Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control

Transcription

1 C. Böhm a R. Findeisen b F. Allgöwer a Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control Stuttgart, March 21 a Institute of Systems Theory and Automatic Control, University of Stuttgart, Pfaffenwaldring 9, 7569 Stuttgart/ Germany {cboehm,allgower}@ist.uni-stuttgart.de b Institute for Automation Engineering Otto-von-Guericke Universität Magdeburg, Universitätsplatz 2, 3916 Magdeburg/ Germany rolf.findeisen@ovgu.de Abstract We consider the problem of controlling continuous-time sector bounded Lur e systems subject to state and input constraints. First, we derive a linear static feedback law which stabilizes Lur e systems with uncertain nonlinearities. The approach is then used to calculate the terminal region and the terminal penalty term for quasi-infinite horizon nonlinear model predictive control (NMPC). The results are further extended to a robustly stabilizing NMPC scheme and finally to an NMPC approach with low online computational demand which is based on the offline calculation of a set of ellipsoids and associated feedback laws. All controllers are calculated by solving linear matrix inequalities and satisfy state and input constraints. To illustrate their effectiveness the controllers are applied to a flexible link robotic arm and their properties are discussed by means of the obtained simulation results. Keywords Nonlinear model predictive control continuous-time Lur e systems constraints linear matrix inequalities robust control Preprint Series Issue No Stuttgart Research Centre for Simulation Technology (SRC SimTech) SimTech Cluster of Excellence Pfaffenwaldring 7a 7569 Stuttgart publications@simtech.uni-stuttgart.de

2 2 C. Böhm a et al. 1 Introduction The analysis of stability of Lur e systems with sector restricted nonlinearities, often referred to as absolute stability, has been studied for decades. The original problem was formulated in [24]. A further early result, namely the famous Popov criterion, has been published in [31]. Later important results have been proposed in [17, 18, 33]. For a review see e.g. [4, 15, 29] which provide a good overview on existing work in the field of stability analysis. Recently, in [2] and [21] linear matrix inequality (LMI) conditions have been derived which guarantee absolute stability of Lur e systems with time-delays. Concerning controller synthesis for Lur e systems with sector restricted nonlinearities, in the following referred to as sector bounded Lur e systems, also many results have been published, see e.g. [3, 22, 27, 3, 34] just to mention a few. Many of those approaches involve LMI techniques which have been proven to be useful for both analysis and controller synthesis of Lur e systems. For example, in [28] and [38] LMIs have been used to solve the H controller synthesis problem for Lur e systems subject to external disturbances. However, the aforementioned controller design methods do not consider state and input constraints which are important and inherent in many applications. To the authors best knowledge there are no approaches capable of considering both state and input constraints explicitly in the controller design for uncertain Lur e systems. This motivates the first goal of this paper, namely the derivation of a new controller design approach using LMI techniques for state and input constrained sector bounded Lur e systems. The resulting controller, which will be a linear static state feedback strategy, is such that it stabilizes nonlinear Lur e systems for all nonlinearities satisfying the sector condition, i.e. it is robust towards model uncertainties. As a second goal we aim to develop nonlinear model predictive control (NMPC) approaches for Lur e systems. NMPC is a control strategy which is based on the repeated online solution of an open-loop optimal control problem, see e.g. [9, 13, 25]. It is capable of dealing with nonlinear multi-input multi-output systems as well as state and input constraints. If certain well known conditions are satisfied, i.e. if the so called terminal region and the terminal penalty term are chosen suitably, NMPC guarantees closed-loop stability, see e.g. [1, 9, 1, 13, 25]. In [5] an approach for Lur e systems has been presented which allows for the calculation of a terminal region and a terminal penalty term via the solution of a set of LMIs. Based on these results an NMPC scheme for Lur e systems is presented in [7] where the nonlinear openloop optimal control problem is approximated by a convex optimization problem subject to LMIs. The solution to this problem, which is solved repeatedly at each sampling instant, is such that an upper bound on the considered infinite horizon cost functional is minimized and the controller is robust towards model uncertainties. In this paper we show the relation of the NMPC approaches [5] and [7] to the static linear feedback controller derived in the first part. Finally, we extend the LMI based NMPC scheme [7] into an offline predictive control approach with significantly reduced computational demand while robustly stabilizing the system. Basically, as in [11, 36, 37] we calculate offline a set of ellipsoids and feedback matrices via the solution of a convex optimization problem subject to LMIs. Online we only determine the ellipsoid in which the current state is lying in and apply the associated feedback control law. Summarizing, the paper presents four new control approaches for constrained and sector bounded continuous-time Lur e systems: One linear state feedback design, two online NMPC approaches and a novel offline predictive control approach that is robustly stabilizing while being computationally efficient. The remainder of the paper is organized as follows: In Section 2 the considered class of sector bounded Lur e systems is introduced. Section 3 derives the first result of this paper, namely a constructive method to design a robustly stabilizing linear feedback controller via the solution of a set of LMIs. Section 4 provides an introduction to NMPC. In Section 5 we present the results of [5] and show the relation to the derived linear controller. Extending these results, the LMI based NMPC scheme [7] is introduced in Section 6. The second contribution of this paper, which is a computationally attractive predictive control strategy based on the offline solution of a set of convex optimization problems subject to LMIs, is proposed in Section 7. In Section 8 we finally apply the considered control strategies to a flexible link robotic arm. The paper concludes with a short summary in Section 9.

3 Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control 3 2 Sector bounded Lur e systems In this paper we consider a subclass of nonlinear systems, namely continuous-time input affine sector bounded Lur e systems (see e.g. [18]), which are described by ẋ = Ax+Gγ(z)+Bu, z = Hx, where A R n n, B R n m, G R n p, and H R p n are constant linear matrices. The vector γ(z) : R p R p consists of p nonlinear functions depending on z. We limit our attention to Lur e systems where the signal z and the nonlinearities γ(z) satisfy a so called sector condition of the form ( βz γ(z) ) T γ(z), (1c) where β = diag(β 1,β 2,...,β p ) with β i R +, i = 1,..., p, see [18]. Figure 1 illustrates the sector condition for p = 1. As can be seen the nonlinearity is bounded to the sector [, β z]. (1a) (1b) βz γ(z) z Fig. 1 Sector condition of Lur e systems with single nonlinearity. The control task is to steer system (1) to the origin while satisfying constraints on the state and the input In the following we assume polytopic constraint sets: Assumption 1 The constraint sets X and U are convex polytopes x(t) X, u(t) U, t. (2) X = {x R n : c T i x x 1,i x = 1,...,r x }, U = {u R m : d T i u u 1,i u = 1,...,r u }, where c T i x R n and d T i u R m and r x and r u the number of state and input constraints. To simplify notation the sets X and U are combined in the constraint set {[ } C x = R u] n+m : c T i x+di T u 1,i = 1,,r, (4) where r = r x + r u. The formulation of the constraint set C in principle allows for the consideration of mixed state and input constraints. However, in this paper we limit our attention to independent state and input constraints as defined in (3a) and (3b), i.e. in (4) di T = for a state constraint and c T i = for an input constraint. In the remainder of the paper we denote the Lur e system as uncertain if the nonlinear function γ(z) is not known exactly but known to be bounded by the sector condition (1c). Furthermore, we refer to a controller as robustly asymptotically stabilizing controller if it stabilizes an uncertain Lur e system asymptotically for all γ(z) satisfying the sector condition. (3a) (3b)

4 4 C. Böhm a et al. 3 Robust control of constrained Lur e systems In this section we design a linear feedback matrix K R m n such that the resulting control law u = Kx (5) robustly asymptotically stabilizes the uncertain Lur e system (1) while satisfying the state and input constraints (3), i.e. independent from the nonlinearity γ(z). The basic idea is to cast the stabilization problem as a (conservative) convex semi-definite program. Due to the state dependency of the feedback law (5) the constraint set C (if K is known) only depends on the states and takes the form C = {x R n : (c T i + d T i K)x 1,i = 1,,r}. (6) From the definition of C it clearly follows that C X. We will work with quadratic Lyapunov functions and control invariant ellipsoids. Thus, the following lemma is helpful and needed to guarantee satisfaction of the constraints: Lemma 1 (Ellipsoid contained in polytope) The ellipsoid E = {x R n : x T Px α} is contained in the set C if and only if Proof The proof can be found in [8] and [1]. (c T i + d T i K)(αP 1 )(c T i + d T i K) T 1, i = 1,,r. (7) Thus, if the conditions of Lemma 1 are satisfied we know that a state x lying in the ellipsoid E lies in the constraint set C and therefore satisfies state and input constraints. Based on this, in the following we derive a novel method which provides a structured way to calculate the feedback matrix K. The basic idea is to formulate the required stability condition, i.e. a Lyapunov inequality, in terms of LMIs by taking the special structure of Lur e systems, namely the sector condition (1c), into account. The sector condition can be expressed in matrix form as [ ] T [ x 1 2 HT β T ][ ] x γ 1 2 βh I. (8) γ Using this, the following theorem provides a constructive way to calculate a stabilizing state feedback matrix K via the solution of a set of LMIs such that the control law (5) robustly asymptotically stabilizes the uncertain Lur e system (1) while satisfying state and input constraints. Theorem 1 (Linear controller) Suppose there exist matrices < X = X T R n n and Y R m n, and constants τ R + and α R +, such that the inequalities [ AX XA T BY Y T B T Gα 2 τ XHT β T ] G T α τ 2 βhx ατi > (9) [ 1 c T i X + dt i Y ] (c T i X +, (1) dt i Y)T X are satisfied for i = 1,...,r. Then the linear feedback law (5) renders the closed-loop of the uncertain system (1) robustly asymptotically stable with K = YX 1 while satisfying state and input constraints. Furthermore, with P = αx 1 the domain of attraction is at least the ellipsoid E = {x R n : x T Px α}. Proof Applying the Schur complement to (1), one obtains 1 (c T i X + d T i Y)X 1 (c T i X + d T i Y) T, i = 1,,r. (11) Since X >, P = αx 1, and K = Y X 1 this inequality is equivalent to 1 (c T i + d T i K)(αP) 1 (c T i + d T i K) T, i = 1,...,r. (12) Thus, it follows from Lemma 1 that E C, i.e. (12) implies that state and input constraints are satisfied for all x E with the feedback law (5). It remains to show that the ellipsoid E is an invariant set under the

5 Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control 5 input u = Kx = Y X 1 x. For this, consider the Lyapunov function candidate V(x) = x T Px. In the case of the Lur e system (1) for asymptotic stability and invariance of E the inequality V = x T (A T P+PA+K T B T P+PBK)x+γ T (z)g T Px+x T PGγ(z) <, (13) has to be satisfied. This condition is equivalent to [ x γ ] T [ A T P+PA+K T B T P+PBK PG G T P ][ x γ ] <. (14) Applying the S -Procedure, see e.g. [8], it follows that (14) and therefore (13) are satisfied for all x and γ satisfying the sector condition (1c) and (8), respectively, if there exists τ R + such that the matrix inequality [ A T P+PA+K T B T P+PBK PG+ τ 2 HT β T ] G T P+ 2 τ βh τi < (15) is satisfied. Multiplying with diag(p 1,I), substituting P 1 and K as defined in the theorem by X, Y and α, and applying the Schur complement one obtains the matrix inequality (9). Thus, V(x) = x T Px is a Lyapunov function and the ellipsoid E is an invariant set under control law (5). Therefore, for all initial conditions x() E the closed-loop is robustly asymptotically stable and the constraints are satisfied at all times. Often it is desirable to maximize the domain of attraction of the controller defined in Theorem 1. This can be achieved by maximizing the volume of the ellipsoid E which is up to a constant αdet(p 1 ). Thus, the optimization problem maximize P,K,α det(αp 1 ) (16) subject to (9)-(1) has to be solved, which can be transformed into a convex optimization problem, see [8, 1] for details. Remark 1 If τ is fixed, conditions (9) and (1) are LMIs and standard solvers, as e.g. sdpt3 [35], can be applied for their solution. In the remainder of the paper we assume τ to be constant and determined offline in a reasonable way. For example, one can solve the optimization problem (16) for several values for τ and apply a line search algorithm to obtain the ellipsoid with maximal volume. The conditions presented in Theorem 1 allow for the calculation of a linear control law for the considered class of Lur e systems. For the controller design the nonlinearities do not have to be known exactly. Thus, the LMI conditions provide a structured way to compute a robustly stabilizing controller for Lur e systems which satisfies state and input constraints. One of the main challenges of the linear feedback is that it might have a rather small region of attraction and that the performance might be very poor. Thus, in the following sections we investigate the results of Theorem 1 from a predictive control perspective. First, we extend the LMI conditions (9) and (1) for the calculation of a stabilizing terminal region and terminal penalty term according to a standard NMPC scheme to be introduced in the next section. Based on these results we then derive an NMPC approach which is robustly stabilizing and which, if slightly modified, can also be applied in an offline NMPC fashion. 4 Stabilizing nonlinear model predictive control This section is intended to review a standard NMPC scheme for continuous-time systems, often referred to as quasi-infinite horizon (QIH) NMPC [1, 9, 13]. Consider for the moment general nonlinear systems with x R n and u R m, subject to state and input constraints ẋ = f(x,u), x() = x (17) u(t) U, x(t) X, t. (18) For simplicity we assume the state and input constraint sets X and U as defined in (3). However, if certain assumptions are satisfied also more general forms of constraint sets could be considered, see e.g. [1].

6 6 C. Böhm a et al. The control task in this section is to stabilize the origin of system (17) while satisfying the constraints (18). One approach to achieve this is NMPC. This control method predicts the future behaviour of the system and optimizes the input based on this prediction. Therefore, we introduce predicted states and inputs, x and ū. The predicted states may differ from the real state x of the considered system (17). In general, the cost functional J, that is minimized over the prediction horizon T p, takes the form J ( x( ),ū( ) ) = t k +T p t k F ( x(s),ū(s) ) ds+e ( x(t k + T p ) ), (19) with the stage cost F and the terminal penalty term E. The open-loop optimal control problem, that is solved repeatedly at the sampling instants t k, which do not necessarily have to be equidistant, is subject to minimize J ( x( ),ū( ) ), ū( ) x(s) = f ( x(s),ū(s) ), x(t k ) = x(t k ), x(s) X, ū(s) U, s [ t k,t k + T p ], x(t k + T p ) E. The solution to the optimization problem is the optimal input trajectory 1 (2a) (2b) ū ( t;x(t k ) ) = argmin ū( ) J( x( ),ū( ) ). (2c) Note that in the stated version the system states are forced to lie within the terminal region E at the end of the prediction horizon T p. The control input applied to system (17) is updated at each sampling instant t k by the repeated solution of the open-loop optimal control problem (2), i.e. the applied control input is given by u(t) = ū (t;x(t k )), t [ t k,t k+1 ). (21) Each optimization at time t k uses the corresponding state measurement x(t k ) as initial condition for the predicted system behavior x(t). Therefore, the NMPC scheme presented provides state feedback at the sampling instants t k. To guarantee stability of the closed-loop system the following assumptions are typically required to hold [1]: Assumption 2 The vector field f : R n R m R n is continuous and satisfies f(,) =. In addition, it is locally Lipschitz in x. Assumption 3 The system (17) has a unique continuous solution for any initial condition in the region of interest and any piecewise continuous and right-continuous input function u( ) : [,T p ] U. Assumption 4 F : X U R is continuous in all arguments with F(,) = and F(x,u) > (x,u) X U \ {,}. It is shown for example in [12] and [13] that closed-loop stability can be guaranteed if the conditions on the terminal penalty term E(x) and the terminal region E stated in Lemma 2 hold. Lemma 2 (NMPC stability) Suppose the open-loop optimal control problem (2) is initially feasible at time t =. If Assumptions 1-4 are satisfied, and if the terminal penalty term E(x) and the terminal region E satisfy (i) E(x) is C 1, E(x) x E, and E() =, (ii) E X is closed and connected, and the origin is in the interior of E, (iii) For all x E there exists a continuous local control law k : R n R m with k() =, such that u = k(x) U and 1 We assume that the minimum is attained, e.g. we use min instead of inf. E x f( x,k(x) ) + F ( x,k(x) ) <, (22)

7 Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control 7 then the (nominal) closed-loop system under the NMPC scheme (17)-(21) converges to the origin as t. In general, it is hard to calculate the terminal penalty term and the terminal region, see e.g. [1, 1]. For certain subclasses of nonlinear systems, however, there exist structured approaches to solve this problem. Here, we consider the class of sector bounded Lur e systems and show that for this class it is possible to solve the considered problem efficiently and in a structured way. In the following section we present the result obtained in [5] for the calculation of a terminal region and a terminal penalty term for Lur e systems. The approach guarantees closed-loop stability of the QIH-NMPC scheme (17)-(21) and can be seen as an extension of the robust controller design method of Theorem 1. 5 Calculation of the terminal region The goal of this section is to outline a structured method to calculate a stabilizing terminal region E and an associated terminal penalty term E(x) for the QIH-NMPC scheme (17)-(21) for the class of sector bounded Lur e systems (1). As shown, an NMPC scheme is stabilizing if the conditions of Lemma 2 on the basis of the Assumptions 1-4 are satisfied. Thus, in the following we assume that the nonlinearity γ(z) is such that the Assumptions 2 and 3 hold, i.e. the vector field f(x,u) = Ax + Gγ(z)+Bu is Lipschitz. In the following we choose the quadratic cost function F(x,u) = x T Qx+u T Ru, (23) with weighting matrices Q = Q T R n n and < R = R T R m m which clearly satisfy Assumption 4. Accordingly, we define the quadratic terminal penalty term with < P = P T R n n and the ellipsoid E(x) = x T Px (24) E = {x X : x T Px α}, (25) serving as terminal region with α R +. For given Q and R the remaining task is to calculate P and α such that conditions (i)-(iii) of Lemma 2 are satisfied. For this, we consider the linear local feedback control law (5). Following Theorem 1, for sector bounded Lur e systems the conditions of Lemma 2 are fulfilled if a set of LMIs is satisfied, which will be adopted in the following. The following theorem provides a structured way to suitably calculate a linear local feedback law, a terminal penalty term E(x) and a terminal region E as defined in (24)-(25) to guarantee convergence of the closed-loop system: Theorem 2 (Terminal region) Suppose the open-loop optimal control problem (2) is initially feasible at time t =. If there exist matrices < X = X T R n n and Y R m n, and scalars τ R + and α R + such that the inequalities AX XA T BY Y T B T Gα 2 τ XHT β T XQ 1 2 Y T R 2 1 G T α τ 2βHX ατi Q 1 2 X αi R 2 1 Y αi [ 1 c T i X + dt i Y (c T i X + dt i Y)T X > (26) ], (27) are satisfied for i = 1,...,r, then system (1) under the NMPC scheme (17)-(21) converges to the origin as t, where P = αx 1 and in (5) K = Y X 1. Proof Convergence of the closed-loop system for the NMPC scheme defined by (17)-(21) can be guaranteed if the conditions (i)-(iii) of Lemma 2 hold. Hence, one has to show that satisfaction of the matrix inequalities in Theorem 2 imply satisfaction of the conditions of Lemma 2. With the choice of E(x) = x T Px, condition (i) of Lemma 2 is obviously satisfied x E since P = αx 1 >. The local control law k(x) in Lemma 2 is chosen as u = k(x) = Kx, where K = Y X 1. From the proof of Theorem 1 we know that satisfaction of the LMIs (27) implies E C X for the terminal region E = {x R n : x T Px α}. Furthermore, it is closed

8 8 C. Böhm a et al. and connected and contains the origin, i.e. condition (ii) of Lemma 2 is satisfied. In addition, it follows from the proof of Theorem 1 that u = k(x) = Kx satisfies the input constraints for all x E and clearly fulfills the requirement k() =. Thus, to proof condition (iii) in Lemma 2 it remains to show that (22) holds x E, which implies that the set E is an invariant set under the control law u = Kx. In the case of the considered class of Lur e systems and the choice of E(x) and F(x,u), (22) becomes which is equivalent to [ x γ x T (A T P+PA+K T B T P+PBK + Q+K T RK)x+γ T (z)g T Px+x T PGγ(z) <, (28) ] T [ A T P+PA+K T B T P+PBK + Q+K T RK PG G T P ][ x γ ] <. (29) Similar to the proof of Theorem 1 we apply the S -Procedure. As a condition for (29) to be satisfied for all x and γ satisfying the sector condition (1c) we obtain that there has to exist a τ R + such that the matrix inequality [ A T P+PA+K T B T P+PBK + Q+K T RK PG+ 2 τ HT β T ] G T P+ τ 2 βh τi < (3) holds. Multiplying with diag(p 1,I), substituting P 1 and K as defined in the theorem by X, Y and α, and applying the Schur complement one obtains the matrix inequality (26) in the theorem, which concludes the proof of condition (iii) of Lemma 2. Therefore, it is shown that Theorem 2 satisfies all conditions of Lemma 2, and system (1) under the NMPC scheme (17)-(21) converges to the origin as t. Remark 2 If Q = and R = conditions (26) and (27) are equivalent to the conditions of Theorem 1 to calculate a robustly stabilizing linear control law. Convergence of the closed-loop system is already achieved if inequality (1c), i.e. the sector condition of the considered system, is satisfied for all x in the state constraint set X, since E X and (22) has to hold for all x E. This implies that also nonlinearities as for example γ(z) = z 3 can be considered as long as they satisfy (1c) in the set of allowed states X. One main advantage of the approach presented is that the LMIs (26)-(27), respectively, are easy to solve. Thus, the terminal region and the terminal penalty term can usually be calculated without facing computational problems. Often it is desirable to maximize the volume of the terminal region E in order to maximize the feasibility region of the NMPC scheme, see e.g. [1]. The procedure to compute a terminal region and a terminal penalty term for a stabilizing NMPC controller usually is the following: A terminal penalty matrix P is chosen and for the chosen P the constant α is computed such that the terminal region E is maximized, see e.g. [1, 9]. The LMIs (26)-(27) provide the simultaneous calculation of P and α. Thus, both degrees of freedom can be used to maximize the terminal region. Since the volume of E is up to a constant αdet(p 1 ), as in Section 3 the corresponding maximization problem becomes maximize α,p,k det(αp 1 ), (31) subject to the LMIs (26)-(27), where in principle as in Section 3 a line search algorithm for τ can be applied. Remark 3 It is well known that in some cases the maximization of the terminal region leads to terminal penalty terms E(x) which dominate the stage cost F(x,u) in the cost functional (19), thus possibly leading to deteriorating control performance. To avoid this, one can impose an additional constraint α α max to the maximization problem (31). Although not needed for the calculation of the terminal region and the terminal penalty term, the QIH-NMPC scheme in this section requires exact knowledge about the nonlinearities of the considered Lur e system. This is necessary in order to precisely predict the future system behaviour in the optimization problem (2). This motivates the derivation of an NMPC approach based on the ideas of [6, 7, 19], in which at each sampling instant a linear feedback law is recalculated which minimizes an upper bound on an infinite horizon cost functional without needing exact knowledge about the nonlinear system dynamics. The controller is calculated via solving a convex optimization problem subject to similar LMI conditions as in Theorem 2 and therefore can be seen as an extension of the results proposed in this section. In the following section this approach, introduced in [7], will be presented.

9 Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control 9 6 Robust NMPC for Lur e systems The NMPC approach presented in the previous section requires the nonlinearities γ(z) in system (1) to be known exactly. Furthermore, the nonlinearities may render the optimization problem (2) non-convex, thus possibly leading to suboptimal solutions, feasibility problems, and high computational effort. To overcome these problems, instead of repeatedly solving the optimization problem (2) online, the basic idea is to extend the results of the previous section such that at each sampling instant t k a linear feedback matrix K k is calculated and applied. Thus, instead of solving a nonlinear open-loop optimal control problem online a convex optimization problem subject to LMIs has to be solved repeatedly [6, 7, 19]. Consequently, the resulting input applied to the possibly uncertain system (1) becomes u(t) = K k x(t), t [t k,t k+1 ]. (32) Following the ideas of [19] and [6], the feedback matrix K k is calculated such that an upper bound on cost function (19) with infinite horizon (T p = ) is minimized at each sampling instant t k. In the following theorem the index k describes the association of the optimization variables with the time instant t k at which the corresponding optimization problem is solved. Note that the result does not require the sampling instants to be equidistant. Theorem 3 (Robust NMPC) Consider system (1). The NMPC controller given by the repeated solution of the optimization problem minimize α k,x k,y k α k (33a) subject to [ ] 1 x T (t k ) > (33b) x(t k ) X k AX k X k A T BY k Yk T BT Gα k τ 2 X kh T β T X k Q 1 2 Yk T R 1 2 G T α k 2 τ βhx k α k τi Q 1 2 X k α k I > (33c) R 2Y 1 k α k I [ 1 c T i X k + di TY ] k (c T i X k + di TY k) T, (33d) X k i = 1,,r, at the sampling instants t k based on the measured state x(t k ) has the following properties with P k = α k X 1 k and K k = Y k X 1 k : (i) The optimization problem (33) is convex for fixed τ. Furthermore, it is feasible at the sampling instant t k+1 if it is feasible at t k. (ii) The solution to the optimization problem (33) minimizes the upper bound V k = x T (t k )P k x(t k ) on the cost functional (19) with T p = at each sampling instant t k. (iii) If the optimization problem is initially feasible, the control law u(t) = K k x(t), t [t k,t k+1 ), (34) robustly asymptotically stabilizes the origin of the uncertain system (1), and the state and input constraints (18) are satisfied for all times t. Proof The proof is divided into three parts establishing the properties (i)-(iii). Part (i): If τ is fixed, the inequalities (33b)-(33d) are LMIs and therefore convexity of the optimization problem (33) follows trivially. Since only (33b) depends on x(t k ), clearly the solution to the optimization problem (33) at the sampling instant t k also satisfies the LMIs (33c) and (33d) at the sampling instant t k+1.

10 1 C. Böhm a et al. From the proof of Theorem 2 we know that satisfaction of (33c) implies satisfaction of (28). Thus, with V k (x) = x T P k x >, Q >, and R > we know that V k (x) = x T (A T P k + P k A+K T k BT P k + P k BK k )x + x T P k Gγ(z)+γ T (z)g T P k x <. (35) It clearly follows that V k (x) is a Lyapunov function and the control law u = K k x, if applied for all times t t k, robustly asymptotically stabilizes the origin of system (1). In particular, from (35) it follows that x T (t k+1 )P k x(t k+1 ) < x T (t k )P k x(t k ). (36) Applying the Schur complement to (33b), substituting X k, and combining the obtained inequality with (36), it follows that x T (t k+1 )P k x(t k+1 ) < x T (t k )P k x(t k ) < α k (37) is satisfied for all t k. Thus, the solution to the optimization problem at the sampling instant t k also satisfies condition (33b) at t k+1. Consequently, by induction it follows that feasibility at t = implies feasibility at all future sampling instants. Part (ii): The integration of (35) from s = t k to infinity with u = K k x leads to V k = x T (t k )P k x(t k ) > x T (s)qx(s)+u T (s)ru(s)ds. (38) t k Hence, V k is an upper bound on cost function (19) with infinite horizon. From Part (i) we know that V k = x T (t k )P k x(t k ) < α k. (39) Thus, minimizing α k implies the minimization of the upper bound V k, see [19] and [6] for details. Part (iii): In order to show robust, asymptotic stability we consider the candidate jump Lyapunov function V k (x) = x T P k x, where P k is recalculated at each sampling instant t k. From (35) it follows that the application of the control law (34) leads to V k < t [t k,t k+1 ). Now we show that at the sampling instants, which are points of possible discontinuities in V k (x), the inequality V k+1 ( x(tk+1 ) ) V k ( x(tk+1 ) ) (4) is satisfied, and thus Lyapunov stability of system (1) is deduced. We know from the proof of Part (i) that the solution to the optimization problem at t k is a feasible, however in general suboptimal, solution at t k+1. Thus, it follows that x T (t k+1 )P k+1 x(t k+1 ) x T (t k+1 )P k x(t k+1 ) (41) is satisfied at each sampling instant t k+1, which is equivalent to (4). To conclude the proof, it remains to establish the constraint satisfaction, which as shown in the proofs of Theorem 1 and 2 directly follows from (33d). The result of Theorem 3 can be interpreted such that at each sampling instant t k an ellipsoid E k is calculated which is invariant under the control law (34). In the time interval t [t k t k+1 ) the controller steers the states of system (1) further into the interior of E k. Therefore, at the following sampling instant a smaller ellipsoid E k+1 can be calculated involving a typically more aggressive gain matrix K k+1. Thus, the recalculation of the feedback matrix reduces conservativeness and leads to improved control performance. Although in many cases more attractive from a computational point of view when compared to the QIH-NMPC approach presented in Section 5, still at each sampling instant a convex optimization problem subject to LMIs has to be solved online. In the following section we extend the results of Theorem 3 to obtain a novel offline predictive control strategy. In order to reduce the online computational burden, a number of ellipsoids E i is calculated offline, and depending on the ellipsoid in which the current state is lying in, the associated control law is chosen online and applied to the system.

11 Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control 11 K 1 K 2 K 3... Fig. 2 Graphical illustration of the key idea of the robust offline NMPC scheme for uncertain sector bounded Lur e systems. 7 Robust offline NMPC Both NMPC schemes presented in Sections 5 and 6 require the control law to be calculated online at each sampling instant. This may lead to high online computational demand which is often a problem at least for fast systems. In [36] and [37] an offline MPC approach with significantly reduced online computational burden for discrete-time linear systems with polytopic uncertainties has been presented. It is based on the offline calculation of a set of nested, invariant ellipsoids with associated control laws. Online a bisection algorithm is executed to determine the smallest ellipsoid in which the current state is lying in, and the associated controller is applied to the plant. This approach has been modified and improved in [2] and [11], and extended to discrete-time nonlinear polynomial systems in [14]. Here, we apply the basic idea of [11, 36, 37], which is the offline calculation of nested, invariant ellipsoids, to the continuous-time case, in particular to the class of Lur e systems. Therefore, we extend the results obtained in Theorem 3 to calculate offline linear state feedback matrices associated to a set of ellipsoids in order to reduce the online computational burden. This allows the application of this method to fast systems. The following algorithm, which is performed offline, delivers a set of N nested, invariant ellipsoids and feedback matrices that are stored in a look-up table for the online implementation. Algorithm 1 (Offline computation) Consider the uncertain system (1) subject to constraints (6). Generate offline sequences P j, K j and α j, j =,,N 1 as follows: 1. Calculate P, K and α by solving the optimization problem (31) in order to obtain a maximal volume ellipsoid E and therefore a maximal feasible region. Store P, K and α in a look-up table. Set j = Choose a suitable state x j in the interior of E j Compute P j, K j and α j by solving the optimization problem (33) with additional constraint E j E j 1, which translates into Store P j, K j and α j. Set j = j + 1. Go to step 2. X j X j 1. (42) Algorithm 1 is executed offline and yields a set of ellipsoids with associated control law, illustrated in Figure 2, by solving a corresponding number of convex optimization problems. The following online algorithm only requires the determination of the ellipsoid in which the current system state is lying. Then the associated control law is applied to the system until the next ellipsoid is reached. Algorithm 2 (Online computation) Consider the look-up table generated by Algorithm 1 and the uncertain system (1) with initial condition x E subject to constraints (6). 1. Perform a bisection algorithm over P j and α j in the look-up table to obtain j which satisfies x T P j x α j and x T P j +1x > α j +1. (43) Set j = j. If j = N 1 go to step 4, else go to step If x T (t)p j+1 x(t) = α j+1, set j = j + 1. If j = N 1 go to step 4, else go to step Apply the control input to system (1). Go to step 2. u(t) = K j x(t) (44)

12 12 C. Böhm a et al. 4. Apply the control input to system (1) for all times. u(t) = K N 1 x(t) (45) Remark 4 Since we consider continuous-time systems Algorithm 2 is formulated in a continuous-time fashion, i.e. it is possible to detect the state x(t) entering the next ellipsoid (step 2) exactly. Therefore, it is not necessary to perform a bisection algorithm at each time t. However, in principle the steps 2 and 3 would have to be executed infinitely many times. In practical applications this is clearly not possible. There, measurements are only available at pre-defined sampling instants. Thus, one has to execute the relevant computational steps always when new measurements are available. Assuming the time between the sampling instants is short enough, step 2 has to be modified only slightly in order to detect the state entering the next ellipsoid. If this cannot be done reliably, one has to remove step 2 and instead execute the bisection algorithm of step 1 at all sampling instants as in the discrete-time case proposed in [37]. In Algorithm 2, which is executed online, no optimization has to be carried out. This reduces the computational complexity of the offline approach significantly when compared to the robust NMPC scheme proposed in the last section. Clearly, as shown in the following the offline approach also leads to closed-loop stability. Theorem 4 (Offline NMPC) The offline NMPC scheme given by the Algorithms 1 and 2 robustly asymptotically stabilizes the uncertain Lur e system (1) while satisfying the state and input constraints (6). Proof From steps 2 and 3 of Algorithm 2 we know that the input u = K j x is applied to system (1) if the state lies in the ellipsoid E j = {x R n : x T P j x α j }. It directly follows from the proof of Theorem 3 that, if the control law was applied to the system for all times, V j = x T P j x is a Lyapunov function and the controller satisfies state and input constraints (6). Consequently, at some time the state enters the ellipsoid E j+1, which lies completely in the ellipsoid E j. According to Algorithm 2 then the control law is switched to u = K j+1 x. By induction it follows that the chosen control strategy steers the states into the smallest ellipsoid E N 1 and the controller u = K N 1 x, which is applied to the plant for all times as soon as x E N 1, finally robustly asymptotically stabilizes the origin of the uncertain system (1). The update of the control law when the state enters the next ellipsoid typically leads to the application of a more aggressive feedback matrix. Due to the state lying closer to the origin, the gain of the feedback matrix can be increased without violating the input constraints. This results in better control performance and faster convergence to the origin. In this section we presented a novel robust offline NMPC strategy for uncertain Lur e systems. The approach does not require computationally expensive online optimizations. Thus, the online computational burden compared to the approach proposed in Section 6 is greatly reduced which makes the controller attractive from an implementational point of view. To illustrate the effectiveness of the results proposed in this paper, in the next section we apply the four presented control approaches in simulation to a flexible link robotic arm. 8 Application example: Flexible link robotic arm We consider the dynamics of a flexible link robotic arm (e.g. [5, 16]), see Figure 3, which are given by the matrices A = 1, B =, (46) and the nonlinear function G T = [ 3.33 ], H = [ 1 ], γ(z) = sinz+z. (47) The model contains the two angles x 1 and x 3, each on one side of the spring, and the corresponding angular

13 Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control 13 x 3 x 4 mg M x 2 K x 1 Fig. 3 Graphical illustration of the flexible link robotic arm. velocities x 2 and x 4, see Figure 3. The input u applied to the system is a torque induced by an electric motor which is used to control the states of the robotic arm. To fulfill the sector condition (1c) the inequality ( βz sin(z) z ) T ( sin(z)+z ) (48) has to be satisfied. This is the case for all β 2. In order to reduce conservativeness for the calculations we choose the smallest possible value β = 2. The constraint set C is defined by the input constraints 1.5 u 1.5 and the state constraints π 2 x 1 π 2 and π 2 x 3 π 2. The control task is to steer the example system from the initial condition x = [1.1 ] T to the origin. Using the model of the flexible link robotic arm we compare the results obtained in this paper, namely the linear robust feedback controller of Section 3, the QIH-NMPC controller derived in Section 5 (see also [5]), and both the online and offline robust NMPC approaches of Section 6 (see also [7]) and Section 7. The simulation time is T sim = 2.5s in all cases. For the calculations of the NMPC approaches we have chosen the weighting matrices Q = diag(1,.1,1,.1) and R = 1. As solver for the LMI problems we apply sdpt3 [35] with the Matlab interface Yalmip [23]. First, we consider the quasi-infinite horizon NMPC scheme of Section 5. In order to obtain a terminal region with maximal volume, we solve the optimization problem (31) in combination with a line search algorithm for τ. Additionally we impose the constraint α 5 to limit the influence of the terminal penalty term on the overall cost. The solution for τ = 2.68 and α = 5 yields the terminal region E = {x R n : x T Px α} and the terminal penalty term E(x) = x T Px defined by the matrix P = (49) With α and P we apply the QIH-NMPC scheme (17)-(21) to the robotic arm. For the simulation the prediction horizon T p = 2s and a constant sampling rate of.25s is chosen. To solve the optimal control problem we apply the NMPC tool OptCon [26, 32] which delivers a piecewise constant input trajectory. The linear robust controller is calculated such that the domain of attraction is maximal, i.e. we solve the optimization problem max α,p,k det(αp 1 ) (5) subject to the LMIs (9)-(1) with τ = 2.68 according to Theorem 1. Furthermore, for the robust NMPC approach we also set τ = The resulting convex optimization problem (33) of Theorem 3 is solved repeatedly with the sampling rate.125s. For the offline approach N = 15 ellipsoids and feedback matrices are computed according to Algorithm 1. Thus, the outer ellipsoid is calculated such that its volume has been maximized which leads to the largest domain of attraction. Then in step 2 some state x 1 on the boundary of the obtained ellipsoid is chosen to N j+1 perform step 3. In the following the states are updated in step 2 according to x j = N x 1, i.e. the states x j move closer to the origin as j increases. Based on the choice of x j in step 3 the remaining ellipsoids are calculated successively. Note that the choice of x j is a design parameter which only has to satisfy x j E j 1. To compare the different control strategies we first evaluate the overall cost

14 14 C. Böhm a et al. J ( x( ),u( ) ) Tsim = x T (s)qx(s)+u T (s)ru(s) ds (51) along the simulated trajectories. Figure 4 shows the cost obtained by the four controllers. As to be expected, the QIH-NMPC scheme leads to the lowest cost. This control method does not posses robustness properties J t Fig. 4 Comparison of cost functional evaluation for different control strategies. Black solid line: Off-line NMPC. Gray solid line: QIH-NMPC. Black dashed line: Robust NMPC. Gray dashed line: Linear controller as in Section 3. and is therefore not conservative, however, it can only be applied straight forwardly if the nonlinearities are known exactly. The performance of the robust NMPC approach is relatively close to the QIH-NMPC and significantly better than the offline NMPC scheme. Although both controllers have been calculated based on similar LMI conditions, this is also not surprising since the online scheme allows re-optimization along the specific trajectory in contrast to the offline one. The comparison with the static linear controller that is computed based on the method proposed in Section 3 reveals clear advantages for the NMPC methods, since the linear controller does not consider the cost functional in the design procedure at all. However, as also confirmed by the behaviour of the exemplary state x 1 and the control input u in Figure 5 the linear controller is not aggressive enough to stabilize the robotic arm within the simulation time. This is due to the conservativeness introduced by the maximization of the domain of attraction. Figure 5 further shows that the offline NMPC approach stabilizes the robotic arm much faster and the control input is more aggressive. This is due to the fact that entering a new ellipsoid results in a more aggressive feedback gain. Figure 5 illustrates that by paying the price of performing some search algorithm in a look-up table online, controller performance can be significantly improved by the offline NMPC scheme when compared to static control approaches. Figure 6 compares the robust NMPC controller with the offline scheme. Although the cost for the offline controller is significantly higher, the obtained state and input trajectories show similar behavior. The offline state trajectories are more aggressive, in contrast the control input is smoother. Summarizing, one can say that if enough computational power is available, the online NMPC approach is preferable. However, if this is not the case the offline algorithm seems to be at least for the robotic arm a serious alternative. Finally, Figure 7 compares the online robust NMPC controller with the QIH-NMPC scheme. As to be expected from the cost function evaluation the trajectories are very similar. The main difference between the two control strategies is that the QIH-NMPC is allowed to exploit the whole available input energy. This results in a more aggressive control input which finally leads to faster convergence and lower cost in the nominal case. The sampling rate of the LMI based robust approach is chosen significantly larger than the one of the QIH-NMPC scheme. A smaller sampling rate does not further improve the performance, in contrast to the QIH-NMPC controller where a larger sampling rate leads to worse performance. Thus, the robust NMPC controller can be applied with larger sampling rates which makes it more attractive from an implementational point of view. We did not analyze the computational demand of the two online approaches which is necessary to solve the associated optimization problems. However, it seems that the LMI based robust approach is faster to solve and therefore seems to be again more attractive to be applied on a real robotic arm.

15 Robust control of constrained sector bounded Lur e systems with applications to nonlinear model predictive control 15 x u t t Fig. 5 Time trajectories of state x 1 and input u for initial condition x = [1.1 ].Black lines: Robust offline NMPC. Gray lines: Linear robust controller. Dashed lines: Input constraints x1.4 u t t Fig. 6 Time trajectories of state x 1 and input u for initial condition x = [1.1 ]. Black lines: Robust offline NMPC. Gray lines: Robust NMPC. Dashed lines: Input constraints x u t t Fig. 7 Time trajectories of state x 1 and input u for initial condition x = [1.1 ]. Black lines: Quasi-infinite horizon NMPC. Gray lines: Robust NMPC. Dashed lines: Input constraints Summary In this paper we presented four control design methods for the class of constrained sector bounded Lur e systems. First, an approach has been derived which allows to calculate a static linear feedback matrix via the solution of a set of LMIs. While satisfying state and input constraints the associated control law stabilizes the Lur e system also when its nonlinearities are uncertain within the considered sector. Building upon this result we presented an LMI based approach to compute the terminal region and the terminal penalty term for a QIH-NMPC scheme. By extending the obtained LMI conditions we further proposed an online robust NMPC scheme where the control law is calculated by repeatedly solving a convex optimization problem subject to

16 16 C. Böhm a et al. LMI conditions. Finally, we derived an offline NMPC scheme which is based on the offline calculation of a number of ellipsoids. The control law is then obtained by online determining the ellipsoid and feedback matrix, respectively, associated to the current state measurement. In the last part of the paper each of the four controllers has been applied to a flexible link robotic arm and some of their specific properties have been discussed and compared. 1 Acknowledgments The authors Christoph Böhm and Frank Allgöwer gratefully acknowledge support by the German Research Foundation (DFG) through the Cluster of Excellence in Simulation Technology (EXC 31) at the University of Stuttgart and the grant AL 316/5 1. References 1. F. Allgöwer, T.A. Badgwell, J.B. Rawlings, and S.J Wright. Nonlinear model predictive control. In Perspectives in Control. Plenary Lectures and Mini-Courses at the 5th European Control Conference ECC 99, pages Springer-Verlag, London, D. Angeli, A. Casavola, G. Franze, and E. Mosca. An ellipsoidal off-line MPC scheme for uncertain polytopic discrete-time systems. Automatica, 44(12): , D. Banjerdpongchai and J.P. How. Parametric robust H 2 control design using iterative linear matrix inequalities synthesis. Journal of Guidance, Control, and Dynamics, 23(1): , D. Banjerdpongchai and H. Kimura. Robust analysis of discrete-time Lur e systems with slope restrictions using convex optimization. Asian Journal of Control, 4(2): , C. Böhm, T. Raff, R. Findeisen, and F. Allgöwer. Calculating the terminal region for Lure systems via LMIs. In Proc. American Control Conference, pages , Seattle, C. Böhm, T. Raff, M. Reble, and F. Allgöwer. LMI-based model predictive control for linear discrete-time periodic systems. In L. Magni, D.M. Raimondo, and F. Allgöwer, editors, Nonlinear Model Predictive Control - Towards New Challenging Applications, Lecture Notes in Control and Information Sciences, pages Springer-Verlag, C. Böhm, S. Yu, R. Findeisen, and F. Allgöwer. Predictive control for Lure systems subject to constraints using LMIs. In Proc. European Control Conference, Budapest, Hungary, S. Boyd, L. El Ghaoui, E. Feron, and V. Balakishnan. Linear Matrix Inequalities in System and Control Theory. SIAM, Philadelphia, H. Chen and F. Allgöwer. A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica, 34(1): , W. Chen and D.J. Ballance. On attraction domain of model predictive control of nonlinear systems with input/state constraints. Technical report, CSC-999, University of Glasgow. 11. B.C. Ding, Y.G. Xi, M.T. Cychowski, and T. O Mahony. Improving off-line approach to robust MPC based-on nominal performance cost. Automatica, 43(1): , R. Findeisen. Nonlinear Model Predictive Control: A Sampled-Data Feedback Perspective. PhD thesis, University of Stuttgart, F.A. Fontes. A general framework to design stabilizing nonlinear model predictive controllers. System and Control Letters, 42(2): , G. Franze, A. Casavola, D. Famularo, and E. Garone. An off-line MPC strategy for nonlinear systems based on SOS programming. In L. Magni, D.M. Raimondo, and F. Allgöwer, editors, Nonlinear Model Predictive Control - Towards New Challenging Applications, Lecture Notes in Control and Information Sciences, pages Springer-Verlag, Y. He and M. Wu. Absolute stability for multiple delay general Lur e control systems with multiple nonlinearities. Journal of Computational and Applied Mathematics, 159(2): , A. Howell and J.A. Hedrick. Nonlinear observer design via convex optimization. In Proceedings of the American Control Conference, pages , Anchorage, Alaska, R.E. Kalman. Lyapunov functions for the problem of Lur e in automatic control. Proceedings of the National Academy of Sciences, 49(2):21 25, H. K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ, 3rd edition, 22.