A new perspective on static and low order anti-windup synthesis

By viewing the anti-windup problem as a decoupled set of subsystems and relating this configuration to a general static anti-windup set-up, LMI conditions are established which guarantee stability and performance of the resulting closed-loop system. The approach taken, and the mapping used for the performance index, are logical and intuitive–-and, it is argued, central to the ‘true’ anti-windup objective. The approach enables one to construct static anti-windup compensators in a systematic and numerically tractable manner. The idea is extended to allow low-order anti-windup compensators to be synthesized, which, while being sub-optimal, can improve transient performance and possess several desired properties (such as low computational overhead and sensible closed-loop pole locations). In addition, low-order anti-windup synthesis is often feasible when the corresponding static synthesis is not.


Introduction
The problem of actuator saturation is well-documented in the literature (the literature on this subject is now vast and any attempt we make to summarize it would inevitably omit valuable contributions made by many scholars) and many researchers have sought to address the problems associated with its presence. It has been tackled from various different perspectives which include synthesizing controllers which directly account for saturation constraints (see e.g. Garcia et al. 1999, De Dona et al. 2000, Turner et al. 2000or Saberi et al. 1996; model predictive control strategies where the control constraints are incorporated into the resulting optimization procedure; and anti-windup techniques, which this paper considers. One of the most successful methodologies used to cope with actuator saturation is anti-windup compensation or conditioning (see e.g. Kothare et al. 1994 or Astrom and Rundqwist 1989). This well-known technique involves a two-step procedure whereby a controller is first synthesized for a nominal, usually linear, system ignoring saturation. An anti-windup compensator is then designed such that, when the control signal undergoes saturation, this compensator becomes active and leads to improved behaviour during this period. This has the advantage that the small signal response of the system is left as intended and that the local performance of the controller is not restricted.
Building upon the success achieved in guaranteeing stability of stable plants subject to actuator saturation, there has been some recent activity aimed at securing performance guarantees for such systems. Although this has always been sought informally, one of the first forays into establishing this notion concretely was made in Mulder et al. (2001) (see also Wada and Saeki (1999) for a similar, but less general, treatment), where the L 2 gain was picked as an appropriate induced system norm to measure an anti-windup compensator's performance. Furthermore, Mulder et al. (2001) gave existence conditions, in terms of a set of linear matrix inequalities (LMIs), by which anti-windup compensators could be synthesized. Mulder et al. (2001) only tackled static anti-windup synthesis: the synthesis of anti-windup compensators which were themselves not dynamic, although their inclusion altered the dynamics of the 'effective' controller.
This line of research was continued in more generality in Grimm et al. (2001 b) and Grimm et al. (2003) where it was shown that, for stable plants, there always exists an anti-windup compensator of order greater than or equal to that of the plant which, in addition, satisfies a finite L 2 gain constraint. The results of Grimm et al. (2001 b) and Grimm et al. (2003) served to complement known results on the existence of globally stabilizing full order anti-windup compensators found in, for example, Teel and Kapoor (1997), Weston and Postlethwaite (2000) and Miyamoto and Vinnicombe (1996). Furthermore, the work in Grimm et al. (2001 b) gave a characterization of all optimal linear anti-windup compensators meeting an L 2 performance constraint and a procedure by which to find a certain, possibly reduced order, compensator. Generally this problem was cast as a non-convex optimization procedure, although in the full-order and static cases the problem is actually convex, which makes the results of Mulder et al. (2001) a special case of the general result in Grimm et al. (2001 b).
Both the works of Mulder et al. (2001) and those of Grimm et al. (2003) were successful at synthesizing compensators which yielded improved performance, particularly when compared to ad hoc methods in existence and when compared to internal model control (IMC), yet in a certain way, they both failed to achieve the true goal of an anti-windup compensator.
The 'true goal' of an anti-windup compensator is, of course, subjective, but a notion of this which has been held informally since the methodology's inception is the notion of a swift return to linear behaviour. This was really only formalized in Teel and Kapoor (1997) (see also Kapoor et al. 1998), although it was first recognized in the work of Miyamoto and Vinnicombe (1996) and, independently, in Weston and Postlethwaite (1998). In all of these works the performance of an anti-windup compensator was based on it returning the output of the system to the nominal, that is, what it would have been had saturation not been present.
In the works of Grimm et al. (2003) and Mulder et al. (2001), the L 2 gain condition is not directly related to the system's return to linear behaviour; indeed there is no direct way of capturing this into the formulae in those papers. Instead, attention there is focused on the minimization of a certain closed-loop induced norm, which is deemed appropriate to capture the performance of the system. In our paper, we re-visit the work of Weston and Postlethwaite (1998) (see also Weston and Postlehwaite 2000) and use the decoupled structure derived in that paper to formulate conditions which encapsulate the return to linear behaviour of a stable system containing an anti-windup compensator. At first we consider purely static antiwindup compensators, which are, from a practical point of view, most desirable. Then these ideas are extended to the sub-optimal synthesis of 'low-order' compensators, which are often feasible for problems for which static compensators are not. It is important to remark that our work assumes that linear behaviour is desirable, and hence a return to linear behaviour marks a return to 'good' performance; in most cases this is a valid assumption.
Other work related to these results, and to this paper, can be found in Crawshaw and Vinnicombe (2000) and Kapoor et al. (1998), both of which take the return to linear behaviour as a central objective. The work of Crawshaw and Vinnicombe (2000) builds on the work of Miyamoto and Vinnicombe (1996) but uses a novel loop-transformation to convert a small gain condition into, effectively, the less conservative circle criterion. Kapoor et al. (1998) use an observer-based anti-windup compensator, a special case of static compensator (early work on this subject was contributed by Walgama and Sternby 1990), and a subspace algorithm to make a certain triplet passive, which ensures recovery of linear behaviour. This latter technique has the advantage of computational simplicity although it does not explicitly tackle an L 2 gain objective, as we propose here.
The notation used in the paper is mainly standard throughout. We denote the L p norm of a time-varying vector yðtÞ 2 R n y as kyk p and the induced L p norm of a possibly non-linear operator Y : Y 1°Y2 from one Lebesgue space to another, as kYk i, p . To avoid notational clutter, we often omit the time variable (t) or Laplace argument (s) of a vector or transfer function if we think that no confusion will arise. Likewise, we often do not distinguish between a system's transfer function and its associated linear operator in the time domain. kyðtÞk ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yðtÞ 0 yðtÞ p is the Euclidean norm of the vector y(t). The distance between a vector y(t) and a compact set, Y, is denoted distðy, YÞ :¼ inf w2Y ky À wk. R iÂj represents the space of all real rational transfer function matrices of dimension i Â j ; RH 1 denotes the set of real rational transfer function matrices, analytic in the closed right-half complex plane and with norm with supremum on the imaginary axis.

Problem formulation 2.1. System description
We consider the stabilizable, detectable and finite dimensional linear-time-invariant (FDLTI) plant subject to input saturation with x p 2 R n p , u m 2 R m , y 2 R q and d 2 R n d . From this we define the following transfer function matrices to represent the disturbance feedforward and feedback parts of G(s) As the work here is seeking global results, we are necessarily forced to assume that G(s) is asymptotically stable; that is GðsÞ 2 RH 1 . This is necessary in the approach we take, as will be clear later.
We assume the following stabilizable, detectable, linear controller has been designed to control the plant G(s) where x c 2 R n c and r 2 R n r . From this we designate the following transfer functions As is the convention in anti-windup design, we assume that the plant input, u m , is given by u m ¼ satðuÞ, where satðuÞ ¼ ½satðu 1 Þ, . . . , satðu m Þ ð5Þ If there is no saturation present (i.e. u m ¼ u), we call the system in which this is the case the nominal closed loop system. Furthermore, we assume that either u m is available for direct measurement, or that a good estimate of it is available. For convenience we define the compact Note that 8 u 2 U, satðuÞ ¼ u. This definition is similar to that in Teel and Kapoor (1997), but not exactly the same as we only consider asymptotically stable plants.
The deadzone function is given by the identity Note that 8 u 2 U, DzðuÞ ¼ 0. Central to our results will be the fact that the deadzone function satisfies the following sector property.
Definition 1: The decentralized non-linearity N ¼ diagð 1 , . . . , m Þ is said to belong to Sector½0, I if the following inequality holds This definition is specialized to the Sector½0, I case from the more general definition given in Khalil (1996, Definition 10.1)-this less general sector bound is sufficient for our purposes.
We make the following assumption on the nominal closed loop system. are in the open left-half complex plane. . The limit lim s!1 ðI À K 2 ðsÞG 2 ðsÞÞ À1 exists.
The first statement ensures the nominal closed-loop system is stable; the second ensures it is mathematically well-posed and, in state-space terms, is equivalent to the existence of the matrices Á :¼ ðI À D p D c Þ À1 and Á Á :¼ ðI À D c D p Þ À1 .

Anti-windup configuration
We consider the problem of designing an antiwindup compensator, YðsÞ, as depicted in figure 1. The anti-windup compensator emits two signals, 1 2 R m and 2 2 R q which enter the control input and the plant output respectively. This is less general than in the work of Grimm et al. (2003), where 2 is permitted to enter the controller state directly (i.e. 2 2 R n c Þ. This is because the current paper has been motivated by engineering problems where it may not always be plausible to inject 2 directly into the controller state equation (for example, in some embedded applications this is frequently the case). There is also some similarity to the work of Kapoor et al. (1998), although only observer-based synthesis is considered there ( 1 ¼ 0).
A novel way of representing most anti-windup configurations was introduced in Weston and Postlethwaite (1998), where one interprets the conditioning of controllers in terms of a single transfer function M(s); this is shown in figure 2. In Weston and Postlethwaite (1998) it was shown that, with all signals labelled identically and noting that DzðuÞ ¼ u À satðuÞ, figure 2 can be re-drawn as figure 3. This configuration reveals an attractive decoupling into nominal linear system, non-linear loop and disturbance filter. Note that if no saturation occurs, then the nominal linear system is all that is required to determine the system's behaviour. However if saturation occurs, the non-linear loop and disturbance filter become active. Using this representation, note that the question of non-linear stability for the complete system is translated into determining whether the non-linear loop is stable. The dynamics of the disturbance filter determine the manner in which the nominal linear behaviour is affected during and after saturation.
The representation in figure 3 was analysed in terms of existing schemes in Weston and Postlethwaite (2000) and ways of analysing stability and performance were also suggested. In particular, stability and performance synthesis for anti-windup compensators of order greater than or equal to that of the plant can be conveniently performed using an LMI formulation of the circle criterion.
Although such a treatment yielded attractive results, from a computational point of view, the method still has deficiencies: normally one would like to keep the dynamics introduced by a conditioning scheme to a minimum. Also the treatment tackled stability of the non-linear loop and performance of the disturbance filter separately, leading to the likelihood of suboptimality. The topic of this paper is to address the synthesis issues of static and low-order anti-windup compensators using the Weston and Postlethwaite (1998) representation. In this formulation, the mapping T : u lin°yd determines the deviation of the conditioned system from the nominal system, so keeping kT k i, p small in some appropriate L p norm yields, in some sense, good anti-windup performance. Note that this type of performance is not captured explicitly in the formulations of Grimm et al. (2001 b) and Mulder et al. (2001): the performance index in these papers is taken as something which is defined on the system as a whole, which allows Grimm et al. (2001 b) to conclude that the anti-windup performance is at best, greater than that of the nominal open and closed loop linear systems. Here, we do not have this constraint, and indeed, our performance index is purely defined on the saturated system. We now formally define the anti-windup problem we will consider in the remainder of the paper.
Definition 1: The anti-windup compensator YðsÞ is said to solve the anti-windup problem if the closed loop system in figure 3 is well-posed and the following hold: (1) If distðu lin , UÞ ¼ 0, 8t ! 0, then y d ¼ 0, 8t ! 0 (assuming zero initial conditions for YðsÞ).
The anti-windup compensator YðsÞ is said to solve strongly the anti-windup problem if, in addition, the following condition is satisfied: (3) The operator T : u lin°yd is well-defined and finite gain L p stable for some integer p 2 ½1, 1Þ.
Remark 1: In this paper, we only deal with the strong version of the anti-windup problem. The weaker version is included to enable the reader to make comparisons with other techniques in the literature, such as that in Teel and Kapoor (1997), partly on which this definition is based. Note that Condition 3 implies Condition 2.
Remark 2: We exclude the consideration of L 1 performance as T could be L 1 stable yet the output could exhibit a limit cycle.
Remark 3: It is important to stress that the work in this paper, and most other papers concerning anti-windup compensation, assumes that the model of the plant is reasonably accurate, that is G 2 is known precisely. It is not trivial to incorporate a robustness analysis into anti-windup compensation design, although some steps in this direction are described by Turner et al. (2003).

Representing M(s)
In order to analyse linear conditioning in terms of figure 3, it is necessary to derive an expression for M(s) in terms of Y. Partitioning Y as from figures 1 and 2, u can be derived, respectively, as Obviously, for the two schemes to be equivalent, we must have From figure 3, note that we must consider M À I for stability of T and G 2 M for the system's recovery after saturation. We will do this using LMIs, but before that we must derive a state-space representation of these two transfer functions. It is easy to derive a non-minimal realization but for numerical robustness we desire a 'minimal realization', such as is given in Appendix A (This realization is minimal iff (if and only if) the realizations for K 2 and G 2 are minimal; if this is not the case then there will be the same number of extra modes as there are in K 2 and G 2 ). Thus we assume that we have the following state-space representation for the transfer functions MðsÞ À I and G 2 ðsÞMðsÞ where Y is a static matrix by virtue of Y 1 and Y 2 being static and the precise form of the state-space matrices is described in Appendix A.

Stability and performance analysis
The aforementioned stability and performance problem can be captured by ensuring that T : u lin°yd is small in some sense. We now state a condition which ensures finite gain L 2 performance of the compensator; furthermore, the result allows the synthesis of an optimal compensator using standard convex optimization routines.
Theorem 1: There exists a static anti-windup compensator Y ¼ ½Y 0 1 Y 0 2 0 2 R ðmþqÞÂm which solves strongly the anti-windup problem for p ¼ 2 if there exist matrices Q > 0, U ¼ diagð 1 , . . . , m Þ > 0, L 2 R ðmþqÞÂm and a positive real scalar such that the following LMI is satisfied Proof: First note that as Y is linear, the first condition in the definition of the anti-windup problem is guaranteed trivially. Next, as Dzð:Þ 2 Sector½0, I, we haveũ Next assume 9vð " it follows that ky d k 2 < ku lin k 2 and hence that kT k i, 2 < ; in other words, the anti-windup problem is solved in the L 2 sense. Next, asũ u 0 Wðu Àũ uÞ ! 0, if then inequality (17) will be satisfied and hence kT k i, 2 < . Thus we consider inequality (18), which after evaluating the derivative with respect to time and substituting for _ " x x " x x, y d and " u u, becomes The LMI in the theorem then follows using standard Schur complement and congruency transformation arguments, as reported, for example, in Mulder et al. (2001) (although the details are not the same).
We must now prove that such a compensator guarantees that the system is well-posed. To assist in proving existence, we use the following lemma, based on a result from Grimm et al. (2001 b) (proof deferred to Appendix C).
Lemma 1: Assume that À2V ÀD DV À VD D 0 < 0 for some diagonal positive definite matrix V > 0. Also assume that the map PðwðtÞÞ: R l°RlÂl is unique for all w(t). Then I þD DPðwðtÞÞ is non-singular for all matrices P 2 &, where Note that in order for the anti-windup system to be well-posed, we must have existence and uniqueness of the equations in the non-linear loop. For this, we must be able to determine u d uniquely from the expression where we have definedD D :¼ ðD 01 þ " D D 1 YÞ. A solution (or solutions) exist iff I þD DKðuðtÞÞ is invertible for all u(t). Note that, in terms of Lemma 1, KðuðtÞÞ 2 &, and hence I þD DKðuðtÞÞ will be invertible for u(t), as the map Kð:Þ: R m°RmÂm is unique, if À2V ÀD DV À VD D 0 < 0 32 M. C. Turner and I. Postlethwaite  for some positive definite diagonal matrix V: but this will be the case if the LMI (14) is satisfied for Y ¼ LU À1 (from looking at the 2,2 term). This has proved that solutions exist, to further prove uniqueness is somewhat harder and is therefore deferred to Appendix D.
If only stability of the anti-windup scheme is required, and finite-gain L 2 performance is not a priority, one can ignore the last two rows and columns of the LMI in Theorem 1 and instead consider the corollary below.
Corollary 1: There exists a static anti-windup compensator Y ¼ ½Y 0 1 Y 0 2 0 2 R ðmþqÞÂm which solves strongly the anti-windup problem for all p 2 ½1, 1Þ if there exist matrices Q > 0, U ¼ diagð 1 , . . . , m Þ > 0 and L 2 R ðmþqÞÂm such that the following LMI is satisfied Furthermore, if this inequality is satisfied, then a suitable Y achieving kT k i, p < , for some > 0 and all integers p 2 ½1, 1Þ, is given by Y ¼ LU À1 .
Proof: The proof of the LMI is much the same as before, except that we do not consider the performance term ky d k 2 À 2 ku lin k 2 in the derivation. To prove that it still satisfies the strong anti-windup problem, we must prove that kT k i, p < , for some > 0. To see this, note that the LMI (22) gives sufficient conditions for the existence of a quadratic Lyapunov function, vð " x xÞ ¼ " x xÞ < Àk " x xk 2 , when u lin ¼ 0. This implies that the origin of (13) withũ u ¼ Dzðu d Þ-the state equation of T -is exponentially stable. Now note that the functions hð " x x, u lin Þ ¼ " are globally Lipschitz in both " x x and u lin (refer to (13) for a description ofũ u and note well-posedness of the interconnection). Then Theorem 6.1 in Khalil (1996) can be applied to establish that kT k i, p < for some > 0 and all p 2 ½1, 1Þ, or, in other words, that the anti-windup problem is solved strongly. oe Remark 4: One may wonder about the point of Theorem 1 in the presence of Corollary 1 as they both solve strongly the anti-windup problems but with Corollary 1 having less constraints and thus being more tractable numerically. However, Corollary 1 gives no direct way of optimizing the size of T , whereas in Theorem 1 this is addressed directly by introducing into the LMI, for the case of p ¼ 2 at least. On the other hand, for large or ill-conditioned problems, the said numerical superiority of the LMI in Corollary 1 may well make it preferable.

Low-order anti-windup compensator synthesis
Thus far the work has focused on synthesizing conditioning schemes of the form ¼ Yũ u, with Y being static. This can often lead to satisfactory performance, but does not use any frequency information in the synthesis of the compensators. In particular, we are normally interested in reacting to saturation at low frequencies i.e. when W 1 ðsÞũ u is large, W 1 ðsÞ being some low-pass filter. For example, it is not desirable, or even physically/computationally possible, to react to saturation caused by a very high frequency sinusoidal disturbance at the controller input, say at 10 12 radians/s. Moreover it is well documented (see Doyle et al. (1987) or ) that static compensation can be infeasible in situations where dynamic compensation may well be feasible.
In contrast, most dynamic schemes available with some sort of stability guarantees, are of order at least equal to that of the plant (for example the IMC scheme, the L 2 scheme of Teel and Kapoor (1997), the normalized coprime factor scheme of Weston and Postlethwaite (2000)). For many applications this can lead to a prohibitively high computational overhead, both for synthesis and implementation. Hence it is logical to address the synthesis of low-order compensators.
In principle, the results of Grimm et al. (2001 b) may be used to derive low order compensators achieving a prescribed L 2 gain (although, as stated in the introduction, this L 2 gain is defined on a different system, and is less helpful than the one we consider here), although the resulting optimization problem is, generally, non-convex, meaning that iterative algorithms for computing fixed-order optimal compensators are prone to becoming stuck at local minima. The approach we propose is definitely sub-optimal, but we have found it to give good results in practice.

A sub-optimal approach
The proposal retains much of the simplicity of the static approach, except we now let our compensator be described by the dynamic equations where Y 1 ðsÞ : where F 1 ðsÞ 2 R mÂm and F 2 ðsÞ 2 R qÂq are transfer function matrices andỸ Y 1 2 R mÂm andỸ Y 2 2 R qÂm are constant matrices. Obviously, Y 1 ðsÞ and Y 2 ðsÞ could be sythesized in an optimal fashion directly, but this could be computationally expensive; lead to unsuitable compensators (large poles, sensitive to parameter variations-well known from H 1 based LMI synthesis); and leave little room for intuition. Instead we propose that F 1 ðsÞ and F 2 ðsÞ be chosen using intuition and experience, and thenỸ Y 1 andỸ Y 2 are synthesized in an optimal way, similar to the pure static synthesis described earlier. Obviously the resulting compensator will be sub-optimal in terms of its L 2 gain but simulation results have shown that using relatively simple choices for F 1 ðsÞ and F 2 ðsÞ, such as first-order low pass filters (which suppress high frequency signals in y d ), very good responses can be obtained. In the scalar case, F 1 ðsÞ and F 2 ðsÞ can be chosen, guided by the graphical circle criterion. Typically, the modulus of the poles of F 1 ðsÞ and F 2 ðsÞ should be less than or equal to those of the controller, otherwise implementational difficulties can arise.

A sub-optimal synthesis routine
As before we obtain that except that this time Y 1 ðsÞ ¼ F 1 ðsÞỸ Y 1 and Y 2 ðsÞ ¼ F 2 ðsÞỸ Y 2 are dynamic. Also as before this implies that A minimal state-space realization for these expressions can be derived as where the matrices (The matrices " A A etc. are different from those in the previous section because they also incorporate the dynamics of YðsÞ in addition) are described in Appendix B andỸ Y is given asỸ Y :¼ ½Ỹ Y 0 1Ỹ Y 0 2 0 . Similar to the static case we have the following theorem.
Proof: The proof is similar, mutatis mutandis, to that of Corollary 1. oe Remark 5: Note that throughout the paper so far, stability and dissipativity have been established through the use of a purely quadratic Lyapunov function, vð " x xÞ ¼ " x x 0 P " x x > 0. In order to reduce the conservatism inherent in such an approach it would be preferable to use a different type of Lyapunov function; in particular a function of the Lur'e type seems appropriate i.e.
vð " x xÞ ¼ " Unfortunately, as pointed out in Weston and Postlethwaite (2000), such a choice of Lyapunov function generally does not lead to a convex problem for anti-windup compensator synthesis (for analysis purposes, more general Lyapunov functions can be used), and hence the determination of a compensator becomes more difficult.

Examples
We demonstrate our results on known examples in the literature and compare our anti-windup solutions with other optimal methods. As most of these techniques require the use of LMIs in the synthesis of compensators we decided to use one LMI solver, the Matlab LMI toolbox (Gahinet et al. 1995), exclusively. This seemed reasonably robust, in a numerical sense, although it did seem to be 'cautious' compared to some solvers. We also tried solving the problems using the SDPHA LMI solver of Potre et al. (1997). Although this tended to be less cautious than the LMI toolbox, and in many cases led to systems with better time responses, it also had instances of finding problems feasible, which were not strictly feasible, thus leading to erroneous results.

Simple dynamic compensator example
We consider the example introduced in Mulder and Kothare (2000) and also considered in  for which static anti-windup compensation is not feasible (in the sense that the LMIs associated with static synthesis are infeasible). This rules out our static results, but we find that our low-order results are feasible. We compare this with the full-order dynamic compensation results of Grimm et al. (2001 b) and the well-known IMC solution, which for stable plants G 2 ðsÞ always leads to a stable, full-order compensator.
The plant is described by C p ¼ ½À0:4 À 0:9, D p ¼ À0:5, The saturation limits are fixed at AE0:5 and the controller is described by For our low-order synthesis we chose our dynamic parts as first-order low-pass filters: These values for F 1 ðsÞ and F 2 ðsÞ were arrived at after several iterations. After solving the LMI in Theorem 2, the optimal value ofỸ Y returned was Figure 4 shows the responses of the various anti-windup compensators. Clearly, the low-order compensator of Theorem 2 shows the most desirable response with a fast settling time and little oscillation. The dynamic compensation of Grimm et al. (2001 b) is also good, but it exhibits a slower return to linear behaviour and a slightly more oscillatory response. The IMC compensation, which had yielded kT k i, 2 ¼ kG 2 k 1 % 10:72, leads to a stable but slow response and obviously the worst behaviour is observed when no anti-windup compensation is present. No static responses are shown due to infeasibility. It is worth noting that in this case the 'low-order' example is not really low order as the dimension of is the same as the number of plant states. Hence the compensator returned by Theorem 2 is of the same order as the full-order dynamic compensator obtained using the technique of Grimm et al. (2001 b). However, it is important to remark that the Grimm et al. (2001 b) compensator leads to a markedly worse reponse than that of Theorem 2; this is because Theorem 2 leads to a direct minimization of kT k i, 2 , whereas the formulae of Grimm et al. (2001 b) concentrate on a closed-loop transfer function.

A more realiztic example: missile auto-pilot
In this section, we consider a more realistic application: a missile auto-pilot. This example was introduced into the wider literature in Rodriguez and Cloutier (1994) to demonstrate the effectiveness of an 'error governor', originally proposed by Kapasouris et al. (1988). Romanchuk (1999) also notes the interest of this example. The plant in question is a simplified model of the dynamics of the roll-yaw channels of a bank-to-turn missile. The controller used is of the LQG/LTR type which yields excellent nominal closed-loop response types in both time and frequency domains. The statespace data of these systems are described in Appendix E. In accordance with Rodriguez and Cloutier (1994), the actuator limits are set at AE8 in both channels. Figure 5 shows the nominal linear responses of the closed-loop system for a pulse of r ¼ ½4:2 À 4:2 0 . Note the excellent decoupling, but observe that the control effort strays outside the set U ¼ co ðÀ8, À 8Þ, ðÀ8, 8Þ, È ð8, 8Þ, ð8, À 8Þg for some time, meaning that we can expect a poor response if actuator saturation were to be introduced. This intuition is verified in figure 6, where it can be observed that the decoupling in the nominal response is destroyed by the introduction of actuator limits. Also note that even in the first channel, tracking performance is lost due to the so-called infeasibility of the reference (a reference is called infeasible if, in the steady state it leads to a control signal which is permanently above the actuator saturation threshold). Figure 7 shows the missile response with the static anti-windup compensation of Grimm et al. (2001 b), which is also equivalent to that described by Mulder et al. (2001). The dynamic compensation is not shown for two reasons: it leads to responses almost identical to the static compensation; also, it leads to numerical problems as the compensators it generated had large poles and hence required a very small sample time for simulation (this problem with large poles also occurs with LMI-based H 1 synthesis). The optimal static compensator leads to an optimal performance index from the reference, r to the objective z ¼ r À y of sup r2L 2 Â ðkzk 2 =krk 2 Þ < ¼ 2:2937 (the dynamic compensation led to a slightly smaller ¼ 1:7834) (these values of are not comparable with those associated with kT k i, 2 ). From the figure it is obvious that this compensation leads to improved performance compared to the uncompensated system, although the second channel still has some significant differences to the linear response. Note that in both channels there is still some error during the pulse due to the infeasibility of the reference. Figure 8 shows the missile responses with the IMC based anti-windup compensation. The performance, as measured by the L 2 induced norm of the mapping of T , is % 376:25. Although the first channel behaves acceptably, the second channel does not and is scarcely better than the uncompensated response.
The response of the first channel is again acceptable and channel 2 exhibits an improved response over the IMC and uncompensated responses. However as with the performance obtained with static scheme of Grimm et al. (2001 b), there is some significant deviation from the linear response. Somewhat surprisingly, the bound on kT k i, 2 is actually higher for the optimal static compensator than for the IMC compensator. Figure 10 shows the missile response with a loworder static compensator. As the designer has several tuning parameters, four or five iterations were performed in order to obtain the best response. For this problem we chose and, according to Theorem 2,Ỹ Y was calculated as This response is the best exhibited by the anti-windup compensators demonstrated here. Both channels behave well when compared to the linear response, the second being almost identical. There is, of course, some deviation due to the infeasibility of the reference variable which results in a temporary error, but return to linear behaviour is swift after this occurrence. The order of this anti-windup compensator is 1, as F 2 ðsÞ and one channel of F 1 ðsÞ was chosen to be static, making it more attractive than the IMC and dynamic compensators of

Conclusion
This paper has considered the static anti-windup synthesis problem from a new perspective, and one which we believe is conceptually appealing and central to the 'true' anti-windup problem. We have also proposed, we believe, the first constructive, numerically tractable procedure for the synthesis of low-order antiwindup compensators, for which stability and performance of the overall system are guaranteed. Simple examples have shown the effectiveness of these schemes, although we think the true worth of these results will be greater when demonstrated on more realistic and complex systems. In such cases, it is often imperative to have an intuitive scheme to which one can consult if there are problems with initial designs. For this reason there needs to be further work on the choice of dynamics for the low order scheme. This is the subject of continuing research. It is still necessary to re-examine some of the more traditional techniques in the light of these new results. Although we think our work on low-order dynamic compensators may help to explain why simple compensators can work in many cases, there is still some way to go before we can justify why some of these techniques sometimes work better than optimal techniques. Some insight to this problem is also found in Mulder and Kothare (2000), where it is shown that quadratic Lyapunov functions are sometimes not always sufficiently general for the anti-windup problem, but, again, more research remains to be conducted.

Appendices
A. State-space derivation for static anti-windup compensator Some tedious algebra yields the state space matrices for a minimal realization of ½M 0 À I ðG 2 MÞ 0 0 as " 4. State-space derivation for dynamic anti-windup compensator As Y 1 ðsÞ ¼ F 1 ðsÞỸ Y 1 and Y 2 ðsÞ ¼ F 2 ðsÞỸ Y 2 , we assign the state-space realizations where x 1 2 R n 1 and x 2 2 R n 2 . Some tedious algebra then yields the state-space matrices for a minimal realization of ½M 0 À I ðG 2 MÞ 0 0 as where " A A 2 R n x Ân x , B 0 2 R n x Âm , " B B 2 R n x ÂðmþqÞ , " C C 1 2 R mÂn x , D 01 2 R mÂm , " D D 1 2 R mÂðmþqÞ , " C C 2 2 R qÂn x , D 02 2 R qÂm , " D D 2 2 R qÂðmþqÞ and n x :¼ n p þ n c þ n 1 þ n 2 .
Sufficiency: It is easy to see that setting S 1 ¼ I is sufficient for (85) to hold.
Necessity: Equation (85) can equivalently be written as I þ ðD D 11 À Xðu b2 ÞÞK 1 ðu a1 Þ ¼ S 1 þ ðD D 11 À Xðu b2 ÞÞK 1 ðSu a1 ÞS 1 ð86Þ A necessary condition for this to hold is that 1 þ ðd ii À x ii ðu b2 ÞÞk 1, i ðu a1, i Þ Defining ii :¼ d ii À x ii ðu b2 Þ, a necessary condition for (87) to hold is that j1 þ ii k 1, i ðu a1, i Þj ¼ j i jj1 þ ii k 1, i ð i u a1, i Þj 8i ð88Þ So assume that i 6 ¼ 1 for some i (this is necessary for u a 6 ¼ u b ), and at first we assume that 0 i < 1 (this is equivalent to u a, i > u b, i , and will be removed later) which, as k i ð:Þ is monotonically non-decreasing, implies that j i u a1, i j < ju a1, i j ð 89Þ jk 1, i ð i u a1, i Þj jk 1, i ðu 1a, i Þj ð90Þ Now we shall consider three cases and show that for 0 i < 1, equation (88) is contradicted in all cases.