- Email: [email protected]
Robust Stability and Constrained Stabilization of Discrete-Time Delay Systems*
Proceedings of the 10-th IFAC Workshop on Time Delay Systems The International Federation of Automatic Control Northeastern University, Boston, USA. June 22-24, 2012
Robust Stability and Constrained Stabilization of Discrete-Time Delay Systems Bahram Shafai, Hanai Sadaka and Rasoul Ghadami Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA. [email protected]
[email protected]
[email protected]
Abstract: This paper revisits the problem of robust stability and stabilization of uncertain time-delay systems. We focus on the class of non-negative discrete-time delay systems and show that it is asymptotically stable if and only if an associated non-negative system without delay is asymptotically stable? . This fact allows one to establish strong result on robust stability and stability radius for this class of systems. An alternative representation of delay systems is also constructed whereby its system matrix is in block companion from. Under the assumption of non-negativity for delay systems, this alternative form represents a conventional non-negative system and similar strong robust stability results are derived. Finally, we consider the problem of constrained stabilization and provide a new LMI feasibility solution for it. This makes it possible to stabilize a general discrete-time delay system such that the closed-loop system admits non-negative structure with desirable properties. 1. INTRODUCTION
for stability, robust stability direct computation of stability radius, and constrained stabilization which resemble to the case of conventional positive systems without delay.
The problem of stability and the control design for delay systems attracted many researchers for the past several decades and excellent books have been published in this area (Gu et al. [2003], Niculescu [2001] and Magdi [2000]). Literature reports various methods for stability and stabilization of delay systems which are either extending the classical transfer function approach (Olgac and Sipahi [2002]) or newly methods based on LMI (Magdi [2000]).
In this paper, we develop a parallel strategy for establishing similar results for the non-negative discrete-time delay systems. In particular, we show that the non-negative discrete-time delay systems is asymptotically stable if and only if an associated non-negative discrete-time system without delay is asymptotically stable. This fact allows one to obtain strong results on robust stability and stability radius for this class of systems. Based on an alternative representation of delay systems, similar results are also derived. Finally, the problem of constrained stabilization is considered. We provide a new LMI feasibility solution for it which makes it possible to stabilize a general discretetime delay system such that the closed-loop system admits non-negative structure with desirable properties.
Among the special types of systems, the class of Metzlerian and nonnegative systems play important role for continuous-time and discrete-time systems (Berman et al. [1989], Kaczorek [2002], Farina and Rinaldi [2000], Haddad and Chellaboina [2004]). The robust stability and constrained non-negative stabilization of regular systems were introduced by Shafai et al. in several papers ( [1991a,b], [1997], see also the references therein) and the results were also published in (Bhattacharyya et al. [1995]). The aim of this paper is to present techniques for robust stability analysis and control of positive time delay systems. Motivated by the problem of observer design for time delay systems (Rami et al. [2007]) and simple stability result of positive delay systems (Buslowicz [2008]) we provide additional insights into the robust analysis and stabilization with positivity constraints. In a recent paper we established strong result for the class of continuous-time Metzlerian delay systems (Shafai and Sadaka [2012]) based on the fact that asymptotic stability with delay is equivalent to the stability of an associated Metzlerian system without delay. Consequently, important results have been derived
2. PRELIMINARY RESULTS Consider the linear discrete-time systems with delays described by x(k + 1) = A0 x(k) +
Ai x(k − i) + Bu(t)
(1)
i=1
y(k) = Cx(k) + Du(k)
(2) n
with the initial conditions x(−k) ∈ R ; k = 1, 2, . . . , l, where l is a positive integer, x(k) ∈ Rn , u(k) ∈ Rm , y(k) ∈ Rp are the state, input and output vectors; respectively, and Ai ∈ Rn×n , i = 0, 1, . . . , l; B ∈ Rn×m ; C ∈ Rp×n , D ∈ Rp×m are system matrices.
? In this paper, it is important to distinguish two terminologies. The discrete-time delay systems in the absent of delay states is referred to as delay-free systems and when the delay arguments are absent, we use the notion “discrete-time systems without delay”.
978-3-902823-04-5/12/$20.00 © 2012 IFAC
l X
The system (1) is asymptotically stable if and only if all roots of the characteristic equation 31
10.3182/20120622-3-US-4021.00058
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
det(zIn −
l X
Ai z −i ) = 0
An immediate consequence of Lemma 1 is the fact that with non-negative matrices Ai , B, C, D; the equivalent system (4), (5) becomes a regular non-negative system.
(3)
i=0
At this point, it is important to distinguish two terminologies with respect to internally positive (non-negative) discrete-time delay systems. Let (1), (2) be non-negative discrete-time delay system, then in the absent of delay state we have a delay-free system i.e. Ai = 0, ∀i = 1, . . . , l; and when the delay arguments in (1) are absent, we use the notion “non-negative discrete-time system without delay.”
have moduli less than 1. T T T T Defining x ¯(k) = x (k) x (k − 1) . . . x (k − l) , the system (1), (2) can equivalently be written as ¯ x ¯(k + 1) = A¯ x(k) + Bu(k) ¯ y(k) = C¯ x ¯(k) + Du(k)
(4) (5)
Lemma 2: The non-negative discrete-time system without delay x(k + 1) = Ad x(k) + Bu(k) (6) y(k) = Cx(k) + Du(k)
where B . . . Al−1 Al 0 ... 0 0 ¯= . , B .. .. .. .. . . . 0 0 0 ... I 0 ¯ =D C¯ = [ C 0 . . . 0 ] , D N ×N ¯ ∈ RN ×m , C¯ ∈ Rp×N , D ¯ ∈ Rp×m , and with A ∈ R , B N = (l + 1)n.
A0 I A= ...
A1 0 .. .
Pl where Ad = i=0 Ai , is asymptotically stable if and only if one of the following equivalent conditions is satisfied: (1) All eigenvalues of Ad lie within the unit disc or Ad is Schur stable i.e. ρ(Ad ) < 1. (2) All principal minors of the matrix I −Ad are positive. (3) The matrix I − Ad is nonsingular M-matrix and [I − −1 Ad ] > 0. (4) All coefficients a ¯i , i = 0, 1, . . . , n − 1 of the characteristic polynomial det(λI − Ad + I) = λn + a ¯n−1 λn−1 + . . . + a ¯1 λ + a ¯0 are positive. (5) There exists a positive diagonal matrix P > 0 such that ATd P Ad − P < 0 or equivalently T −P Ad P < 0 P Ad −P
The stability and stabilization of the above system have been established by many researchers through the classical techniques using (3) or by applying Lyaponov Theory on (1) leading to a set of LMI conditions. Our goal is to provide simple alternative stability and robust stability results which can effectively be used in the so-called constrained stabilization problem. To be more specific, we intend to give a technique to stabilize the above general linear discrete-time delay system with the condition that the closed-loop system becomes stable and admits a special structure. The special class of systems considered here is positive systems, which has attracted control community from different angles. The delay-free continuous-time positive systems known as Metzlerian systems have several nice properties through the associated Metzler matrix, which is closely related to the class of M-matrices. A Metzler matrix is defined by its non-negative off-diagonal elements. However, for convenience in connection to stability study, it is usually defined in strict sense; namely, by additional assumption that its diagonal elements are negative. If we denote M to be (strictly) Metzler matrix, then its entries are mij > 0, mii < 0 and −M becomes an M-matrix. In this paper we shall concentrate on the class of positive systems for discrete-time case, in which the associated system matrix belong to the class of non-negative or strictly positive Matrices, i.e. if A is non-negative matrix, then its entries aij ≥ 0. For a detailed exposition of conventional non-delay positive systems one should refer to Berman et al. [1989], Kaczorek [2002] and Farina and Rinaldi [2000].
Remark 1: It should be pointed out that all of the equivalent conditions in Lemma 2 apply equally to the matrix A associated with the equivalent delay systems (4), (5) when considering non-negative discrete-time delay systems for (1), (2). It is also not difficult to show that (Kaczorek [2004], [2007]) det(zIN − A) = det(z l+1 In −
l X
Ai z
l−i
)
i=0 N −1
= z N + αN −1 z + . . . + α1 z + α0 (7) and asymptotic stability with respect to the equivalent system is guaranteed if and only if all roots of the characteristic polynomial (7) have moduli less than 1, or A is Schur stable i.e. ρ(A) < 1. 3. ROBUST STABILITY OF NON-NEGATIVE DELAY SYSTEMS The equivalent stability conditions of previous section suggest that the asymptotic stability of non-negative delay system (1), (2) is equivalent to asymptotic stability of its corresponding system (4), (5) without delay. This is also evident from (7) in Remark 1. Motivated by this and a recent result (Rami et al. [2007]), we were able to establish robust stability and constrained stabilization results in Shafai and Sadaka [2012] for continuous-time Metzlerian delay system. Here, we provide similar results for discretetime case.
Definition 1: The discrete-time delay system (1), (2) is called internally positive if for any initial condition x(−k) n and all inputs u(k) ∈ Rm + , we have x(k) ∈ R+ and p y(k) ∈ R+ for all k ∈ Z+ , where Z+ denotes the set of non-negative integers. Lemma 1: The discrete-time delay system (1), (2) is intern×n nally positive if and only if Ai ∈ R+ , i = 0, 1, . . . , l; B ∈ n×m p×n p×m R+ , C ∈ R+ , D ∈ R+ . 32
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
0, 1, . . . , l; where δi1 , Ei1 are redefined as δi , Ei ; respectively. In this case the number of uncertain parameters δi matches with the number of system matrices Ai and the associated vector set of unknown parameters becomes T δ = [ δ0 δ1 . . . δl ] . For simplicity, in this paper we assume qi = q and δir = δr for all i = 0, 1, . . . , l in (13), then q X Ai (δ) = Ai + δr Eir , i = 0, 1, 2, . . . , l (14)
Theorem 1: The non-negative discrete-time delay systems represented by (1), (2) is asymptotically stable if and only n if there exists a strictly positive vector v ∈ R+ such that [Ad − I]v < 0,
where
Ad =
l X
Ai .
(8)
i=0
Furthermore, its asymptotic stability is guaranteed by the associated non-negative discrete-time systems without delay x(k + 1) = Ad x(k) + Bu(k) (9) y(k) = Cx(k) + Du(k)
r=1 T
where δ = [ δ1 δ2 . . . δq ] . Thus, for single and multiple perturbations, δ can be represented as the vector of uncertain parameters confined within a prescribed set of interest Ω, i.e. δ ∈ Ω.
Proof: It is shown in Rami et al. [2007], Shafai and Sadaka [2012] that continuous-time Metzlerian delay system x(t) ˙ = M0 x(t) +
l X
Mi x(t − τi )
Assuming that bounds are available for δr , then it can be shown that the set of possible Ai (δ) matrices {Ai (δ) : δ ∈ Ω} is a polytope. It is possible to develop robust stability results when perturbation matrices Eir are of unity rank using classical techniques or LMI-based method.
(10)
i=1
is asymptotically stable if and only if there exist a positive Pl n vector v ∈ R+ such that Ac v < 0 where Ac = i=0 Mi and consequently the equivalence of asymptotic stability with and without delays was established. Taking advantage of the relationship between continuous-time and discrete-time systems −1
Ad = (I − Ac ) (I + Ac )
This will particularly be simple when (12) represents nonnegative discrete-time delay systems. In such a case, the robust stability of (12) is equivalent to robust stability of its corresponding systems without delay
(11) x(k + 1) =
and substituting (11) in (8) , we need to show that Xv < 0, −1 where X = [(I − Ac ) (I + Ac ) − I]. The last expression −1 can be simplified to X = 2(I − Ac ) Ac . Since Ac is strictly stable Metzlerian, −Ac and obviously I − Ac are −1 M-matrices. Thus (I − Ac ) > 0 and using Ac v < 0, it −1 follows that Xv = 2(I − Ac ) Ac v < 0.
or alternatively it is equivalent to robust stability of (4) N with A ∈ RN + replaced by A(δ) ∈ R+ and its submatrices n Ai (δ) ∈ R+ given by (13). So, let us impose the constraint that the uncertain delay system (12) is non-negative delay system i.e. the matrices n×n Ai (δ) ∈ R+ , i = 0, 1, 2, . . . , l are non-negative for all δ, n×m and B ∈ R+ . Then we have the following result, which can directly be stated with the aid of Corollary 1. Lemma 3: Let the system (12) be uncertain non-negative discrete-time delay system. Then it is robustly stable independent of delay if and only if the corresponding system without delay, i.e. x(k + 1) = Ad (δ)x(k) + Bu(k)
Now, consider the general discrete-time delay systems (1) with parametric uncertainties described by l X
Ai (δ)x(k)
i=0
Corollary 1: The non-negative discrete-time delay system represented by (1), (2) is asymptotically stable if and only if all equivalent conditions of Lemma 2 are satisfied with respect to its non-negative discrete-time system without n delay (9). Furthermore, there exists v ∈ R+ such that (Ad − I)v < 0, which is an additional equivalent condition for its asymptotic stability.
x(k + 1) =
l X
(15)
is robustly stable, where Ai (δ)x(k − i) + Bu(k)
(12) Ad (δ) =
i=0
l X
Ai (δ)
(16)
i=0
where Ai (δ), ∀ i = 0, 1, 2, . . . , l are uncertain matrices defined by affine multiple parameter perturbations of the form qi X Ai (δ) = Ai + δir Eir , r = 1, . . . , qi ; i = 0, . . . , l (13)
is a nonnegative matrix for all δ ∈ Ω. As pointed out above, if Ai (δ) depends affine linearly on δ as given by (13), then under certain mild conditions various vertex-type results can be employed to check the robust stability of the uncertain non-negative delay systems. The simplest possible polytope of matrices is considered to be an interval matrix. Since the interval matrices comprise of a closed and bounded convex set, they can be represented as a convex hull of its vertex matrices. Therefore, one can relate interval and affine uncertainties. Consequently, if A(δ) is represented as interval uncertainty
r=1
where δir are unknown scalers and Eir are given matrices specifying the structure of the perturbations, and T δi = [ δi1 δi2 . . . δiqi ] is the ith sub-vector of uncertain parameters δir , r = 1, . . . , qi confined within a prescribed set of interest Ωi , i.e. δi ∈ Ωi , i = 0, 1, 2, . . . , l. When r = 1, we have an affine single perturbation structure and Ai (δ) reduces to Ai (δ) = Ai + δi Ei , ∀ i = 33
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Next, we consider the general discrete-time delay systems (1) with a more general type of uncertainty structure described by
through the affine perturbation structure leading to interval non-negative delay system or the uncertain nonnegative delay system is directly defined as interval nonnegative delay system, we can write (12) as l X x(k + 1) = [Ai , A¯i ]x(k − i) + Bu(k) (17)
x(k + 1) =
l X
Ai (∆i )x(k − i) + Bu(k)
(20)
i=0
i=0
where Ai ’s in (1) are subjected to affine perturbations of the form
then the following result can be stated Theorem 2: The non-negative interval delay system (17) is robustly stable if and only if the following single nonnegative system without delay, i.e. l X A¯i x(k + 1) = A¯d x(k) + Bu(k), A¯d =
Ai (∆i ) = Ai + Di ∆i Ei
(21)
where Di ∈ Rn×di , Ei ∈ Rei ×n represent the structure of the uncertainties and the matrices ∆i ∈ Rdi ×ei are unknown uncertainty matrices. The robust stability results for (20), (21) is not trivial when general type of discretetime delay systems is considered. Although, stability and robust stability tests in terms of LMIs are available (Gu et al. [2003], Niculescu [2001], and Magdi [2000]), the problem of computing robust stability radius is more complex even in case of conventional delay-free systems. In general, the computation of complex and real stability radii, rC and rR requires the solution of a complicated global optimization problem (Qiu et al. [1995]). However, for the class of positive systems (continuous and discrete cases), the complex and real stability radii coincide and can be computed by closed form expression (Shafai et al. [1997], Hinrichsen and Son [1998]). Further development of this result for general and special class of continuous-time delay systems performed in Sadaka et al. [2007]. Here we concentrate on non-negative discrete-time delay systems and show that the stability radii can simply be computed with the aid of Theorem 1 or alternatively through the equivalent representation (4). It is also important to point out that in computing stability radius, one should always assume that the system without uncertainty is stable.
i=0
is asymptotically stable, and the stability can be verified by checking the positivity of leading principal minors of I− A¯d . Furthermore, the equivalent interval representation of ¯ non-negative interval delay systems (4) with A(δ) = [A, A] ¯ ¯ where A and A are constructed by Ai and Ai for all i = 0, 1, . . . , l, is robustly stable if and only if A¯ satisfies all equivalent conditions stated in Lemma 2. Proof: With the aid of Corollary 1 and Lemma 3, we know that non-negative interval delay systems (17) is asymptotically stable if and only if the non-negative interval discrete-time system without delay x(k+1) = [Ad , A¯d ]x(k) is asymptotically stable. It has also been established in Shafai et al. [1991b] that the robust stability of nonnegative interval systems x(k + 1) = [Ad , A¯d ]x(k) is equivalent to the asymptotic stability of the system with the upper interval, namely x(k + 1) = A¯d x(k). The proof of the second part follows in a similar fashion, namely with ¯ respect to equivalent representation (4), (5); A(δ) = [A, A] represents a non-negative interval matrix and its stability is guaranteed by the stability of A¯ alone.
Theorem 4: Let the uncertain delay system (20) with (21) be a non-negative discrete-time delay system and assume that it is asymptotically stable independent of delay without uncertainty. Then the real and complex stability radii of the uncertain non-negative delay systems (20) coincide and it is given by the following formula n×di ei ×n if Di = Dj ∈ R+ or Ei = Ej ∈ R+ far all i, j = 0, 1, . . . , l 1
(22) rC = rR = Pl −1
max Ei (I − i=0 Ai ) Di
It is also possible to provide robust stability bounds on one or more uncertain parameters assuming the system without uncertainty is stable. One such a result is stated below under a very restricted class of uncertainty structure. More general uncertainty structures will be considered subsequently. Theorem 3: Let the non-negative discrete-time systems (12) be defined with the uncertain structure (13) such that δir are all equal ∀ i = 1, . . . , l and ∀ r = 1, . . . , q defined by r unknown Pq parameter . Furthermore, assume that Ei and H = r=1 Eir represent dyadic (unity rank) matrices with all of whose entries are either 0 or 1, or all elements are ones. Then this special class of non-negative discrete-time system x(k + 1) = (Ad + H)x(k) + Bu(k) (18)
i
(19)
Proof: Using Lemma 3 and Theorem 1, robust stability of (20) is equivalent to robust stability of uncertain nonnegative discrete-time systems without delays. So (9) with uncertainty structure (21) can be written as x(k + 1) = P Pl (Ad + Di ∆i Ei )x(k) + Bu(k) where Ad = i=0 Ai . Assuming Di = Dj or Ei = Ej , one can use Shafai et al. [1997], Hinrichsen and Son [1998] to validate (22).
where S = I −Ad and Si is a matrix constructed from S by replacing its i-th row (column) with the i-th row (column) of H.
Corollary 2: Let the non-negative discrete-time delay system (20) without uncertainty be asymptotically stable independent of delay and let the uncertainty be associated with the system matrix of the delay-free state i.e.
with ρ(Ad ) < 1 is robustly stable if and only if det S P < det Si
Proof: The proof uses determinant properties and requires a lengthy derivation, and it is omitted.
x(k + 1) = (A0 + D0 ∆E0 )x(k) +
l X i=0
34
Ai x(k − i)
(23)
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Applying this result to (26) and (27) leads to the following conclusion.
Then the real and complex stability radius of (23) coincide and it is given by the following formulas depending on the characterization of ∆,
Corollary 3: Let the equivalent representation of nonnegative discrete-time delay system (4), (5) be asymptotically stable independent of delay. Furthermore, let the polynomial matrix P (z) in (26) and its perturbed version P∆ (z) in (27) be associated with block companion matrix N ×N A ∈ R+ of (4) and its perturbed version A(∆) as defined above; respectively. Then the real and complex stability radii with respect to the uncertain equivalent representation of the system coincide and it is given by the following formula if Di = Dj or Ei = Ej for all i, j = 0, 1, . . . , l. 1 rC = rR = (30) maxi kEi P −1 (1)Di k
(1) Let k.k2 denote the Euclidean norm and let ∆ ∈ Rd0 ×e0 then 1
rC = rR = (24) Pl −1
E0 (I − i=0 Ai ) D0 (2) Let ∆ be defined by the set ∆ = {S◦∆ : Sij ≥ 0} with k∆k = max{δij : δij 6= 0} where [S ◦ ∆]ij = Sij δij represents the Schur product. Then 1 (25) rC = rR = Pl −1 ρ E0 (I − i=0 Ai ) D0 S Finally, let us consider the equivalent representation of nonnegative discrete-time system (4), (5). If we include the uncertainty structure (21) in the block canonical structure ×N of the matrix A ∈ RN in (4), then one can not directly + apply the formulas derived in Theorem 4 or Corollary 2. To overcome this problem, we provide additional insights and establish connection to the derived stability radius formula through the equivalent representation.
Proof: Although one can provide a lengthy proof to establish this result through polynomial matrix framework, we can simply show this with the aid of Theorem 4. Note that P (z) evaluated at z = 1 leads to P (1) = I − Pl i=0 Ai and (30) is equivalent to (22) of Theorem 4. This confirms that the stability radius of non-negative discretetime systems (20), (21) is the same as the one with respect to equivalent representation (4), (5).
Let us associate the non-negative polynomial matrix P (z) = Iz
l+1
l
− Al z − . . . − A1 z − A0
(26) 4. CONSTRAINED STABILIZATION
to the block canonical matrix A defined in (4). Similar to the spectrum and spectral radius defined for matrices, we define the spectral of P (z) by σ(P (·)) = {λ ∈ C : det P (λ) = 0} where σ(P (·)) denote the set of all latent roots of the characteristic equation det P (z) = 0. Similarly, the spectral radius is defined by ρ(P (·)) = sup{|λ| : λ ∈ σ(P (·))} The perturbed matrix A defined by A(∆) is constructed through the submatrices Ai (∆i ) = Ai + Di ∆i Ei , which leads to the perturbed polynomial matrices: P∆ (z) = Iz
l+1
The reported results of the previous sections enable one to consider the constrained stabilization problem for general time-delay systems. there are two main reasons for performing this task. First, the positive systems have elegant properties and, by imposing positivity constraint in control design, one can carry over those interesting properties for the closed-loop systems. Second, if the original system is positive to begin with, it is important to maintain positivity when controlling the system. The designers perform this task due to the requirement imposed by application. So, let the stable feedback control law of the form
l
− (Al + Dl ∆l El )z − . . . − (A1 + D1 ∆1 E1 )z
−(A0 + D0 ∆0 E0 ).
−1
(z)Dj ,
∀i, j = 1, . . . , l.
l X
Ki x(k − i)
(31)
i=1
(27)
It is well-known that a polynomial matrix P (·) of the form (26) is Schur stable if σ(P (·)) < 1. Thus, in the following development, we assume that the unperturbed polynomial is Schur stable. We also define auxiliary transfer function matrices Gij (z) associated with the triples {Ei , P (z), Dj } as follows Gij (z) = Ei P
u(k) = K0 x(k) +
be applied to the general discrete-time delay system (1), (2). Then we obtain the following dynamical equation for the closed-loop system x(k + 1) = (A0 + BK0 )x(k) +
l X
(Ai + BKi )x(k − i)(32)
i=1
(28) The goal is to design the control law (31) for system (1), (2) such that the closed-loop system (32) becomes stable and constrained to be non-negative. The following theorem provides an answer to this problem
Lemma 4: Let the polynomial matrix (26) be stable and non-negative. Then ρ(Al + Al−1 + . . . + A0 ) < 1.
Theorem 5: The closed-loop system (32) is asymptotically stable and non-negative if and only if one of the following equivalent conditions is satisfied
Using the above lemma and the well-known fact that in general rC ≤ rR , the authors of (Hinrichsen et al. [2003]) have shown that if Di = Dj or Ei = Ej then for nonnegative polynomial matrices we have 1 = rR (29) rC = max kGii (1)k
(1)
l X
(Ai + BKi ) is stable non-negative matrix and Ai +
i=0
BKi ≥ 0, ∀i = 0, 1, . . . , l.
i
35
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
n (2) There exists a vector w ∈ R+ and the gain matrices m×n Ki ∈ R such that l X [ (Ai + BKi ) − I]w < 0 (33)
REFERENCES K. Gu,V.L. Kharitonov and J. Chen, “Stability of TimeDelay Systems.” Birkhauser, 2003. S.-I. Niculescu, “Delay Effects on Stability: a Robust Control Approach.” Springer-Verlag, 2001. S. Mahmoud Magdi , Robust Control and Filtering for Time-Delay Systems, Marcel Dekker Inc., NY, 2000. H. Sadaka, B. Shafai, R. Sipahi, J. Chen, “An Alternative Characterization of Robust Stability and Stability Radius for Linear Time Delay Systems,” 46th IEEE Conference on Decision and Control, New Orleans, pp. 2112 - 2116, 2007. N. Olgac and R. Sipahi, “An exact Method for the stability Analysis of Time Delayed LTI Systems,” IEEE Transactions on Automatic Control, 47(5), pp. 793 - 797, 2002. A. Berman, M. Neumann and R. J. Stern, Nonnegative Matrices in Dynamic Systems. Wiley, New York, 1989. T. Kaczorek, Positive 1D and 2D systems. Springer, 2002. L. Farina, and S. Rinaldi, Positive Linear System: Theory and Applications, Wiley, 2000. W.M. Haddad and V. Chellaboina, “Stability Theory for Nonnegative and Compartmental Dynamical Systems With Time Delay,” Syst. and Control Letters, 2004. B. Shafai and C.V. Hollot, “Nonnegative Stabilization of Interval Discrete Systems,” in Control of Uncertain Dynamic Systems, S.P. Bhattacharyya and L.H. Keel eds., CRC Press, pp. 471-490, 1991a. B. Shafai, K. Perev, D. Cowley, and Y. Chehab,“ Necessary and Sufficient Condition for Stability of Nonnegative Interval Discrete Systems,” IEEE Trans. Automat. Contr., pp.742 - 746, 1991b. S.P. Bhattacharyya, H. Chapellat, and L.H. Keel,“Robust Control: The Parametric Approach,”Prentice Hall,1995. M. Ait Rami, U. Helmke and F. Tadeo, “Positive Observation Problem for Linear Time-Delay Positive Systems,” Proc. of the 15th Mediterranean Conference on Control and Automation, pp. 1-6, Athens-Greece, 2007. M. Buslowicz, “Simple Stability Conditions for Linear Positive Discrete-Time Systems with Delay,” Bulletin of Polish Academy of Sciences, pp. 325-328, 2008. B. Shafai, H. Sadaka,“Robust Stability and Stabilization of Uncertain Delay System,” to appear in Proc. of American Control Conference, Montreal, Canada, 2012. T. Kaczorek,“Stability of Positive Discrete-Time Systems with Time Delay,” 8th World Multiconference on Systems, Cybernetics and Informatics, pp. 321-324, 2004. T. Kaczorek, Polynomial and Rational Matrices, Springer, 2007. L. Qiu, B. Bernhardsson, A. Rantzer, E. Davison, B. Young, and J. Doyle, “A Formula for Computation of the Real Stability Radius,” Automatica, Vol. 31, No. 6, pp. 879-890, 1995. B. Shafai, J. Chen, and M. Kothandaraman, “Explicit Formula for Stability Radii of Nonnegative and Metzelerian Matrices,” IEEE Transactions on Automatic Control, 42(2), pp. 265-270, 1997. D. Hinrichsen and N. K. Son, “Stability Radius of DiscreteTime Systems under Affine Parameter Perturbations,” International Journal of Robust and Nonlinear Control, 8 pp. 1169-1188, 1998. D. Hinrichsen, N. K. Son, P. H. A. Ngoc “Stability Radii of Positive Higher Order Difference Systems,” Systems and control letters, pp. 377-388, 2003.
i=0
(3) There exists a positive definite diagonal matrix P such that l l X T X (Ai + BKi ) P (Ai + BKi ) − P < 0 (34) i=0
i=0
or equivalently one of the following LMIs is satisfied l X T −P Ai + BKi P i=0 < 0 (35) X l P Ai + BKi −P i=0
X l
−Q
Q
l X
Ai + BKi
i=0
Ai + BKi Q
−Q
T
< 0 (36)
i=0
Furthermore, the feedback control law (31) guarantees the non-negative stability of the closed-loop system if the following LMI has a feasible solution with respect to Q ∈ n×n m×n R and Gi ∈ R where K = [ K0 K1 . . . Kl ] −1 with Ki = Gi Q . l X T T T −Q QAd + Gi B i=0 < 0 (37) l X −Ad Q + B Gi −Q i=0
Ai Q + BGi ≥ 0, Q > 0 where Ad =
l X
(38)
Ai and Q = diag{qj , j = 1, 2, . . . , n}.
i=0
Proof: The proof of this result is constructive. The equivalence of conditions 1) - 3) follows directly from Lemma 2. The conditions 1) and 3) are translated to the condition (35) which can equivalently be written as (36) by applying congruent transformation to LMI (35). Then simple algebraic manipulations leads to (37) and (38). 5. CONCLUSION This paper considered the problem of robust stability and stabilization of delay systems with positivity constraint. Several robust stability results as well as closed form formula for robust stability radius are given and connections to previous results published in the literature are established. Based on the stability conditions for the class of non-negative discrete-time systems, a procedure for the constrained stabilization of general linear discrete-time delay systems is given. This makes it possible to stabilize a general discrete-time delay system such that the closedloop system admits non-negative structure with desirable properties. We formulated and solved the problem by using an LMI. 36
Robust Stability and Constrained Stabilization of Discrete-Time Delay Systems Bahram Shafai, Hanai Sadaka and Rasoul Ghadami Department of Electrical and Computer Engineering, Northeastern University, Boston, MA 02115 USA. [email protected]
[email protected]
[email protected]
Abstract: This paper revisits the problem of robust stability and stabilization of uncertain time-delay systems. We focus on the class of non-negative discrete-time delay systems and show that it is asymptotically stable if and only if an associated non-negative system without delay is asymptotically stable? . This fact allows one to establish strong result on robust stability and stability radius for this class of systems. An alternative representation of delay systems is also constructed whereby its system matrix is in block companion from. Under the assumption of non-negativity for delay systems, this alternative form represents a conventional non-negative system and similar strong robust stability results are derived. Finally, we consider the problem of constrained stabilization and provide a new LMI feasibility solution for it. This makes it possible to stabilize a general discrete-time delay system such that the closed-loop system admits non-negative structure with desirable properties. 1. INTRODUCTION
for stability, robust stability direct computation of stability radius, and constrained stabilization which resemble to the case of conventional positive systems without delay.
The problem of stability and the control design for delay systems attracted many researchers for the past several decades and excellent books have been published in this area (Gu et al. [2003], Niculescu [2001] and Magdi [2000]). Literature reports various methods for stability and stabilization of delay systems which are either extending the classical transfer function approach (Olgac and Sipahi [2002]) or newly methods based on LMI (Magdi [2000]).
In this paper, we develop a parallel strategy for establishing similar results for the non-negative discrete-time delay systems. In particular, we show that the non-negative discrete-time delay systems is asymptotically stable if and only if an associated non-negative discrete-time system without delay is asymptotically stable. This fact allows one to obtain strong results on robust stability and stability radius for this class of systems. Based on an alternative representation of delay systems, similar results are also derived. Finally, the problem of constrained stabilization is considered. We provide a new LMI feasibility solution for it which makes it possible to stabilize a general discretetime delay system such that the closed-loop system admits non-negative structure with desirable properties.
Among the special types of systems, the class of Metzlerian and nonnegative systems play important role for continuous-time and discrete-time systems (Berman et al. [1989], Kaczorek [2002], Farina and Rinaldi [2000], Haddad and Chellaboina [2004]). The robust stability and constrained non-negative stabilization of regular systems were introduced by Shafai et al. in several papers ( [1991a,b], [1997], see also the references therein) and the results were also published in (Bhattacharyya et al. [1995]). The aim of this paper is to present techniques for robust stability analysis and control of positive time delay systems. Motivated by the problem of observer design for time delay systems (Rami et al. [2007]) and simple stability result of positive delay systems (Buslowicz [2008]) we provide additional insights into the robust analysis and stabilization with positivity constraints. In a recent paper we established strong result for the class of continuous-time Metzlerian delay systems (Shafai and Sadaka [2012]) based on the fact that asymptotic stability with delay is equivalent to the stability of an associated Metzlerian system without delay. Consequently, important results have been derived
2. PRELIMINARY RESULTS Consider the linear discrete-time systems with delays described by x(k + 1) = A0 x(k) +
Ai x(k − i) + Bu(t)
(1)
i=1
y(k) = Cx(k) + Du(k)
(2) n
with the initial conditions x(−k) ∈ R ; k = 1, 2, . . . , l, where l is a positive integer, x(k) ∈ Rn , u(k) ∈ Rm , y(k) ∈ Rp are the state, input and output vectors; respectively, and Ai ∈ Rn×n , i = 0, 1, . . . , l; B ∈ Rn×m ; C ∈ Rp×n , D ∈ Rp×m are system matrices.
? In this paper, it is important to distinguish two terminologies. The discrete-time delay systems in the absent of delay states is referred to as delay-free systems and when the delay arguments are absent, we use the notion “discrete-time systems without delay”.
978-3-902823-04-5/12/$20.00 © 2012 IFAC
l X
The system (1) is asymptotically stable if and only if all roots of the characteristic equation 31
10.3182/20120622-3-US-4021.00058
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
det(zIn −
l X
Ai z −i ) = 0
An immediate consequence of Lemma 1 is the fact that with non-negative matrices Ai , B, C, D; the equivalent system (4), (5) becomes a regular non-negative system.
(3)
i=0
At this point, it is important to distinguish two terminologies with respect to internally positive (non-negative) discrete-time delay systems. Let (1), (2) be non-negative discrete-time delay system, then in the absent of delay state we have a delay-free system i.e. Ai = 0, ∀i = 1, . . . , l; and when the delay arguments in (1) are absent, we use the notion “non-negative discrete-time system without delay.”
have moduli less than 1. T T T T Defining x ¯(k) = x (k) x (k − 1) . . . x (k − l) , the system (1), (2) can equivalently be written as ¯ x ¯(k + 1) = A¯ x(k) + Bu(k) ¯ y(k) = C¯ x ¯(k) + Du(k)
(4) (5)
Lemma 2: The non-negative discrete-time system without delay x(k + 1) = Ad x(k) + Bu(k) (6) y(k) = Cx(k) + Du(k)
where B . . . Al−1 Al 0 ... 0 0 ¯= . , B .. .. .. .. . . . 0 0 0 ... I 0 ¯ =D C¯ = [ C 0 . . . 0 ] , D N ×N ¯ ∈ RN ×m , C¯ ∈ Rp×N , D ¯ ∈ Rp×m , and with A ∈ R , B N = (l + 1)n.
A0 I A= ...
A1 0 .. .
Pl where Ad = i=0 Ai , is asymptotically stable if and only if one of the following equivalent conditions is satisfied: (1) All eigenvalues of Ad lie within the unit disc or Ad is Schur stable i.e. ρ(Ad ) < 1. (2) All principal minors of the matrix I −Ad are positive. (3) The matrix I − Ad is nonsingular M-matrix and [I − −1 Ad ] > 0. (4) All coefficients a ¯i , i = 0, 1, . . . , n − 1 of the characteristic polynomial det(λI − Ad + I) = λn + a ¯n−1 λn−1 + . . . + a ¯1 λ + a ¯0 are positive. (5) There exists a positive diagonal matrix P > 0 such that ATd P Ad − P < 0 or equivalently T −P Ad P < 0 P Ad −P
The stability and stabilization of the above system have been established by many researchers through the classical techniques using (3) or by applying Lyaponov Theory on (1) leading to a set of LMI conditions. Our goal is to provide simple alternative stability and robust stability results which can effectively be used in the so-called constrained stabilization problem. To be more specific, we intend to give a technique to stabilize the above general linear discrete-time delay system with the condition that the closed-loop system becomes stable and admits a special structure. The special class of systems considered here is positive systems, which has attracted control community from different angles. The delay-free continuous-time positive systems known as Metzlerian systems have several nice properties through the associated Metzler matrix, which is closely related to the class of M-matrices. A Metzler matrix is defined by its non-negative off-diagonal elements. However, for convenience in connection to stability study, it is usually defined in strict sense; namely, by additional assumption that its diagonal elements are negative. If we denote M to be (strictly) Metzler matrix, then its entries are mij > 0, mii < 0 and −M becomes an M-matrix. In this paper we shall concentrate on the class of positive systems for discrete-time case, in which the associated system matrix belong to the class of non-negative or strictly positive Matrices, i.e. if A is non-negative matrix, then its entries aij ≥ 0. For a detailed exposition of conventional non-delay positive systems one should refer to Berman et al. [1989], Kaczorek [2002] and Farina and Rinaldi [2000].
Remark 1: It should be pointed out that all of the equivalent conditions in Lemma 2 apply equally to the matrix A associated with the equivalent delay systems (4), (5) when considering non-negative discrete-time delay systems for (1), (2). It is also not difficult to show that (Kaczorek [2004], [2007]) det(zIN − A) = det(z l+1 In −
l X
Ai z
l−i
)
i=0 N −1
= z N + αN −1 z + . . . + α1 z + α0 (7) and asymptotic stability with respect to the equivalent system is guaranteed if and only if all roots of the characteristic polynomial (7) have moduli less than 1, or A is Schur stable i.e. ρ(A) < 1. 3. ROBUST STABILITY OF NON-NEGATIVE DELAY SYSTEMS The equivalent stability conditions of previous section suggest that the asymptotic stability of non-negative delay system (1), (2) is equivalent to asymptotic stability of its corresponding system (4), (5) without delay. This is also evident from (7) in Remark 1. Motivated by this and a recent result (Rami et al. [2007]), we were able to establish robust stability and constrained stabilization results in Shafai and Sadaka [2012] for continuous-time Metzlerian delay system. Here, we provide similar results for discretetime case.
Definition 1: The discrete-time delay system (1), (2) is called internally positive if for any initial condition x(−k) n and all inputs u(k) ∈ Rm + , we have x(k) ∈ R+ and p y(k) ∈ R+ for all k ∈ Z+ , where Z+ denotes the set of non-negative integers. Lemma 1: The discrete-time delay system (1), (2) is intern×n nally positive if and only if Ai ∈ R+ , i = 0, 1, . . . , l; B ∈ n×m p×n p×m R+ , C ∈ R+ , D ∈ R+ . 32
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
0, 1, . . . , l; where δi1 , Ei1 are redefined as δi , Ei ; respectively. In this case the number of uncertain parameters δi matches with the number of system matrices Ai and the associated vector set of unknown parameters becomes T δ = [ δ0 δ1 . . . δl ] . For simplicity, in this paper we assume qi = q and δir = δr for all i = 0, 1, . . . , l in (13), then q X Ai (δ) = Ai + δr Eir , i = 0, 1, 2, . . . , l (14)
Theorem 1: The non-negative discrete-time delay systems represented by (1), (2) is asymptotically stable if and only n if there exists a strictly positive vector v ∈ R+ such that [Ad − I]v < 0,
where
Ad =
l X
Ai .
(8)
i=0
Furthermore, its asymptotic stability is guaranteed by the associated non-negative discrete-time systems without delay x(k + 1) = Ad x(k) + Bu(k) (9) y(k) = Cx(k) + Du(k)
r=1 T
where δ = [ δ1 δ2 . . . δq ] . Thus, for single and multiple perturbations, δ can be represented as the vector of uncertain parameters confined within a prescribed set of interest Ω, i.e. δ ∈ Ω.
Proof: It is shown in Rami et al. [2007], Shafai and Sadaka [2012] that continuous-time Metzlerian delay system x(t) ˙ = M0 x(t) +
l X
Mi x(t − τi )
Assuming that bounds are available for δr , then it can be shown that the set of possible Ai (δ) matrices {Ai (δ) : δ ∈ Ω} is a polytope. It is possible to develop robust stability results when perturbation matrices Eir are of unity rank using classical techniques or LMI-based method.
(10)
i=1
is asymptotically stable if and only if there exist a positive Pl n vector v ∈ R+ such that Ac v < 0 where Ac = i=0 Mi and consequently the equivalence of asymptotic stability with and without delays was established. Taking advantage of the relationship between continuous-time and discrete-time systems −1
Ad = (I − Ac ) (I + Ac )
This will particularly be simple when (12) represents nonnegative discrete-time delay systems. In such a case, the robust stability of (12) is equivalent to robust stability of its corresponding systems without delay
(11) x(k + 1) =
and substituting (11) in (8) , we need to show that Xv < 0, −1 where X = [(I − Ac ) (I + Ac ) − I]. The last expression −1 can be simplified to X = 2(I − Ac ) Ac . Since Ac is strictly stable Metzlerian, −Ac and obviously I − Ac are −1 M-matrices. Thus (I − Ac ) > 0 and using Ac v < 0, it −1 follows that Xv = 2(I − Ac ) Ac v < 0.
or alternatively it is equivalent to robust stability of (4) N with A ∈ RN + replaced by A(δ) ∈ R+ and its submatrices n Ai (δ) ∈ R+ given by (13). So, let us impose the constraint that the uncertain delay system (12) is non-negative delay system i.e. the matrices n×n Ai (δ) ∈ R+ , i = 0, 1, 2, . . . , l are non-negative for all δ, n×m and B ∈ R+ . Then we have the following result, which can directly be stated with the aid of Corollary 1. Lemma 3: Let the system (12) be uncertain non-negative discrete-time delay system. Then it is robustly stable independent of delay if and only if the corresponding system without delay, i.e. x(k + 1) = Ad (δ)x(k) + Bu(k)
Now, consider the general discrete-time delay systems (1) with parametric uncertainties described by l X
Ai (δ)x(k)
i=0
Corollary 1: The non-negative discrete-time delay system represented by (1), (2) is asymptotically stable if and only if all equivalent conditions of Lemma 2 are satisfied with respect to its non-negative discrete-time system without n delay (9). Furthermore, there exists v ∈ R+ such that (Ad − I)v < 0, which is an additional equivalent condition for its asymptotic stability.
x(k + 1) =
l X
(15)
is robustly stable, where Ai (δ)x(k − i) + Bu(k)
(12) Ad (δ) =
i=0
l X
Ai (δ)
(16)
i=0
where Ai (δ), ∀ i = 0, 1, 2, . . . , l are uncertain matrices defined by affine multiple parameter perturbations of the form qi X Ai (δ) = Ai + δir Eir , r = 1, . . . , qi ; i = 0, . . . , l (13)
is a nonnegative matrix for all δ ∈ Ω. As pointed out above, if Ai (δ) depends affine linearly on δ as given by (13), then under certain mild conditions various vertex-type results can be employed to check the robust stability of the uncertain non-negative delay systems. The simplest possible polytope of matrices is considered to be an interval matrix. Since the interval matrices comprise of a closed and bounded convex set, they can be represented as a convex hull of its vertex matrices. Therefore, one can relate interval and affine uncertainties. Consequently, if A(δ) is represented as interval uncertainty
r=1
where δir are unknown scalers and Eir are given matrices specifying the structure of the perturbations, and T δi = [ δi1 δi2 . . . δiqi ] is the ith sub-vector of uncertain parameters δir , r = 1, . . . , qi confined within a prescribed set of interest Ωi , i.e. δi ∈ Ωi , i = 0, 1, 2, . . . , l. When r = 1, we have an affine single perturbation structure and Ai (δ) reduces to Ai (δ) = Ai + δi Ei , ∀ i = 33
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Next, we consider the general discrete-time delay systems (1) with a more general type of uncertainty structure described by
through the affine perturbation structure leading to interval non-negative delay system or the uncertain nonnegative delay system is directly defined as interval nonnegative delay system, we can write (12) as l X x(k + 1) = [Ai , A¯i ]x(k − i) + Bu(k) (17)
x(k + 1) =
l X
Ai (∆i )x(k − i) + Bu(k)
(20)
i=0
i=0
where Ai ’s in (1) are subjected to affine perturbations of the form
then the following result can be stated Theorem 2: The non-negative interval delay system (17) is robustly stable if and only if the following single nonnegative system without delay, i.e. l X A¯i x(k + 1) = A¯d x(k) + Bu(k), A¯d =
Ai (∆i ) = Ai + Di ∆i Ei
(21)
where Di ∈ Rn×di , Ei ∈ Rei ×n represent the structure of the uncertainties and the matrices ∆i ∈ Rdi ×ei are unknown uncertainty matrices. The robust stability results for (20), (21) is not trivial when general type of discretetime delay systems is considered. Although, stability and robust stability tests in terms of LMIs are available (Gu et al. [2003], Niculescu [2001], and Magdi [2000]), the problem of computing robust stability radius is more complex even in case of conventional delay-free systems. In general, the computation of complex and real stability radii, rC and rR requires the solution of a complicated global optimization problem (Qiu et al. [1995]). However, for the class of positive systems (continuous and discrete cases), the complex and real stability radii coincide and can be computed by closed form expression (Shafai et al. [1997], Hinrichsen and Son [1998]). Further development of this result for general and special class of continuous-time delay systems performed in Sadaka et al. [2007]. Here we concentrate on non-negative discrete-time delay systems and show that the stability radii can simply be computed with the aid of Theorem 1 or alternatively through the equivalent representation (4). It is also important to point out that in computing stability radius, one should always assume that the system without uncertainty is stable.
i=0
is asymptotically stable, and the stability can be verified by checking the positivity of leading principal minors of I− A¯d . Furthermore, the equivalent interval representation of ¯ non-negative interval delay systems (4) with A(δ) = [A, A] ¯ ¯ where A and A are constructed by Ai and Ai for all i = 0, 1, . . . , l, is robustly stable if and only if A¯ satisfies all equivalent conditions stated in Lemma 2. Proof: With the aid of Corollary 1 and Lemma 3, we know that non-negative interval delay systems (17) is asymptotically stable if and only if the non-negative interval discrete-time system without delay x(k+1) = [Ad , A¯d ]x(k) is asymptotically stable. It has also been established in Shafai et al. [1991b] that the robust stability of nonnegative interval systems x(k + 1) = [Ad , A¯d ]x(k) is equivalent to the asymptotic stability of the system with the upper interval, namely x(k + 1) = A¯d x(k). The proof of the second part follows in a similar fashion, namely with ¯ respect to equivalent representation (4), (5); A(δ) = [A, A] represents a non-negative interval matrix and its stability is guaranteed by the stability of A¯ alone.
Theorem 4: Let the uncertain delay system (20) with (21) be a non-negative discrete-time delay system and assume that it is asymptotically stable independent of delay without uncertainty. Then the real and complex stability radii of the uncertain non-negative delay systems (20) coincide and it is given by the following formula n×di ei ×n if Di = Dj ∈ R+ or Ei = Ej ∈ R+ far all i, j = 0, 1, . . . , l 1
(22) rC = rR = Pl −1
max Ei (I − i=0 Ai ) Di
It is also possible to provide robust stability bounds on one or more uncertain parameters assuming the system without uncertainty is stable. One such a result is stated below under a very restricted class of uncertainty structure. More general uncertainty structures will be considered subsequently. Theorem 3: Let the non-negative discrete-time systems (12) be defined with the uncertain structure (13) such that δir are all equal ∀ i = 1, . . . , l and ∀ r = 1, . . . , q defined by r unknown Pq parameter . Furthermore, assume that Ei and H = r=1 Eir represent dyadic (unity rank) matrices with all of whose entries are either 0 or 1, or all elements are ones. Then this special class of non-negative discrete-time system x(k + 1) = (Ad + H)x(k) + Bu(k) (18)
i
(19)
Proof: Using Lemma 3 and Theorem 1, robust stability of (20) is equivalent to robust stability of uncertain nonnegative discrete-time systems without delays. So (9) with uncertainty structure (21) can be written as x(k + 1) = P Pl (Ad + Di ∆i Ei )x(k) + Bu(k) where Ad = i=0 Ai . Assuming Di = Dj or Ei = Ej , one can use Shafai et al. [1997], Hinrichsen and Son [1998] to validate (22).
where S = I −Ad and Si is a matrix constructed from S by replacing its i-th row (column) with the i-th row (column) of H.
Corollary 2: Let the non-negative discrete-time delay system (20) without uncertainty be asymptotically stable independent of delay and let the uncertainty be associated with the system matrix of the delay-free state i.e.
with ρ(Ad ) < 1 is robustly stable if and only if det S P < det Si
Proof: The proof uses determinant properties and requires a lengthy derivation, and it is omitted.
x(k + 1) = (A0 + D0 ∆E0 )x(k) +
l X i=0
34
Ai x(k − i)
(23)
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
Applying this result to (26) and (27) leads to the following conclusion.
Then the real and complex stability radius of (23) coincide and it is given by the following formulas depending on the characterization of ∆,
Corollary 3: Let the equivalent representation of nonnegative discrete-time delay system (4), (5) be asymptotically stable independent of delay. Furthermore, let the polynomial matrix P (z) in (26) and its perturbed version P∆ (z) in (27) be associated with block companion matrix N ×N A ∈ R+ of (4) and its perturbed version A(∆) as defined above; respectively. Then the real and complex stability radii with respect to the uncertain equivalent representation of the system coincide and it is given by the following formula if Di = Dj or Ei = Ej for all i, j = 0, 1, . . . , l. 1 rC = rR = (30) maxi kEi P −1 (1)Di k
(1) Let k.k2 denote the Euclidean norm and let ∆ ∈ Rd0 ×e0 then 1
rC = rR = (24) Pl −1
E0 (I − i=0 Ai ) D0 (2) Let ∆ be defined by the set ∆ = {S◦∆ : Sij ≥ 0} with k∆k = max{δij : δij 6= 0} where [S ◦ ∆]ij = Sij δij represents the Schur product. Then 1 (25) rC = rR = Pl −1 ρ E0 (I − i=0 Ai ) D0 S Finally, let us consider the equivalent representation of nonnegative discrete-time system (4), (5). If we include the uncertainty structure (21) in the block canonical structure ×N of the matrix A ∈ RN in (4), then one can not directly + apply the formulas derived in Theorem 4 or Corollary 2. To overcome this problem, we provide additional insights and establish connection to the derived stability radius formula through the equivalent representation.
Proof: Although one can provide a lengthy proof to establish this result through polynomial matrix framework, we can simply show this with the aid of Theorem 4. Note that P (z) evaluated at z = 1 leads to P (1) = I − Pl i=0 Ai and (30) is equivalent to (22) of Theorem 4. This confirms that the stability radius of non-negative discretetime systems (20), (21) is the same as the one with respect to equivalent representation (4), (5).
Let us associate the non-negative polynomial matrix P (z) = Iz
l+1
l
− Al z − . . . − A1 z − A0
(26) 4. CONSTRAINED STABILIZATION
to the block canonical matrix A defined in (4). Similar to the spectrum and spectral radius defined for matrices, we define the spectral of P (z) by σ(P (·)) = {λ ∈ C : det P (λ) = 0} where σ(P (·)) denote the set of all latent roots of the characteristic equation det P (z) = 0. Similarly, the spectral radius is defined by ρ(P (·)) = sup{|λ| : λ ∈ σ(P (·))} The perturbed matrix A defined by A(∆) is constructed through the submatrices Ai (∆i ) = Ai + Di ∆i Ei , which leads to the perturbed polynomial matrices: P∆ (z) = Iz
l+1
The reported results of the previous sections enable one to consider the constrained stabilization problem for general time-delay systems. there are two main reasons for performing this task. First, the positive systems have elegant properties and, by imposing positivity constraint in control design, one can carry over those interesting properties for the closed-loop systems. Second, if the original system is positive to begin with, it is important to maintain positivity when controlling the system. The designers perform this task due to the requirement imposed by application. So, let the stable feedback control law of the form
l
− (Al + Dl ∆l El )z − . . . − (A1 + D1 ∆1 E1 )z
−(A0 + D0 ∆0 E0 ).
−1
(z)Dj ,
∀i, j = 1, . . . , l.
l X
Ki x(k − i)
(31)
i=1
(27)
It is well-known that a polynomial matrix P (·) of the form (26) is Schur stable if σ(P (·)) < 1. Thus, in the following development, we assume that the unperturbed polynomial is Schur stable. We also define auxiliary transfer function matrices Gij (z) associated with the triples {Ei , P (z), Dj } as follows Gij (z) = Ei P
u(k) = K0 x(k) +
be applied to the general discrete-time delay system (1), (2). Then we obtain the following dynamical equation for the closed-loop system x(k + 1) = (A0 + BK0 )x(k) +
l X
(Ai + BKi )x(k − i)(32)
i=1
(28) The goal is to design the control law (31) for system (1), (2) such that the closed-loop system (32) becomes stable and constrained to be non-negative. The following theorem provides an answer to this problem
Lemma 4: Let the polynomial matrix (26) be stable and non-negative. Then ρ(Al + Al−1 + . . . + A0 ) < 1.
Theorem 5: The closed-loop system (32) is asymptotically stable and non-negative if and only if one of the following equivalent conditions is satisfied
Using the above lemma and the well-known fact that in general rC ≤ rR , the authors of (Hinrichsen et al. [2003]) have shown that if Di = Dj or Ei = Ej then for nonnegative polynomial matrices we have 1 = rR (29) rC = max kGii (1)k
(1)
l X
(Ai + BKi ) is stable non-negative matrix and Ai +
i=0
BKi ≥ 0, ∀i = 0, 1, . . . , l.
i
35
10-th IFAC Workshop on Time Delay Systems Boston, USA. June 22-24, 2012
n (2) There exists a vector w ∈ R+ and the gain matrices m×n Ki ∈ R such that l X [ (Ai + BKi ) − I]w < 0 (33)
REFERENCES K. Gu,V.L. Kharitonov and J. Chen, “Stability of TimeDelay Systems.” Birkhauser, 2003. S.-I. Niculescu, “Delay Effects on Stability: a Robust Control Approach.” Springer-Verlag, 2001. S. Mahmoud Magdi , Robust Control and Filtering for Time-Delay Systems, Marcel Dekker Inc., NY, 2000. H. Sadaka, B. Shafai, R. Sipahi, J. Chen, “An Alternative Characterization of Robust Stability and Stability Radius for Linear Time Delay Systems,” 46th IEEE Conference on Decision and Control, New Orleans, pp. 2112 - 2116, 2007. N. Olgac and R. Sipahi, “An exact Method for the stability Analysis of Time Delayed LTI Systems,” IEEE Transactions on Automatic Control, 47(5), pp. 793 - 797, 2002. A. Berman, M. Neumann and R. J. Stern, Nonnegative Matrices in Dynamic Systems. Wiley, New York, 1989. T. Kaczorek, Positive 1D and 2D systems. Springer, 2002. L. Farina, and S. Rinaldi, Positive Linear System: Theory and Applications, Wiley, 2000. W.M. Haddad and V. Chellaboina, “Stability Theory for Nonnegative and Compartmental Dynamical Systems With Time Delay,” Syst. and Control Letters, 2004. B. Shafai and C.V. Hollot, “Nonnegative Stabilization of Interval Discrete Systems,” in Control of Uncertain Dynamic Systems, S.P. Bhattacharyya and L.H. Keel eds., CRC Press, pp. 471-490, 1991a. B. Shafai, K. Perev, D. Cowley, and Y. Chehab,“ Necessary and Sufficient Condition for Stability of Nonnegative Interval Discrete Systems,” IEEE Trans. Automat. Contr., pp.742 - 746, 1991b. S.P. Bhattacharyya, H. Chapellat, and L.H. Keel,“Robust Control: The Parametric Approach,”Prentice Hall,1995. M. Ait Rami, U. Helmke and F. Tadeo, “Positive Observation Problem for Linear Time-Delay Positive Systems,” Proc. of the 15th Mediterranean Conference on Control and Automation, pp. 1-6, Athens-Greece, 2007. M. Buslowicz, “Simple Stability Conditions for Linear Positive Discrete-Time Systems with Delay,” Bulletin of Polish Academy of Sciences, pp. 325-328, 2008. B. Shafai, H. Sadaka,“Robust Stability and Stabilization of Uncertain Delay System,” to appear in Proc. of American Control Conference, Montreal, Canada, 2012. T. Kaczorek,“Stability of Positive Discrete-Time Systems with Time Delay,” 8th World Multiconference on Systems, Cybernetics and Informatics, pp. 321-324, 2004. T. Kaczorek, Polynomial and Rational Matrices, Springer, 2007. L. Qiu, B. Bernhardsson, A. Rantzer, E. Davison, B. Young, and J. Doyle, “A Formula for Computation of the Real Stability Radius,” Automatica, Vol. 31, No. 6, pp. 879-890, 1995. B. Shafai, J. Chen, and M. Kothandaraman, “Explicit Formula for Stability Radii of Nonnegative and Metzelerian Matrices,” IEEE Transactions on Automatic Control, 42(2), pp. 265-270, 1997. D. Hinrichsen and N. K. Son, “Stability Radius of DiscreteTime Systems under Affine Parameter Perturbations,” International Journal of Robust and Nonlinear Control, 8 pp. 1169-1188, 1998. D. Hinrichsen, N. K. Son, P. H. A. Ngoc “Stability Radii of Positive Higher Order Difference Systems,” Systems and control letters, pp. 377-388, 2003.
i=0
(3) There exists a positive definite diagonal matrix P such that l l X T X (Ai + BKi ) P (Ai + BKi ) − P < 0 (34) i=0
i=0
or equivalently one of the following LMIs is satisfied l X T −P Ai + BKi P i=0 < 0 (35) X l P Ai + BKi −P i=0
X l
−Q
Q
l X
Ai + BKi
i=0
Ai + BKi Q
−Q
T
< 0 (36)
i=0
Furthermore, the feedback control law (31) guarantees the non-negative stability of the closed-loop system if the following LMI has a feasible solution with respect to Q ∈ n×n m×n R and Gi ∈ R where K = [ K0 K1 . . . Kl ] −1 with Ki = Gi Q . l X T T T −Q QAd + Gi B i=0 < 0 (37) l X −Ad Q + B Gi −Q i=0
Ai Q + BGi ≥ 0, Q > 0 where Ad =
l X
(38)
Ai and Q = diag{qj , j = 1, 2, . . . , n}.
i=0
Proof: The proof of this result is constructive. The equivalence of conditions 1) - 3) follows directly from Lemma 2. The conditions 1) and 3) are translated to the condition (35) which can equivalently be written as (36) by applying congruent transformation to LMI (35). Then simple algebraic manipulations leads to (37) and (38). 5. CONCLUSION This paper considered the problem of robust stability and stabilization of delay systems with positivity constraint. Several robust stability results as well as closed form formula for robust stability radius are given and connections to previous results published in the literature are established. Based on the stability conditions for the class of non-negative discrete-time systems, a procedure for the constrained stabilization of general linear discrete-time delay systems is given. This makes it possible to stabilize a general discrete-time delay system such that the closedloop system admits non-negative structure with desirable properties. We formulated and solved the problem by using an LMI. 36