Controller Design for Delay-Independent Stability of Multiple Time-Delay Systems via Déscartes s Rule of Signs

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1 Controller Design for Delay-Independent Stability of Multiple Time-Delay Systems via Déscartes s Rule of Signs Ismail Ilker Delice Rifat Sipahi,1 Department of Mechanical and Industrial Engineering, Northeastern University, Boston, MA USA. ( delicei@coe.neu.edu, rifat@coe.neu.edu). Abstract: A general class of multi-input linear time-invariant (LTI) multiple time-delay system (MTDS) is investigated in order to obtain a control law which stabilizes the LTI-MTDS independently of all the delays. The method commences by reformulating the infinite-dimensional analysis as a finitedimensional algebraic one without any sacrifice of accuracy and exactness. After this step, iterated discriminant allows one to construct a single-variable polynomial, coefficients of which are the controller gains. This crucial step succinctly formulates the delay-independent stability (DIS) condition of the controlled MTDS based on the roots of the single-variable polynomial. Implementation of the Déscartes s rule of signs then reveals, without computing these roots, the sufficient conditions on the controller gains to make the LTI-MTDS delay-independent stable. Case studies are provided to demonstrate the effectiveness of the proposed methodology. Keywords: Multiple time-delay systems, Delay-independent Stability, Controller synthesis, Iterated discriminant. 1. INTRODUCTION Time-Delay Systems (TDS) arise in many applications from diverse areas such as economy, biology, machine tool chatter, population dynamics, communication systems; see Loiseau et al. (2009); Gu et al. (2003) and the references therein for other applications. Asymptotic stability analysis of and control synthesis for TDS are nontrivial tasks even for the linear time-invariant (LTI) problems, primarily due to the infinite dimensional nature of the TDS. Considerable number of results are reported along these lines, see for instance, Stépán (1989); Moon et al. (2001); Fridman and Shaked (2002); Gu et al. (2003); Sipahi and Olgac (2005); Silva et al. (2001); Michiels et al. (2002); Fridman et al. (2003); Wu and Ren (2004); Michiels and Niculescu (2007); Gündeş et al. (2007). In this paper, we study one of the open problems in the literature, namely the control synthesis of LTI multiple time-delay systems (MTDS), departing from the frequency domain analysis. The objective here is to find controllers that render the stability of LTI-MTDS insensitive to any delays in the closed-loop, that is, LTI-MTDS becomes delay-independent stable (DIS). We investigate this problem on the general class of multi-input LTI systems, ẋ(t) = A x(t) + B u(t), (1) where A R N N and B R N L are the constant system and control matrices, respectively; system (1) is assumed to be controllable, x(t) R N is the state vector, and the controller u(t) is affected by multiple nonnegative delays τ l 1 This research has been supported in part by the award from the National Science Foundation ECCS and Dr. R. Sipahi s start-up funds available at Northeastern University. u(t) = L K l x(t τ l ) R L, (2) l=1 where K l R L N, l = 1,..., L, are the control laws, and we define K = [K 1,..., K L ] R L L N. The control synthesis of the general multi-input LTI-MTDS given by (1)-(2), i.e., the selection of matrix K that stabilizes (1) for some delays τ l is a challenging task, and is addressed in both frequency-domain (Michiels et al., 2002) and in timedomain (Moon et al., 2001; Fridman and Shaked, 2002; Fridman et al., 2003). In this paper, we go one step further and investigate the design methods that reveal K matrix, which render the LTI-MTDS stable independent of all the delays. Similar problems are investigated in the context of H control design based on Lyapunov-Krasovskii framework, see Başer (2003) and the references therein. In this paper, we approach this non-trivial design problem from the frequency-domain stability analysis, which eventually leads to practical and time-efficient algebraic design tools that do not require to solve Linear Matrix Inequalities (LMI). The essence of our approach is as follows. It is known that the imaginary eigenvalues (s = jω) of (1) may cause stability reversals/switches at some delays τ (Datko, 1978). For a given K, if such eigenvalues do not exist for any τ l, and if the delay-free system is asymptotically stable (when all τ l = 0), then the controlled system (1)-(2) is DIS. While there are techniques to test DIS of TDS for a given K, see for instance Kamen (1980); Thowsen (1982); Hertz et al. (1984); Chen et al. (1995); Gu et al. (2001); Wei et al. (2008); Souza et al. (2009); Wu and Ren (2004); Wang and Hu (1999); Wei et al. (2008), application of these techniques to design the unknown K in the presence of multiple delays is an open problem. The reason for this is that the existing work is either

2 applicable for single- or two-delay cases; or feasible for particular subclasses of two-delay problems with no commensurate degrees in delays; or based on frequency sweeping conditions which are time-consuming, graphical-based and not applicable to the general system (1)-(2). Recently, the authors introduced a new approach based on algebraic tools to test DIS property of (1)-(2) (Delice and Sipahi, 2009). Given K, one can check DIS condition of (1)-(2) directly without frequency sweeping and graphical display. This methodology is in principle able to obtain a single-variable polynomial, roots of which declare the necessary and sufficient DIS conditions of (1)-(2). The approach is also flexible and can now be used to relax the controller gains, which eventually become the coefficients of the single-variable polynomial. At this point, Déscartes s rule of signs proves to conveniently reveal the sufficient DIS conditions in designing K, without even solving the roots of the polynomial, but by only inspecting its coefficients. Consequently a practical approach for designing K guaranteeing DIS in (1)-(2) becomes possible. Notations used in the text are standard. We use bold face font for matrices, vectors and sets. R +, R 0+, Z + and N denote the set of positive real numbers, nonnegative real numbers, positive integer numbers and natural numbers (including zero), respectively. Open right half, open left half and imaginary axis of complex plane C are represented by C +, C and jr, respectively, and C + = jr C +. We use s C for the Laplace variable; R(s) for the real part of s and I(s) for the imaginary part of s. M stands for the determinant of a square matrix M; indicates a fixed value of a variable. T = {T l } L l=1 = (T 1, T 2,..., T L ) is the pseudo-delay vector; τ = {τ l } L l=1 is the delay vector and c l is the commensurate degree of τ l. R Tl (p 1, p 2 ) stands for resultant of multinomials p 1 (T ) and p 2 (T ) with eliminating T l. We omit arguments when no confusion occurs. 2. PRELIMINARIES Characteristic function of the system in (1)-(2) is given by: f(s, τ, K) = K P k=0 P k (s, K) e s L l=1 υ kl τ l, (3) where P k are polynomials in terms of s and the entries of K, and K P Z +, υ kl N. The characteristic function (3) possesses infinitely many roots due to the presence of transcendental terms. Since delays cannot pervade through the highest order derivative of the states in (1), characteristic function (3) represents a retarded class LTI-TDS (Stépán, 1989), and thus {υ 0l } L l=1 = 0. Definition 1. Given K, MTDS (1)-(2) is DIS if and only if f(s, τ, K) 0, s C +, τ R L 0+. (4) Notice that checking (4) for a given K is a difficult task, but designing K such that (4) holds is even more difficult given that all or some of the entries of K are free parameters. In the main results, we will demonstrate how we achieve the latter. Exploiting the continuity property of the roots of (3) on C is often preferred in order to reduce the difficulties in analyzing the stability of LTI-MTDS (Datko, 1978). Stability of (3) may change only when the roots cross the imaginary axis. In other words, for detecting the stability transitions, one should analyze the characteristic function on the imaginary axis by setting s = jω, ω R 0+. Given K, the ω values, where s = jω is a root of (3) for some delay values, define the crossing frequency set (CFS) Ω = {ω R 0+ f(jω, τ, K)= 0, for some τ R L 0+}. (5) At this point, we convert the infinite dimensional characteristic function (3) to a finite dimensional characteristic function via the exact Rekasius transformation (Rekasius, 1980), e τ l s := 1 T l s 1 + T l s, s = jω, T l R, l = 1,..., L. (6) Upon substitution of (6) into (3) and with a manipulation to remove the fractions, we obtain the transformed characteristic function, g(ω, T, K) = f(jω, τ, K) L (1+jT l ω) c l, which is a function of ω, T and K. e τ l s := 1 jt l ω 1+jT l ω, l = 1,..., L. l=1 Corollary 1. (Sipahi and Olgac (2005)). Let Ω be the crossing frequency set of g(ω, T, K). The identity Ω Ω holds. Remark 1. Since Ω Ω holds, we prefer to study Ω from the algebraic equation (7) instead of studying Ω from the transcendental equation (3). This is central for the main results below. We finally present Déscartes s rule of signs, which we need in the proof of our main results. Theorem 2. (Sturmfels (2002)). The number of positive real roots of a polynomial is at most the number of sign changes in its coefficient sequence, which is the sequence of the coefficients sorted with respect to ascending/descending powers of the polynomial variable. It is noted that zero (missing) coefficients are ignored when counting the number of sign changes in a sequence. For instance, a sequence +, 0,, 0, +, 0, + has two sign changes (Sturmfels, 2002). Using Theorem 2, the set of conditions, which make the signs of the coefficients of a polynomial identical to each other, can be computed without solving the roots of the polynomial. These conditions guarantee that the polynomial has no positive real roots. Remark 2. If ω = 0 is a root of (3), then system (1) is not DIS, and this possibility can be checked and treated by Fazelinia et al. (2007) in the case of τ l. In the remaining of the text, we neglect such degeneracies, since τ l is not a practical case in control applications. Mathematical details of analyzing τ l are left for another study (Sipahi and Delice, 2010), see also Michiels and Niculescu (2007) for L = 2 delay-sensitivity analysis. We also note that ω = 0 can be a root of (3) when τ = 0. We prevent this possibility as well, by requiring that the delay-free system is asymptotically stable, that is, A + B L l=1 K l being Hurwitz stable should be satisfied as a necessary condition for DIS. This condition automatically guarantees that a feasible K exists. 3. MAIN RESULT In the sequel, we present the theoretical framework, which enables a practical and direct computation of the controller (7)

3 gains that render (1) DIS. For this purpose, we first define the resultant and discriminant, and next present the theorems with their proofs. For a given K, the characteristic equation (7) can be written as g(ω, T ) = g R (ω, T ) + j g I (ω, T ) = 0. (8) When (8) holds, its real g R and imaginary part g I are concurrently zero and g R = g I = c L i=0 c L i=0 a i (T 1,..., T L 1 ) T i L = 0, a cl 0, (9) b i (T 1,..., T L 1 ) T i L = 0, b cl 0. (10) Now, we utilize the theory of resultant to eliminate T L, without loss of generality, from the two polynomials g R = 0 and g I = 0 (Gelfand et al., 1994). A 2c L -order Sylvester matrix is constructed by eliminating T L, and its determinant is a cl a cl 1... a a cl a cl 1.. a 1 a R TL (g R, g I ) = a 1 a 0. (11) b cl b cl 1... b b cl b cl 1.. b 1 b b 1 b 0 When polynomials g R and g I have common zeros, it is necessary that the resultant is zero. Corollary 3. (Delice and Sipahi (2009)). Let all the ω, T 1,..., T L 1 roots of the resultant R TL (g R, g I ) constitute the set V = {(ω, T 1,..., T L 1 ) C L R TL (g R, g I ) = 0}, and let all (ω, T ) R L+1 roots of g(ω, T ) = 0 define the set V = {(ω, T ) R L+1 g(ω, T ) = 0}, then the projections of all the points in V onto ω, T 1,..., T L 1 C L space are a subset of V. Next, we provide the definition of discriminant. Definition 2. (Gelfand et al. (1994); Sturmfels (2002)). Let F = F (µ l ) = F (ν, µ 1, µ 2,...), then the discriminant of the polynomial F with respect to µ l is defined as D µl (F ) R µl (F, F/ µ l ). (12) We are now ready to present the theorems which allow us design the controller gains to make (1) DIS. Theorem 4. For a given K, the minimum and maximum positive real zeros of the iterated discriminant D(ω) D T1 (D T2 (... DTL 1 (R TL (g R, g I )) )), (13) that correspond to T R L are the exact finite lower and upper ω bounds of the crossing frequency set Ω, respectively. ω Proof 1. As per Corollary 3, all ω that give rise to s = jω solution in (3) also satisfy R TL (g R, g I ) = 0 for some ω, T 1,..., T L 1 where a mapping to T L exists through (9)- (10). It is therefore sufficient to seek and ω by studying R ω TL (g R, g I ) = 0. For the minima/maxima of ω to exist, it is necessary that ω/ T L 1 = 0, which can be written for regular points by invoking the implicit function theorem (Courant, 1988) ω = R T L / T L 1 T L 1 R TL / ω = 0. (14) The regular points of R TL (g R, g I ) = 0 satisfy R TL / ω 0, hence we have R TL / T L 1 = 0. One can now search for the common solutions between R TL = 0 and R TL / T L 1 = 0. Among these solutions lie the and ω for some T 1,..., T L 1. For this search, one can eliminate ω T L 1 by constructing R TL 1 (R TL, R TL / T L 1 ) via (11). Notice that and ω solutions are now embedded into the solutions of R ω TL 1 = 0 with corresponding T 1,..., T L 2, and mappings to T L 1 and T L via R TL = 0 and g = 0, respectively. If and ω exist, then it is necessary that ω/ T ω L 2 = 0, which can be analyzed for regular points with the same logic used in (14). The repetition of the same procedure until only the parameter ω survives and all the T l are eliminated leads to the following single-variable polynomial in ω, D(ω) R T1 (R T2 (... RTL 1 (R TL, R TL / T L 1 )) )), which is (13) as per Definition 2. The minima/maxima, and ω, if they exist, are among the roots of D(ω). Then the entries ω of T R L can be sequentially sought via the mappings R T2 = 0, R T3 = 0,..., g = 0. Consequently, minimum and maximum positive real zeros of the polynomial D(ω) that correspond to T R L give rise to the exact lower and upper ω bounds ω of the CFS, respectively, and these zeros are finite (Delice and Sipahi, 2009). We note that the above theorem is valid at singular points of the resultants as well, except at those singular points arising from repeated factors of R Tl = 0. When R Tl has repeated factors, Theorem 4 needs a slight modification as per Abhyankar (1990). We state that this modification does not affect the subsequent developments we present below. Excluding ω = 0 roots of D(ω) (see Remark 2), it can be proven that D(ω) is even in terms of ω. After relaxing the controller law K, this polynomial reads K ω D(ω) = α 2k (K) ω 2k, (15) k=0 where α 2k (K) coefficients are in terms of the controller gains in K, and K ω Z +. Theorem 5. MTDS in (1)-(2) is stable independent of delays in the L-D delay domain if all α 2k (K) in (15) have the same sign, and A + B L l=1 K l is Hurwitz stable. Proof 2. According to Déscartes s rule of signs in Theorem 2, if all the coefficients of the even polynomial (15) have the same sign, then there exists no positive real ω roots of (15). Having no positive real roots of (15) indicates that all ω solutions are complex conjugates since D(ω) is an even polynomial. When there exists no positive real roots, we have Ω = from Theorem 4. Since Ω = Ω as per Corollary 1 and Ω (CFS) generates the stability transitions, CFS being empty set indicates that there are no stability transitions for all delays τ R L +, and the entire L-D delay domain exhibits the delay-free system s stability behavior, which is stable by construction. Note that Theorem 5 requires us to inspect the coefficients of the polynomial D(ω) without solving the roots of D(ω). This choice leads to sufficient conditions, however, it offers a practical control synthesis approach constructed by algebraic tools. This is the main contribution of the article, and solving D(ω) for finding both necessary and sufficient conditions are left to future work. Remark 3. It is shown in Michiels et al. (2002) for single delay problems that for a TDS to be DIS, it is necessary that A matrix

4 is Hurwitz. Connection of this result to the DIS design of (1)-(2) indicates that the same condition holds for A in (1). Checking this condition, however, can be by-passed in our study since Ω = guarantees that A is Hurwitz. 4.1 Case Study 1: 4. CASE STUDY Consider the MTDS in (1) given by A = [ ] 0 1, B = 6 a 1 [ ] 1 0, (16) 0 1 where a 1 = 7.1, and the controller is given by 2 u(t) = K l x(t τ l ), (17) l=1 [ ] [ ] where K 1 =, K k =. The characteristic 0 k 2 function of the closed loop system is f(s, τ, K) = s s + 6 k 1 e τ1 s k 2 s e τ2 s, (18) and it is easy to see that the delay-free system (when τ 1 = τ 2 = 0) is stable for k 1 < 6 and k 2 < 7.1. Our approach commences with the manipulation in (7) for the two delays. We then find (8), and then construct R T2 via (11) by eliminating T 2. Next, the discriminant of R T2 is calculated with respect to T 1, D T1 (R T2 ). This operation is the iterated discriminant procedure introduced in Theorem 4, and it leads to a single-variable polynomial (ignoring ω = 0 as noted earlier) given by 6 D T1 (R T2 ) = D(ω) = α 2k (k 1, k 2 ) ω 2k, (19) k=0 where α 2k (k 1, k 2 ) are listed in the Appendix. Implicit functions α 2k (k 1, k 2 ) are drawn on k 1 k 2 domain next, see Figure 1. Since α 12 = 1 > 0, the shaded region in Figure 1 is found by imposing the positivity of all α 2k as well as by maintaining the stability of the delay-free system. As per Theorem 5, we conclude that (k 1, k 2 ) pairs chosen from the shaded region guarantee that system in (16) with delayed state-feedback law in (17) is delay-independent stable. In order to validate our result in Figure 1, the numerical toolbox DDE-BIFTOOL (Engelborghs, 2000) is implemented on the same system (16)-(17). Although DDE-BIFTOOL is not designed for DIS test, we proceed to a case study where τ 1 and τ 2 are chosen as 100. The rightmost root distribution of (16)- (17) is found with respect to k 1 k 2, and is depicted in Figure 2 using color coding that indicates the number of unstable roots. The white region corresponds to the case when this number is zero, that is, when the closed loop system is stable. Although Figure 2 is not conclusive to fully validate Figure 1, it provides a certain level of confidence. We now analyze the effects of damping ratio in the open loop system to the shaded DIS region in Figure 1. The boundaries of the DIS regions are extracted for different a 1 values and are depicted in Figure 3. When a 1 = 7.1, a 1 = 4.5, and a 1 = 3.4, the corresponding damping ratios are ξ > 1 (solid black curve), ξ = (dashed red curve), and ξ = (dotted blue curve), respectively. Controller gains chosen from the closed regions in Figure 3 make the system delay-independent stable for the given a 1 parameter or equivalently the damping ratio. Inspection of Figure 3 shows that DIS regions in the space of controller gains are bounded. These results are consistent with the earlier work (Michiels and Niculescu, 2007) on bounded sets of stabilizing gains. We finally present in Figure 4 the real part σ of the right most root with color code. In this figure, the boundary of the DIS region is displayed. With a second-order system assumption, Fig. 2. Comparison of the proposed method (color curves) and DDE-BIFTOOL result (gray shaded regions) for τ 1 = 100 and τ 2 = 100 on k 1 k 2 domain. Gray color coding indicates the number of unstable roots. White region indicates stability. Fig. 1. Boundaries formed by α 2k (k 1, k 2 ) coefficients. Controller gains from the shaded region render the system delay-independent stable (DIS). Fig. 3. DIS regions are obtained for a 1 = 7.1 (outer curve, damping ratio ξ > 1), a 1 = 4.5 (dashed red curve, damping ratio ξ = ), and a 1 = 3.4 (inner curve, damping ratio ξ = 0.694). Controller gains from the closed regions render the system delay-independent stable for a given a 1 parameter.

5 Reference 2 ts wn e PD + s 2xw w 2 2 n n Output Fig. 6. Block diagram of closed-loop system, ξ > 0, ω n > 0. Fig. 4. DDE-BIFTOOL result in Case 1 for τ 1 = 0.1 and τ 2 = 0.15 on k 1 k 2 domain. Gray color coding indicates the real part σ of the rightmost root. it is easy to see that settling time 4/σ of the closed-loop system improves for some controller gain pairs chosen from the enclosed DIS region. This is an interesting result as it shows that a closed-loop system can be made DIS while still improving its settling time performance. 4.2 Case Study 2: We consider the same MTDS in case study 1, but this time, we take the controller law as [ ] K =. (20) k k 2 0 Following the procedure in case study 1, the boundaries α 2k (k 1, k 2 ) = 0 and the delay-free system s stability conditions (black color) are drawn in Figure 5. As per Theorem 5, we state that (k 1, k 2 ) pairs chosen from the shaded region guarantee that system in (16) is delay-independent stable. 4.3 Case Study 3: Our methodology is also applicable to single delay DIS problems. Consider the block diagram in Figure 6. The characteristic function of the closed-loop system is f(s, τ, K) = s ξ ω n s + ωn 2 + (k p + k d s) ωn 2 e τ s, (21) where ξ > 0, ω n > 0; k p and k d are the proportional and derivative gains of the PD controller, respectively. It is easy to see that the delay-free system is asymptotically stable for k p > 1 and k d > 2 ξ/ω n. Let ω n = 1 and follow the procedure as in case study 1 to obtain D(ω) (ignoring ω = 0 as noted earlier) D(ω) = ω 4 + ( 2 k 2 d + 4 ξ 2 ) ω k 2 p. (22) As per Theorem 5, we conclude that the closed-loop system in Figure 6 is delay independent stable if k p < 1, k d < 2 ξ and ξ > We further analyze the effect of natural frequency on DIS condition. Let ω n = 5, then the DIS condition is found as k p < 1, k d < 0.4 ξ and ξ > Notice that the condition on the proportional gain of the PD controller does not change with the natural frequency, and the range of the derivative gain of the controller changes inversely proportional to the magnitude of the natural frequency. Finally, note that sufficient amount of damping ratio, ξ > , is needed for DIS, independently of the natural frequency. Remark 4. Given the complications in assessing DIS of linear time-invariant multiple time-delay systems, our procedure based on Theorem 5 is efficient. It solves the control synthesis problem under 0.3 seconds on average for all the three cases. 5. CONCLUDING REMARKS A new approach is presented to synthesize control laws that render the most general multi-input linear time-invariant multiple time-delay system (MTDS) delay-independent stable (DIS). This is achieved with Rekasius substitution and iterated discriminants. The approach leads to computationally efficient practical tools to compute the set of controller gains with sufficient DIS conditions. APPENDIX The coefficients of (19) with 4-digit precision are as follows, α 0 (k 1, k 2 ) = k k k k k k = (k 1 6) 2 (k 1 + 6) 4 > 0, α 2 (k 1, k 2 ) = 2 k k k k k k k 1 k k 2 1 k k 3 1 k k 4 1 k Fig. 5. Implicit functions of α 2k (k 1, k 2 ) coefficients and delayfree system stability condition (black color). Controller gains from the shaded region render the system DIS. ( = 0.01 (k 1 + 6) k k k 1 ) +300 k1 2 k k2 2 k k ,

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