Performance Analysis for Discrete-Time Linear Systems with Saturated Linear Feedback: a Nonlinear Saturation-Dependent Gain-Scheduling Approach

In this paper, we propose a new technique for the performance analysis of discrete-time linear systems controlled by a saturated linear control law. Two performance indices, the computation of invariant sets and the L 2 performance, are considered. The main contributions of the paper are the following: i) a new linear parameter varying system framework is presented to model the saturated system, ii) a nonlinear saturation-dependent auxiliary feedback matrix is considered, iii) new sufficient conditions for the performance analysis are proposed. It is shown that the conditions can be expressed as a set of linear matrix inequalities. Furthermore, it is shown that the conditions are guaranteed to be less conservative than existing solutions in the literature. Three numerical examples are presented to illustrate the effectiveness of the proposed method.


Introduction
Saturation is probably the most commonly encountered non-linearity in control engineering. Analyzing the system performance that can be achieved un-Email address: hoai-nam.nguyen@telecom-sudparis.eu (Hoai-Nam Nguyen). der the input saturation is of great importance, and has received the attention of many researchers, for example [10], [15], [22], [26], and references therein.
With the absolute stability analysis tools, such as the circle and Popov criteria, several methods have been proposed for the performance analysis of saturated systems such as the estimation of the domain of attraction [9], [16], L 2 and/or L ∞ disturbance rejection [21]. In [14], a piecewise quadratic estimate of the domain of attraction for a continuous-time saturated system is considered. The main idea is to use an initial ellipsoidal estimation obtained by means of the circle criterion.
One of the most relevant methods to the performance analysis of saturated systems is based on a linear difference inclusion (LDI) framework. The idea is to express the saturated linear feedback law as a convex hull of a group of 2 m linear feedback laws, where m is the control input dimension. Using this framework, in [11], [4], [2], an estimation of the domain of attraction is obtained, and in [6], [24], the problem of L 2 gain analysis is addressed. In conjunction with the LDI framework, various Lyapunov functions were developed, for example, quadratic Lyapunov functions [11], [12], saturation-dependent Lyapunov function [4], [24], composite Lyapunov function and max quadratic Lyapunov function [13]. However all the existing results were obtained by using a linear saturation-independent auxiliary feedback law.
In this paper we present an approach to the performance analysis of discretetime saturated linear systems using the LDI framework. Given a system with m saturated control inputs, we show how to select an LDI in such a way the performance is optimized. Our idea is to use a nonlinear saturation-dependent auxiliary feedback law, whereby the real-time information on the severity of saturation is fully exploited. The linear parameter varying modeling framework is used to model the resulting system. The obtained conditions are converted into linear matrix inequalities (LMI) constraints.
The conference contribution [20] touches on the contents of this paper.
The paper is structured as follows. Section 2 describes the problem formulation and some preliminaries. Section 3 is dedicated to the main results of the paper.
Three numerical examples with comparison to earlier solutions are evaluated in Section 4, before drawing the conclusions in Section 5.
Notation: A positive-definite (semi-definite) matrix P is denoted by P ≻ 0 (P ⪰ 0). 0, I, 1 are, respectively, the zero matrix, the identity matrix, and the all-ones vector of appropriate dimensions. For a given P ⪰ 0, E(P ) represents the following ellipsoid For a given matrix H of appropriate dimension, L(H) is used to denote the following symmetric polyhedron The inequalities are to be interpreted element-wise.
For symmetric matrices, the symbol ( * ) denotes each of its symmetric block.

Problem Formulation
Consider the following discrete-time linear system The saturation function sat(u) : R m → R m is defined as The objective of this paper is to carry out systematically an analysis of system (1), under a given linear state feedback law The following two problems will be considered (1) In the absence of w, we would like to compute an invariant set Ω as large as possible so that if x(k) ∈ Ω, we have x(k + 1) ∈ Ω, ∀k ≥ 0. (2) With a given bound on the L 2 norm of w, i.e., ∥w∥ 2 2 ≤ β, we would like to determine a number α > 0 as small as possible, so that under the condition x(0) = 0, we have ∥z∥ 2 ≤ α∥w∥ 2 . Performing this analysis for each β ∈ (0, ∞), we obtain an estimate of the nonlinear L 2 gain.
It is assumed that all eigenvalues of A + BK are in the interior of the unit circle.

Previous Works: LDI Modeling Framework
In the following we recall the linear differential inclusion (LDI) modeling framework, which was proposed in [1], [11], [12]. This framework can be considered as a generalization of the circle criterion [16] for the saturation nonlinearity.
Denote e j as the jth standard basis of R m , i.e., Associated to S l , ∀l = 1, 2 m , consider the following scalars v l(j) , ∀j = 1, m, The following lemma is taken from [22]. It has been proposed originally in [1]. It will be used to model the saturation non-linearity (2).

Lemma 1: [22]
With v s(j) defined as in (4), the following equation holds where K j denotes the jth row of K, j = 1, m.
For example, if m = 1, we have Define D as the set of m × m diagonal matrices whose diagonal elements are either 1 or 0. For example, if m = 2 then There are 2 m elements in D. Define E l , l = 1, 2 m as an element in D.
In Lemma 1, if we select v l(j) = v j = H j x, l = 1, 2 m , j = 1, m, then we obtain the following result, which is proposed in [11] Lemma 2: [11] Let K, H ∈ R m×n . For all x ∈ L(H), one has For the single input case, Lemma 1 and Lemma 2 are the same if a linear auxiliary feedback gain is chosen for v 2 (1) . For the multi-input case, Lemma 1 provides an extra degree of freedom. Hence for the performance analysis, conditions based on Lemma 1 are less conservative than that are based on Lemma 2. However, this comes with a cost of a higher computational complexity.
Substituting (6) in (1), one obtains Hence (1) can be modeled as an uncertain time-varying system ∀x : −1 ≤ Hx ≤ 1, whereby the parameters λ l , l = 1, 2 m are unknown and time-varying. It was shown in [11] that for the estimation of the domain of attraction, conditions using (7) are less conservative than that are based on the circle criterion or the vertex analysis. Furthermore, as it is proved in [11], Lemma 2 provides a necessary and sufficient conditions for an ellipsoid to be invariant for the single input case, i.e., m = 1.
In [4], it was noticed that the parameters λ l , l = 1, 2 m reflects the severity of the saturation function. Consequently, λ l , l = 1, 2 m are functions of x. To see this, consider the case m = 1, and assume H = 0. In this case, ∀x, −1 ≤ Hx = 0 ≤ 1. Using (6), one obtains Similarly, if we assume H = 1 2 K, then using (6), one gets, Using the available information of λ l , l = 1, 2 m , a saturation-dependent Lyapunov function was proposed in [4] to estimate the domain of attraction, and in [24] to estimate the L 2 gain. The conditions are proved to be less conservative than that are based on quadratic Lypanov function.
The auxiliary feedback matrix H in [4], [11], [24] is a decision variable. It can be optimized to provide a less conservative estimation of the domain of attraction and/or of the L 2 gain. However, in [11], [4], [24], H is a constant matrix, and does not depend on λ l , l = 1, 2 m . Hence, the real-time information on the severity of saturation is not exploited. The main objective of this paper is to show that by selecting a nonlinear saturation-dependent auxiliary feedback matrix H, a significant improvement in the performance analysis can be obtained. For this purpose, some preliminary results are recalled in the next section.

Preliminaries
The following double sum positivity problem of the form will be dealt several times in this paper, where the coefficients λ l satisfy Remark 1: It is well known [23] that Lemma 4 is less conservative Lemma 3, i.e., if there exist matrices Γ l 1 l 2 satisfying (10), they also satisfy (9). The main advantage of (9) with respect to (10) is that (9) has a fewer number of LMI constraints than (10). Hence the computational complexity is reduced.
Lemma 5: Given matrices P, G of appropriate dimension with P ≻ 0. Then, see [5] ( Lemma 6: For given matrices F ∈ R n f ×n , and P ⪰ 0, E(P, 1) ⊆ L(F ) if and only if, see [10] 1 where e j is the jth standard basic of R n f .
Concerning the LMIs, we will make use of the following results.
Property 1 (Congruence): Let P and Q are matrices of appropriate dimension, where P = P T , and Q is a full rank matrix. It holds that 3 Performance Analysis

LPV Modeling
In this section, we present an LPV framework to model the saturated system (1). As will be shown, our LPV model allows to consider a nonlinear saturation-dependent auxiliary feedback matrix in contrast to the LPV model (7) in [4], [24].
Substituting (3) to (1), one obtains Using Lemma 1, there exist λ l (k) and v l(j) (k), l = 1, 2 m , j = 1, m such that (1) and Using (17), (16) can be rewritten as The auxiliary variable v(k) can be considered as a control input for the system (18). Hence the problem of carrying out the performance analysis for the system (15) becomes the problem of selecting the optimal input v(k) to obtain the best performance for (18).

Remark 2:
If Lemma 2 is used to model the saturation non-linearity instead of Lemma 1, then (18) can still be used to model (15) but with the following matrices A l , B l , and v where F l , G l are unknown matrices that will be treated as decision variables.
Remark 3: In the literature [11], [24], [18], only a linear saturation-independent control law v(k) = Hx(k) was considered with H = F G −1 . Clearly, this is a particular case of (20) with F l = F and G l = G, ∀l = 1, 2 m . The nonlinear saturation dependent control law (20) takes the real time information of the saturation into account. As will be shown in the examples, a less conservative estimate of the performance is obtained.
Substituting (20) into (16), one obtains the following closed-loop system where It should be stressed that system (15) can be modeled as (22) only for x such that Define the following saturation-dependent Lyapunov function with where P l ⪰ 0, l = 1, 2 m are unknown matrices that will be treated as decision variables.

Invariant Set Computation
In this section, we are interested in computing an invariant set as large as possible in the absence of w. For this purpose, the following definitions are recalled [19].

Definition 1 (Domain of Attraction):
A set Ω is said to be inside the domain of attraction for system (15), if for any initial condition x(0) ∈ Ω, one has lim k→∞ x(k) = 0.
Clearly from Definition 2, if a set is invariant, then it contains the origin in its interior. It is also clear that if a set is invariant then it is inside the domain of attraction, but not the other way around, i.e., if a set is inside the domain of attraction, then it is not necessarily invariant.
The following theorem provides the theoretical support of the algorithm proposed to calculate an invariant set for the system (15).
Theorem 1: Consider the system (15). Assume that there exist matrices P l ⪰ 0, F l , G l , l = 1, 2 m satisfying the following matrix inequalities then, the set V (k, x(k)) ≤ 1 is invariant.
Proof: It was showing that for all x satisfying (23), (22) can be used to model (15). Using Lemma 6, condition (23) is equivalent Recall that e j is the jth standard basic of R m2 m−1 . With Schur complement, one gets, ∀j = 1, m2 m−1 Note that (29) is (26). In remains to show that V (k, x(k + 1)) ≤ 1 is invariant.

Rewrite (27) as
with r = 1, 2 m , and Combining (33), (34), relaxed LMI conditions can be formulated using Lemmas 3, or 4 as follows Corollary 1: If there exist matrices P l ⪰ 0, G l , F l , l = 1, 2 m such that (33) hold and then the set The proof of corollary 1 is straightforward.
In the interest of the size of the domain of attraction, which is proportional logdet(P l ), the set Since (38) and/or (39) are a convex SDP problem, they can be solved efficiently using free available LMI parser such as CVX [8], or Yalmip [17]. In the following, we refer to the optimization problems (38), and (39), respectively, as algorithm 1 and algorithm 2.
Remark 4: Note that the number of LMIs in (38) and (39) increases exponentially as the number of the system input m increases.

L 2 Performance Analysis
In this section, we are interested in estimating the L 2 gain for the system (1), . This L 2 gain is defined as follows.
Definition 3 (L 2 gain): For a given γ > 0, the system (1), (3) is said to be with a L 2 gain less than γ, if for the zero initial condition, one has for all nonzero w ∈ W .
Recall that the set W is where β > 0 is a given constant.
The following theorem establishes a sufficient condition to estimate the L 2 gain for the system (1).
If (44) holds, then it follows that Note that (22) is asymptotically stable for states near the origin. It follows that lim k→∞ x(k) = 0. Hence lim k→∞ V (k, x(k)) = 0. With the zero initial condition, condition (45) becomes It is concluded that the system (1) has L 2 gain performance γ.
Rewrite the left hand side of (44) as and the right hand side of (44) as Combining (46), (47), one obtains Thus, with Schur complement, one gets Using Schur complement, one obtains Pre-and post-multiplication of (48) by This condition is (41). Consequently, Thus, using Lemma 5, one obtains Using Schur complement, this condition is equivalently rewritten as Using Lemma 4, one gets The proof is complete. 2 By using Lemma 2 to model the saturation non-linearity and by setting F l = F, G l = G in the conditions of Theorem 3, one recover Theorem 1 in [24].
Hence the result in [24] is a particular case of ours.

Condition (42) holds if and only if
with r = 1, 2 m , and Combining (52) and Lemma 3 or Lemma 4, one obtains the following corollary.
For further use, we refer to the optimization problems (56), and (57), respectively, as algorithm 3 and algorithm 4.

Remark 3:
In the unsaturated linear system case, it is well known [7] that the parameter β has no impact on the L 2 gain γ. However, in the presence of the saturation non-linearity, this is no longer the case, i.e., γ is a function of β. Using (51), it is clear that this function is non-increasing, i.e., γ 1 ≤ γ 2 if β 1 ≥ β 2 .

Examples
Three examples are considered in this section. The CVX toolbox was used to solve SDP optimization problems.

Example 1
This example is taken from [4]. Consider the following system The LQ controller with Q = I, and R = 0.  Fig. 1 shows the intersection of two ellipsoids E(P 1 , 1) and E(P 2 , 1) (solid cyan and solid violet). For comparison, Fig. 1 also shows the intersection of two ellipsoids (dashed yellow and dashed red) obtained by using Theorem 1 in [4], and the ellipsoid obtained by using [11] (dash-dot green). We can see that the estimate of the invariant set obtained by using the nonlinear saturationdependent auxiliary feedback gain is larger than that by the linear saturation independent ones.
For different initial conditions, Fig. 2 presents the state trajectories in the phase plane.  Fig. 1. Invariant sets for our approach (solid cyan and solid violet), for [4] (dashed yellow and dashed red), and for [11] (dash-dot green) for example 1.

Example 3
To further illustrate the performance of our approach, we consider the following multi-input system x(k + 1) =     Fig. 4 also shows the intersection of four ellipsoids (dashed lines) obtained by using Theorem 1 in [4], and the ellipsoid with [11] (dash-dot green). Finally, for different initial conditions, Fig. 5 presents the state trajec- tories in the phase plane.

Conclusion
In this paper, a novel approach to the performance analysis of a saturated linear system is proposed. The main contribution of the paper is a new nonlinear saturation-dependent auxiliary feedback law. Using the linear parameter varying system modeling framework, sufficient conditions for the computation of