Giorgio Valmorbida
email: giorgio.valmorbida@centralesupelec.fr

Morten Hovd
email: morten.hovd@itk.ntnu.no

Quadratic programming with ramp functions and fast online QP-MPC solutions ⋆

Keywords: PWA systems, Ramp functions, Model Predictive Control

A novel method is proposed for solving quadratic programming problems arising in model predictive control. The method is based on an implicit representation of the Karush-Kuhn-Tucker conditions using ramp functions. The method is shown to be highly efficient on both small and fairly large Quadratic Program problems, can be implemented using simple computer code, and has modest memory requirements.

Introduction

Model Predictive Control (MPC) has been a great success in industry [START_REF] Qin | A survey of industrial model predictive control technology[END_REF], and since its initial development in the 1970's (see, e.g. [START_REF] Richalet | Model predictive heuristic control: Application to industrial processes[END_REF]) it has found application in a wide range of industrial processes. The main feature of MPC distinguishing it from classical control design methods such as Linear Quadratic Regulator is the ability to take into account constraints in both inputs, state, and outputs. However, the strong abilities of MPC do not come without a cost. In the conventional MPC formulation, an optimization problem has to be solved online, and for large systems and/or systems requiring high sampling rates, the computational load may become prohibitive. Many approaches have been studied in order to reduce the computational requirements of MPC, including input blocking [START_REF] Cagienard | Move blocking strategies in receding horizon control[END_REF] and utilizing structure in the optimization problem (e.g. [START_REF] Rao | Application of interior-point methods to model predictive control[END_REF]). This paper will make no attempt at covering all such approaches. This article proposes a new approach to solving strictly convex Quadratic Programs (QP) problems resulting from MPC problem formulations, based on an implicit problem formulation via ramp functions. Numerical results are provided that demonstrate that the QP problems are solved very fast, for a wide range of problem sizes. The MPC problem is represented exactly, without any simplifications or approximations. The computer code required to implement the solution method is very simple, and the computer memory requirement is modest. The paper is organized as follows. Section 2 provides preliminary results, and describes the MPC formulation used in the paper. Section 3 provides the main result, and in Section 4 the application of the main result for efficiently solving QP problems is described. Section 5 demonstrates the efficiency of the method on a range of MPC problems. A discussion is provided in Section 6, focusing on infeasibility detection and infeasibility handling, while Section 7 concludes the paper. Notation. For a vector y ∈ R n , y i indicates its ith component. For a matrix M ∈ R n×m , M (i,j) denotes its (i, j) component, M (i,•) denotes its ith row, and M (•,j) indicates its jth column. Let D N denote the set of diagonal matrices of dimension N , the set of positive (semi-)definite matrices as S n >0(≥0) = {M ∈ R n×n | M = M ⊤ , M >(≥)0}, and A ⊆ {1, 2, . . . , N } defines the matrix I A ⊂ D N , with I A(i,i) ∈ {0, 1} and I A(i,i) = 1 if i ∈ A, I A(i,i) = 0 if i / ∈ A. We also denote A c = {1, 2, . . . , N } \ A. We denote e i the ith vector of the canonical basis of R N .

MPC formulation and preliminary results

QP-MPC formulation

Consider the discrete state-space model

x k+1 = Ax k + Bu k (1) with x k ∈ X ⊆ R n and u k ∈ U ⊆ R m , ∀k ∈ Z ≥0
, where X and U are polytopic sets. Given the objective function

J(u, x k ) = x ⊤ k+N P T x k+N + N -1 i=0 x ⊤ k+i P i x k+i + u ⊤ k+i R i u k+i (2) where u = [u ⊤ k • • • u ⊤ k+N -1 ] ⊤ , and P i ∈ S n ≥0 , R i ∈ S n >0 ∀i = 1, . . . , N -1, and P T ∈ S n ≥0 . As discussed in [1, Sec 2.1], the MPC problem is formulated as: minimize u 1 2 u ⊤ Hu + x ⊤ k F ⊤ u (3) subject to Gz ≤ S u x k + w
where the matrices H, F, G,and S u depend on the constraints defining X , U, and on the matrices defining system (1) and the objective function [START_REF] Cagienard | Move blocking strategies in receding horizon control[END_REF]. Since R i ∈ S n >0 , we have H ∈ S n >0 thus an invertible matrix. Also following [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF], the variable change

z = u + H -1 F x k results in minimize z 1 2 z ⊤ Hz (4)
subject to Gz ≤ Sx k + w where S = S u + GH -1 F . In this paper we search for solutions to the above problem based on the solution of an implicit equation in terms of ramp functions. Some properties of the ramp functions as detailed below Definition 1 The ramp function r(y) is given by

r(y) = 0 if y < 0 y if y ≥ 0 (5) Lemma 1 The ramp function is the only function satis- fying, ∀y ∈ R (r(y) -y)r(y) = 0 (6a) r(y) ≥ 0 (6b) (r(y) -y) ≥ 0. ( 6c
)
Proof. First note that the ramp function can be expressed as the unique solution to the convex optimization problem parameterized in y (thus depending on y) with strictly convex objective function as follows minimize

r 1 2 (r -y) 2 subject to r ≥ 0. ( 7
)
With the Lagrangian associated to the optimization problem, L(r, λ) = 1 2 (r -y) 2 -λr, we obtain the Karush-Kuhn-Tucker (KKT) conditions (r -y) -λ = 0; λr = 0; r ≥ 0; λ ≥ 0 which are necessary for optimality and also sufficient since the problem is strictly convex. To obtain a description in the variables (y, r) one can use λ = (r -y) above to obtain r ≥ 0, (r -y) ≥ 0, r(r -y) = 0. ■ For y ∈ R m let us define the function r : R m → R m , the vector-valued ramp function, as

r(y) = r(y 1 ) r(y 2 ) • • • r(y m )
⊤ Remark 1 Note that, due to the piecewise definition of the ramp function in (5), we have that vector r can also be expressed as the product r(y) = I A y where I A ∈ D m , with its diagonal elements verifying

I A(i,i) ∈ {0, 1}.
Clearly, the set A ⊆ {1, 2, . . . , N } depends on y. ⌟

Main results

This section shows how a Linear Complementarity Problem (LCP) can be associated to a QP-MPC to obtain a piece-wise affine (PWA) representation implicitly defined in terms of the vector-valued ramp function r. Section 4 details how this implicit PWA representation can be exploited to quickly calculate solutions to the QP-MPC problem.

Theorem 1 The solution of the QP-MPC is a PWA function on the variable x, given by

u(x) = -H -1 F ⊤ x -H -1 G ⊤ r(y) (9a)
y = -Sx + (I -GH -1 G ⊤ )r(y) -w. (9b)
Proof. The following are the KKT conditions associated to (3) as presented in [1, eq.(15)]

Hz + G ⊤ λ = 0 (10a) λ i (S (i,•) x + w i -G (i,•) z) = 0 (10b) λ ≥ 0 (10c) Sx + w -Gz ≥ 0. ( 10d
)
From (10a) we obtain z = -H -1 G ⊤ λ, and using this expression in (10b)-(10d), gives the following LCP constraints

λ i (S (i,•) x + w i + G (i,•) H -1 G ⊤ λ) = 0 λ ≥ 0 Sx + w + GH -1 G ⊤ λ ≥ 0.
The above set of constraints can be rewritten as

λ i λ i --S (i,•) x + (I -GH -1 G ⊤ ) (i,•) λ -w i = 0 λ ≥ 0 λ --Sx + (I -GH -1 G ⊤ )λ -w ≥ 0.
By defining

y = -Sx + (I -GH -1 G ⊤ )λ -w, (11)
we observe that the above set of inequalities and the complementarity constraint become

λ i (λ i -y i ) = 0, λ ≥ 0, (λ -y) ≥ 0.
According to Lemma 1, the set of multipliers λ is given by the ramp function of variable y, namely

λ = r(y) (12)
then, from [START_REF] Rao | Application of interior-point methods to model predictive control[END_REF], we have that y satisfies the implicit algebraic equation (9b), while (9a) follows from the change of variables u = z -H -1 F x, from (10a) and [START_REF] Richalet | Model predictive heuristic control: Application to industrial processes[END_REF]. ■

From the above theorem, we have that, given x, the computation of the control action, u, can be carried out by solving the implicit equation in (9b), yielding y. From this solution, the Lagrange multipliers λ can be computed according to [START_REF] Richalet | Model predictive heuristic control: Application to industrial processes[END_REF]. The computation of u(x) in (9a) thus boils down to the solution of the implicit equation.

The next section proposes an algorithm to solve this implicit equation by exploiting the fact that the ramp function can be written as a matrix multiplication r(y) = I A(x) y where we indicate the dependence of the set of active constraints A on the value of x. Indeed, if the solution, depending on x, gives y i < 0 then i / ∈ A. If instead, y i ≥ 0, then i ∈ A.

Implementation

It is clear from Theorem 1 that the KKT conditions for the MPC optimization problem is an LCP. This is well known both for QPs in general (see, e.g. [START_REF] Murty | Linear Complementarity, Linear and Nonlinear Programming[END_REF]), and within the MPC literature (e.g. [START_REF] Camacho | Constrained generalized predictive control[END_REF][START_REF] Herceg | Multi-Parametric Toolbox 3.0[END_REF]). However, this knowledge has not been utilized to devise an algorithm for solving the MPC QP problem, using ramp functions to take advantage of the structure and continuity of the problem.

In this section, we present an algorithm to solve the algebraic equation (9b). The algorithm exploits the fact that the ramp function can be expressed as a matrix multiplication by a diagonal matrix as pointed out in Remark 1. That is, that (9b) can be written as

y -(I -GH -1 G ⊤ )I A y = -Sx -w (13)
for some matrix I A to be determined. The set A corresponds to the set of active constraints, namely to the set for which the values of multipliers λ are not zero. Therefore, according to the ramp function defining the multipliers, if y i < 0 then the corresponding value of the multiplier is λ i = 0, and in this case, I A(i,i) = 0, that is, the the corresponding diagonal element of I A is zero.

If instead y i ≥ 0 then I A(i,i) = 1. The following definition gives the definition of the pairs (y, I A ) allowing to establish an equivalence between (9b) and ( 13).

Definition 2 For a given x, a pair is said (y, I A ) compatible if (y, I A ) satisfy (13) and

I A(i,i) = 1 if y i ≥ 0 and I A(i,i) = 0 if y i < 0.
Based on the above observations, the solution to the implicit equation ( 9b) is obtained whenever we satisfy [START_REF] Skogestad | Matlab distillation column model[END_REF] with a compatible pair (y, I A ). We shall denote A(x) as the set of active constraints for a given x defining the right hand side of (9b). To search for a solution to (9b), we propose below an algorithm that searches for the set of active constraints by adding elements to or removing elements from the set A, thus modifying matrix I A one element at the time, and updating the solutions y using the matrix inversion lemma, which is recalled below.

Lemma 2 (Matrix Inversion Lemma) Let Q ∈ R N ×N be an invertible matrix, we have

(Q+q u q c q v ) -1 = Q -1 -Q -1 q u (q -1 c +q v Q -1 q u ) -1 q v Q -1
where q c ∈ R \ {0} is a scalar, q u ∈ R N , and q ⊤ v ∈ R N , that is q u is a column vector and q v is a row vector.

Remark 2 Noting that (q -1 c + q v Q -1 q u ) is a scalar, it is clear that the update of Q -1 when adding the rank one term Q -1 q u q v Q -1 requires only sums and multiplications and a single division by a scalar. ⌟

Let us define

Q(A) = I A c + GH -1 G ⊤ I A
and observe that the solution to the linear system ( 13) is obtained following

y -(I -GH -1 G ⊤ )I A y = -Sx -w Q(A)y = -Sx -w y = Q(A) -1 (-Sx -w) .
Consider the case where i / ∈ A and define A +i = A∪{i}, namely i enters the set A. Since I A+i = I A + e i e ⊤ i , and

I A c +i = I A c -I A c (•,i) e ⊤ i , we have Q(A +i ) = Q(A) -(I A c -GH -1 G ⊤ ) (•,i) e ⊤ i .
The above matrix Q(A +i ) presents a sum of Q(A) with a rank one matrix as in Lemma 2, with, respectively

q c = -1, q v = e ⊤ i and q u = (I A c -GH -1 G ⊤ ) (•,i)
. Following the matrix inversion lemma, we have the update of the inverse by adding i to A as

Q(A +i ) -1 =Q(A) -1 -(-1 + e ⊤ i Q(A) -1 (I A c -GH -1 G ⊤ ) (•,i) ) -1 × Q(A) -1 (I A c -GH -1 G ⊤ ) (•,i) e ⊤ i Q(A) -1 . By defining v(A, i) = Q -1 (A)(I A c -GH -1 G ⊤ ) (•,i) , we obtain Q(A +i ) -1 =Q(A) -1 (14) -(-1 + v(A, i) i ) -1 v(A, i) Q(A) -1 (i,•) .
We also have the update of y, denoted y + [i] as

y + [i] =Q(A +i ) -1 (-Sx -w) =Q(A) -1 (-Sx -w) -(-1 + e ⊤ i Q(A) -1 (I A c -GH -1 G ⊤ ) (•,i) ) -1 × Q(A) -1 (I A c -GH -1 G ⊤ ) (•,i) e ⊤ i Q(A) -1 (-Sx -w) =y -(-1 + e ⊤ i Q(A) -1 (I A c -GH -1 G ⊤ ) (•,i) ) -1 × Q(A) -1 (I A c -GH -1 G ⊤ ) (•,i) y i , that is, y + [i] = y -y i (-1 + v(A, i) i ) -1 v(A, i).
Similarly, consider the case where i ∈ A and define

A -i = A \ {i}, namely i is removed from the set A, Q(A -i ) = Q(A) + e ⊤ i (I A c -GH -1 G ⊤ ) (•,i)
The above matrix Q(A -i ) presents a sum of Q(A) with a rank one matrix as in Lemma 2, with, respectively q c = 1, and, as above, q v = e ⊤ i and q u = (I A -GH -1 G ⊤ ) (•,i) . thus, similarly to above, the following expressions for the inverse and solution updates

Q(A -i ) -1 =Q(A) -1 (15) -(1 + v(A, i) i ) -1 v(A, i) Q(A) -1 (i,•) , y -[i] =y -y i (1 + v(A, i) i ) -1 v(A, i). ( 16
)
Given the above steps to update the inverse and y, we use the following criteria to choose which elements to add to or to remove from the current set of active constraints A. These criteria are based on elements of the pair (y, A)

• If y i < 0 for i ∈ A, then constraint i is removed from A and the inverse of Q(A -i ) and an update of y are computed. • If y i ≥ 0 for i / ∈ A, then constraint i is added to A and the inverse of Q(A +i ) and an update of y are computed.

The order in which we carry out the inclusion and removal of elements in A is as follows • Remove first from A constraints corresponding to negative y i . If there are more then one such constraint, remove first the one corresponding to the most negative y i . • Then add to A a constraint that is not a member of A and corresponds to a positive value of y i . If there is more than one such constraint, add first the one corresponding to the largest y i .

The above steps are described in Algorithm 1 below. The solution of the algebraic loop in (9b) is obtained with Algorithm 1 and its use in the QP-MPC to obtain the control input u k from the vector u as detailed in Algorithm 2.

Algorithm 1 Solution to the algebraic equation ( 9b)

Require: GH -1 G, A, invQ, y satisfying y = invQ(-Sx -w) while (A, y) not compatible do if ind(sign(y, -1)) ∩ A ̸ = ∅ then L ← ind(sign(y, -1)) ∩ A i ← ind(min(y, L)) v ← invQ(I A c -GH -1 G ⊤ ) (•,i) A ← A \ {i} q 0 ← 1 else if ind(sign(y, 1)) ̸ = A then L ← ind(sign(y, 1)) \ A i ← ind(max(y, L)) v ← invQ(I A c -GH -1 G ⊤ ) (•,i) A ← A ∪ {i} q 0 ← -1 end if invQ ← invQ -(q 0 + v i ) -1 v (invQ) (i,•) y ← y -y i (q 0 + v i ) -1 v end while return A, invQ, y
Algorithm 2 Solution to the QP-MPC (3)

Require: GH -1 G, S, w, F as in (9b) and ( 3)

while MPC running do Obtain x k invQ ← I A ← ∅ y ← (-Sx k -w) y ← Algorithm1(GH -1 G, A, invQ, y) u(x k ) ← -H -1 F ⊤ x k -H -1 G ⊤ r(y)
Apply u(x k ) to the plant end while

Numerical Results

To illustrate the performance of the proposed approach, numerical results are provided for MPC applied to three systems of different size. The simulation times are reported for simulating 100 timesteps using the implicit QP solution approach described in the previous section, and compared to the simulation times when using qpOASES [START_REF] Ferreau | An online active set strategy to overcome the limitations of explicit MPC[END_REF][START_REF] Ferreau | qpOASES: A parametric active-set algorithm for quadratic programming[END_REF] (starting from the same initial state). The simulations are performed in Matlab on a Windows PC. The calculation times can vary a little from run to run, since Windows is not a real time operating system. The simulations are therefore repeated multiple times, and the average time is reported.

The solver qpOASES is a highly regarded QP solver for MPC problems. It takes advantage of the observation that the state x on the right hand side of the constraints in ( 3) is unlikely to change very much from one timestep to the next. At time k, a parametric active set approach is used to follow a homotopy path from the optimal solution for x k-1 to the optimal solution for x k . In the examples, the upper and lower constraints for u are separated from the other constraints in (3), and given to qpOASES separately, as recommended in [START_REF] Ferreau | qpOASES: A parametric active-set algorithm for quadratic programming[END_REF]. At each timestep except the initial one, the hot start functionality in qpOASES is used.

The two optimization solvers give essentially the same results, and they would be indistinguishable in plots showing the closed-loop behavior of the systems. Therefore we do not illustrate the time responses of the closed loop. Instead, the focus here are on the simulation times, which document the efficiency of the proposed method. In each example, constant weights P i = P and R i = R are used in (2), and the terminal weight P T is set equal to the solution of the algebraic Riccati equation for the weights P and R.

Example 1 Consider the double integrator example from [START_REF] Hovd | On the design of exact penalty functions for MPC using mixed integer programming[END_REF]. The system dynamics are given by

x k+1 = 1 1 0 1 x k + 1 0.3 u k
with the input constrained between -1 and 1, and each state constrained between -5 and 5, P = I, and R = 1.

The prediction horizon N = 10 is used, and the system with control is simulated for 100 time steps. The average time for 5 runs starting from x 0 = [5 -2] ⊤ when solving the QP using ramp functions is 0.0020s, whereas the average time when using qpOASES is 0.0023s. ⌟

Example 2 We now study the four-state system from [START_REF] Hovd | On the design of exact penalty functions for MPC using mixed integer programming[END_REF][START_REF] Hovd | Handling state and output constraints in MPC using time-dependent weights[END_REF]. The discrete-time dynamics are given by First the model is balanced and reduced to 25 states.

x k+1 =        0.
Next, the reduced model is converted to discrete time. Both the original system and the reduced model equations and initial state are not listed here for space reasons1 .

Using a prediction horizon N = 30, the resulting QP problem has 1562 constraints and 120 degrees of freedom. Simulating 100 timesteps when solving the QP using ramp functions took on average 0.0055s, while the average time for qpOASES was 0.0401s. ⌟ Table 2 summarizes indicators for the performance. Note, in particular, that few of the time steps required the maximum number of iterations in Algorithm [START_REF] Bemporad | The explicit linear quadratic regulator for constrained systems[END_REF]. Note also the sparsity of matrix Q(A) -1 , which is smaller for the example with more states. 1 Order of the system (n), prediction horizon (N ). All examples ran 100 time-steps from the reported initial conditions. For Algorithm 2 the table reports the maximum number of iterations of Algorithm 1 within a time step, time steps in which the maximum number iterations of Algorithm 1 were executed, average number of iterations of Algorithm 1 for the 100 time-steps, maximum number of active constratins (as percentage of total), percentage non-zero elements in Q -1 (A), average time step for the 100 time steps and maximum time taken within one time-step (corresponding to the first time k = 1).

Discussion

Relationship to active set methods. Algorithm 1 updates the set of active constraints at each iteration, and may thus be categorized as an active set method. However, conventional active set methods solve the KKT conditions (in the form of a set of linear equations) for each candidate active set tried, in order to determine a step direction and thereby find which constraint to add or drop. In contrast, Algorithm 1 updates invQ through a rank 1 update, and the constraint to be added or dropped is found simply from looking at the values of the elements of the y vector, taking into account whether the corresponding constraint is in the active set.

Memory requirements for Algorithm 1

The computer memory required for implementing Algorithm 1 is clearly very modest, consisting mainly of the storage of GH -1 G ⊤ ), S, and w, which is the minimum required to define the LCP, as well as the leading m rows of H -1 F ⊤ and H -1 G ⊤ that are required for calculating u from x and λ (see Theorem 1). In addition, Q -1 and the set of active constraints A needs to be stored (and updated), but storing these as sparse matrices can save significant memory, since, as noted above, the num-ber of active constraints cannot exceed the number of degrees of freedom. Indeed, for the above numerical examples, a full matrix Q -1 would contain n 2 c elements, where n c = 66, n c = 316 and n c = 1562 respectively for examples 1, 2 and 3. However, since only a few of these elements are not zero, as detailed in Table 2, column Worst sparsity of Q -1 (A), the sparse storage allows to reduce the storage space to a small fraction of the dimensions of Q -1 (A).

Initial data for Algorithm 1

One possible input data for the solution of the implicit equation using Algorithm 1 is A = ∅ corresponding to no active constraints, giving

Q A = I, hence y = -Sx -w.
This initial input is clearly very efficient when there are no active constraints. However, with the proposed rules for removing or adding constraints to the active set given in Section 4, the numerical experiments showed that the solution method is still very efficient, and the number of solver iterations within one timestep have not been found to exceed the number of (actually) active constraints by much.

Typically, the value of the state will not change much from one timestep to the next, and it may therefore seem advantageous to start the optimization from the active set at the previous timestep. However, it is important to take advantage of the fact that Q -1 typically is a highly sparse matrix. It is observed that when (over multiple time steps) hot starting the optimization from the active set at the previous time step, then Q -1 will gradually fill up with elements of magnitude around the order of eps, i.e., elements that should have been set to zero when adding and later removing constraints. The small non-zero values are due to finite precision in the calculations. This both slows down the calculations and for large problems significantly increases the requirement for computer memory. The problem is easily handled by cleaning Q -1 of such very small elements (e.g., using the Matlab function clean), but it is found that this takes more time than starting each iteration with an empty active set, corresponding to Q -1 = I. However, there are situations where it may be possible to do the cleaning of tiny elements from Q -1 without incurring significant time loss, especially if parallel computing is an option:

• There is actually a sequence of calculations that need to be carried out within each timestep: first measurements are obtained, next the model is updated from measurements (typically using state estimation), only thereafter are the MPC calculations performed. It may therefore be possible to clean Q -1 while the model is updated. • In some cases there may be ample time for calculations within each sample interval, but it is desired to minimize the time delay within each sample interval associated with the MPC calculations. In such a situation, the cleaning of Q -1 can be performed after the MPC calculations and after the new input has been implemented, but before the end of the sample interval.

Infeasibility detection

To detect infeasibily we have relied on the heuristics of checking the value of (-1 + v(A, i) i ) when adding constraints. Indeed, we observe that these values become very small when an inconsistent set of constraints is selected.

Infeasibility has been investigated for all examples above by scaling the initial state until the problem becomes infeasible. Using a threshold value of 10 -13 for (-1 + v(A, i) i ), infeasibility was correctly determined in all three examples, to within an accuracy of three decimal places in the scaling factor.The appropriate threshold value will depend on, and should be significantly larger than, the machine precision. Clearly, setting the threshold too large will result in the algorithm declaring infeasibility unnecessarily. Even though we implemented this procedure for infeasibility detection, it has not been detailed Algorithm 1.

Soft Constraints

From an application point of view, a quite different issue is what should happen when an MPC problem is infeasible. The basic options are either to have some backup functionality bringing the system to a safe state (which often means shutting down the system entirely), or to design the MPC to "make the best of it", trying to minimize whatever damage might be caused by operation in an infeasible region of the state space. Such damage minimization can be achieved by using soft constraints, by adding slack variables to constraints where violations are physically possible and operationally (temporarily) tolerable, and adding corresponding penalty terms to the objective function.

A desirable property is for the soft constraints to be exact. This is achieved by a sufficiently large weight on a penalty term linear in the slack variable. What is sufficiently large can be calculated as described in [START_REF] Hovd | On the design of exact penalty functions for MPC using mixed integer programming[END_REF]. Quadratic terms in the penalty function do not affect whether the soft constraint is exact, and quadratic terms are therefore sometimes dropped. However, when solving the MPC QP using ramp functions, the Hessian matrix needs to be invertible (positive definite), and hence weights on quadratic terms in the penalty functions are required.

Precision of the Lagrange Multipliers and complementarity KKT conditions

Although of little importance in practical applications, it is interesting to observe that the complementarity constraint of the KKT conditions is fulfilled with very high accuracy for the proposed method. That is, the λ's for the inactive constraints are identically zero (or of magnitude similar to machine precision), whereas conven-tional optimization routines will give λ's for inactive constraints in the range of some tolerance specificationoften in the range of 10 -8 .

Need for Rank 2 updates

The expression for the control law (9) in Theorem 1 is obtained from the general convex QP formulation (4).

Recall that Algorithm 1 uses rank 1 updates to compute a solution to [START_REF] Murty | Linear Complementarity, Linear and Nonlinear Programming[END_REF]. We point out that Algorithm 1, combined with the infeasibility detection described above, has been effectivelly applied to QP problems arising from MPC formulations discussed in Section 2 and illustrated in examples in in Section 5.

Although, in the authors' experience, Algorithm 1 works very well for QP problems arising from MPC formulations, strictly convex QP formulations exist for which a modification of the algorithm is required. This modification copes with the need for rank two updates. To illustrate such a case, consider the problem (4) with S = 0 and

H = 11 9 9 11 , G =        1 0 0 -1 -1 √ 2 -1 √ 2 -3 √ 10 -1 √ 10        , w =        -0.5 -0.8 -1 2 √ 2 -0.15 √ 10        .
Since there are only two degrees of freedom in the corresponding QP, the maximum number of strongly active constraints is two. Starting Algorithm 1 with an empty active set, A = ∅, constraint 2 is the first to add to the active set, and then constraint 4 should be added. The active set A = {2, 4}, results in y = 0.2833 5.2444 -0.0589 5.0772 ⊤ .

which, following Algorithm 1, is not a compatible solution. Thus, constraint 1 should be added to the active set -but since the problem has only two degrees of freedom, this would lead to a rank defect Q matrix. Therefore, one of the constraints in the active set has to be removed when introducing constraint 1 into the active set, even though all constraints in the active set have positive y values, namely a rank 2 update.

The following procedure can be used to identify the constraint to be removed. Let the index i denote the constraint to be added to the active set, and note that the problem only arises when the number of constraints already in the active set equals m. The current (nonoptimal) solution point can be calculated from (9a) and the current value of y. Clearly, the solution point needs to move in a direction for which the value of constraint i decreases. Start with v 0 = -G ⊤ (i,:) as the candidate direction for changing the solution point, and set S 0 = I.

Obviously, the solution point cannot move in a direction in which the values of the constraints in the active set increase.

(1) For k = 1 : m, define j as the constraint index (row index in G) for element k of A.

(2) Select a constraint index k such that G (j,:) v k-1 ≥ 0.

(3) Set v k = (I -r ⊤ j (r j r ⊤ j ) -1 r j )v k-1 , where r T j = S j-1 G ⊤ (j,:) and S j = (I -r ⊤ j (r j r ⊤ j ) -1 r j )S j-1 . Note that r j r ⊤ j is a scalar, so the inversion is actually a scalar division. (4) Repeat from (2) until G (j,:) v k-1 ≤ 0 for all remaining unselected constraints.

The constraint to be removed from A is then the first constraint to become inactive when moving the solution point in the direction v m . If v m = 0 the problem is infeasible.

Following these steps in the above example, we find that constraint 2 should be removed from the active set. It is easily checked that A = {1, 4} is optimal.

The corresponding rank 2 update to Q -1 can be carried out either using the matrix inversion lemma directly, or applying two rank 1 updates, by first removing the constraint to leave the active set.

Conclusions and Perspectives

A novel method for solving QPs arising from MPC problems has been proposed. The method is shown to be efficient for a wide range of problem sizes, and can be implemented using short and simple computer code. The method is currently limited to strictly convex QP problems, semi-definite Hessian matrices cannot be accommodated.

Future work will address this limitation, and also attempt to extend the method to LP-MPC. Finally, both for general QPs and for QP-MPC in particular, we did not study the convergence of the proposed algorithm, this is a topic for future research.

Fig. 1 .

 1 Fig. 1. Solution times in seconds per timestep for a single run of Algorithm 1 and qpOASES, for each example.

Figure 2

 2 Figure2shows the simulation times for 100 simulation runs, each of length 100 timesteps, for Algorithm 1 and qpOASES. For run i, i = {1, . . . , 100} the same initial conditions are used for Algorithm 1 and qpOASES. The initial conditions are randomly generated, although only feasible initial conditions are retained for comparison of simulation times. Algorithm 1 more often results in shorter simulation times.

Fig. 2 .

 2 Fig. 2. Solution times in seconds for simulation runs of length 100 timesteps for Algorithm 1 and qpOASES, for each example.

  Total execution time for 100 time steps of Algorithm 2 and qpOASES considering the same initial conditions (both from the average of several runs).

		Algorithm 2	qpOASES
	Ex. Total time (ms)	Total time (ms)
	1	2.0	2.3
	2	4.1	9.0
	3	5.5	40.1
	Table 2

They can be obtained for morten.hovd@itk.ntnu.no on request, together with the weights used in the MPC objective function.

⋆ This work was supported by the French-Norwegian exchange programme AURORA under contract 294676. This research is also funded in part by ANR via project HANDY, number ANR-18-CE40-0010.


Ariádne L J Bertolin

Ricardo C L F Oliveira

Giorgio Valmorbida
email: giorgio.valmorbida@centralesupelec.fr

Pedro L D Peres
email: peres@dt.fee.unicamp.br

Output feedback control of discrete-time Lur'e systems through FIR Zames-Falb multipliers

Keywords: Discrete-time Lur'e systems, FIR Zames-Falb multipliers, Absolute stability, Output-feedback control, Linear matrix inequalities Discrete-time Lur'e systems, FIR Zames-Falb multipliers, Absolute stability, Output-feedback control, Linear matrix inequalities Discrete-time Lur'e systems, FIR Zames-Falb multipliers, Absolute stability, Output-feedback control, Linear matrix inequalities Discrete-time Lur'e systems, FIR Zames-Falb multipliers, Absolute stability, Output-feedback control, Linear matrix inequalities Discrete-time Lur'e systems, FIR Zames-Falb multipliers, Absolute stability, Output-feedback control, Linear matrix inequalities Discrete-time Lur'e systems, FIR Zames-Falb multipliers, Absolute stability, Output-feedback control, Linear matrix inequalities Discrete-time Lur'e systems, FIR Zames-Falb multipliers, Absolute stability, Output-feedback control, Linear matrix inequalities Discrete-time Lur'e systems, FIR Zames-Falb multipliers, Absolute stability, Output-feedback control, Linear matrix inequalities

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Lur'e systems are feedback interconnections between a linear time-invariant (LTI) plant with a static nonlinear function (Khalil, 2002, Chapter 7), as presented in Figure 1. Under assumptions on the nonlinearity, as sector and slope bounds, conditions for absolute stability can be formulated in time domain, where the stability certificates are Lyapunov functions [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF][START_REF] Park | A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF]Bertolin et al., 2022a;[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], or in the frequency domain, where the stability certificates are LTI systems called multipliers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Turner | On the existence of stable, causal multipliers for systems with sloperestricted nonlinearities[END_REF][START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Ahmad | Convex LMI approach for stability of critically stable systems with slope-restricted nonlinearities[END_REF]Haddad andBernstein, 1994, 1993;Carrasco et al., 2012a[START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF][START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF][START_REF] Carrasco | LMI search for rational anticausal Zames-Falb multipliers[END_REF][START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF].

For slope-bounded nonlinearities, the Zames-Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] O'shea | A combined frequency-time domain stability criterion for autonomous continuous systems[END_REF][START_REF] O'shea | An improved frequency time domain stability criterion for autonomous continuous systems[END_REF][START_REF] Turner | Zames-Falb multipliers: don't panic[END_REF] are the most effective method to assess stability. A positive realness condition must be verified for the overall system and the ℓ 1 norm of the impulse response of the multiplier must be bounded. For discrete-time Lur'e systems, the requirements on the bound of the ℓ 1 norm (called star norm) for infinite impulse response ZF multipliers [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF] and positive realness conditions are expressed in terms of linear ma-★ Supported by the Brazilian agencies: Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09, CNPq and grant 2019/10947-1 of São Paulo Research Foundation (FAPESP). trix inequalities (LMIs). These formulations consider a statespace realization of the multiplier. Although being effective to some extent, the ℓ 1 bound constraint can be conservative, and requires a scalar search. Moreover, the conditions apply only to odd nonlinearities. The finite impulse response (FIR) ZF multipliers [START_REF] Wang | A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers[END_REF][START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] offer a simpler solution to bound the ℓ 1 norm of the impulse response and the positive realness condition is still expressed with an LMI constraint. Besides, FIR multipliers handle odd or nonodd nonlinearities, and can be causal, anti-causal or noncausal.

Regarding control design for discrete-time Lur'e systems, most of the existing works are based on Lur'e-Lyapunov functions (Bertolin et al., 2022a;[START_REF] Kim | Robust static and fixedorder dynamic output feedback control of discrete-time parametric uncertain Luré systems: Sequential SDP relaxation approaches[END_REF]Louis et al., 2015b,a), with the exception of [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]) that used star-norm bounds for the ℓ 1 constraint and positive realness to derive synthesis conditions. In this paper, the aim is to propose a method for the output feedback control of discrete-time Lur'e systems based on FIR ZF multipliers. An iterative algorithm, where an LMI optimization problem is solved at each iteration, is first presented for absolute stability analysis and its extension to cope with control design is immediate. A control law using state or output feedback, that may also include the feedback of the output of the nonlinearity, can be designed jointly with the FIR ZF multiplier. Odd or non-odd nonlinearities, causal or anti-causal FIR multipliers of any given order can be considered. Numerical examples from the literature illustrate the performance of the proposed method.

INTRODUCTION

Lur'e systems are feedback interconnections between a linear time-invariant (LTI) plant with a static nonlinear function (Khalil, 2002, Chapter 7), as presented in Figure 1. Under assumptions on the nonlinearity, as sector and slope bounds, conditions for absolute stability can be formulated in time domain, where the stability certificates are Lyapunov functions [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF][START_REF] Park | A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF]Bertolin et al., 2022a;[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], or in the frequency domain, where the stability certificates are LTI systems called multipliers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Turner | On the existence of stable, causal multipliers for systems with sloperestricted nonlinearities[END_REF][START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Ahmad | Convex LMI approach for stability of critically stable systems with slope-restricted nonlinearities[END_REF]Haddad andBernstein, 1994, 1993;Carrasco et al., 2012a[START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF][START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF][START_REF] Carrasco | LMI search for rational anticausal Zames-Falb multipliers[END_REF][START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF].

For slope-bounded nonlinearities, the Zames-Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] O'shea | A combined frequency-time domain stability criterion for autonomous continuous systems[END_REF][START_REF] O'shea | An improved frequency time domain stability criterion for autonomous continuous systems[END_REF][START_REF] Turner | Zames-Falb multipliers: don't panic[END_REF] are the most effective method to assess stability. A positive realness condition must be verified for the overall system and the ℓ 1 norm of the impulse response of the multiplier must be bounded. For discrete-time Lur'e systems, the requirements on the bound of the ℓ 1 norm (called star norm) for infinite impulse response ZF multipliers [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF] trix inequalities (LMIs). These formulations consider a statespace realization of the multiplier. Although being effective to some extent, the ℓ 1 bound constraint can be conservative, and requires a scalar search. Moreover, the conditions apply only to odd nonlinearities. The finite impulse response (FIR) ZF multipliers [START_REF] Wang | A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers[END_REF][START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] offer a simpler solution to bound the ℓ 1 norm of the impulse response and the positive realness condition is still expressed with an LMI constraint. Besides, FIR multipliers handle odd or nonodd nonlinearities, and can be causal, anti-causal or noncausal.

Regarding control design for discrete-time Lur'e systems, most of the existing works are based on Lur'e-Lyapunov functions (Bertolin et al., 2022a;[START_REF] Kim | Robust static and fixedorder dynamic output feedback control of discrete-time parametric uncertain Luré systems: Sequential SDP relaxation approaches[END_REF]Louis et al., 2015b,a), with the exception of [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]) that used star-norm bounds for the ℓ 1 constraint and positive realness to derive synthesis conditions. In this paper, the aim is to propose a method for the output feedback control of discrete-time Lur'e systems based on FIR ZF multipliers. An iterative algorithm, where an LMI optimization problem is solved at each iteration, is first presented for absolute stability analysis and its extension to cope with control design is immediate. A control law using state or output feedback, that may also include the feedback of the output of the nonlinearity, can be designed jointly with the FIR ZF multiplier. Odd or non-odd nonlinearities, causal or anti-causal FIR multipliers of any given order can be considered. Numerical examples from the literature illustrate the performance of the proposed method.

INTRODUCTION

Lur'e systems are feedback interconnections between a linear time-invariant (LTI) plant with a static nonlinear function (Khalil, 2002, Chapter 7), as presented in Figure 1. Under assumptions on the nonlinearity, as sector and slope bounds, conditions for absolute stability can be formulated in time domain, where the stability certificates are Lyapunov functions [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF][START_REF] Park | A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF]Bertolin et al., 2022a;[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], or in the frequency domain, where the stability certificates are LTI systems called multipliers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Turner | On the existence of stable, causal multipliers for systems with sloperestricted nonlinearities[END_REF][START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Ahmad | Convex LMI approach for stability of critically stable systems with slope-restricted nonlinearities[END_REF]Haddad andBernstein, 1994, 1993;Carrasco et al., 2012a[START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF][START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF][START_REF] Carrasco | LMI search for rational anticausal Zames-Falb multipliers[END_REF][START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF].

For slope-bounded nonlinearities, the Zames-Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] O'shea | A combined frequency-time domain stability criterion for autonomous continuous systems[END_REF][START_REF] O'shea | An improved frequency time domain stability criterion for autonomous continuous systems[END_REF][START_REF] Turner | Zames-Falb multipliers: don't panic[END_REF] are the most effective method to assess stability. A positive realness condition must be verified for the overall system and the ℓ 1 norm of the impulse response of the multiplier must be bounded. For discrete-time Lur'e systems, the requirements on the bound of the ℓ 1 norm (called star norm) for infinite impulse response ZF multipliers [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF] and positive realness conditions are expressed in terms of linear ma-★ Supported by the Brazilian agencies: Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09, CNPq and grant 2019/10947-1 of São Paulo Research Foundation (FAPESP). trix inequalities (LMIs). These formulations consider a statespace realization of the multiplier. Although being effective to some extent, the ℓ 1 bound constraint can be conservative, and requires a scalar search. Moreover, the conditions apply only to odd nonlinearities. The finite impulse response (FIR) ZF multipliers [START_REF] Wang | A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers[END_REF][START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] offer a simpler solution to bound the ℓ 1 norm of the impulse response and the positive realness condition is still expressed with an LMI constraint. Besides, FIR multipliers handle odd or nonodd nonlinearities, and can be causal, anti-causal or noncausal.

Regarding control design for discrete-time Lur'e systems, most of the existing works are based on Lur'e-Lyapunov functions (Bertolin et al., 2022a;[START_REF] Kim | Robust static and fixedorder dynamic output feedback control of discrete-time parametric uncertain Luré systems: Sequential SDP relaxation approaches[END_REF]Louis et al., 2015b,a), with the exception of [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]) that used star-norm bounds for the ℓ 1 constraint and positive realness to derive synthesis conditions. In this paper, the aim is to propose a method for the output feedback control of discrete-time Lur'e systems based on FIR ZF multipliers. An iterative algorithm, where an LMI optimization problem is solved at each iteration, is first presented for absolute stability analysis and its extension to cope with control design is immediate. A control law using state or output feedback, that may also include the feedback of the output of the nonlinearity, can be designed jointly with the FIR ZF multiplier. Odd or non-odd nonlinearities, causal or anti-causal FIR multipliers of any given order can be considered. Numerical examples from the literature illustrate the performance of the proposed method. Lur'e systems are feedback interconnections between a linear time-invariant (LTI) plant with a static nonlinear function (Khalil, 2002, Chapter 7), as presented in Figure 1. Under assumptions on the nonlinearity, as sector and slope bounds, conditions for absolute stability can be formulated in time domain, where the stability certificates are Lyapunov functions [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF][START_REF] Park | A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF]Bertolin et al., 2022a;[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], or in the frequency domain, where the stability certificates are LTI systems called multipliers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Turner | On the existence of stable, causal multipliers for systems with sloperestricted nonlinearities[END_REF][START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Ahmad | Convex LMI approach for stability of critically stable systems with slope-restricted nonlinearities[END_REF]Haddad andBernstein, 1994, 1993;Carrasco et al., 2012a[START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF][START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF][START_REF] Carrasco | LMI search for rational anticausal Zames-Falb multipliers[END_REF][START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF].

For slope-bounded nonlinearities, the Zames-Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] O'shea | A combined frequency-time domain stability criterion for autonomous continuous systems[END_REF][START_REF] O'shea | An improved frequency time domain stability criterion for autonomous continuous systems[END_REF][START_REF] Turner | Zames-Falb multipliers: don't panic[END_REF] are the most effective method to assess stability. A positive realness condition must be verified for the overall system and the ℓ 1 norm of the impulse response of the multiplier must be bounded. For discrete-time Lur'e systems, the requirements on the bound of the ℓ 1 norm (called star norm) for infinite impulse response ZF multipliers [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF] and positive realness conditions are expressed in terms of linear ma-★ Supported by the Brazilian agencies: Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09, CNPq and grant 2019/10947-1 of São Paulo Research Foundation (FAPESP). trix inequalities (LMIs). These formulations consider a statespace realization of the multiplier. Although being effective to some extent, the ℓ 1 bound constraint can be conservative, and requires a scalar search. Moreover, the conditions apply only to odd nonlinearities. The finite impulse response (FIR) ZF multipliers [START_REF] Wang | A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers[END_REF][START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] offer a simpler solution to bound the ℓ 1 norm of the impulse response and the positive realness condition is still expressed with an LMI constraint. Besides, FIR multipliers handle odd or nonodd nonlinearities, and can be causal, anti-causal or noncausal.

Regarding control design for discrete-time Lur'e systems, most of the existing works are based on Lur'e-Lyapunov functions (Bertolin et al., 2022a;[START_REF] Kim | Robust static and fixedorder dynamic output feedback control of discrete-time parametric uncertain Luré systems: Sequential SDP relaxation approaches[END_REF]Louis et al., 2015b,a), with the exception of [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]) that used star-norm bounds for the ℓ 1 constraint and positive realness to derive synthesis conditions. In this paper, the aim is to propose a method for the output feedback control of discrete-time Lur'e systems based on FIR ZF multipliers. An iterative algorithm, where an LMI optimization problem is solved at each iteration, is first presented for absolute stability analysis and its extension to cope with control design is immediate. A control law using state or output feedback, that may also include the feedback of the output of the nonlinearity, can be designed jointly with the FIR ZF multiplier. Odd or non-odd nonlinearities, causal or anti-causal FIR multipliers of any given order can be considered. Numerical examples from the literature illustrate the performance of the proposed method. Lur'e systems are feedback interconnections between a linear time-invariant (LTI) plant with a static nonlinear function (Khalil, 2002, Chapter 7), as presented in Figure 1. Under assumptions on the nonlinearity, as sector and slope bounds, conditions for absolute stability can be formulated in time domain, where the stability certificates are Lyapunov functions [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF][START_REF] Park | A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF]Bertolin et al., 2022a;[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], or in the frequency domain, where the stability certificates are LTI systems called multipliers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Turner | On the existence of stable, causal multipliers for systems with sloperestricted nonlinearities[END_REF][START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Ahmad | Convex LMI approach for stability of critically stable systems with slope-restricted nonlinearities[END_REF]Haddad andBernstein, 1994, 1993;Carrasco et al., 2012a[START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF][START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF][START_REF] Carrasco | LMI search for rational anticausal Zames-Falb multipliers[END_REF][START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF].

For slope-bounded nonlinearities, the Zames-Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] O'shea | A combined frequency-time domain stability criterion for autonomous continuous systems[END_REF][START_REF] O'shea | An improved frequency time domain stability criterion for autonomous continuous systems[END_REF][START_REF] Turner | Zames-Falb multipliers: don't panic[END_REF] are the most effective method to assess stability. A positive realness condition must be verified for the overall system and the ℓ 1 norm of the impulse response of the multiplier must be bounded. For discrete-time Lur'e systems, the requirements on the bound of the ℓ 1 norm (called star norm) for infinite impulse response ZF multipliers [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF] trix inequalities (LMIs). These formulations consider a statespace realization of the multiplier. Although being effective to some extent, the ℓ 1 bound constraint can be conservative, and requires a scalar search. Moreover, the conditions apply only to odd nonlinearities. The finite impulse response (FIR) ZF multipliers [START_REF] Wang | A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers[END_REF][START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] offer a simpler solution to bound the ℓ 1 norm of the impulse response and the positive realness condition is still expressed with an LMI constraint. Besides, FIR multipliers handle odd or nonodd nonlinearities, and can be causal, anti-causal or noncausal.

Regarding control design for discrete-time Lur'e systems, most of the existing works are based on Lur'e-Lyapunov functions (Bertolin et al., 2022a;[START_REF] Kim | Robust static and fixedorder dynamic output feedback control of discrete-time parametric uncertain Luré systems: Sequential SDP relaxation approaches[END_REF]Louis et al., 2015b,a), with the exception of [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]) that used star-norm bounds for the ℓ 1 constraint and positive realness to derive synthesis conditions. In this paper, the aim is to propose a method for the output feedback control of discrete-time Lur'e systems based on FIR ZF multipliers. An iterative algorithm, where an LMI optimization problem is solved at each iteration, is first presented for absolute stability analysis and its extension to cope with control design is immediate. A control law using state or output feedback, that may also include the feedback of the output of the nonlinearity, can be designed jointly with the FIR ZF multiplier. Odd or non-odd nonlinearities, causal or anti-causal FIR multipliers of any given order can be considered. Numerical examples from the literature illustrate the performance of the proposed method. Lur'e systems are feedback interconnections between a linear time-invariant (LTI) plant with a static nonlinear function (Khalil, 2002, Chapter 7), as presented in Figure 1. Under assumptions on the nonlinearity, as sector and slope bounds, conditions for absolute stability can be formulated in time domain, where the stability certificates are Lyapunov functions [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF][START_REF] Park | A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF]Bertolin et al., 2022a;[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], or in the frequency domain, where the stability certificates are LTI systems called multipliers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Turner | On the existence of stable, causal multipliers for systems with sloperestricted nonlinearities[END_REF][START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Ahmad | Convex LMI approach for stability of critically stable systems with slope-restricted nonlinearities[END_REF]Haddad andBernstein, 1994, 1993;Carrasco et al., 2012a[START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF][START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF][START_REF] Carrasco | LMI search for rational anticausal Zames-Falb multipliers[END_REF][START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF].

For slope-bounded nonlinearities, the Zames-Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] O'shea | A combined frequency-time domain stability criterion for autonomous continuous systems[END_REF][START_REF] O'shea | An improved frequency time domain stability criterion for autonomous continuous systems[END_REF][START_REF] Turner | Zames-Falb multipliers: don't panic[END_REF] are the most effective method to assess stability. A positive realness condition must be verified for the overall system and the ℓ 1 norm of the impulse response of the multiplier must be bounded. For discrete-time Lur'e systems, the requirements on the bound of the ℓ 1 norm (called star norm) for infinite impulse response ZF multipliers [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF] trix inequalities (LMIs). These formulations consider a statespace realization of the multiplier. Although being effective to some extent, the ℓ 1 bound constraint can be conservative, and requires a scalar search. Moreover, the conditions apply only to odd nonlinearities. The finite impulse response (FIR) ZF multipliers [START_REF] Wang | A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers[END_REF][START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] offer a simpler solution to bound the ℓ 1 norm of the impulse response and the positive realness condition is still expressed with an LMI constraint. Besides, FIR multipliers handle odd or nonodd nonlinearities, and can be causal, anti-causal or noncausal.

Regarding control design for discrete-time Lur'e systems, most of the existing works are based on Lur'e-Lyapunov functions (Bertolin et al., 2022a;[START_REF] Kim | Robust static and fixedorder dynamic output feedback control of discrete-time parametric uncertain Luré systems: Sequential SDP relaxation approaches[END_REF]Louis et al., 2015b,a), with the exception of [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]) that used star-norm bounds for the ℓ 1 constraint and positive realness to derive synthesis conditions. In this paper, the aim is to propose a method for the output feedback control of discrete-time Lur'e systems based on FIR ZF multipliers. An iterative algorithm, where an LMI optimization problem is solved at each iteration, is first presented for absolute stability analysis and its extension to cope with control design is immediate. A control law using state or output feedback, that may also include the feedback of the output of the nonlinearity, can be designed jointly with the FIR ZF multiplier. Odd or non-odd nonlinearities, causal or anti-causal FIR multipliers of any given order can be considered. Numerical examples from the literature illustrate the performance of the proposed method. Lur'e systems are feedback interconnections between a linear time-invariant (LTI) plant with a static nonlinear function (Khalil, 2002, Chapter 7), as presented in Figure 1. Under assumptions on the nonlinearity, as sector and slope bounds, conditions for absolute stability can be formulated in time domain, where the stability certificates are Lyapunov functions [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF][START_REF] Park | A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF]Bertolin et al., 2022a;[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], or in the frequency domain, where the stability certificates are LTI systems called multipliers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Turner | On the existence of stable, causal multipliers for systems with sloperestricted nonlinearities[END_REF][START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Ahmad | Convex LMI approach for stability of critically stable systems with slope-restricted nonlinearities[END_REF]Haddad andBernstein, 1994, 1993;Carrasco et al., 2012a[START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF][START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF][START_REF] Carrasco | LMI search for rational anticausal Zames-Falb multipliers[END_REF][START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF].

For slope-bounded nonlinearities, the Zames-Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] O'shea | A combined frequency-time domain stability criterion for autonomous continuous systems[END_REF][START_REF] O'shea | An improved frequency time domain stability criterion for autonomous continuous systems[END_REF][START_REF] Turner | Zames-Falb multipliers: don't panic[END_REF] are the most effective method to assess stability. A positive realness condition must be verified for the overall system and the ℓ 1 norm of the impulse response of the multiplier must be bounded. For discrete-time Lur'e systems, the requirements on the bound of the ℓ 1 norm (called star norm) for infinite impulse response ZF multipliers [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF] trix inequalities (LMIs). These formulations consider a statespace realization of the multiplier. Although being effective to some extent, the ℓ 1 bound constraint can be conservative, and requires a scalar search. Moreover, the conditions apply only to odd nonlinearities. The finite impulse response (FIR) ZF multipliers [START_REF] Wang | A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers[END_REF][START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] offer a simpler solution to bound the ℓ 1 norm of the impulse response and the positive realness condition is still expressed with an LMI constraint. Besides, FIR multipliers handle odd or nonodd nonlinearities, and can be causal, anti-causal or noncausal.

Regarding control design for discrete-time Lur'e systems, most of the existing works are based on Lur'e-Lyapunov functions (Bertolin et al., 2022a;[START_REF] Kim | Robust static and fixedorder dynamic output feedback control of discrete-time parametric uncertain Luré systems: Sequential SDP relaxation approaches[END_REF]Louis et al., 2015b,a), with the exception of [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]) that used star-norm bounds for the ℓ 1 constraint and positive realness to derive synthesis conditions. In this paper, the aim is to propose a method for the output feedback control of discrete-time Lur'e systems based on FIR ZF multipliers. An iterative algorithm, where an LMI optimization problem is solved at each iteration, is first presented for absolute stability analysis and its extension to cope with control design is immediate. A control law using state or output feedback, that may also include the feedback of the output of the nonlinearity, can be designed jointly with the FIR ZF multiplier. Odd or non-odd nonlinearities, causal or anti-causal FIR multipliers of any given order can be considered. Numerical examples from the literature illustrate the performance of the proposed method. Lur'e systems are feedback interconnections between a linear time-invariant (LTI) plant with a static nonlinear function (Khalil, 2002, Chapter 7), as presented in Figure 1. Under assumptions on the nonlinearity, as sector and slope bounds, conditions for absolute stability can be formulated in time domain, where the stability certificates are Lyapunov functions [START_REF] Gonzaga | Stability analysis of discrete-time Lur'e systems[END_REF][START_REF] Ahmad | A less conservative LMI condition for stability of discrete-time systems with slope-restricted nonlinearities[END_REF][START_REF] Park | A revisited Popov criterion for nonlinear Lur'e systems with sector-restrictions[END_REF][START_REF] Park | Stability criteria of sector-and slope-restricted Lur'e systems[END_REF]Bertolin et al., 2022a;[START_REF] Park | A less conservative stability criterion for discrete-time Lur'e systems with sector and slope restrictions[END_REF], or in the frequency domain, where the stability certificates are LTI systems called multipliers [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Turner | On the existence of stable, causal multipliers for systems with sloperestricted nonlinearities[END_REF][START_REF] Turner | Discrete-time systems with slope restricted nonlinearities: Zames-Falb multiplier analysis using external positivity[END_REF][START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Ahmad | Convex LMI approach for stability of critically stable systems with slope-restricted nonlinearities[END_REF]Haddad andBernstein, 1994, 1993;Carrasco et al., 2012a[START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF][START_REF] Carrasco | Zames-Falb multipliers for absolute stability: From O'Shea's contribution to convex searches[END_REF][START_REF] Safonov | Computer-aided stability analysis renders Popov criterion obsolete[END_REF][START_REF] Carrasco | LMI search for rational anticausal Zames-Falb multipliers[END_REF][START_REF] Carrasco | LMI searches for anticausal and noncausal rational Zames-Falb multipliers[END_REF].

For slope-bounded nonlinearities, the Zames-Falb (ZF) multipliers [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF][START_REF] O'shea | A combined frequency-time domain stability criterion for autonomous continuous systems[END_REF][START_REF] O'shea | An improved frequency time domain stability criterion for autonomous continuous systems[END_REF][START_REF] Turner | Zames-Falb multipliers: don't panic[END_REF] are the most effective method to assess stability. A positive realness condition must be verified for the overall system and the ℓ 1 norm of the impulse response of the multiplier must be bounded. For discrete-time Lur'e systems, the requirements on the bound of the ℓ 1 norm (called star norm) for infinite impulse response ZF multipliers [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF] trix inequalities (LMIs). These formulations consider a statespace realization of the multiplier. Although being effective to some extent, the ℓ 1 bound constraint can be conservative, and requires a scalar search. Moreover, the conditions apply only to odd nonlinearities. The finite impulse response (FIR) ZF multipliers [START_REF] Wang | A complete and convex search for discrete-time noncausal FIR Zames-Falb multipliers[END_REF][START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] offer a simpler solution to bound the ℓ 1 norm of the impulse response and the positive realness condition is still expressed with an LMI constraint. Besides, FIR multipliers handle odd or nonodd nonlinearities, and can be causal, anti-causal or noncausal.

Regarding control design for discrete-time Lur'e systems, most of the existing works are based on Lur'e-Lyapunov functions (Bertolin et al., 2022a;[START_REF] Kim | Robust static and fixedorder dynamic output feedback control of discrete-time parametric uncertain Luré systems: Sequential SDP relaxation approaches[END_REF]Louis et al., 2015b,a), with the exception of [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]) that used star-norm bounds for the ℓ 1 constraint and positive realness to derive synthesis conditions. In this paper, the aim is to propose a method for the output feedback control of discrete-time Lur'e systems based on FIR ZF multipliers. An iterative algorithm, where an LMI optimization problem is solved at each iteration, is first presented for absolute stability analysis and its extension to cope with control design is immediate. A control law using state or output feedback, that may also include the feedback of the output of the nonlinearity, can be designed jointly with the FIR ZF multiplier. Odd or non-odd nonlinearities, causal or anti-causal FIR multipliers of any given order can be considered. Numerical examples from the literature illustrate the performance of the proposed method. Notation: For a symmetric matrix, 𝐴 > 0 ( 𝐴 < 0) means that 𝐴 is positive (negative) definite. For matrices or vectors ( 𝑇 ) indicates the transpose, He( 𝐴) = 𝐴 + 𝐴 𝑇 and 𝑑𝑖𝑎𝑔( 𝐴 1 , . . . , 𝐴 𝑛 ) represents a block diagonal matrix formed by the square matrices (or elements) 𝐴 1 , . . . , 𝐴 𝑛 . The symbol ★ represents a term induced by symmetry in a square matrix, ∼ indicates a statespace realization of a transfer function and Re{𝑀 ( 𝑗𝜔)} means the real part of 𝑀 ( 𝑗𝜔). The identity and the zero matrices are denoted, respectively, by 𝐼 and 0. B ⊥ represents a basis for the null space of B, i.e., BB ⊥ = 0. Throughout the text the dimensions of the matrices may be omitted for simplicity (being inferred from the context).

PROBLEM DEFINITION

Consider the discrete-time nonlinear Lur'e system given by

       𝑥(𝑘 + 1) = 𝐴𝑥(𝑘) + 𝐵 𝜙 𝜙(𝑧(𝑘)) + 𝐵 𝑢 𝑢(𝑘) 𝑧(𝑘) = 𝐶 𝑧 𝑥(𝑘) + 𝐷 𝑧 𝜙 𝜙(𝑧(𝑘)) 𝑦(𝑘) = 𝐶 𝑦 𝑥(𝑘) (1)
where 𝑥 ∈ R 𝑛 is the state, 𝑢 ∈ R 𝑛 is the control input, 𝑧 ∈ R, 𝜙 : R → R, and 𝑦 ∈ R 𝑛 is the measured output. Matrices 𝐴, 𝐵 𝜙 , 𝐵 𝑢 , 𝐶 𝑧 , 𝐶 𝑦 , and 𝐷 𝑧 𝜙 are real and have appropriate dimensions.

The nonlinearity 𝜙 is a monotonic function (odd or non-odd), that verifies the slope bound

0 ≤ 𝜙( ẑ) -𝜙(𝑧) ( ẑ -𝑧) ≤ Λ (2)
for all 𝑧, ẑ ∈ R, 𝑧 ≠ ẑ, and 𝜙( ẑ), 𝜙(𝑧) ∈ [0, Λ] where Λ is a given positive scalar.

System (1) is assumed to be well-posed, that is, the algebraic loop involving 𝑧 and 𝜙(𝑧) has a unique solution. A sufficient condition for well-posedness is given by

1 + Λ𝐷 𝑧 𝜙 > 0.
(3)

The main goal of this paper is to design a stabilizing static output-feedback controller for system (1). Considering the control law 𝑢 = 𝐾𝑦 + 𝐾 𝜙 𝜙(𝑧), the state-space representation for the closed-loop system is given by

� 𝑥(𝑘 + 1) = 𝐴 𝑐𝑙 𝑥(𝑘) + 𝐵 𝑐𝑙 𝜙(𝑧(𝑘)) 𝑧(𝑘) = 𝐶 𝑧 𝑥(𝑘) + 𝐷 𝑧 𝜙 𝜙(𝑧(𝑘)) (4)
with 𝐴 𝑐𝑙 = 𝐴 + 𝐵 𝑢 𝐾𝐶 𝑦 and 𝐵 𝑐𝑙 = 𝐵 𝜙 + 𝐵 𝑢 𝐾 𝜙 .

As an important result that is used to derive the proposed conditions, the Finsler's Lemma (de Oliveira and Skelton, 2001) is reproduced below. Lemma 1. (Finsler's lemma). Consider matrices Q ∈ R ℓ×ℓ and B ∈ R 𝑚×ℓ , with 𝑟𝑎𝑛𝑘 (B) < ℓ and BB ⊥ = 0. Then, the following conditions are equivalent:

i) B 𝑇 ⊥ QB ⊥ < 0; ii) ∃X ∈ R ℓ×𝑚 such that Q + XB + B 𝑇 X 𝑇 < 0.

Stability conditions

By performing a loop transformation, stability analysis of the closed-loop system (4) can be investigated through the existence of a ZF multiplier 𝑀 (𝑧) [START_REF] Zames | Stability conditions for systems with monotone and slope-restricted nonlinearities[END_REF], such that Re{𝑀 (𝑒 𝑗 𝜔 ) G𝑐𝑙 (𝑒 𝑗 𝜔 )} > 0, ∀𝜔 ∈ [0, 2𝜋] (5) is verified, with G𝑐𝑙 (𝑒 𝑗 𝜔 ) = (1 + Λ𝐺 𝑐𝑙 (𝑒 𝑗 𝜔 )). Consider, without loss of generality, 𝑀 (𝑧) ∼ (𝐴 𝑚 , 𝐵 𝑚 ,𝐶 𝑚 , 1), with state space realization of 𝑀 (𝑧) G𝑐𝑙 (𝑧) given by

𝐴 𝐼 = � 𝐴 𝑝 0 𝐵 𝑚 𝐶 𝑝 𝐴 𝑚 � , 𝐵 𝐼 = � 𝐵 𝑝 𝐵 𝑚 𝐷 𝑝 � 𝐶 𝐼 = � 𝐶 𝑝 𝐶 𝑚 � , 𝐷 𝐼 = 𝐷 𝑝 , where 𝐴 𝑝 = 𝐴 𝑐𝑙 , 𝐵 𝑝 = 𝐵 𝑐𝑙 , 𝐶 𝑝 = Λ𝐶 𝑧 and 𝐷 𝑝 = (1 + Λ𝐷 𝑧 𝜙 ).
Then condition (5) can be verified through the following lemma [START_REF] Ahmad | LMI searches for discrete-time Zames-Falb multipliers[END_REF]. Lemma 2. System 𝑀 (𝑧) G𝑐𝑙 (𝑧) ∼ (𝐴 𝐼 , 𝐵 𝐼 ,𝐶 𝐼 , 𝐷 𝐼 ) is stable and positive real if there exists a positive definite matrix

𝑃 = 𝑃 𝑇 , such that � 𝐴 𝑇 𝐼 𝑃𝐴 𝐼 -𝑃 𝐴 𝑇 𝐼 𝑃𝐵 𝐼 -𝐶 𝑇 𝐼 ★ 𝐵 𝑇 𝐼 𝑃𝐵 𝐼 -𝐷 𝑇 𝐼 -𝐷 𝐼 � < 0. (6)

FIR multipliers

As presented in the previous subsection, besides the positive realness condition, 𝑀 (𝑧) must belong to the class of ZF multipliers to certify the absolute stability of the closed-loop Lur'e system (4). In this work, the following causal FIR multiplier is adopted

𝑀 (𝑧) = 𝑛 � 𝑖=0 𝑚 𝑖 𝑧 -𝑖 (7)
where 𝑛 𝑀 ≥ 1 is the multiplier order and 𝑚 0 = 𝐷 𝑚 = 1.

This particular structure for 𝑀 (𝑧) simplifies the constraint to bound the ℓ 1 norm 𝑀 (𝑧). As a matter of fact, such condition can be established in terms of linear programming involving the coefficients 𝑚 𝑖 , as presented in the next lemma for both odd and non-odd nonlinearities [START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF]. Lemma 3. 𝑀 (𝑧) given in ( 7) is a ZF multiplier if i)

𝜎 = 𝑛 � 𝑖=0 𝑚 𝑖 ≥ 0, (8)
with 𝑚 𝑖 ≤ 0, for 𝑖 = 1, . . . , 𝑛 𝑀 and 𝑚 0 = 1 for non-odd nonlinearities. ii)

𝜎 = 𝑛 � 𝑖=0 𝑚 + 𝑖 + 𝑛 � 𝑖=0 𝑚 - 𝑖 ≤ 2, (9)
where 𝑚 𝑖 = 𝑚 + 𝑖 -𝑚 - 𝑖 , and 𝑚 + 𝑖 , 𝑚 - 𝑖 ≥ 0, for 𝑖 = 1, . . . , 𝑛 𝑀 , 𝑚 + 0 = 1 and 𝑚 - 0 = 0 for odd nonlinearities. In this paper, the search for 𝑀 (𝑧) ∼ (𝐴 𝑚 , 𝐵 𝑚 ,𝐶 𝑚 , 1) is simplified by adopting the following controllable canonical form

𝐴 𝑚 =            0 1 0 • • • 0 0 0 1 • • • . . . . . . . . . . . . . . . . . . 0 0 0 • • • 1 0 0 0 • • • 0            , 𝐵 𝑚 =           0 0 . . . 0 1           , 𝐶 𝑚 =         𝑚 𝑛 𝑚 (𝑛 -1) . . . 𝑚 1         𝑇 . (10)
The search for anti-causal FIR ZF multipliers can be performed by transforming 𝐴 , 𝐵 , 𝐶 and 𝐷 as proposed in [START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF]. Although in [START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] a non-causal multiplier can be computed by solving a convex optimization problem, the resulting conditions are not easily extended for control design, which is the main purpose of this paper.

MAIN RESULTS

As the main contribution of this work, the present section deals with the design of the static output-feedback gains 𝐾 and 𝐾 associated to the closed-loop system (4).

Theorem 1. Let the positive integer 𝑛 and the matrices 𝑋 ∈ R ( + )×( + ) , 𝑖 = 1, 3, 4, 𝑋 2 ∈ R ( + )×1 , with 𝑋 4 of full rank, be given. If there exist

𝑋 ∈ R ( + )×( + ) , 𝑖 = 1, 3, 4, 𝑋 2 ∈ R 1×( + ) , 0 < 𝑃 = 𝑃 ∈ R ( + )×( + ) , 𝐾 ∈ R × , 𝐾 ∈ R ×1 and 𝐶 ∈ R 1× such that Q + He �        𝑋 1 𝑋 2 𝑋 3 𝑋 4        ���� � 𝑋 1 𝑋 2 𝑋 3 𝑋 4 � � �� � � < 0, (11)
where

Q =        -𝑃 -𝐶 0 𝐴 ★ -2𝐷 0 𝐵 ★ ★ 𝑃 -𝐼 ★ ★ ★ 0       
, and the conditions of Lemma 3 (𝜙 odd or non-odd) are verified, then the gains 𝐾 and 𝐾 stabilize the closed-loop system (4) and 𝑀 (𝑧) given in ( 7) is a FIR ZF multiplier.

Proof. Consider

B ⊥ = � 𝐼 -𝑋 -1 4 𝑋 1 -𝑋 -1 4 𝑋 2 -𝑋 -1 4 𝑋 3 �
as basis for the null space of 𝑋, that is, 𝑋B ⊥ = 0. Invoking Lemma 1, condition (11) is equivalent to B ⊥ QB ⊥ < 0, that can be rewritten as

      -𝑃 -𝐶 0 ★ -2𝐷 0 ★ ★ 𝑃       + He �        ( 𝑋 -1 4 𝑋 1 ) ( 𝑋 -1 4 𝑋 2 ) ( 𝑋 -1 4 𝑋 3 )        � 𝐴 𝐵 -𝐼 � � < 0. (12)
The above inequality is, again, in the form ii) of Lemma 1. Then, computing a basis for the null space of � 𝐴 𝐵 -𝐼 � , the following condition, equivalent to (6), can be obtained

� 𝐴 𝑃𝐴 -𝑃 𝐴 𝑃𝐵 -𝐶 𝐵 𝑃𝐴 -𝐶 𝐵 𝑃𝐵 -2𝐷 � < 0, (13)
proving the positive realness of the interconnected system. The feasibility of (8) (non-odd) or ( 9) (odd) assures that 𝑀 (𝑧) is a ZF multiplier. As a consequence, the closed-loop system is stable.

The main advantage of Theorem 1 is that the variables 𝑃, 𝐶 and the matrices of the system appear affinely in the conditions. Thus, the stabilizing gains can be directly computed, without resorting to change of variables or any other algebraic techniques, which in general are difficult to be applied for static output-feedback control, even for linear time-invariant systems. Moreover, the multiplier 𝑀 (𝑧) can be of any given order 𝑛 ≥ 1. However, note that inequality ( 11) is convex because 𝑋 is a fixed matrix. Consequently, due to the lack of a general approach for choosing suitable matrices 𝑋 , 𝑖 = 1, . . . , 4, the results obtained through this sufficient condition may be conservative. A strategy to overcome this difficulty is the application of relaxations, as shown in the next theorem.

Theorem 2. Let 𝑋 = � 0 0 -𝜈𝐼 𝐼 � , where 0 < 𝜈 < 2(1+Λ𝐷 ) -1 and the change of variables 𝑀 (𝑧) ∼ (𝐴 , 𝐵 , 𝐶 , 1), for

𝐶 = � 𝑚 𝑚 ( -1) • • • 𝑚 1 � = � (𝜌 ( -1) 𝑚 ) (𝜌 ( -2) 𝑚 ( -1) ) • • • 𝑚 1 � , (14)
and

𝐴 = 𝐴 = 𝜌𝐴 + 𝐵 K𝐶 𝐵 = 𝐵 = 𝜌𝐵 + 𝐵 K 𝐶 = 𝜌𝐶 (15)
with 𝜌 > 0, K = 𝜌𝐾 and K = 𝜌𝐾 . Then, the conditions of Theorem 1 always have a feasible solution with a sufficiently small value of 𝜌.

Proof. The ZF multiplier 𝑀 (𝑧) ∼ (𝐴 , 𝐵 , 𝐶 , 1) can be equivalently rewritten as,

𝑀 (𝑧) = 𝐶 (𝑧𝐼 -𝐴 ) -1 𝐵 + 1 = 𝐶 (𝑧𝐼 -𝜌𝐴 ) -1 𝐵 + 1.
(16) Then, by ( 16) and from the choices made in ( 15), one has

𝐴 = � 𝜌𝐴 + 𝐵 K𝐶 0 𝜌𝐵 𝐶 𝜌𝐴 � , 𝐵 = � 𝜌𝐵 + 𝐵 K 𝐵 (1 -Λ𝐷 ) � , 𝐶 = � 𝜌𝐶 𝐶 � .
With 𝐴 , 𝐵 , 𝐶 in inequality (11), and fixing 𝑋 1 = 𝑋 2 = 0 and

𝑋 3 = -𝜈𝑋 4 = 0.5𝐼, one has         -𝑃 -𝐶 0 𝐴 ★ -2𝐷 0 𝐵 ★ ★ 𝑃-𝜈𝐼 0 ★ ★ ★ -1 𝐼         < 0.
With a sufficiently small value of 𝜌, 𝐾 = 𝐾 = 0 and 𝐶 = 0 (note that the conditions of Lemma 3 remain feasible), and applying a Schur complement, one has

      𝑃 0 0 ★ 2𝐷 -𝜈𝐵 𝐵 0 ★ ★ 𝜈𝐼 -𝑃       > 0.
Blocks (1,1) and (3,3) of the above inequality are feasible with 0 < 𝑃 < 𝜈𝐼. Finally, block (2,2) is feasible with the choice

0 < 𝜈 < 2(1 + Λ𝐷 ) -1 , because 2𝐷 -𝜈𝐵 𝐵 = 2(1 + Λ𝐷 ) -𝜈(1 + Λ𝐷 ) 2 𝐵 𝐵 = 2 -𝜈(1 + Λ𝐷 ) > 0, since (1 + Λ𝐷 ) > 0 by assumption (3).
Note that, adopting the initialization proposed by Theorem 2, Theorem 1 always produces a feasible solution thanks to the parameter 𝜌, which is a relaxation factor in matrices 𝐴 , 𝐵 , 𝐶 of the closed-loop system. With the proposed change of variables, 𝜌 appears affinely in condition (11). This way, 𝜌 can be maximized as an objective function to find the stabilizing gains 𝐾 and 𝐾 and also a valid multiplier 𝑀 (𝑧) which certificates the stability of the original Lur'e system (𝜌 = 1). While 𝜌 < 1 the resulting matrices 𝑋 , 𝑖 = 1, . . . , 4 can be used as new values of 𝑋 , 𝑖 = 1, . . . , 4, thus generating a new feasible solution. This is possible because ( 11) is structured in the form ii) of Finsler's Lemma, i.e., Q +He(XB) < 0. Then, as He(XB) = He(B 𝑇 X 𝑇 ), the choice B = X 𝑇 guarantees a feasible solution in a new test, ensuring that the value of 𝜌 cannot decrease.

The iterative procedure is presented in Algorithm 1 (the index (•) 𝑖𝑡 represents the value of the variable at iteration 𝑖𝑡). The input parameters are 𝑖𝑡 𝑚𝑎𝑥 , which defines a stopping criterion, 𝑛 𝑀 , multiplier order, and 𝑋, the initial matrix required by Theorem 1. If 𝜌 ≥ 1 at some iteration, then the stability of the closed-loop Lur'e system is certified. Unfortunately, although the sequence of values for 𝜌 is non-decreasing, the convergence to a value greater than 1 cannot be assured.

Algorithm 1 Iterative procedure for stabilization Input parameters: 𝑖𝑡 𝑚𝑎𝑥 , 𝑛 𝑀 , 𝑋; Make the changes of variables as in ( 14) and ( 15); 𝑖𝑡 ← 1; while 𝑖𝑡 ≤ 𝑖𝑡 𝑚𝑎𝑥 do maximize 𝜌 𝑖𝑡 subject to: (8) and ( 11) (𝜙 non-odd) or ( 9) and ( 11) (𝜙 odd) ; if 𝜌 𝑖𝑡 ≥ 1 then return 𝜌 𝑖𝑡 , 𝐶 𝑚,𝑖𝑡 and 𝑋 end 𝑋 = 𝑋 𝑇 ; 𝑖𝑡 = 𝑖𝑡 + 1; end Next theorem states that, if 𝜌 > 1 is obtained, then the controller gains and the FIR ZF multiplier 𝑀 (𝑧) for the original system (𝜌 = 1) can be computed as a convex combination of the solutions of two consecutive iterations. Theorem 3. Let 𝐴 𝑝 , 𝐵 𝑝 , 𝐶 𝑝 and 𝐶 𝑚 be as given in Theorem 2 and consider the existence of two consecutive solutions of conditions of Lemma 3 and (11) provided by Algorithm 1, respectively at iterations 𝑘 and 𝑘 + 1, such that

Q 𝑘 + He( 𝑋 𝑘 𝑋 𝑇 𝑘-1 ) < 0, 𝜌 𝑘 < 1, (17) Q 𝑘+1 + He( 𝑋 𝑘+1 𝑋 𝑇 𝑘 ) < 0, 𝜌 𝑘+1 > 1. (18)
Then, the controller gains 𝐾 and 𝐾 𝜙 and the ZF multiplier 𝑀 (𝑧) for the original system (i.e., for 𝜌 = 1) are obtained from a convex combination of the solutions at iterations 𝑘 and 𝑘 + 1.

Proof. Consider the nonnegative scalars 𝜁 𝑘 and 𝜁 𝑘+1 such that,

𝜁 𝑘 + 𝜁 𝑘+1 = 1, 𝜁 𝑘 = 𝜌 𝑘+1 -1 𝜌 𝑘+1 -𝜌 𝑘 , 𝜁 𝑘+1 = 1 -𝜌 𝑘 𝜌 𝑘+1 -𝜌 𝑘 .
Multiplying (17) by 𝜁 𝑘 and (18) by 𝜁 𝑘+1 , and summing up, one has

𝜁 𝑘 Q 𝑘 + He( 𝑋 𝑘 𝑋 𝑇 𝑘-1 ) + 𝜁 𝑘+1 Q 𝑘+1 + He( 𝑋 𝑘+1 𝑋 𝑇 𝑘 ) = 𝜁 𝑘 Q 𝑘 + 𝜁 𝑘+1 Q 𝑘+1 + He 𝜁 𝑘 𝑋 𝑘 𝑋 𝑇 𝑘-1 + 𝜁 𝑘+1 𝑋 𝑘+1 𝑋 𝑇 𝑘 < 0. (19)
Regarding the term = 𝐶 𝑝 (𝜁 𝑘 𝐶 𝑚,𝑘 + 𝜁 𝑘+1 𝐶 𝑚,𝑘+1 ) = 𝐶 𝑝 C 𝑚 .

𝜁 𝑘 Q 𝑘 + 𝜁 𝑘+1 Q 𝑘+1 ,
Since (19) holds and the convex combination also preserves the positive realness condition, the ZF multiplier associated with 𝜌 = 1 given by 𝑀 (𝑧) ∼ (𝐴 𝑚 , 𝐵 𝑚 , C 𝑚 , 1) assures closedloop stability with the gains 𝐾 = 𝐾 and 𝐾 𝜙 = 𝐾 𝜙 .

EXAMPLES

In this section, numerical examples are presented to illustrate the performance of Algorithm 1 (named A1) in stability analysis (fix 𝐾 = 𝐾 𝜙 = 0), and in control design with state, namely with 𝐶 𝑦 = 𝐼, and output feedback. Both FIR multipliers for odd and non-odd nonlinearities are investigated. The initial choices proposed for A1 are 𝑖𝑡 𝑚𝑎𝑥 = 250 and 𝜈 = (1 + Λ𝐷 𝑧 𝜙 ) -1 . Although the proposed method could deal with any given order for the ZF multipliers, 𝑛 𝑀 = 2 has been fixed in all tests for simplicity.

In the experiments, A1 is compared with the method from Bertolin et al. (2022b) (B22), that performs a line search of parameter 𝛼 ∈ (0, 1) (related with a bound to the ℓ 1 norm of the ZF multiplier) as follows: i) A bisection is used to find the maximum value of Λ, considering 10 equally spaced points for 𝛼 ∈ (0, 1); ii) For the value of 𝛼 that provided the best Λ in the previous step (defined by 𝛼 ★ ), two new bisections are performed, first considering 10 equally spaced points in the interval 𝛼 ∈ [0.75𝛼 ★ , 1.25𝛼 ★ ], obtaining a new 𝛼 ★ and then using 50 equally spaced points for 𝛼 ∈ [0.9𝛼 ★ , 1.1𝛼 ★ ] (70 searches in total). The implementation was made in Matlab-2022 using the Yalmip parser [START_REF] Löfberg | YALMIP: A toolbox for modeling and optimization in MATLAB[END_REF]) and the Mosek solver [START_REF] Andersen | The MOSEK interior point optimizer for linear programming: An implementation of the homogeneous algorithm[END_REF]), on a computer with Windows 10 operating system and Core i7 processor.

Experiment 1 -Stability analysis

Seven Lur'e systems from the literature are investigated. Systems 1 to 6 are from [START_REF] Carrasco | Convex searches for discrete-time Zames-Falb multipliers[END_REF] and system 7 is from [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF]. The aim of the experiment is to search for the maximum value of Λ such that each system remains stable. For this purpose, the search for the maximum value is performed by a bisection procedure in Λ with precision given by 10 -3 . Using a transformation on the system matrices, A1 can provide ZF multipliers 𝑀 (𝑧) that are causal (𝑐) or anticausal (𝑎𝑐). Table 1 shows the results obtained using algorithm A1, and also (for comparison) the results from Bertolin et al. (2022b) (B22) also considering second order ZF multipliers, and from Carrasco et al. (2020) (C20) -noncausal FIR ZF multipliers with 𝑛 𝑓 = 𝑛 𝑏 , i.e., same number of causal and anticausal terms, presented in parenthesis.

As can be noted in Table 1, C20 provides the best results for stability analysis, both in terms of larger values for Λ and smaller computational effort. Neither A1 nor B22 compete with C20 in terms of stability analysis but, differently from C20, both can be extended to cope with the problem of control design for discrete-time Lur'e systems, as presented next. In a Table 1. Maximum values of Λ in stability analysis obtained by A1, B22 for causal (𝑐) and anti-causal (𝑎𝑐) multipliers, and C20 (noncausal 𝑛𝑐), considering 𝜙 odd and non-odd (shaded). The numerical complexity can be inferred from the time (𝑡) in seconds required to test each system. The value of 𝑛 = 𝑛 of C20 is given in parenthesis. comparison between A1 and B22, one can note that A1 presents superior values of Λ on almost all systems for both causal and anti-causal ZF multipliers. Moreover, the computational time spent by A1 is substantially smaller (order of hundred times) than the one required by B22, due to the scalar search performed on 𝛼 ∈ (0, 1) by the method from B22.

Experiment 2 -Control design

Since the main objective of this work is the synthesis of output feedback control laws for discrete-time Lur'e systems, this experiment evaluates the performance of A1 for the control design of state (𝑠 𝑓 ) and output (𝑜 𝑓 ) feedback gains. For that, matrices 𝐵 are given by (as in [START_REF] Bertolin | An LMI approach for stability analysis and output-feedback stabilization of discrete-time Lur'e systems using Zames-Falb multipliers[END_REF])

(1, 6, 7) : 𝐵 = 0 1 , (2, 3) : 𝐵 = 0 1 0 0 (4) : 𝐵 = 0 1 0 0 0 , (5) : 𝐵 = 0 1 0 , and 𝐶 = 𝐶 . As in the analysis experiment, the maximum value of Λ for each system is computed through a bisection procedure. In all the cases, A1 is first used with the conditions that ensure stability for non-odd nonlinearities. After convergence, the value of 𝑋 obtained at the last iteration is used as 𝑋 to initialize the algorithm in the search for the largest Λ associated with odd nonlinearities. This strategy can always be used since the feasibility of (8) implies that (9) always admits a solution. Moreover, with this initialization, the values of Λ cannot be smaller than the ones previously obtained for nonodd nonlinearities, resulting in less computational effort for A1 when dealing with the odd case.

The results (maximum values of Λ and time demanded 𝑡, in seconds) are shown in Table 2. As can be noted, when compared with B22, A1 demands considerably lower computational times for all the cases, being also superior in terms of the achievable values of Λ for almost all the systems investigated. Moreover, A1 can deal with both odd and non-odd nonlinearities while B22 is restricted tohe case of odd nonlinear functions. It is worth to mention that other values of Λ (possibly greater) could be obtained by increasing the order 𝑛 of the ZF multiplier.

CONCLUSIONS

An LMI-based iterative method has been proposed for the stability analysis of discrete-time Lur'e systems through FIR ZF multipliers of any given order, with immediate extension to cope with static output and state feedback control. Both odd and non-odd nonlinearities are considered. Anti-causal multipliers can also be obtained with a transformation on the system matrices. Future work will investigate noncausal FIR ZF multipliers with arbitrary number of causal and anti-causal terms for both stability analysis and control design using integral quadratic constraints (IQC) based strategies.

Table 2. Maximum values of Λ with odd and non-odd (shaded) in control design obtained by A1 and B22 for causal ( ) and anti-causal ( ) multipliers, considering state ( ) and output ( ) feedback. The numerical complexity can be inferred from the time ( ) in seconds required to test each system. The notation +100 means that values of Λ larger than 100 have been obtained.

  and positive realness conditions are expressed in terms of linear ma-★ Supported by the Brazilian agencies: Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09, CNPq and grant 2019/10947-1 of São Paulo Research Foundation (FAPESP).

  Output feedback control of discrete-time Lur'e systems through FIR Zames-Falb multipliers ★ Ariádne L. J. Bertolin * Ricardo C. L. F. Oliveira * Giorgio Valmorbida * * Pedro L. D. Peres *

  Output feedback control of discrete-time Lur'e systems through FIR Zames-Falb multipliers ★ Ariádne L. J. Bertolin * Ricardo C. L. F. Oliveira * Giorgio Valmorbida * * Pedro L. D. Peres * 1. INTRODUCTION

  Output feedback control of discrete-time Lur'e systems through FIR Zames-Falb multipliers ★ Ariádne L. J. Bertolin * Ricardo C. L. F. Oliveira * Giorgio Valmorbida * * Pedro L. D. Peres * 1. INTRODUCTION

  and positive realness conditions are expressed in terms of linear ma-★ Supported by the Brazilian agencies: Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09, CNPq and grant 2019/10947-1 of São Paulo Research Foundation (FAPESP).

  Output feedback control of discrete-time Lur'e systems through FIR Zames-Falb multipliers ★ Ariádne L. J. Bertolin * Ricardo C. L. F. Oliveira * Giorgio Valmorbida * * Pedro L. D. Peres * 1. INTRODUCTION

  and positive realness conditions are expressed in terms of linear ma-★ Supported by the Brazilian agencies: Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09, CNPq and grant 2019/10947-1 of São Paulo Research Foundation (FAPESP).

  Output feedback control of discrete-time Lur'e systems through FIR Zames-Falb multipliers ★ Ariádne L. J. Bertolin * Ricardo C. L. F. Oliveira * Giorgio Valmorbida * * Pedro L. D. Peres * 1. INTRODUCTION

  and positive realness conditions are expressed in terms of linear ma-★ Supported by the Brazilian agencies: Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09, CNPq and grant 2019/10947-1 of São Paulo Research Foundation (FAPESP).

  Output feedback control of discrete-time Lur'e systems through FIR Zames-Falb multipliers ★ Ariádne L. J. Bertolin * Ricardo C. L. F. Oliveira * Giorgio Valmorbida * * Pedro L. D. Peres * 1. INTRODUCTION

  and positive realness conditions are expressed in terms of linear ma-★ Supported by the Brazilian agencies: Coordenac ¸ão de Aperfeic ¸oamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001, Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09, CNPq and grant 2019/10947-1 of São Paulo Research Foundation (FAPESP).

Fig. 1 .

 1 Fig. 1. Block diagram of a Lur'e system.

  one has for the block 𝐴 𝐼 𝜁 𝑘 𝜌 𝑘 𝐴 𝑝 + 𝐵 𝑢 K𝑘 𝐶 𝑦 0 𝜌 𝑘 𝐵 𝑚 𝐶 𝑝 𝐴 𝑚 + 𝜁 𝑘+1 𝜌 𝑘+1 𝐴 𝑝 + 𝐵 𝑢 K𝑘+1 𝐶 𝑦 0 𝜌 𝑘+1 𝐵 𝑚 𝐶 𝑝 𝐴 𝑚 = 𝐴 𝑝 + 𝐵 𝑢 𝜁 𝑘 K𝑘 + 𝜁 𝑘+1 K𝑘+1 𝐶 𝑦 0 𝐵 𝑚 𝐶 𝑝 𝐴 𝑚 = 𝐴 𝑝 + 𝐵 𝑢 𝐾𝐶 𝑦 0 𝐵 𝑚 𝐶 𝑝 𝐴 𝑚 and similarly for 𝐵 𝐼 and 𝐶 𝐼 𝜁 𝑘 𝜌 𝑘 𝐵 𝜙 + 𝐵 𝑢 K𝜙,𝑘 𝐵 𝑚 𝐷 𝑝 + 𝜁 𝑘+1 𝜌 𝑘+1 𝐵 𝜙 + 𝐵 𝑢 K𝜙,𝑘+1 𝐵 𝑚 𝐷 𝑝 = 𝐵 𝜙 + 𝐵 𝑢 𝜁 𝑘 K𝜙,𝑘 + 𝜁 𝑘+1 K𝜙,𝑘+1 𝐵 𝑚 𝐷 𝑝 = 𝐵 𝜙 + 𝐵 𝑢 𝐾 𝜙 𝐵 𝑚 𝐷 𝑝 , 𝜁 𝑘 𝜌 𝑘 𝐶 𝑝 𝐶 𝑚,𝑘 + 𝜁 𝑘+1 𝜌 𝑘+1 𝐶 𝑝 𝐶 𝑚,𝑘+1