Robust linear static antiwindup
with probabilistic certificates
Abstract
In this paper, we address robust static antiwindup compensator design and performance analysis for saturated linear closed loops in the presence of nonlinear probabilistic parameter uncertainties via randomized techniques. The proposed static antiwindup analysis and robust performance synthesis correspond to several optimization goals, ranging from minimization of the nonlinear input/output gain to maximization of the stability region or maximization of the domain of attraction. We also introduce a novel paradigm accounting for uncertainties in the energy of the disturbance inputs.
Due to the special structure of linear static antiwindup design, wherein the design variables are decoupled from the Lyapunov certificates, we introduce a significant extension, called scenario with certificates (SwC), of the socalled scenario approach for uncertain optimization problems. This extension is of independent interest for similar robust synthesis problems involving parameterdependent Lyapunov functions. We demonstrate that the scenario with certificates robust design formulation is appealing because it provides a way to implicitly design the parameterdependent Lyapunov functions and to remove restrictive assumptions about convexity with respect to the uncertain parameters. Subsequently, to reduce the computational cost, we present a sequential randomized algorithm for iteratively solving this problem. The obtained results are illustrated by numerical examples.
I Introduction
Antiwindup designs correspond to control systems augmentations in light of actuator saturations, to mitigate the negative effects of the input nonlinearity. Their development has a history dating back to the era of analog controllers, more than half a century ago, and the most effective techniques are well illustrated in [33, 37, 34, 18]. When robustness to parameter uncertainties must be taken into account, only recent results on suitable antiwindup constructions become available, all of them formulated in the deterministic robust control context. Some relevant examples correspond to [26, 36, 31, 14, 30, 19, 17, 22, 20], where several successful solutions differing in nature and architecture have been proposed. While these available robust antiwindup solutions arise from different approaches and paradigms to the robust antiwindup problem, they mostly share the common feature of arising from a deterministic approach wherein a constant but unknown parameter belongs to a (known) compact set. Then, by suitable relaxations of the assumptions at the price of increased conservativeness, these sets are convexified to obtain numerically tractable approaches to the analysis and design problem. In this paper we follow a radically different paradigm, arising from randomized methods for performance analysis and control design.
Randomized and probabilistic methods for control received a growing attention in the systems and control community in recent years [35]. These methods deal with the design of controllers for systems affected by possibly nonlinear, structured and unstructured uncertainties. One of the key features of these methods is to break the curse of dimensionality, i.e., uncertainty is “lifted” and the resulting controller satisfies a given performance for “almost” all uncertainty realizations. In other words, in this framework, we accept a “small” risk of performance violation.
One of the successful methods that have been developed in the area of randomized and probabilistic methods is the socalled scenario approach, which provides an effective tool for solving control problems formulated in terms of robust optimization [6]. In this case, the sample complexity, which is the number of random samples that should be drawn according to a given probabilistic distribution, is derived a priori, and it depends only on the number of design parameters , and probabilistic parameters called accuracy and confidence .
In parallel with these methods, sequentialbased approaches have been developed, see for instance the recent sequential probabilistic validation techniques proposed in [1] and references therein. In particular, in [11] an algorithm is proposed which, at each iteration, constructs a candidate controller, whose performance is then validated through a Monte Carlo approach. If the controller does not enjoy the required probabilistic performance specification, a new controller is designed based on new sample extractions. At each step of the sequence, a reducedsize scenario problem is solved. This method is usually effective in practical applications, even if its sample complexity cannot be determined a priori. These methods may be used in specific control problems such as designing a common quadratic Lyapunov function. In these cases, however, the fact that a single common Lyapunov function should hold for all possible uncertainties leads to an overly conservative design. The same drawback is known in classical robust control, where the design of a common quadratic Lyapunov function requires an exponential number of computations [2, 9]. For these reasons, parameterized Lyapunov functions have been developed and used in many robust control problems subject to uncertainty [3, Chapter 19.4].
Within the above surveyed context, the contribution of this paper is twofold. In the first part of the paper (Section II), we develop a new framework, denoted as scenario with certificates, which is very effective in dealing with parameterdependent Lyapunov functions. This framework continues the research originally proposed in [29] for feasibility problems in the context of randomized methods. The main idea in this approach is to distinguish between design variables and certificates and has the advantage, compared to classical robust methods, that no explicit parameterization (linear or nonlinear) of the Lyapunov functions is required. In other words, the method is based on a “hidden” parameterization of the Lyapunov functions, and has the clear advantage to reduce the conservatism compared to the methods based on the design of common Lyapunov functions.
In the second part of the paper (Sections III and IV), we show the application of the scenario with certificates approach to antiwindup design and analysis in the presence of timeinvariant uncertainty. In particular, we concentrate on a specific antiwindup scheme for linear saturated plantcontroller feedbacks: static direct linear antiwindup design (see, e.g., [37, Part II]). Direct linear antiwindup corresponds to augmenting a linear saturated control design with a linear gain driven by the excess of saturation and injecting suitable correction “antiwindup” signals at the state and output equation of the predesigned windupprone linear controller. Several different performance optimization tasks are considered, and we present different alternatives in the subsections of Section III. A notable one, which is novel to the antiwindup field and arises naturally from the proposed probabilistic context, is the one (in Section IIIB) where the design minimizes an upper bound of the area spanned by the nonlinear gain curve, accounting for uncertain (but probabilistically known) energy of the external disturbance acting on the saturated closed loop. Each proposed performance metric is shown together with a robust performance analysis result that is not limited to the antiwindup context but is applicable to any uncertain linear closed loop subject to saturation in the classical LFT form. In all the above contexts, we will show that the probabilistic approach allows to reduce conservatism as well as to cope with uncertainty entering nonlinearly in the problem description, without overbounding it. The latter case has instead already been treated in various examples in the literature, where the tradeoff between the robust and deterministic approach is usually referred to as probability degradation function, see e.g. [35, Ex. 11.1 and 12.1].
Preliminary results in the direction of this paper were presented in [16, 15]. In particular, in [16] the results were based on the classical scenario optimization approach. In that formulation, both the certificates and the design variables were treated as optimization variables over the whole operating region, thus leading to a conservative solution, and  sometimes  to infeasibility. The scenario with certificates solution proposed here was then introduced in [15], where we also provided preliminary results on the design of antiwindup compensators minimizing the nonlinear gain.
As compared to these preliminary results, in this paper we fully exploit the potential of the proposed randomized approach towards the design of static antiwindup gains arising from suitable performance/robustness tradeoffs. More specifically, after analyzing indepth the formal properties and the algorithmic solutions for the novel randomized approach, we apply it to the robust design of antiwindup compensators within different problem settings; namely, we address the minimization of the nonlinear gain, the minimization of the area spanned by the nonlinear gain curve, the minimization of the reachable set and the maximization of the domain of attraction for closedloop saturated systems. For each of the above problems, we provide several discussions and a suitable simulation example (in Section IV).
The paper is ended by some concluding remarks.
Notation
In the remainder of the paper, the following notation and definitions are adopted:

the norm of a scalar valued signal , defined for , is

denotes the Euler number;

given a square matrix , ;

given a matrix , denotes the row of .
Ii Scenario with certificates
In this section, we briefly recall the scenario approach in dealing with convex optimization problems in the presence of uncertainty, and subsequently introduce a novel framework that we name scenario with certificates (SwC).
Iia The scenario approach
The socalled scenario approach [6] has been developed to deal with robust convex optimization problems of the form
(RO)  
where, for given within the uncertainty set , are convex functions of the optimization variable , the domain is a convex and compact set in and the uncertainty set is not necessarily compact. Furthermore, we assume that is a continuous (possibly nonlinear) function of for any given .
Following the probabilistic approach discussed for instance in [35, 7] a probabilistic description of the uncertainty is considered over . That is, we formally assume that is a random variable with given probability distribution with support . Such a probability distribution may describe the likelihood of each occurrence of the uncertainty or a userdefined weight for all possible uncertain situations. Then, independent identically distributed (iid) samples are extracted according to the probability distribution of the uncertainty over .
These samples are used to construct the following scenario optimization (SO) problem, based on instances (scenarios) of the uncertain constraints
(SO)  
Problem (SO) can be seen as a probabilistic relaxation of problem (RO), since it deals only with a subset of the constraints considered in (RO), according to the probability distribution of the uncertainty. However, under rather mild assumptions on problem (RO), by suitably choosing , this approximation may in practice become negligible in some probabilistic sense. Specifically, can be selected depending on the level of “risk” of constraint violation that the user is willing to accept. To this end, the violation probability of the design is defined as
(1) 
where denotes the probability with respect to the distribution of the random variable . Similarly, the reliability of the design is given by
Then the following result has been proven in [10].
Proposition 1.
We note that nonuniqueness of the optimal solution can be circumvented by imposing additional “tiebreak” rules in the problem, see, e.g., Appendix A of [6]. Also, in [8] it is shown that the feasibility assumption can be removed at the expense of substituting with in .
From Equation (2), explicit bounds on the number of samples necessary to guarantee the “goodness” of the solution have been derived. The bound provided in [1] shows that, if, for given , the sample complexity is chosen to satisfy the bound
(4) 
then the solution of problem (SO) satisfies with probability . This bound improves by a constant factor upon previous bounds, see e.g. [8], and it shows that problem (SO) exhibits linear dependence in and , and logarithmic dependence on . Note however that, from a practical viewpoint, it is always preferable to numerically solve the one dimensional problem of finding the smallest integer such that .
IiB Scenario with certificates
The classical scenario approach previously discussed deals with uncertain optimization problems where all variables are to be designed. On the other hand, in the design with certificates approach we distinguish between design variables and certificates . In particular, we consider now a function , which is assumed to be jointly convex in and for given (where and are supposed to be nonempty), and construct the following robust optimization problem with certificates
(RwC)  
where the set is defined as
(5) 
The key observation that is at the basis of the approach developed in this section is that the set is convex in for any given , as formally shown in Theorem 1 below.
Remark 1.
[Common vs. parameterdependent certificates] As discussed in the Introduction, problem (RwC) corresponds to searching for socalled parameterdependent certificates, in the sense that a different certificate is allowed for every instance of the uncertainty , that is . This is very different from the approach frequently adopted when dealing with uncertain systems, based on the design of common certificates. This would result in a robust problem of the form
(CO)  
where the common certificate should be the same for all possible values of . Clearly, if the spread of the uncertainty is large, it is unreasonable to expect the same certificate to hold for all . For instance, in the classical case when the certificates correspond to Lyapunov functions for proving stability, the difference between the two approaches lies on the difference between common Lyapunov functions and parameterdependent ones. In particular, for this problem, different solutions have been proposed in the robust control literature, which are based on explicit parameterizations (e.g. linear or bilinear) of the function , see for instance [3]. One of the main novelties of the probabilistic approach discussed in this paper is the fact that no explicit parameterization is necessary.
In [29], an approach to handle parameterdependent linear matrix inequalities (LMIs) has been introduced, and a solution for feasibility problems, based on uncertainty randomization and on an iterative ellipsoidal algorithm, has been derived. The approach considers different certificates for each sampled value of the random uncertainty. In the same paper, the conservatism reduction is illustrated by means of a numerical example showing that traditional robustness methods based on common Lyapunov functions fail.
In the current work, we follow along this line of research, and propose to approximate problem (RwC) introducing the following scenario with certificates problem, based again on a multisample extraction
(SwC)  
Note that, contrary to problem (SO), in this case a new certificate variable is created for every sample , , that is . To analyze the properties of the solution , we note that, in the case of SwC, the reliability and violation probabilities of design are given by
We now state the main result regarding the scenario optimization with certificates.
Theorem 1.
Proof.
We first prove convexity of the set . To see this, consider . Then, there exist such that
Consider now , with , and let . From convexity of with respect to both and it immediately follows that
hence , which proves convexity.
Now, observe that the condition is equivalent to requiring
so that problem (RwC) is equivalent to
(7)  
Note that, from the convexity of , it follows that the function is convex in for given ; see also [5, p. 113]. Hence, problem (7) is a robust convex optimization problem. Then, we construct its scenario counterpart
(8)  
where the subscript for the variables highlights that the different minimization problems are independent. Finally, we note that (8) immediately rewrites as problem (SwC). ∎
We remark that problem (SwC) has separate constraints, one for each , and each constraint involves a different certificate. However, notice that the dimension of the certificates does not enter in the righthand side of the probability bound (6) in Theorem 1. Hence, the sample complexity of problem (SwC) is smaller than that of the scenario counterpart of the problem with common certificates (CO), in which both and play the role of design variables. On the other hand, the complexity of solving problem (SwC) is higher, since the number of optimization variables significantly increases, because a different variable is introduced for every sample . This increase in complexity is not surprising, being problem (RwC) much more difficult than problem (CO). In particular, we remark that, in the case when the constraints are linear matrix inequalities, then the scenario problem can be reformulated as a semidefinite program by combining the LMIs into a single LMI with blockdiagonal structure. It is known, see [4], that the computational cost of this problem with respect to the number of diagonal blocks is of the order of . The sequential method discussed in the next section aims at improving the computational efficiency by reducing the number of scenarios.
IiC Sequential randomized algorithm for SwC
Motivated by the computational burden of the SwC solution, in this section, we present a sequential randomized algorithm that alleviates the load by solving a series of reducedsize problems. The algorithm is a minor modification of [11, Algorithm 1], which was introduced for the standard scenario approach, and it is based on separate design and validation steps. The design step requires the solution of the reducedsize SwC problem. In the validation step, contrary to [11] where only functional evaluations are required, the feasibility problems (3) and (4) need to be solved. However, it should be pointed out that the latter problems are of small size, and can be solved independently, and hence parallelized. The sequential procedure is presented in Algorithm 1, and its theoretical properties are stated in the subsequent lemma. Its proof follows the same lines of that in [11, Theorem 1 and Algorithm 1], and is omitted for brevity. It should be stressed, however, that in [11] the sequential approach was not applied to SwC, but to standard scenario optimization.
Sequential Algorithm for SwC

Initialization
set the iteration counter . Choose the desired probabilistic levels , and the desired number of iterations 
Update
set and where is the smallest integer s.t. 
Design

draw iid (design) samples

solve the following reducedsize SwC problem

if the last iteration is reached ,
return

Lemma 1.
Assume that, for any multisample extraction, problem (3) is feasible and attains a unique optimal solution. Then, given accuracy level and confidence level , let
(11) 
where , with , is a finite hyperharmonic series. Then, the probability that at iteration Algorithm 1 returns a solution with violation greater than is at most , i.e.,
(12) 
Remark 2.
The dimension of the system that can be handled by the algorithm depends not only on , but also on the desired probabilistic accuracy and confidence. The paper [11] considers a realworld example of a hard disk drive consisting of design parameters and uncertain parameters. It is shown that the sequential approach provides results even for very tiny values of accuracy and confidence, contrary to the oneshot solution, i.e. the one considering all the constraints at once.
In the second part of this paper, we introduce the problem of robust gain minimization for linear antiwindup systems. The SwC approach appears to be well suited for such a design problem, for several reasons: i) the nominal design can be formulated in terms of linear matrix inequalities, ii) the uncertainty set can in principle be of any size and shape, and iii) the optimization variables can be easily divided in design variables for the antiwindup augmentation and certificates for stability and performance guarantees, iv) the number of uncertain parameters can in principle be arbitrarily large and any functional dependence is allowed.
Iii Antiwindup compensator design
Consider the linear uncertain continuoustime plant with inputs subject to saturation
(13)  
where is the plant state, is the control input, is an external input (possibly comprising references and disturbances), is the performance output, is the measured output and denotes random uncertainty within the set . We denote by the nominal value of the uncertain parameters.
As customary with linear antiwindup design [37], we assume that a linear controller has been designed, based on the nominal system, in order to induce suitable nominal closedloop properties when interconnected to plant (III)
(14) 
where is the controller state, typically comprises references (but may also contain disturbances), is the controller output and is an extra input available for antiwindup action. The controller (14) is typically designed in such a way that the socalled unconstrained closedloop system given by (III), (14), , is nominally asymptotically stable and satisfies some nominal or robust performance requirements.
Consider now the (physically more reasonable) saturated interconnection , where the entry of is , denoting the input by . When the input saturates, the closed loop system composed by the feedback loop between (III) and (14) is no longer linear and may exhibit undesirable behavior, usually called controller windup. Then, one may wish to use the free input to design a suitable static antiwindup compensator of the form
(15) 
This signal can be injected into the right hand side of the controller dynamics (14) to recover stability and performance of the unconstrained closedloop system.
When lumping together the plantcontrollerantiwindup components (III), (14), (15), , one obtains the socalled antiwindup closedloop system, a nonlinear control system which can be compactly written using the state as in (16) (at the top of the next page),
(16) 
(17a)  
(17b)  
(17c) 
(18) 
where denotes the deadzone function, i.e., , and all the matrices are uniquely determined by the data in (III), (14), (15) (see, e.g., the full authority antiwindup section in [37] for explicit expressions of these matrices).
The compact form in (16) may be used to represent both the saturated closed loop before antiwindup compensation, by selecting , or the closed loop with antiwindup compensation, by performing some nonzero selection of .
Iiia gain minimization
First, we analyze system (16) for the nominal case, that is when no uncertainty is present and is a singleton coinciding with the nominal value of the parameters.
In this nominal case, the results in [12, 24, 23, 33] and references therein generalize the wellknown sector conditions originating from absolute stability theory, into a socalled generalized sector condition, stating that given any matrix , it holds that for all satisfying . This condition is a powerful tool because it enables us to provide a nonglobal homogeneous characterization of the stability and performance properties of the nonlinear closed loop (16) by way of an extension of absolute stability theory. In particular, in (17) the generalized sector condition provides guarantees on the derivative of a quadratic Lyapunov function in a suitable (ellipsoidal) sublevel set (see (19) below) contained in the region where (this is guaranteed by (17c)). Here, parameters , and can be optimized by way of a convex semidefinite program. More formally, we recall the following stability and performance analysis result from [23, Theorem 2].
Proposition 2 (Regional stability/performance analysis).
Given a scalar , consider the nominal system, that is let . Assume that the semidefinite programming (SDP) problem (17) in the variables , , and is feasible. Then:
As suggested in [23], one may use the result of Proposition 2 to compute an estimate of the nominal nonlinear gain curve (see [27]), namely a function such that for each in the feasibility set of (17) and for each satisfying , the zero initial state solution to (16) satisfies
To do so, it is possible to sample the nonlinear gain curve by selecting suitable positive values and, for each , solving (c), after replacing . Then, the gain curve estimate can be constructed by interpolating the points , .
Following the derivations in [12] (which generalize the global results of [28]), one may notice that the product appears in a linear way in equation (17b) and, for a fixed value of , the synthesis of a static antiwindup gain minimizing the nonlinear gain can be written as a convex optimization problem, as stated next.
Proposition 3 (Regional stability/performance synthesis).
Given the plantcontroller pair (III), (14), and a scalar , consider the nominal system, that is is a singleton. Assume that the SDP problem
is feasible. Then, selecting the static antiwindup gain as
(22) 
the antiwindup closedloop system (III), (14), (15), or its equivalent representation in (16) satisfies properties (a)(c) of Proposition 2.
Remark 3.
The static linear antiwindup architecture (15) adopted in Proposition 3 and in the rest of this paper is only one among many possible choices (see, e.g., [37]). In particular, when using direct linear antiwindup designs, an alternative appealing approach is given by the design of a plantorder linear filter (namely, of the same order of the plant) generalizing the static selection in (15). Such a dynamic generalization of (15) was shown in [21] to be important to guarantee global exponential stability in the presence of saturation. However, this fact was later deemphasized once the above mentioned generalized sector condition was introduced (see [24], which provides the nonglobal extension of the results in [21]). More specifically, nonglobal guarantees of stability in the presence of saturations/deadzones is a fundamental tool to establish exponential stability properties of the origin for nonasymptotically stable plants that are stabilized through a saturated control input.
Remark 4.
The separation between the Lyapunov certificate and the optimization variables in (18) is only possible when adopting the static architecture in (15), which makes the robust extensions provided below reasonably simple. Extensions to the dynamic plantoder antiwindup case is possible only if one adopts certain conservative convex relaxations of the nonconvex robust conditions, along similar directions to those well surveyed, for example, in [13].
Consider now the uncertain case, when the system matrices in (16) defining the dynamics of and are continuous (possibly nonlinear) functions of the uncertainty , which is considered to be time invariant. Then, the interest is in finding robust solutions to the analysis and design problems discussed before. For instance, in the analysis case, one could search for common certificates in (c) such that is minimized over (17) for all . This approach is pursued in [16], where scenario results are used to find probabilistic guaranteed estimates. Note that the use of a common Lyapunov function is well justified when the uncertainty is, for instance, timevarying. However, as discussed in Section IIC, an approach based on common certificates is in general very conservative in the case of timeinvariant uncertainty, and one would be more interested in finding parameterdependent certificates. To do this, we would need to solve the following robust optimization problem with certificates
(23)  
A similar rationale can be applied to robustify the antiwindup synthesis problem of Proposition 3. As a matter of fact, when the system matrices are uncertain, one meets similar obstructions to those highlighted as far as analysis was concerned. Again, instead of looking for common Lyapunov certificates as in [16], we write the following RwC problem
(24)  
Note that both problems (23) and (24) are difficult nonconvex semiinfinite optimization problems, due to the fact that one has to determine the certificates as functions of the uncertain parameter . A classical approach in this case is to assume a specific dependence (generally affine) of the certificates on the uncertainty. Instead, in this paper we adopt a probabilistic approach, assuming that is a random variable with given probability distribution over , and apply the SwC approach discussed in Section IIC. This allows us to find an implicit dependence on of the certificates. This is in the spirit of the original idea proposed in [29]. The following two theorems, whose proofs come straightforwardly from Propositions 1 and 2, exploit the SwC approach to address the robust nonlinear gain estimation and synthesis for saturated systems.
(25) 
In particular, our first antiwindup theorem provides a convex optimization procedure to obtain probabilistic information about the worst case nonlinear gain. To this end, we fix an upper bound for and define two scalars and in denoting, respectively, an acceptable level of probability of constraint violation and a level of confidence. Then, inspired by (c), we apply Theorem 1 with the design variables and the certificates given, respectively, by and and the number of samples selected, based on bound (6), to satisfy
(26) 
Then the following result is a straightforward consequence of Theorem 1 and Proposition 2.
Theorem 2 (Probabilistic performance analysis).
Given scalars , and , select satisfying (26), fix and .
Our second antiwindup theorem allows for robust randomized synthesis using the SwC approach and follows parallel steps to those of Theorem 2 by combining Theorem 1 with Proposition 3. To this end, and following (3), we choose the design variables and the certificates as follows: and . Indeed, the variables must include the quantities and used to determine the antiwindup gain in (22): these variables must be the same over all sample extractions so that a unique antiwindup gain can be determined. Then the following holds combining Theorem 1 with Proposition 3.
Theorem 3 (Probabilistic antiwindup synthesis).
Given scalars , and , select satisfying (26), fix and .
For completeness, in (25) we report the SwC problem based on the application of Theorem 3. Similarly, the SwC problem based on the application of Theorem 2 can be constructed following the same rationale and it is not reported here due to space limitations.
Remark 5.
The reformulation of the SwC approach for nonlinear gain analysis and antiwindup synthesis in Theorems 2 and 3 is appealing from an engineering viewpoint. As a matter of fact, since the instances of the system matrices are extracted according to the probability distribution of the uncertainty, this solution provides a view of what may happen in most of practical situations. Moreover, we stress that the proposed formulation does not constrain the unknown Lyapunov matrices ’s to be the same for all the sampled perturbations. Instead, it allows them to vary among different samples. This is possible because the ’s (as well as the ’s) are only instrumental for the computation of the robust compensator. Note that, unlike system matrices, e.g., the ’s, which are uncertain by definition, the certificates are unknown but they are not random variables.
IiiB Uncertain disturbance energy
The previous approach can be further modified by observing that the design is valid only for a given value of , correspondnig to an upper bound on the disturbance energy (see the guarantees in Theorems 2 and 3). However, randomization makes it possible to change the perspective of antiwindup augmentation, by considering also as an uncertain variable, to take into account the knowledge of a certain known probability distribution of the energy of the disturbances acting on the system.
When considering an uncertain disturbance energy, rather than minimizing the gain at a specific value of , we may consider to minimize the curve over a compact range of values of , possibly being relevant for the specific distribution.
To this end, we represent the gain curve estimate by a polynomial curve over the considered interval, that is,
(27) 
Then we may compute an upper bound on the area spanned by the gain as follows
This leads to the following RwC problem
(28)  
where we used and, according to parametrization (27),
Notice that the above design problem is still a convex optimization problem, with the only difference that the cost function is not a single value of (for a nominal ), but a set of values of ’s. Namely, we want to minimize the area underlying the curve parameterized as a polynomial curve. The problem constraints can still be formulated as LMI’s.
The following result is a straightforward generalization of the construction in Theorem 3 for this new optimization goal.
Theorem 4 (Probabilistic synthesis with uncertain energy).
Given scalars , select satisfying (26), fix and .
Remark 6.
Notice that, when also the input is uncertain, an additional tuning knob appears, namely , which characterizes the tradeoff between computational load and conservativeness. On the one hand, more additional parameters mean a more difficult optimization problem. On the other hand, if the gain curve is well approximated by the selected polynomial expansion, the upper bound is tight. From practical experience [37], the gain curves are typically sigmoidal or exponential functions. Then small values of are already enough to obtain good results (usually, from to ). Notice that other basis functions whose integral is linearly parameterized can be suitably selected, without any conceptual change.
(29a)  
(29b)  
(29c) 
(30a)  
(30b)  
(30c) 
IiiC Optimized domain of attraction and reachable set
Similar derivations to the ones of the previous sections can be obtained by focusing on different performance goals, as well characterized in [12] (see also [24]). In particular, two performance goals which have been well characterized within the context of the use of generalized sectors for saturated systems correspond to: i) maximizing the size of a quadratic estimate of the domain of attraction of the origin in the absence of disturbances (that is, ), ii) minimizing the best quadratic estimate of the reachable set from zero initial conditions and in the presence of a bounded disturbance .
The goal of this section is then to briefly overview the possible extensions of the results in Theorems 3 and 4 to these two cases. The following two propositions establish the baseline results, proven in [12, 24] for the nominal case.
Proposition 4 (Domain of attraction).
Given the plantcontroller pair (III), (14) and a matrix , consider the nominal system, that is is a singleton. Assume that the SDP problem
is feasible. Then, selecting the static antiwindup gain as in (22), the nonlinear algebraic loop in (16) is well posed and for any initial condition in the set
(32) 
the (unique) solution to the antiwindup closed loop with satisfies .
Proposition 5 (Reachable set).
In light of the results summarized above, we can formulate robust optimal design and analysis exploiting the constraints (29) and (30), respectively, and leading to randomized analysis and synthesis tools. These are stated below in two theorems whose formulations parallel the one of Theorem 3. Analysis results can also be easily stated, paralleling the formulation in Theorem 2, but are omitted due to their straightforward nature, and to avoid overloading the exposition.
Theorem 5 (Robust domain of attraction).
Given scalars , select satisfying (26), fix and .
If for a selection of and a scalar the scenario approximation (SwC) of problem (4) is feasible and attains a unique optimal solution, then for any initial condition in the set (32), any solution of the uncertain system (16) with antiwindup static compensator (22) and with satisfies for all ,
with level of confidence no smaller than .
In Theorem 5 we characterize properties of the scenario approximation (SwC) of problem (4) with the certificates . Then, according to the definition in (5), it becomes clear that constraints (29) are imposed with certificates depending on the uncertainty , which lead to reduced conservativeness. An interesting feature arising from these dependent certificates in (29) is that the rightmost constraint in (29a) implies that is a uniform lower bound on all certificates . Stated otherwise, this implies that , namely set is a subset of all the stability regions obtained for each one of the extracted samples . Then, differently from classical deterministic approaches, although the set is a guaranteed region of robust stability, it is not necessarily a forward invariant set (whereas for each we know that is a forward invariant set).
A similar (but somewhat converse) comment applies to the robust reachable set studied in the theorem below, wherein the rightmost inequality in (30a) implies that for each we have , namely set is a superset of all the reachable set estimates obtained from the scenario approximation of (5).
Theorem 6 (Robust reachable set).
Given scalars , and , select satisfying (26), fix and .
Remark 7.
Notice that Theorem 6 proposes a selection of the antiwindup gain that minimizes a suitable measure of the size of the reachable set for a specific selection of the bound on the norm of the disturbance . It is then possible to follow similar derivations to those given in Section IIIB with the goal of providing a suitably weighted optimal selection of the antiwindup gain performed by focusing on the size of the reachable set in the presence of an unknown norm of the disturbance, for which probabilistic information is available. Then one may quantify the “size” of the reachable set for each value of by a suitable parametrization similar to the right hand side of (27), and finally minimize some net performance metric taking into account the whole range of possible occurrences of the norm of the disturbance . Since this extension is straightforward, it is not discussed in greater detail.
Iv Simulation examples
The following numerical results are obtained using Matlab R2015a on a 64 bit Windows 8.1 computer eq