Tractable stability analysis for systems containing repeated scalar slope‐restricted nonlinearities

This paper proposes an LMI‐based approach for studying the stability of feedback interconnections of a finite dimensional LTI system and a nonlinear element that consists of several identical scalar nonlinearities that have restrictions on their sector and slope. The results are based on the integral quadratic constraint stability analysis framework and other recent results that give a sharp characterisation of stability multipliers for monotone, repeated scalar nonlinearities. Several examples show the effectiveness of the proposed approach; the lack of conservatism in the results is noteworthy. Copyright © 2013 John Wiley & Sons, Ltd.


INTRODUCTION
This paper re-visits the much-studied stability problem depicted in Figure 1 in which P .s/ is a finite dimensional LTI system andˆ../ W L m 2e 7 ! L m 2e is a static time-invariant nonlinear element. Depending on the information available aboutˆ../, different criteria may be used to study the stability of the interconnection. Commonly, the Circle and Popov criteria are used to assess the stability of such systems, but when further information about the nonlinearity is known, and depending on the computational resources available, various other criteria may be used instead. In order to keep the paper reasonably succinct, we do not review all available approaches, but the interested reader is encouraged to consult [1][2][3][4][5][6] and references therein.
Of all the various stability criteria available for the study of the system in Figure 1, one of the most promising was proposed in the late 1960s by Zames and Falb [7]. Zames and Falb proved that 973 will be assumed that the repeated nonlinearities are both slope restricted and sector bounded, in a similar way to [16]. While it is easy to see that slope restriction implies a sector bound of the same size, it may be the case that the nonlinearity in question satisfies a 'tighter' sector bound and thus information about this sector bound can be used to reduce conservatism in the results. While the development of the MIMO results takes inspiration from the scalar case, the precise details change in order to cope with the multivariable nature of the problem, and in particular, the results appeal to the recent work of [9] in order to obtain appropriate conditions on the Zames-Falb multiplier. The paper also uses the IQC machinery to develop the results and, from one perspective, may be viewed as an improvement on the work of [13] in which the multiplier is chosen more efficiently and from the larger set proposed by [9]. A preliminary version of this paper appeared in [20].
Notation: Notation is standard throughout. H ij denotes the ij 'th element of the matrix H ; H i denotes the i'th row/column of the matrix H (meaning discernible from context). The L 2 norm of a vector-valued function x.t/ is defined as kxk 2 WD q R 1 0 kx.t/k 2 dt where kxk denotes a vector's Euclidean norm. The space of m-dimensional vector-valued functions with finite L 2 norm is denoted by L m 2 and the extended space L m 2e . In instances when the dimension of the space is not important, we simply write x 2 L 2 and x 2 L 2e , respectively. Similarly, the L 1 norm of a scalarvalued function ı.t/ is defined as kık 1 WD R 1 0 kı.t/kdt, and the space where this norm is finite is denoted by L 1 . With some abuse of notation, we say that a transfer function matrix H.s/ 2 L 1 if the impulse responses of its elements H ij .t / are all in L 1 . The space of real rational m n transfer function matrices, bounded on the imaginary axis, is denoted by RL m n 1 ; the subspace of RL m n 1 , which is analytically continuous in the right half complex plane, is denoted by RH m n 1 . An operator H is said to be bounded if kH.u/k 6 kuk for all u 2 L 2e and some > 0.

NONLINEARITIES AND INTEGRAL QUADRATIC CONSTRAINTS
Consider Figure 1 in which P .s/ is the finite dimensional LTI part of the system with state-space realisation where A p 2 R n n , B p 2 R n m , C p 2 R m n and D p 2 R m m . In this paper, we assume thatˆ../ belongs to the class of so-called repeated static scalar nonlinearities, N RS , defined below. .
The oddness assumption (meaning the nonlinearity is symmetric) in item (iv) is actually not necessary but will be assumed here for brevity of presentation. This paper will consider several nonlinearities belonging to this class. Of particular interest are the so-called sector bounded nonlinearities, where a static nonlinearity ../ is said to belong to the sector OE0,ˇ ( ../ 2 SectorOE0,ˇ) if, for someˇ> 0, the following inequality holds. Similarly, ../ is said to have slope restriction OE0,˛ ( ../ 2 @OE0,˛) if, for some˛> 0, the following inequality holds.
It is straightforward to see that ../ 2 @OE0,˛ ) ../ 2 SectorOE0,˛, but the converse is not true. It may be that a nonlinearity satisfying a certain slope restriction, @OE0,˛, may satisfy tighter sector bounds, that is, ../ 2 SectorOE0,ˇ, withˇ<˛. It is emphasised that many nonlinearities that are encountered in practice are both sector and slope bounded; the saturation and deadzone nonlinearities are two common examples. For such nonlinearities, the sector and slope bounds are the same, but this is not always the case. Finally, a nonlinearity ../ is said to be monotonically non-decreasing if the following inequality holds: . .x/ .y//.x y/ > 0 8x, y 2 R The classes of nonlinearity considered in the remainder of the paper are now formally introduced.

Remark 1 N RS
=ˇi s simply the class of repeated scalar nonlinearities in which each scalar nonlinearity is sector bounded with the same bound. It is easy to see that typical nonlinearities such as multivariable saturations/deadzones can be 'loop shifted' into this form. Likewise, N RS = is the set of repeated scalar nonlinearities in which each entry is slope restricted by the same bound. Again, many common nonlinearities can be 'loop shifted' to be in this form. A sub-class of N RS = was studied in [21]. N RS =ˇi s simply the intersection of the two previous classes of nonlinearities. Note that, trivially, N RS = D N RS =˛; that is, the slope restriction implies the same sector bound. However, it is possible for the sector bound to be smaller than the slope restriction; that is, N RS =ˇ N RS = D N RS =˛i n general; that is, usingˇmay reduce conservatism.

Remark 2 N RS
M is basically the class of monotone non-decreasing nonlinearities considered in [7] but generalised to the repeated scalar case ( [9]). The significance of this class of nonlinearities is that it can be proved, under mild conditions, that if a nonlinearityˆ../ 2 N RS = , then loop shifting can be used to transform it into an equivalent nonlinearity of the form Q ../ 2 N RS M , effectively meaning that stability results available for the class N RS M can be used for the class N RS = . In our work, this means that the new Zames-Falb multipliers derived by [9] can be applied to the class of nonlinearities considered in this paper. This is important because it allows us to obtain results of reduced conservatism compared with [13] and in fact is easier to use in obtaining convex conditions. See [6] for a more comprehensive discussion of loop shifting.
The main objective of this paper is to obtain tractable results that can be used to assess the stability of the feedback interconnection described by Figure 1. In order to do this, extensive use is made of the IQC analysis framework introduced in [5,12] and extended in [13,22]. In general, it is said that a nonlinearityˆ../ satisfies the IQC defined by ….j!/ ifˆis such that where O y.j!/ and O u.j!/ are the Fourier Transforms of y.t/ and u.t / Dˆ.y.t //, respectively. The structure of ….j!/ depends on the properties of the nonlinearityˆ../. There are several different IQCs that will be used in this paper.
1. The Zames-Falb IQC. The 'Zames-Falb' IQC used here is a combination of the one presented in [13] with the relaxation suggested by [9]. More precisely, ifˆ../ 2 N RS = , it satisfies the IQC defined by … D … ZF .j!/ where and M.s/ is a multiplier such that where This class of multiplier, M.s/, is somewhat broader than that in [13] because the improvement suggested by [9] does not require the multiplier M.s/ to be symmetric: only the row and column dominance conditions in (5) and (6) are stipulated. This improvement is both important in lessening conservatism of the results and also in allowing our new results to be cast as a quasi-convex optimisation problem. 2. The Circle IQC. It is well known (see, e.g. [5]) that ifˆ../ 2 N RS =ˇ, it satisfies the IQC given by … D … Č , where and V is any positive definite diagonal matrix. Note that becauseˆ2 N RS actuallyˆ2 N RS =˛Á , this multiplier can be used with any slope restricted nonlinearity. 3. The Popov IQC. Following [22], ifˆ../ 2 N RS =ˇ, it satisfies the IQC given by … D … P where where ƒ is an (indefinite) diagonal matrix. As with the Circle IQC, the Popov IQC can also be used in the analysis of slope-restricted nonlinearities.
One of the attractive features of IQC analysis [5,12,13] is that if a nonlinearity satisfies more than one IQC, these IQCs can be combined. Hence, whenˆ2 N RS =ˇ, it satisfies the IQC defined by … RS =ˇW D … ZF C … P C … Č , which is given later.  (5) and (6) are satisfied, ƒ is a diagonal matrix and V is a positive definite diagonal matrix. Theorem 1 of [22] then allows the following theorem to be stated.

Theorem 1
Consider the interconnection in Figure 1 where P .s/ 2 RH 1 andˆ../ W L m 2e 7 ! L m 2e 2 N RS =ˇf or some˛> 0 andˇ> 0. Assume that the closed-loop system is well posed. Then, the system is stable if the following inequality is satisfied.

Remark 3
Strictly speaking, 'stable' here means that L 2 boundedness of the exogenous signals, w and v, implies L 2 boundedness of all loop signals; that is, 'stability' of Figure 1 means 'L 2 stability' of Figure 1. However, as P .s/ 2 RH m m 1 andˆ../ is static, L 2 stability of Figure 1 actually implies global asymptotic stability of the origin of the unforced system.

Remark 4
In this form, Theorem 1 is quite difficult to verify efficiently; one needs to ensure inequality (10) holds for M.s/ of a very general structure. In [12,13,23], the approach advocated has been to fix a structure of M.s/ such that inequalities (5) and (6) hold and then to check inequality (10); no real guidance on how to choose M.s/ was given, and the results in [13] basically reduce to the checking of ad hoc structures for M.s/, which, although possible and useful for some systems, is rather time consuming in general. In [14,16], it was shown how, by restricting the structure of M.s/ to be causal and of order equal to that of P .s/, it was possible to choose M.s/ much more systematically using LMIs and a line search. Those results were confined to the case of SISO systems; here, we generalise those results to MIMO systems with repeated scalar nonlinearities.

MAIN RESULTS
The aim of this section is to provide a tractable way of determining whether the system in Figure 1 is stable whenˆ2 N RS =ˇ. If the sector information is discarded, that is, ifˆ2 N RS = and the Popov and Circle IQC's are ignored, the results will reduce to a multivariable generalisation of those in [14]. Related results will be discussed after the main result has been presented

Preliminary results
There are a number of preliminary results required for proving the main results.
Proof Follows by direct calculation.

Lemma 1
Consider a transfer function matrix H.s/ 2 RH m m 1 . Then, kH ij k 1 6 ij for some i, j if there exist positive definite matrices Y j > 0 and scalars j > 0 and j > 0 such that the following matrix inequalities hold for all i, j 2 ¹1, : : : , mº: Proof This result is a slight generalisation of that proved in [24] -see appendix for proof.

Lemma 2
Consider the real matrix H 0 2 R m m and positive scalars i > 0 8i 2 ¹1, : : : , mº. The inequalities hold if and only if there exists a real matrix R 2 R m m such that This lemma is a non-symmetric generalisation of a result of [21]. Symmetry is not required, and the proof follows easily from that in [21]. Note that this result allows an inequality involving a nonlinear function of a matrix variable H 0 to be verified by a set of linear conditions given in items 1-3.

Main result
The following is the main result of the paper.

Remark 5
For fixed˛> 0,ˇ> 0 and j > 0 for all j 2 ¹1, : : : , mº, conditions (16)-(22) form a system of linear matrix inequalities, which can be solved efficiently using modern software. Thus, if the sector/slope bounds are known a priori, it is relatively easy to verify stability. If, instead, it is necessary to compute the maximum sector/slope size for which the system remains stable, this can be performed by combining the LMIs in (16)-(22) with a bisection over˛(fixingˇD Ä˛for some constant Ä 2 OE0, 1) and a further search over j . The key point is that Theorem 2 gives a relatively efficient, if potentially conservative, way of determining the stability of Figure 1.

Proof of Theorem 2
The proof of Theorem 2 basically requires us to translate the frequency domain inequality in Theorem 1 and the L 1 constraints in inequalities (5) and (6) into matrix inequalities. There are three distinct parts of the proof, which, as will be shown, are coupled through some matrix variables.

Part 1: Frequency domain condition
It is necessary to convert the frequency domain inequality in Theorem 1 into a set of matrix inequalities. This part of the proof is shared with that reported in [14,15], and so it will only be sketched. First note that inequality (10) Using this state-space realisation and the KYP Lemma [25], it then follows that inequality (23) is satisfied (and hence, inequality (10) is satisfied) if there exists a symmetric matrix P such that Under the additional assumption that P > 0, and noting that because A H 2 R n n , it follows that P 2 R 2n 2n and that P is nonsingular. Hence, partitioning P and Q WD P 1 into sub-matrices of dimension n n Ä Q 11 Q 12 Q 0 12 Q 22 Ä P 11 P 12 P 0 12 P 22 and introducing the matrices it follows that using the congruence transformation diag.… 1 , I / on inequality (24) and carrying out some algebra similar to [14] yield matrix inequality 2 6 6 6 6 6 6 4 ?

<0
(27) Applying the congruence transformation diag.Q 1 11 , I , I / D diag.S 11 , I , I / to inequality (27) and defining yield inequality (16). Part 2: The L 1 conditions The previous part of the proof was similar to that given in [14,16] and was included for completeness. However, in the multivariable case, the multiplier M.s/ needs to satisfy the rather more complex 'L 1 ' conditions in inequalities (5) and (6). These need more careful consideration than the scalar case presented in [14]. First note that if kH ij k 1 6 ij , ij > 0 8i, j 2 ¹1, : : : , mº sufficient conditions for inequalities (5) and (6) to hold are given by ij 8i 2 ¹1, : : : , mº Defining Thus, sufficient conditions for inequalities (5) and (6)  Furthermore, a sufficient condition for inequalities (32) to hold is given by direct application of Lemma 1; that is, inequalities (13) and (14) must hold. Note that there are problematic products of matrix variables Y j and the multiplier state-space matrices A H and B H . To remove these products, and again, at the price of some conservatism, let Y j WD P 22 8j 2 ¹1, : : : , mº. This then gives the inequalities: Furthermore, applying the congruence transformation diag.Q 12 , I / to each of the j inequalities (39) gives the j inequalities Ä Next, from (25), it follows that Q 12 P 22 D Q 11 P 12 . Using the inequalities in (41) and noting further from (25) that Q 11 P 12 Q 0 12 D Q 11 .I P 11 Q 11 / the j inequalities are obtained Then, using the congruence transformation diag.Q 1 11 , Using (28)-(29) then yields the inequalities (17) in the theorem. In a similar way, applying the congruence transformation diag Q 1 11 Q 12 , I , I to each of the inequalities (40) gives after similar working to the above 2 4 j .S 11 P 11 / 0 Part 3: Positive definiteness of P In Part 1 of the proof, it was assumed that P > 0. Here, we show that satisfaction of the LMIs indeed guarantees that this is indeed the case. Note that P > 0 is equivalent to … 0 1 P … 1 > 0, which can be written as This is equivalent, by the Schur complement, to P 11 S 11 > 0, which is guaranteed by inequality (18).

Remark 6
Conservatism: Compared with the SISO case described in [14], there are more potential sources of conservatism in the multivariable results derived here. As with the SISO case, conservatism is introduced by the restriction that M.s/ be stable, causal and of order equal to P .s/. However, additional conservatism is introduced by bounding each kH ij k 1 6 ij : only one such bound is introduced in the SISO case. Additionally, to obtain conditions that are convex, it was necessary to stipulate Y j D P 22 8j , which introduces yet more conservatism into the results. As with the SISO case, the goal here is to trade conservatism with tractability: again, we note that in its original form (Theorem 1), the search over M.s/ is an infinite-dimensional optimisation problem and thus very difficult to solve. Noting the remarks in [26], however, the conservatism in the 'L 1 ' conditions can sometimes detrimentally influence conservatism. Convexity: As with the SISO case, a remark about convexity is warranted. Note that the inequalities in Theorem 1 are linear for fixed j and for given sector/slope bounds. Although it was initially thought that j was not too influential on the results ( [14]), recent work has shown that this is not to be the case ( [26]) and thus some search over j appears to be beneficial. When the slope/sector bounds are unknown, it appears reasonable to fixˇD Ä˛, and then Theorem 1 reduces to an LMI plus a bisection over˛and a search over j .

Related results
Theorem 2 applies to nonlinearities of the class N RS =ˇ; that is, the sector and slope bounds are not necessarily the same. As observed earlier, in certain nonlinearities of interest, the sector and slope bounds are identical. In this case, it is natural to ponder the existence of improved results. In fact, ifˆ2 N RS =˛( sector bound given by slope restriction), the Circle IQC used in the derivation of Theorem 2 can be relaxed.
Proof Item (i) has its origins in the work of [21] who stated it for the case where N RS 1=? . A more general proof is given here. First, let ….y/ WD˛y ˆ.y/, with each element of …../ given by .y i / D˛y i .y i /; then, inequality (i) is equivalent to 982 M. C. TURNER, M. L. KERR AND J. SOFRONŶ where the shorthand i WD .y i / and i WD .y i / has been used. Now, as i i > 0 and as By symmetry of V , this becomes This expression will be non-negative if ../ and ../ are monotonically non-decreasing functions. Note that as ../ is slope restricted in OE0,˛, it is monotonically non-decreasing. Furthermore, because i D˛.y i / .y i /, monotonicity of holds if the following expression is non-negative for all y 1 , y 2 2 R: .y 1 , y 2 / WD . .y 1 / .y 2 // .y 1 y 2 / (55) D .˛.y 1 y 2 / . .y 1 / .y 2 /// .y 1 y 2 / (56) However, as 2 @OE0,˛ by assumption, then .y 1 , y 2 / > 0, and hence ../ is monotonically nondecreasing. In turn, this implies thatˆ.y/ 0 V ….y/ is non-negative and hence inequality (i) holds. It is easy to see that inequality (i) implies inequality (ii) and hence the proof is complete. Therefore ifˆ../ 2 N RS =˛, that is, if the slope restriction and sector bound are identical,ˆ../ satisfies the IQC (2) defined by ….j!/ D … C DD .j!/ with where V D V 0 is such that V i i > P m j D1,j ¤i jV ij j 8i. This condition on V is weaker than the requirement for V to be diagonal, and lower conservatism in the results may be expected.
Thus, in the case thatˆ../ 2 N RS =˛, it satisfies the IQC defined by … D Q … RS =˛D … ZF C … P C … C DD . In a similar way to that described in the previous subsection, the following theorem can then be proved. (59)

Remark 7
Theorems 2 and 3 make use of the Popov multiplier … P in their derivations. From a state-space perspective, this requires the linear subsystem P .s/ to be strictly proper (i.e. D p Á 0). If the linear subsystem is not strictly proper, then results based on just the Zames-Falb multiplier may be derived. In this case, the 'passivity' inequality in Proposition 2 of [14] can be combined with inequalities (17)- (22) from Theorem 2 to obtain a direct multivariable extension of [14]. However, it has been noted ( [16,17]) that the inclusion of a Popov term appears to offer some useful numerical flexibility -this will be demonstrated in the next section.

EXAMPLES
This section compares the results derived here to what appears to be the state-of-the-art in the literature, namely those of [4]. Note that the results of [4] incorporate, as special cases, the Popov criterion and several other stability criteria. It has already been demonstrated that the results of [4] are less conservative than those of Haddad and Kapila [1], Suykens et al. [3] and Chen and Wen [2]. As noted in [4,14], Park's method is also convex, making it relatively easy to compute a solution. In order to apply both the results here and the results of Park, it is necessary to stipulate a relationship between the sector bounds and the slope restrictions. ‡ For simplicity, we letˆ../ 2 N RS =˛a nd attempt to maximise the value of˛for which the relevant criteria guarantee stability. It is important to note that the linear constant gain˛I 2 N RS =˛, so an upper bound on the maximum achievable˛can be determined by simply searching for the maximum˛such that the eigenvalues of A p C˛B p C p have strictly negative real part. This is important because this gives one an idea of how conservative the results derived here are. Table I shows a comparison of the maximum sector/slope sizes obtained using the results of Theorem 3, Park's results [4] and the upper bound, for a number of examples. Some of these examples have been taken from the literature, but most have been randomly generated. Rather than giving a long list of state-space matrices in this paper, the reader is directed to [27] for Matlab code for the generation of the examples and results. The values of D j 8i 2 ¹1, : : : , mº are also given: some limited search of was performed, but these values should not be considered optimal in any way. The results calculated using Theorem 3 all use a proper (rather than strictly proper) Zames-Falb multiplier, which seems to be useful in numerical routines, despite the assertion that it should not [26]. Two cases of Theorem 3 are given: the first without the Circle and Popov terms, that is, … D … ZF ; and the second with the Circle and Popov terms, that is, … D … ZF C … C DD C … P .
The largest slope/sector sizes for which stability holds for a given example are highlighted in yellow in Table I  7, 8 and 9, both the method of [4] and Theorem 3 (with and without the Popov multiplier) are able to guarantee stability for the slope/sector size predicted by linear analysis. For examples 6, 11 and 25, the method of [4] and Theorem 3 are also able to guarantee stability for the same slope/sector size predicted by linear analysis, but now the Popov multiplier must be included in Theorem 3 in order to obtain these slope/sector sizes. For the remainder of the examples, it appears that the largest sector/slope sizes for which stability holds are guaranteed by Theorem 3 (sometimes with and without the Popov multiplier). These results clearly show that, in many examples, application of Theorem 3 leads to a less conservative estimate of the sector/slope size the system is able to tolerate before instability results. Note that, apart from example 28, it appears that the results given here are typically equal or superior to [4] in the MIMO cases. It is interesting to understand why this might be the case, and it is probably because, in the SISO case, it is the conservatism of the L 1 inequalities that 'limits' the effectiveness of Theorem 2, whereas in the MIMO case, the 'passivity' condition (inequality (27)) can often be difficult to satisfy: With the results of Theorem 2, there is more freedom in satisfying this condition than that in the equivalent condition in [4].

CONCLUSION
This paper has proposed a new, reasonably tractable approach for assessing the stability of multivariable feedback interconnections consisting of a linear part and a nonlinearity in which all elements have identical sector bounded, slope-restricted characteristics. Various examples, both