Anti-windup synthesis for systems with rate-limits using Riccati equations

This article develops an anti-windup (AW) scheme for systems with rate-limited actuators. The main results show how a full-order AW compensator can be synthesised using an algebraic Riccati equation and several free parameters. A further result then shows how the free parameters may be chosen to influence, in an intuitive way, the local gain of the system and the size of the region of attraction. The results are of interest due to their low computational burden and hence their applicability to the systems of large state dimension.


Introduction
Many of the constrained input literature address the problem of input saturation whereby the magnitude of the control vector is subjected to element-wise constraints. Not surprisingly, researchers have devoted much attention to this problem due to its prevalence in most engineering applications. The magnitude saturation problem led practitioners and researchers to develop compensators which could be retrofitted (Edwards and Postlethwaite 1998) to existing controllers to assist the original controller in maintaining stability and performance during periods of saturation. These compensators have become known as 'antiwindup' (AW) compensators and many synthesis techniques have been proposed (some of the notable examples are Kothare, Campo, Morari, and Nett (1994), Miyamoto and Vinnicombe (1996), Mulder, Kothare, and Morari (2001), Saeki and Wada (2002), Crawshaw (2003), Herrmann, Turner, and Postlethwaite (2004), Turner and Postlethwaite (2004), Cao and Lin (2006), Kerr, Villota, and Jayasuriya (2007). It is evident that at present, at least in principle, the control engineer who faces magnitude saturation problems has tools to enable him/her to address the problem, with some of the more modern AW schemes guaranteeing stability and performance.
Magnitude limits on the control vector are not the only class of input nonlinearity which may be present in an otherwise linear system. Another significant class of nonlinearity which is well known to cause problems is input rate saturation. Although there are various ways of modelling this phenomenon, in essence, rate saturation (or rate-limiting as it is often called) limits, normally in an element-wise fashion, the rate of change of the control vector. This type of nonlinearity is not as widespread as the saturation nonlinearity, as, for example, electrical systems typically are less prone to suffer from this problem. Nevertheless, rate-limited actuators can be a significant problem in many systems, particularly those with mechanical actuators, and in some applications ratelimiting can have devastating consequences. Some of the more notorious examples of the catastrophic effects of rate-limiting are the crashes of the SAAB Grippen (Anonymous 1994) and Boeing V22 Osprey aircraft (Bolkcom 2005), and the meltdown of the Chernobyl nuclear power plant (Stein 2003).
Recently, some attempts at applying AW schemes to the rate saturation problem have been made (Barbu, Reginatto, Teel, and Zaccarian 1999;Hu, Teel, and Zaccarian 2005;Tarbouriech, Garcia, and Glattfelder 2006;Tarbouriech, Queinnec, and Garcia 2006). Although the development of these schemes can be regarded as a step towards bridging the gap between the need to provide rigorous guarantees of the closedloop system's behaviour during rate saturation, and the need to develop techniques which are useful in practice, these schemes have tended to be lacking in intuition and have not been developed enough for the practising engineer. One of the main advantages of using AW conditioning schemes is that the linear performance (i.e. the performance when rate saturation does not occur) is not directly restricted by the conditioning method, giving full freedom in the design of the linear controller. Moreover, the AW compensation schemes advocated in Weston and Postlethwaite (2000), which we follow here, are designed independently of the linear controller and thus the same AW compensator can be used with multiple controllers. This can be useful in some practical systems where the linear controller may change, depending on the operating condition.
The AW technique developed for systems with rate-limits in this article is based on the framework introduced in Weston and Postlethwaite (2000) and the Riccati equation synthesis method proposed in Sofrony, Turner, and Postlethwaite (2007). In order to make the method of Sofrony et al. (2007) applicable to rate-limited systems we use fairly standard results on sector narrowing -see Hindi andBoyd (1998), Pittet, Tarbouriech, andBurgat (1997) and Wu and Soto (2004), for example -where the deadzone nonlinearity is described locally by a sector of reduced size, leading to a tighter bound. By combining these various results, an AW design framework which is both rigorous and intuitive is developed. In particular, the AW technique proposed contains several tuning variables which can be adjusted in a reasonably straightforward manner in order to trade-off features important in practical AW problems. In fact, the results in this article were developed with the aim of implementing AW compensators on a sophisticated experimental aircraft; these results are reported elsewhere (Brieger et al. 2007(Brieger et al. , 2008Sofrony, Turner, Postlethwaite, Brieger, and Leissling 2006). This article is organised as follows. In the next section, a discussion of linear systems subject to ratelimited actuators will be given. Following this, the AW problem for rate-limited actuators will be introduced and a method for solving this based on the Riccati-based technique of Sofrony et al. (2007) will be proposed. A simulation example then follows and this article ends with some concluding remarks.

Notation
Notation is mainly standard throughout this article. The notation HðsÞ 2 RH 1 indicates that the transfer function H(s) is real, rational and is analytic in the right-half-complex plane. We make extensive use of the saturation function defined as sat " u ðuÞ :¼ sat m, " um ðu m Þ 2 6 6 4 3 7 7 5 and sat i, " The deadzone function is also frequently used and is defined as This article considers mainly local AW compensation which requires consideration of a region of attraction (ROA). For a nonlinear system _ x ¼ f ðxÞ with equilibrium point x ¼ 0, the ROA is defined as The L 2 norm of a vector x(t) is defined as xðtÞ 0 xðtÞdt and the induced L 2 norm (or L 2 gain) of a nonlinear operator, H, as Later in this article, it is convenient to work with a local L 2 gain which is defined as where W is some subset of the input space of H. It is clear that kHk i,2,W kHk i,2 .
2. Linear systems with rate-limited actuators

System modelling
This article is concerned with systems which take the form shown in Figure 1 which depicts a closed-loop system that is linear except for the rate-limit nonlinearity. The stabilisable, detectable finite-dimensional linear time invariant (FDLTI) plant has state-space realisation where x 2 R n p is the plant's state vector, y 2 R q is the vector of measured outputs and u r 2 R m is the input to the plant. The controller K(s) is also FDLTI and produces a control signal u 2 R m from a reference input r 2 R n r and the measured output y. Disturbances have not been included, although are handled with little extra effort (Weston and Postlethwaite 2000). The input to the plant and the controller output are not necessarily identical due to the rate-limit nonlinearity, Èð:Þ : R m°Rm , present at the plant input. This ratelimit nonlinearity represents the restrictions on the rates of the actuator and is well known to complicate the stability and performance analysis of a system. Unlike the position saturation nonlinearity, rate saturation is a dynamic nonlinearity and several methods have been proposed to model it (Rundquist and Stahl-Gunnarsson 1996;Saberi and Stoorvogel 1999;Hu et al. 2005;. Following Rundquist and Stahl-Gunnarsson (1996) and others, in the work here it is useful to model the rate-limit nonlinearity as shown in Figure 2. This model of the rate-limiter features a (decentralised) magnitude saturation in a feedback loop with a set of integrators and a (decentralised) set of gains H ¼ diagðh 1 , . . . , h m Þ. The rate-constraint values, which are now interpreted as magnitude saturation constraints in Figure 2, are defined as " d i , i 2 1, 2, . . . , m f g . The system's dynamic equations are as follows: This model of a rate-limit is appealing because, when the rate-limits have not been reached (i.e. when the saturation block in Figure 2 acts as the identity) the model behaves as a bank of low-pass filters with cut-off frequency 1/h i in each channel. However, when the ratelimits have been reached, the rate-limit model can be interpreted as a restricted tracking problem: that is the output u r attempts to follow u but with a restricted rate.
Remark 1: The time constants 1 h i must typically be greater than the dominant frequency components of u in order to prevent the linear dynamics of the actuator from interfering too much with the nominal control system design. This is often a valid practical assumption but if not, some account of these linear dynamics must be taken during nominal linear controller design.

AW architecture and the equivalent system
When the behaviour of the rate-limit nonlinearity is described by Equation (4), the only nonlinearity present in the system depicted in Figure 1 is the saturation nonlinearity internal to the rate-limit. By augmenting the plant and controller as shown in Figure 3, one can translate the rate-limit problem into an equivalent magnitude limit problem. In Figure 3, the 'augmented' controller and 'augmented' plant are respectively given by A state-space representation ofGðsÞ is easily derived as The augmented plant now includes dynamics absorbed from the rate-limit model and thus has a larger statevector ½x 0 , x 0 r 0 2 R n p þm ; its poles are now those of the plant G(s) plus m poles at the origin. The rate-limit problem can thus be thought of as a standard magnitude limit problem with the new 'plant',G, driven by the signal d m and the new 'controller',K, producing the 'control' signal d.
Figures 4 and 5 imply that the rate saturation problems can be treated in similar ways to magnitude limited problems, and indeed similar AW architectures can be used. Figure 4 shows the architecture used in this article; the 'augmented' plant,G, and controller, K, are those introduced earlier and the signal y ¼ ½ y 0 x 0 r 0 2 R qþm represents the 'augmented' measurement from the plantG. The AW compensator, Â (s), is driven by the signald : which is a signal internal to the rate-limit. The outputs from the AW compensator are labelled d d 2 R m and y d 2 R qþm which are injected at the augmented controller's input and output, respectively. It is important to mention that this architecture could be difficult to realise for physical rate-limits but as many systems contain software rate-limits to protect the physical rate-limits, it would be straightforward to apply to the software limits.

AW compensation
The diagram in Figure 4 is somewhat difficult to interpret and, aside from using the AW compensator Â (s) to guarantee the stability of the system, the remaining design goals are unclear. However, with the AW compensator parameterised as it is possible to re-draw Figure 4 as Figure 5. In this diagram, the subscript 'lin' denotes a signal which is produced from a purely linear system and the subscript 'd' denotes a signal produced by a system driven by the deadzone function, 'disturbing' the linear behaviour. This transformation was first proposed in Postlethwaite (1998, 2000) (see also Miyamoto and Vinnicombe (1996); Teel and Kapoor (1997)) and reveals an attractive decoupling of the system into: nominal (unsaturated) linear loop, nonlinear loop and disturbance filter. Two interesting features are evident from the diagram: (i) Assuming that the nominal (unsaturated) loop is stable, then providedGðsÞMðsÞ 2 RH 1 , the nonlinear stability problem reduces to that of guaranteeing stability of the nonlinear loop. (ii) The deviation of the system from nominal linear performance is quantified by the 'size' of signalỹ d .
From the above, it follows that the AW problem can be defined in terms of the nonlinear operator T p : d lin°yd Sofrony, Turner, and Postlethwaite 2005a), the 'gain' of which dictates how much the linear performance degrades in the presence of (rate) saturation.

Problem set-up
In the magnitude limit case, GðsÞ GðsÞ, and therefore providing that GðsÞ 2 RH 1 global stability can be established if global stability of the nonlinear loop can be established for certain M(s); it has been proven (see e.g. Grimm et al. (2003); Sofrony et al. (2007)) that global quadratic stability is possible if and only if the nominal linear plant is bounded real, i.e. kGðsÞk 1 5 for some positive scalar . The rate-limit case is not quite as simple; even if G(s) is stable, becausẽ GðsÞ ¼ ½GðsÞ 0 =s I=s, it follows thatGðsÞ = 2 RH 1 and therefore not bounded real. This implies that we must seek local stability. Additionally, the only way to ensure that the so-called disturbance filter is stable is to introduce zeros in M(s) to 'cancel' the unstable poles inGðsÞ. One convenient way to do this is to let M(s) be part of a coprime factorisation ofGðsÞ ¼ NðsÞMðsÞ À1 . In this case, a state-space realisation for our AW compensator can be given as where F is a matrix chosen to ensureÃ þBF is Hurwitz and is effectively the free parameter in our AW synthesis routine. In this case a state-space realisation of T p , which represents the mapping from our linear 'control signal', d lin , to the perturbation to the output, y d , can be written as However, this cancelling of the unstable poles of G(s) causes problems in ensuring that the nonlinear loop is globally stable. To circumvent this, we drop the stipulation of global stability and instead seek to ensure that the system is locally exponentially stable. This also makes practical sense, as in most control problems we are not interested in ensuring stability for all possible states; rather we seek stability for a set of states around an operating point. Thus formally we can pose the following problem.
Problem 1: Rate-limit AW compensation The objective of the rate-limit AW problem is to design a compensator of the form given in Equation (8) such that: (a) When d lin 0, the operator T p is locally exponentially stable; that is lim t!1x ðtÞ ¼ 0 for allxðtÞ 2 X & A. (b) Whenxð0Þ 0, kT p k i,2,W for some 4 0 and some set W: As our AW compensator in Equation (8) is a function of the state-feedback gain F, the above problem essentially reduces to choosing F such that T p is locally exponentially stable and kT p k i,2,W . Further note that, by the structure of our AW compensator d m divides the space of the input signal into linear unconstrained region (i.e. d 2 D) and a saturated region (i.e. d = 2 D)), then the AW compensator will remain inactive and linear performance will be not be disturbed. Also, in essence the local exponential stability assumption ensures that if distðd lin , DÞ 2 L 2 and d lin 2 W, thenỹ d 2 L 2 , ensuring an asymptotic return to linear behaviour if saturation occurs -see Teel and Kapoor (1997) or Rantzer (2000) for more details.

Local AW
The concept we make use of to enable local results to be derived is a long-standing tool in constrained input control, perhaps first being introduced in Hindi and Boyd (1998) and, since then, used frequently in the AW literature (e.g. Wu and Soto 2004;Sofrony et al. 2005).
The idea is as follows. It is well known that the deadzone nonlinearity Dzðd Þ 2 Sector½0, I for all d 2 R m . Now assume that d i d o,i for some d 0,i for all i, and define where d 0,i 4 " d i . If this is the case for all time t40, it is evident that Dzðd Þ 2 Sector½0, K where K ¼ diagð 1 , 2 , . . . , m Þ and i ¼ ðd o,i À " d i Þ=d o,i . It is possible to simplify this further by letting ¼ max i i , which implies that dzðd Þ 2 Sector½0, I for all d 2 Sðd 0 Þ. Therefore by restricting our attention to d 2 Sðd 0 Þ where d 0 5 1, which in turn implies 2 (0,1), the deadzone is guaranteed to inhabit a narrower sector than Sector½0, I ]; that is Note that for any arbitrary large d 0 there always exists an 2 (0,1) such that Dzðd Þ 2 Sector½0, I (in fact, global results are recovered if d 0,i ¼ 1 8i 2 1, . . . , m f g , and hence Dzðd Þ 2 Sector½0, I ). Note that by assuming Dzðd Þ 2 Sector½0, I , we are effectively restricting our attention to only a subset of possible signals d and thus results derived on this basis will be necessarily local, rather than global.

Region of attraction
As we are interested in deriving local results, it is desirable to estimate the region in which our proposed AW compensation technique provides local asymptotic stability. In other words, when deriving an AW compensator we also seek an ROA to accompany it, with it preferably being as large as possible. In general, computation of the ROA for nonlinear systems is a nontrivial task and over the years many authors (Fuller 1969;Teel 1999;Barbu et al. 1999;Cao, Lin, and Ward 2002;Gomes da Silva Jr, Tarbouriech, and Reginatto 2002;Gomes da Silva Jr, Tarbouriech, and Garcia 2003;Queinnec, Tarbouriech, and Garcia 2006;) have proposed methods which can be used to estimate the ROA for a saturated system.
Quite frequently in saturated systems analysis, an exact description of the ROA is not required and here we are content to consider a ROA with a crude geometry. In the saturated systems literature (as in Lin, Mantri, and Saberi (1998) and Gomes da Silva et al. (2002), for example), an ellipsoidal approximation is typically used: The largest such ellipsoid can be estimated (see e.g. Suarez, Alvaerz-Ramirez, and Solis-Daun (1997) and Turner and Postlethwaite (2001)) using a simple formula which will be given later. The salient feature of this type of representation is that it will allow a transparent relationship between AW design parameters and a subset of the ROA, E & X & A to be observed. It must be mentioned that the states of interest when characterising the region of guaranteed stability are the states of our AW compensatorx, as stability of our linear loop is guaranteed to be global.

AW solution
The objective of the AW compensator proposed is to find a Â(s) such that . The map T p : d lin°yd is sufficiently small. . The nonlinear loop is locally asymptotically stable.
We have seen that finding such a Â(s) can be reduced to finding an appropriate coprime factorisatioñ GðsÞ ¼ NðsÞMðsÞ À1 , and that this, in turn, reduces to finding an appropriate state feedback gain F. A systematic way of computing F, based on the Riccati equation approach reported in Sofrony et al. (2005aSofrony et al. ( , 2007, is given in the theorem below. Theorem 3.1: There exists a full order AW compensator Â ¼ ½Â 0 1 Â 0 2 0 2 R ðqþ2mÞÂm , as stated in (8), which solves Problem 1 if there exist matrices P ¼ P 0 40, W ¼ diagðw 1 , . . . , w m Þ 4 0 and positive real scalars and such that the following Riccati equation is satisfied: where the matrix Z must satisfy the following inequality: Furthermore, if Equation (13) is satisfied, a suitable Â is obtained by calculating the matrix F as A subset of the ROA is given by the ellipsoid where and w i is the ith diagonal element of W and B i is the ith column vector of B, for some i 2 1, . . . , m f g .
Proof: A similar theorem was briefly introduced in Sofrony et al. (2005a), although a slightly different proof will be provided here. It is constructed in several stages as detailed below: (i) The main inequality. Consider the function The final term 2d 0 Wðd Àd Þ arises from the assumption that d 2 Sðd Þ 8t ! 0 and hence that 2d 0 W ðd Àd Þ ! 0. Now consider two cases: (1) Assume thatxð0Þ ¼ 0 and d lin 2 W, where This implies that d 2 Sðd 0 Þ 8t ! 0 and thus if inequality (18) holds then this implies d dtx 0 Px þ kCxk 2 À 2 kd lin k 2 5 0: Integration with respect to time from t ¼ 0 to t ¼ 1 gives xð1Þ 0 Pxð1Þ Àxð0Þ 0 Pxð0Þ þ kỹ d k 2 2 À 2 kd lin k 2 2 5 0, ð21Þ which in turn implies that kỹ d k 2 À kd lin k 2 5 0 and hence kT k i,2,W 5 , i.e. condition (b) of Problem 1 is satisfied.
(2) Next assume that for some c40. Furthermore, let c be sufficiently small to ensure that d 2 Sðd 0 Þ. Then it follows that 2d 0 Wðd ÀdÞ ! 0 and inequality (18) implies that and thus VðxÞ is a Lyapunov function for the nonlinear loop in Figure 5, which ensures local asymptotic stability for allxð0Þ 2 E. Thus the system is locally asymptotically stable and hence, if inequality (18) holds, condition (a) of Problem 1 is satisfied.
Thus, it remains for us to prove that satisfaction of inequality (18) is equivalent to the expressions given in the theorem.
(ii) Evaluation of the main inequality. The remainder of the proof shows how we can re-write the cost function L, Equation (18), as the conditions given in Theorem 3.1. Evaluating Equation (18) results in an inequality containing several cross terms ind, d lin andx. Using similar ideas to those proposed in Sofrony, Turner, and Postlethwaite (2005), it is possible to address this issue and avoid LMI formulations. Expanding (18) and substituting d ¼ d lin À d d gives We now group the cross-product terms in three steps: (I) Cross-product terms involving d lin andd are grouped as follows: Combining the above with (24), a cost function containing no cross-product terms between d lin and d is obtained. Using Equation (9) to expand _x and noticing that d d ¼ Fx: where Z ¼ 2W À À2 2 W 2 : (II) Cross-product terms involvingd andx are grouped as follows: Note that Z must be positive definite in order to have a well-posed problem. This condition arises from the necessity of making the quadratic term ÀkðZ 1 2d À Z À 1 2 ðB 0 P À WF ÞxÞk 2 negative for any pair (d,x). It can easily be shown that if Z50, this is not always guaranteed. By replacing this new group of terms, a new cost function is obtained. This cost function has no crossproduct terms betweend andx.
(iii) Calculation of the ROA. In part (i) it was shown that the system is locally asymptotically stable in a set E & X where X is a subset of the ROA. As mentioned earlier, we estimate ellipsoidal subsets of the ROA and in particular are interested in calculating the largest such ellipsoid, " E, which has the following form: For the estimation of the ROA it is assumed that d lin ¼ 0 and, to ensure that the deadzone is within a reduced sector, that the statesx belong to the set Thus to calculate an ellipsoidal ROA we seek the largest set such that or equivalently we seek the largest set International Journal of Control Thus ifx 2 " E it follows that d 2 Sðd 0 Þ and Dzðd Þ 2 Sector½0, I ; hence the system is locally stable. To calculate " E, we follow a method used in, for example, Suarez et al. (1997) and Turner and Postlethwaite (2001). The largest ellipsoid ROA can be posed a maximisation problem: which is equivalent to where F i is the ith row of F. The largest such ellipsoid can then be calculated using Substituting for F yields which completes the proof. oe Remark 2: It can be seen that as gets close to unity, the size of c max and hence the size of " E increases, for fixed and W. Thus by making arbitrarily close to unity, the size of the ROA can be made arbitrarily large for some appropriate choice of and W.

Improved tuning procedure
The main issue in using Theorem 3.1 for AW synthesis is that the Riccati equation (13) is not guaranteed to be solvable for arbitrary , and W and hence the design could be complex. However, using Equation (14), which imposes the inequality constraint Z40, it is possible to rewrite Theorem 3.1 in a manner where a solution to the Riccati equation (13) is always guaranteed.
Corollary 1: There exists a full order AW compensator Â ¼ ½Â 0 1 Â 0 2 0 2 R ðqþ2mÞÂm which solves strongly the AW problem if there exists a matrix P ¼ P 0 40 and positive real scalars 2 (0,1), and 40 such that the following Riccati equation is satisfied: Furthermore, if Equation (47) is satisfied, a suitable Â achieving kT k i,2,W 5 is obtained by calculating the matrix gain F as follows: where ¼ ffiffi 1 q . An estimate to the ROA is given by the ellipsoid Proof: The proof will show how, by redefining certain parameters, the expressions in Theorem 3.1 can be rewritten as those in Corollary 1. As before, and are assumed to be positive scalars, and W is chosen to be W ¼ !I, ! 4 0. First note that using inequality (14), we obtain a bound on À2 as À2 I 52 ð1 À Þ 2 W À1 ) 9k 2ð0, 2Þ s:t: À2 ¼ k The extra term k is included in order to allow us to rewrite the inequality as an equality. Substituting this into Equation (13) yields the Riccati equatioñ where R is defined as R ¼ ð2 À kÞ ð1ÀÞ 2 ! À1 I. This allows us to replace R by R ¼ I where which yields Riccati equation (47) in Corollary 1. Similarly, using the expressions for and À2 in Equation (15) allows us to write Similarly, substituting this expression for F into Equation (45) allows a c max to be calculated as Note that the expression for the Riccati equation is simpler than before (it is only a function of ) but the expression for the and the ROA are still not clear. However, note from Equation (50), the effect of k can be offset to some extent by tuning W ¼ ! I appropriately. Thus, if one simply fixes k ¼ 1, it follows that the expressions for F and c max reduce to those given in the theorem.
Remark 3: The usefulness of Corollary 1 is in its ease of tuning: the three parameters ( 2 ð0, 1Þ, !40 and 40) of Theorem 3.1 have been replaced by just two ( 2 (0,1) and 40). Furthermore, the Riccati equation in Corollary 1 is just a function of and is just a standard LQR-type Riccati equation which will always have a solution P ¼ P 0 40 as ðÃ,B,CÞ are assumed stabilisable and detectable.
This new formulation is appealing because the Riccati equation is purely a function of and an explicit bound on the local L 2 gain is given by . Another interesting feature of this new simplified strategy is that it allows precise mathematical conclusions to be drawn about the trade-off between the local performance level and the ROA: it can be observed that Equations (52) and (54) embody the trade-off between the performance and ROA as they both involve . In particular, it is interesting to see how the ROA's size alters as a function of . Note that the ROA could equivalently be expressed as EðÞ It is difficult to characterise the size of this ellipsoid, but one way of doing so is by the sum of the length of its semi-axes, which is equal to the trace of E À1 ðÞ, which can be written as So the key parameter which determines the 'size' of the ROA is the matrix P(). We will establish how this varies as a function of . First, multiply Equation (47) by to get Defining QðÞ :¼ PðÞ 4 0 we now havẽ Now if ð 1 2C ,ÃÞ is detectable, it follows (Saberi, Lin, and Teel 1996;Teel 1999;Hu et al. 2005) that From this it also follows that dQ ii ðÞ=d 4 0. Thus PðÞ ¼ QðÞ and P ii ðÞ ¼ Q ii ðÞ are monotonically increasing functions of . Hence tr½E À1 ðÞ ¼ max i " d 2 i tr½Q À1 ðÞQ À1 ii ðÞ is a monotonically decreasing function of which implies that as increases, the 'size' of the ROA decreases. Furthermore, in the limit as tends to zero ( ! 1, i.e. global results), the ROA becomes the whole state-space.
Also note that we can write F as and thus we can see that as becomes smaller, F becomes smaller (in the sense that its norm reduces) and as ! 0, F ! 0 and we recover the IMC solution; this also corresponds to ¼ 1. In other words, we have the 'worst' performance for the IMC AW case (which is not surprising sinceGðsÞ is not bounded real).
In conclusion we can see that meaning that there is a clear trade-off between minimising the local L 2 gain and maximising the size of the ROA.
Remark 4: The tuning procedure presented above states that for a given -and therefore for a given performance bound and ROA -there exists a family of compensator gains, F, parameterised by 2 (0,1). The parameter can be seen to play a similar role to the free parameter in the low-high-gain design of Lin et al. (1998) and one may expect some performance enhancement when is chosen close to unity. However, it is difficult to specify precisely what this performance enhancement is. Experience suggests that when is chosen close to unity (i.e. F is 'high gain') noticeably better time domain performance is observed when the states are close to the origin, but that this can worsen dramatically for states closer to the edge (but within) the ROA. When is chosen closer to zero (i.e. F is 'low gain'), the difference between the small and large signal performance of the compensators does not seem so noticeable and appears to allow the compensators to preserve stability for regions beyond the guaranteed ROA.

Example
In this section, the effectiveness of the results are shown through a well-known academic example. Consider the problem of a missile auto-pilot as presented in Rodriguez and Cloutier (1994), Romanchuk (1999), Sofrony et al. (2007), which is a third order two-input-two-output dynamic linear system. As required, the feedback interconnection of the plant and controller when no rate-limits are present is internally stable and well posed. However the 'real' nonlinear system has actuators with rate-constraints of AE3ðdeg=sÞ imposed in both channels. Figure 6(a) shows the nominal linear response of the missile for a pulse r ¼ ½5 À 5. Notice the decoupled outputs and swift but well-damped rise-times in both channels. However, it must be mentioned that the rate of the control signal exceeds the imposed limits and consequently the system with rate-limited actuators could exhibit a loss in performance and perhaps stability. Figure 6(b) confirms this and shows clearly how the system loses its decoupling and tracking properties when rate-limits are introduced.
To limit the performance degradation during periods of rate saturation an AW compensator is introduced. As mentioned earlier, for the rate-saturated case, the task of choosing appropriate values of the design parameters presented in Theorem 3.1 is not as simple as in Sofrony et al. (2005), but the approach is still intuitive. The designer's task can be simplified by using the tuning algorithm proposed in Section 3.5; we will concentrate on this approach.
Corollary 1 provides a simple way of characterising the performance level and stability region of the AW compensator. Notice that the available parameters are and , where gives a measure of nonlinear performance and defines the polyhedral set of allowed signals d, i.e. Sðd 0 Þ ¼ d : Àd 0 " d " d 0 f g . The main goal of our design is to synthesise an AW compensator which guarantees stability in a set as large as possible while preserving performance as much as possible. As shown earlier, in order to achieve good performance (i.e. kT p k i,2,W small), it is desirable to choose large; from Equation (54) it can be concluded that for an enlarged ROA must be small. In other words, the trade-off between performance and ROA is encapsulated in the design parameter .
The main issue with the approach used in this article is that it is local in nature: performance and stability guarantees are ensured only if d 2 Sðd 0 Þ. On the other hand, the conditions presented are only sufficient, and a system which does not guarantee d 2 Sðd 0 Þ may still be stable within a large ROA (or perhaps the whole state space). Even if we assume that the AW compensator can guarantee global (or semiglobal) stability, degradation of the closed-loop compensated system's performance may still be present as the norm bound of the map T p no longer holds. In fact, after some analysis, it can be inferred that as the system moves away from the pre-established ROA, the gain kT p k i,2,W increases, yielding an increase on the system's deviation from linear dynamics and hence a degradation of the prescribed nonlinear performance.
To illustrate the design of AW compensators using Corollary 1, two compensators are designed: ¼ 0:9 is fixed and takes values of low ¼ 10 À4 and high ¼ 10 for the 'low' and 'high' performance compensators, respectively. Figure 7(a) depicts missile responses obtained with the low and high AW compensators, respectively and a reference pulse with amplitude of AE5½deg. Clearly, in both cases the system's response is enhanced by the use of AW compensation: rate saturation levels are reduced, tracking properties are retained and return to linear behaviour is well behaved. It is somewhat more difficult to conclude which compensator delivers the best performance; for the low AW compensator the second channel exhibits fast tracking performance while the first channel is sluggish. On balance, however, one might say that the 'high-performance' compensator does indeed perform better for this small input demand. Figure 7(b) illustrates the missile responses obtained with the two compensators when the reference demands are increased to AE30½deg. Such a large reference demand causes the system to depart from its guaranteed region of stability (i.e. d = 2 Sðd, d o Þ) and while stability is not lost, performance does degrade. As the region of stability for the high-performance compensator is smaller than that of the low-performance compensator, it is not surprising that its performance deteriorates more. Although both compensators suffer a lack of tracking ability in the first channel (due to the restricted rate and short duration of the pulse demand), the low-performance compensator retains its tracking ability in the second channel and exhibits a reduced phase-lag in the first channel. However, the high-gain compensator is unable to preserve this tracking performance and the second channel fails to follow that of the linear system well. In conclusion, this example appears preferable to accept a loss of small-signal performance in exchange for better preservation of large-signal performance.

Conclusion
This article has proposed an AW scheme for systems with rate-limited actuators. The proposed synthesis algorithms have been constructed using ideas which are reasonably well known in the AW literature, although the Riccati-based solution is novel and superior from a computational perspective. A key feature of the proposed AW algorithms is that they are fairly easy to tune and feature a parameter, , which captures the trade-off between local performance and the size of a subset of the ROA.
The proposed algorithms have been demonstrated on a simple example, but their real use is envisaged for complex, high-order systems where LMI-based methods may have difficulty. These algorithms were used for the design of the AW compensators for investigating rate-limiting on the ATTAS aircraft. Details of the flight tests are reported in Brieger et al. (2006Brieger et al. ( , 2007Brieger et al. ( , 2008 but, broadly speaking, it was found that the methods described within this article are useful for designing AW compensators for complex, practical systems.