality (6) is satisfied. Definition 6. Lyapunov invariants are characteristics of linear systems that coincide for dynamically equivalent systems.

Let U = (U (n))n∈N be any admissible for system (1) feedback control. Then, for the closed-loop system (3), we can define the Lyapunov spectrum   λ(A + BU ) = λ1 (A + BU ), . . . , λs (A + BU ) ∈ Rs≤

and the Lyapunov irregularity coefficient σL (A + BU ). Definition 10. The Lyapunov spectrum and the Lyapunov irregularity coefficient of system (3) are called simultaneously proportionally locally assignable if there exist  > 0 and δ > 0 such that for all     µ = µ1 , . . . , µs ∈ Oδ λ(A) and σ ∈ [0, δ) there exists an admissible for system (1) feedback control U , such that estimate   (7) U ∞ ≤  max σ, |λ1 (A) − µ1 |, . . . , |λs (A) − µs | and equalities λ(A + BU ) = µ, σL (A + BU ) = σ are satisfied.

The notion of “dynamically equivalent systems” is wellknown in the theory of Lyapunov exponents of linear differential systems (Adrianova (1995)) and has been used to reduce such a system to a simpler one (for example an upper triangular). In this theory, a crucial role is played by the fact that equivalent systems have the same Lyapunov spectrum (see, for example, Barreira and Pesin (2002), (Ga˘ıshun, 2001, p. 125)). Thus, the Lyapunov spectrum is a Lyapunov invariant. In addition, Lyapunov invariants are such properties of linear systems as stability, asymptotic stability, exponential stability, and many others. Now we consider two more characteristics of system (2) and prove that they are Lyapunov invariants.   Definition 7. Let λ1 (A), . . . , λs (A) be the Lyapunov spectrum of system (2). The number σL (A) =

s 

i=1 s 

In this paper, we present sufficient conditions for the solvability of the problem of the simultaneous proportional local assignability of the Lyapunov spectrum and the Lyapunov irregularity coefficient for discrete-time systems.

λi (A) − det A

First, we prove an auxiliary result. . . Lemma 11. Let nk = k!, Ik = {nk + 1, nk + 2, . . . , nk+1 },

n−1  1   = λi (A) − lim inf ln det A(k) n→∞ n i=1

ψ(1) = 1,

k=1

Then

is called the Lyapunov irregularity coefficient of system (2). System (2) is said to be regular if its Lyapunov irregularity coefficient is equal to zero. Lemma 8. The Lyapunov irregularity coefficient is a Lyapunov invariant.   Proof. Let L = L(n) n∈N be a Lyapunov sequence, such that the Lyapunov transformation (4) establishes the dynamic equivalence between systems (2) and (5). Then, λ(A) = λ(C) and σL (C) =

s  i=1

=

s  i=1

lim inf n→∞

λi (A) − lim inf n→∞

1 n

k=1

n−1 1 ψ(j) = −1, n j=1

lim sup n→∞

for n ∈ Ik .

n−1 1 ψ(j) = 1. n j=1

(8)

Proof. Since −1 ≤ ψ(n) ≤ 1 for all n ∈ N, it follows that lim inf n→∞

n−1 1 ψ(j) ≥ −1, n j=1

lim sup n→∞

n−1 1 ψ(j) ≤ 1. n j=1

(9)

For each m ∈ N, we have the inequalities n2m−1 n −1 n2m  −1 1   1 2m ψ(j) ≤ 1+ ψ(j) −1 ≤ n2m j=1 n2m j=1 j=n +1

λi (C) − det C n−1 

ψ(n) = (−1)k

2m−1

  lndet L(k + 1)A(k)L−1 (k)

=

n−1  1   ln det L(k + 1)L−1 (k) = σL (A) − lim n→∞ n

=

k=1

n2m−1 1  

n2m

1 

j=1

n2m

j=n2m−1 +1

(−1)





n2m−1 − n2m + n2m−1 + 1

=−1+ 42

1+

n2m −1

2 2(2m − 1)! + 1 2n2m−1 + 1 < −1 + , = −1 + n2m (2m)! m

IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018 Svetlana N. Popova / IFAC PapersOnLine 51-32 (2018) 40–45

therefore, lim

m→∞

1 n2m

n2m −1 j=1

and the function ψ(·) is defined in Lemma 11. Then, the inclusion νs (n) ∈ [µs − λs (A) − σ, µs − λs (A)] is valid, therefore   |νs (n)| ≤ |µs − λs (A)| + σ ≤ 2 max |µs − λs (A)|, σ . Let us denote   α = 2 max σ, |µ1 − λ1 (A)|, . . . , |µs − λs (A)| . Then, we have the estimates

ψ(j) = −1.

Taking into account the first inequality (9), we obtain the first equality (8). To prove the second equality (8) we use the inequalities n2m+1 −1 n2m+1 −1  n2m   1 1  1≥ 1 ψ(j) ≥ (−1) + n2m+1 j=1 n2m+1 j=1 j=n +1

R − I∞ = sup max |eνj (n) − 1| n∈N j=1,...,s  |νj (n)|  ≤ sup max e − 1 ≤ eα − 1 ≤ e2β − 1 = δ,

2m

1





−n2m + n2m+1 − n2m − 1 n2m+1 3 2(2m)! + 1 2n2m + 1 >1− . =1− =1− n2m+1 (2m + 1)! 2m + 1 From these inequalities and from the second inequality (9) we obtain the required equality. =

n∈N j=1,...,s

which entail the inclution R ∈ Rδ . Observe that the function f (t) = (et − 1)/t is strictly increasing for t > 0 therefore and  for any α  ∈ (0, 2β) the following inequality eα − 1 /α < e2β − 1 /(2β) holds. It implies that

Now we are ready to formulate and prove the main theorem. Theorem 12. If system (1) is uniformly completely controllable and the free system (2) is regular, then system (3) has the property of simultaneous proportional local controllability of the Lyapunov spectrum and the Lyapunov irregularity coefficient.

eα − 1 <

e2β − 1 α. 2β

Consequently,

  V ∞ ≤ lR − I∞ ≤ l eα − 1 (13) e2β − 1 . α = l1 α. ≤l 2β We apply the inverse transformation y = L(n)x to system (11) and obtain the system   x(n + 1) = A(n) + B(n)V (n)L(n) x(n). Let . U (n) = V (n)L(n), n ∈ N, then A(n) + B(n)U (n)   −1 = L (n + 1) C(n) + L(n + 1)B(n)V (n) L(n).   Since the sequence V (n) n∈N is an admissible control for   system (10) and L(n) n∈N is a Lyapunov sequence, it   follows that A(n)+B(n)U (n) n∈N is a Lyapunov sequence   and U (n) n∈N is an admissible control for system (1). Let L∞ ≤ l0 . Then, from estimate (13), we obtain .  U ∞ ≤ l0 V ∞ ≤ l0 l1 α = α 2   =  max σ, |µ1 − λ1 (A)|, . . . , |µs − λs (A)| .

Proof. We reduce system (2) by the Perron transformation (4) (Ga˘ıshun, 2001, p. 70) to a system (5) with  the upper triangular matrix C = C(n) n∈N , C(n) = {cij (n)}si,j=1 . System (5) is regular by Corollary 9. Using Lyapunov theorem on the regularity of the triangular system (Ga˘ıshun, 2001, p. 67), we find that the Lyapunov spectrum of this system is formed by the exact mean values of the diagonal elements of the matrix C(·) and coincides with the Lyapunov spectrum of system (2). Without loss of generality we assume that n−1 1 λj (A) = λj (C) = lim ln |cjj (n)|, j = 1, . . . , s. n→∞ n k=1

The transformation (4) transforms system (1) into the system y(n + 1) = C(n)y(n) + L(n + 1)B(n)u, (10) which is uniformly completely controllable (Halanay and Ionescu, 1994, Proposition 3, p. 34). We apply Theorem 3 to this system and obtain that there exist δ > 0 and  l > 0, such that for every sequence R = R(n) n∈N ∈ Rδ there exists an admissible for (10) feedback control V =   V (n) n∈N , such that V ∞ ≤ lR − I∞ and the system   y(n + 1) = C(n) + L(n + 1)B(n)V (n) y(n) (11) is dynamically equivalent to the system z(n + 1) = C(n)R(n)z(n). (12) Let β = ln(1 + δ)/2. We take   any number  σ ∈ [0, β), any sequence µ = µ1 , . . . , µs ∈ Oβ λ(A) , and define a sequence R = R(n) n∈N by the equality   R(n) = diag eν1 (n) , . . . , eνs (n) , where νj (n) = µj − λj (A), j = 1, . . . , s − 1,  σ νs (n) = µs − λs (A) + ψ(n) − 1 , 2

43

Of this manner, the inequality (7) is satisfied.

Systems (3) and (12) are dynamically equivalent, so the equalities λ(A + BU ) = λ(CR), σL (A + BU ) = σL (CR) are valid. Let us prove that λ(CR) = µ. To this end, we consider a truncated system (14) η(n + 1) = G(n)η(n), n ∈ N, η ∈ Rs−1 , whose matrix G is obtained from the matrix CR by deleting the last row and the last column. Since the matrix G(n) = {gij (n)}s−1 i,j=1 is upper triangular and its diagonal elements have exact mean values, system (14) is a regular one and its Lyapunov spectrum consists of the numbers 43

IFAC CAO 2018 44 Yekaterinburg, Russia, October 15-19, 2018 Svetlana N. Popova / IFAC PapersOnLine 51-32 (2018) 40–45

n

1 λj (G) = lim ln |cjj (n)eνj | n→∞ n j=1

Let i ∈ {1, . . . , s−1}. Suppose that for all j ∈ {i+1, . . . , s} the inequality λ[zjs ] ≤ µs is proved. Let us prove that λ[zis ] ≤ µs . Indeed,   −1 s s λ h−1 i fij zj ≤ λ[hi ] + λ[zj ]

= λj (A) + νj = µj , j = 1, . . . , s − 1. Let η 1 (·), . . . , η s−1 (·) be a normal fundamental basis of solutions of system (14), such that λ[η j ] = µj , j = 1, . . . , s − 1. For each n ∈ N and j ∈ {1, . . . , s − 1}, we denote by z j (n) the vector for which the first (s − 1) coordinates coincide with the corresponding coordinates of the vector η j (n), and the last coordinate is 0. Then z 1 (·), . . . , z s−1 (·) are linearly independent solutions of system (12), and λ[z j ] = µj , j = 1, . . . , s − 1. Let us find the Lyapunov exponent of the solution z s (·) of system (12) with the initial condition z s (1) = es . First, we write down the exact formulae for the coordinates z1s , . . . , zss of this solution. We introduce the notation C(n)R(n) = F (n) = {fij (n)}si,j=1 , n ∈ N, hj (n) =

n−1 

fjj (k),

n−1 1 ln |cii (k)| − νi + λ[zjs ] n→∞ n

= − lim

k=1

≤ −λi (A) − µi + λi (A) + µs = µs − µi . Since µs − µi ≥ 0, we have the inequality (Ga˘ıshun, 2001, pp. 50–51) n−1 

λ

k=1

 s h−1 (k)f (k)z (k) ≤ µs − µi . ij j i

It follows from (15) that λ[zis ] ≤ λ[hi ] +

max

j=i+1,...,s

n−1 

λ

≤ µi + µ s − µ i = µs

k=1

for all i ∈ {1, . . . , s − 1}.

i=1,...,s

Let us prove that the solutions z 1 (·), . . . , z s (·) form a normal fundamental basis of  system (12). We take an arbitrary nonzero sequence c1 , . . . , cs and calculate the Lyapunov exponent of the linear combination z(n) = s  ci z i (n).

The coordinate zss (·) is a solution of the scalar linear homogeneous equation zss (n + 1) = fss (n)zss (n), therefore, the equality n−1  fss (k)zss (1) = hs (n), n ∈ N, zss (n) =

i=1

If the coeffitient cs is not equal to 0, then for the s-th coordinate of the solution z(·) we have zs (n) = cs zss (n) for all n ∈ N. It follows that µs ≥ λ[z] ≥ λ[zs ] = λ[zss ] = µs , therefore z(n) ≡ 0 and λ[z] = max{λ[z i ] : ci = 0} in this case.

k=1

holds.

For each i ∈ {1, . . . , s−1}, the coordinate zis (·) is a solution of the linear nonhomogeneous equation s  s s zi (n + 1) = fii (n)zi (n) + fij (n)zjs (n),

If cs = 0, then the s-th coordinate of the solution z(·) is identically equal to 0, and the first (s − 1) coordinates of this solution coincide with the linear combination s−1  η(n) = ci η i (n) of the solution of system (14). Since

j=i+1 condition zis (1)

= 0, therefore and satisfies the initial from the formula for solutions of a linear nonhomogeneous equation (see, for example, Banshchikova and Popova (2016)) we have s n−1   s (15) h−1 zis (n) = hi (n) i (k)fij (k)zj (k).

i=1

η 1 (·), . . . , η s−1 (·) form a normal basis of system (14), it follows that η(n) ≡ 0 only in the case c1 = · · · = cs−1 = 0 and λ[z] = λ[η] = max{λ[η i ] : ci = 0} = max{λ[z i ] : ci = 0}.

j=i+1 k=1

Thus, the basis z 1 (·), . . . , z s (·) has the property of incompressibility and for this reason it is normal. Therefore, for the Lyapunov spectrum of system (3) the equality λ(A + BU ) = λ(CR) = µ holds.

Now we calculate the Lyapunov exponents of the coordinates of the solution z s (·). For the last coordinate we have equalities n−1  1    λ[zss ] = λ[hs ] = lim sup ln fss (k) = fss n→∞ n

It remains for us to prove that σL (A + BU ) = σ. Indeed,

k=1

n−1 1 = lim sup ln |css (k)eνs (k) | n→∞ n

1 = lim n→∞ n

k=1



Consequently, λ[z s ] = max λ[zis ] = λ[zss ] = µs .

j = 1, . . . , s.

k=1

k=1 n−1 

s h−1 i (k)fij (k)zj (k)

σL (A + BU ) = σL (CR) =

s  j=1

λj (CR) − det(CR)

n−1 s  1   = µj − lim ln |cjj (k)| n→∞ n j=1 j=1 s 

n−1 1 ln |css (k)| + lim sup νs (k) n→∞ n k=1

= λs (A) + µs − λs (A) n−1  1 σ + lim sup ψ(k) − 1 = µs . 2 n→∞ n

− lim inf n→∞

k=1

44

k=1

1 n

n−1 s   k=1 j=1

νj (k)



IFAC CAO 2018 Yekaterinburg, Russia, October 15-19, 2018 Svetlana N. Popova / IFAC PapersOnLine 51-32 (2018) 40–45

=

s  j=1

−

µj −

s  j=1

σ 1 lim inf 2 n→∞ n

λj (A) −

n−1  k=1

s   j=1

µj − λj (A)

 ψ(k) − 1 = σ.

and therefore the closed loop-system (18) is unstable for all ν ∈ (0, δ). In that case we may destabilize system (16) by a small feedback.



If we choose σ = 0, then the closed-loop system (18) remains regular. If σ > 0, then we get the irregular system (18).

Example. Consider a model of digital positioning system, described in (Kwakernaak and Sivan, 1972, Example 6.2, pp. 447 – 448). After discretization, we obtain the following discrete-time linear system: x(n + 1) = A(n)x(n) + b(n)u(n), (16) where     1 0, 08015 0, 003396 A(n) ≡ A = , b(n) ≡ b = . 0 0, 6313 0, 06308

REFERENCES Adrianova, L.Y. (1995). Introduction to Linear Systems of Differential Equations. Translations of mathematical monographs, 0065-9282; vol. 146. American Mathematical Soc., Providence, R.I. Babiarz, A., Banshchikova, I., Czornik, A., Makarov, E., Niezabitowski, M., and Popova, S. (2017a). Necessary and sufficient conditions for assignability of the Lyapunov spectrum of discrete linear time-varying systems. IEEE Transactions on Automatic Control. Accepted for publication. Babiarz, A., Banshchikova, I., Czornik, A., Makarov, E., Niezabitowski, M., and Popova, S. (2017b). Proportional local assignability of Lyapunov spectrum of linear discrete time-varying systems. SIAM Journal on Control and Optimization. Unpublished. Babiarz, A., Czornik, A., Makarov, E., Niezabitowski, M., and Popova, S. (2017c). Pole placement theorem of discrete time-varying linear systems. SIAM Journal on Control and Optimization, 55(2), 671–692. Banshchikova, I.N. and Popova, S.N. (2016). On the spectral set of a linear discrete system with stable Lyapunov exponents. Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 26(1), 15–26. Barreira, L. and Pesin, Y.B. (2002). Lyapunov Exponents and Smooth Ergodic Theory. University Lecture Series. American Mathematical Society. Dickinson, B. (1974). On the fundamental theorem of linear state variable feedback. IEEE Transactions on Automatic Control, 19(5), 577–579. Elaydi, S.N. (1999). An Introduction to Difference Equations. Springer-Verlag, New York. Ga˘ıshun, I.V. (2001). Discrete-Time Systems. Institut Matematiki NAN Belarusi, Minsk. Gohberg, I., Kaashoek, M., and Kos, J. (1996). Classification of linear time-varying difference equations under kinematic similarity. Integral Equations and Operator Theory, 25(4), 445–480. Halanay, A. and Ionescu, V. (1994). Time-Varying Discrete Linear Systems: Input-Output Operators, Riccati Equations, Disturbance Attenuation. Springer, Berlin. Klamka, J. (1991). Controllability of Dynamical Systems. Kluwer Academic Publishers, Dordrecht. Kwakernaak, H. and Sivan, R. (1972). Linear Optimal Control Systems, volume 1. Wiley-interscience, New York. Makarov, E.K. and Popova, S.N. (2012). Controllability of Asymptotic Invariants of Time-Dependent Linear Systems. Belorusskaya nauka, Minsk. Sontag, E.D. (2013). Mathematical Control Theory: Deterministic Finite Dimensional Systems, volume 6. Springer Science & Business Media.

For this system, the condition   0, 003396 0, 008451862 det[b, Ab] = det ≈ −0, 0004 = 0 0, 06308 0, 039822404

of complete controllability is satisfied and the length of the segment of uniform complete controllability is K = 2 (see (Elaydi, 1999, p. 433), (Ga˘ıshun, 2001, pp. 238–239)). We note that the free system x(n + 1) = Ax(n), n ∈ N, x ∈ R2 , (17) is regular and the Lyapunov spectrum λ(A) consists of logarithms of the diagonal elements of the matrix A, i.e. λ1 (A) = ln 0, 6313 < 0, λ2 (A) = 0. The free system is stable but not asymptotically (Elaydi, 1999, p. 187). Its solution with the initial condition x(0) = x0 e1 is constant for all x0 ∈ R. Closing system (16) by the linear feedback u(n) = U (n)x(n),   where U (n) = u1 (n), u2 (n) ∈ R1×2 , we obtain the following closed-loop system   x(n + 1) = A + bU (n) x(n) (18)   1 + 0, 003396u1 (n) 0, 08015 + 0, 003396u2 (n) x(n). = 0, 06308u1 (n) 0, 6313 + 0, 06308u2 (n)

Since for system (16) the conditions of Theorem 12 are fulfilled, it follows that there  exist δ > 0 and > 0, such that for any sequence µ = µ1 , µ2 ∈ Oδ λ(A) and any number σ ∈ [0, δ), there exists control U (n), such that   U ∞ ≤ max σ, |µ1 − λ1 (A)|, |µ2 − λ2 (A)| and (19) λ(A + bU ) = µ, σL (A + bU ) = σ.

Let us take σ ∈ [0, δ), µ1 = λ1(A), µ  2 = ν, where |ν| < δ. Then, the condition µ ∈ Oδ λ(A) is satisfied. We can construct the corresponding control U (n) = Uν,σ (n), such   that Uν,σ ∞ ≤ max σ, |ν| and (19) is satisfied.

If ν < 0, then the closed-loop system (18) with the control U (n) = Uν,σ (n) has the following Lyapunov spectrum λ1 (A + bUν,σ ) = ln 0.6313 < 0, λ2 (A + bUν,σ ) = ν < 0, and therefore the closed loop-system (18) is asymptotically stable for all ν ∈ (−δ, 0). We may stabilize system (16) by a small feedback. If ν > 0, then λ1 (A + bUν,σ ) = ln 0, 6313 < 0,

45

λ2 (A + bUν,σ ) = ν > 0, 45