Coursehero >> California >> UC Irvine >> ENG 08

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

American 2005 Control Conference June 8-10, 2005. Portland, OR, USA WeB08.4 The Ef cient Computation of Polyhedral Invariant Sets for Linear Systems with Polytopic Uncertainty B. Pluymers , J.A. Rossiter , J.A.K. Suykens , B. De Moor Katholieke Universiteit Leuven Department of Electrical Engineering, ESAT-SCD-SISTA E-Mail : {bert.pluymers, johan.suykens, bart.demoor}@esat.kuleuven.ac.be Internet : http://www.esat.kuleuven.ac.be/scd/ University of Shef eld Department of Automatic Control and Systems Engineering E-Mail : j.a.rossiter@shef eld.ac.uk Internet : http://www.shef.ac.uk/acse/ Abstract In this paper the concept of maximal admissable set (MAS), introduced by Gilbert et al [3] for linear timeinvariant systems, is extended to linear systems with polytopic uncertainty under linear state feedback. It is shown that by constructing a tree of state predictions using the vertices of the uncertainty polytope and by imposing state and input constraints on these predictions a polyhedral robust invariant set can be constructed. The resulting set is proven to be the maximal admissable set. The number of constraints de ning the invariant set is shown to be nite if the closed loop system is quadratically stable (i.e. has a quadratic Lyapunov function). An algorithm is also proposed that ef ciently computes the polyhedral set without exhaustively exploring the entire prediction tree. This is achieved through the formulation of a more general invariance condition than that proposed in Gilbert et al (1991) and by the removal of redundant constraints in intermediate steps. The ef ciency and correctness of the algorithm is demonstrated by means of a numerical example. is proposed to construct low-complexity robust polyhedral invariant sets for uncertain linear systems driven by a linear feedback controller. A set de ned by component-wise bounds in a similarly transformed state space is considered and invariance is imposed by demanding that the PerronFrobenius norm of the closed loop system matrices is smaller than 1. However, this leads to conservative invariant sets and in some cases no invariant set can be obtained. Another, but also conservative, approach is the construction of ellipsoidal invariant sets. We refer to [5] and [7] for recent results in this direction. This paper proposes an ef cient algorithm that constructs the maximal admissable set [3] for linear systems with polytopic model uncertainty, which are controlled by a linear feedback controller and are subject to linear state and input constraints. A more general invariance condition than that proposed in [3, p. 1010, Theorem 2.2] is proposed, leading to an increased ef ciency of our algorithm compared to [3, p. 1011, Algorithm 3.2] and enabling the extension towards linear systems with polytopic uncertainty. This paper is organised as follows. In Section II the problem is formulated, after which, in Section III, a theoretical approach is taken towards the construction of the polyhedral invariant set. In Section IV a generalized invariance condition is formulated leading to an ef cient algorithm for constructing a solution satisfying this invariance condition. The new algorithm is then applied to a numerical example in Section V demonstrating the ef ccacity of the proposed algorithm. Section VI then gives some conclusions and Section VII concludes by pointing out several areas of future research. II. PROBLEM FORMULATION Consider the linear time-varying (LTV) system xk+1 = (k)xk yk = Cxk (1a) (1b) I. INTRODUCTION The notion of invariant sets arises in many problems concerning analysis of dynamical systems, controller design and the construction of Lyapunov functions. An overview of the concept of set invariance and many references can be found in the overview paper [2]. A systematic way for constructing polyhedral sets for linear systems was initially proposed in [3]. The proposed algorithm constructs an invariant set by iteratively adding additional constraints until invariance is obtained. This paper extends these results towards linear systems with polytopic model uncertainty. A number of contributions have been published in this direction. In [1] describes the construction of controllability sets for linear systems with polytopic model uncertainty and polytopic disturbances. These sets do not take a given controller into account, but rather guarantee that for each state inside the set, some control action exists that steers the system further inside the set with a given convergence rate. In [4] and related works, theoretical results related to invariant sets for uncertain systems with disturbances are discussed, but no general algorithms for the setting considered in this paper have been proposed. In [6] a method 0-7803-9098-9/05/$25.00 2005 AACC 804 with xk Rnx denoting the state of the system at discrete time k, yk Rny denoting the output of the system. The time-varying matrix (k) belongs to a given uncertainty polytope = R nx nx L L term polyhedral invariant sets, we will use it throughout this paper to denote polytopic invariant sets of arbitrary dimension. III. THEORETICAL APPROACH = i=1 i i , i=1 i = 1, i 0 . (2) (3) The following theorem, though not practically executable, directly provides a solution to P1. Theorem 1: The set S de ned as S = {x|ASi x bSi }, where AS0 = Ay C, and for i = 1, . . . , ASi 1 1 . . ASi = , . ASi 1 L is a valid solution to P1. Proof: It follows from the de nition of S0 and the fact that by de nition S S0 , that Cx Y, x S, which shows that S is feasible. In a next step we prove that if x0 S that yk Y for k = 1, . . . , . This can be seen by observing that x0 Sk and by then making a convex combination of the L matrix blocks in ASk and bSk with i (k 1), i = 1, . . . , L as weights, with i (k) de ned as L in (k) = i=1 i (k) i . By recursively making convex combinations of the L matrix blocks of the resulting two matrices A and b with weights i (k 2) down to i (0), we eventually get the set of inequalities AS0 (k 1) (k 2) . . . (1) (0)x0 bS0 . (10) i=0 The output is subject to linear constraints yk Y = {y|Ay y by } , k = 0, . . . , , with 0 Y, which is equivalent with by 0. We will assume that the system is robustly asymptotically stable n Si with Si = (8) b S0 = b y , bSi 1 . = . , . bSi 1 lim (k) ,k=0,...,n 1 x0 =1 max xn 2 = 0. (4) b Si (9) The aim is to nd the set S of initial states x0 for which all corresponding outputs y(0), . . . , y( ) satisfy the output constraints Y. This problem includes the problem of nding the set of allowable initial conditions x0 for which a linear system xk+1 = A(k)xk +B(k)uk with polytopic model uncertainty [A(k) B(k)] , controlled by a linear state feedback controller uk = F xk , satis es linear state and input constraints X = {x|Ax x bx } and U = {u|Au u bu }. This can be seen by replacing (k), C, Ay and by in (1)-(3) with A(k) + B(k)F, Inx nx , [AT (Au F )T ]T and [bT bT ]T . x xu We rst give a formal de nition of the concept of robust positive invariance and then formalize the problem that is solved in this paper. De nition 1 (Robust Positive Invariance): Given a system (1)-(2) satisfying (4) then S Rnx is a robust positive invariant set if x S, x S, . (5a) De nition 2 (Feasibility): An invariant set S for a system (1)-(2) is feasible with respect to constraints (3) if Cx Y, x S. (6) It is clear that if a set is invariant and feasible, all initial states x0 within that set guarantee that all corresponding future outputs will stay within the imposed constraint set Y. The problem tackled in this paper is the following : Problem 1 (P1): Given a system (1)-(2) satisfying (4), state constraints (3), nd a feasible and robust positive invariant set S of polyhedral form S = {x Rnx |AS x bS } . (7) In the following sections we refer to this problem as P1. Remark 1: Strictly speaking, polyhedrons are 3dimensional polytopes and hence both terms cannot be interchanged. However, due to the broad use of the This proves for k = 1, . . . , that xk S0 and hence that yk Y if x0 S Sk . We now prove that x1 S, x0 S. Assume that x0 S and that x1 S. This / would mean that there exists a k 0 for which ASk and bSk contain a row de ning a linear inequality that is violated by x1 , which would in turn mean that for certain values of (i) , i = 1, . . . , k the corresponding output yk+1 would violate the output constraint. This contradicts the fact that was previously proven that all future outputs satisfy the output constraints. We therefore must conclude that also x1 S. This proves robust positive invariance of S, which proves that S is a valid solution to P1. Remark 2: By means of a similar argumentation it is possible to prove that the above set S is the largest possible feasible and robust positive invariant set for the given system and constraints. Indeed, one can see that any state outside S leads to a future output that violates the output constraints for at least one realization of the uncertainty, which then means that any other feasible robust positive invariant set S cannot contain any states outside S and that therefore S S. Remark 3: Although Theorem 1 constructs S using linear inequalities it is not guaranteed that the set S is polyhedral, 805 since an in nite number of constraints is used. Only when a nite number of constraints is suf cient, S will be polyhedral. The following theorem shows when S can be described by a nite number of constraints. Theorem 2: Considering the following de nitions a = AS0 , bmin = min bS0 (i), i (11a) (11b) (11c) can easily be veri ed that when the closed loop system is quadratically stable (i.e. has a quadratic Lyapunov function) an ellipsoidal invariant set E = {x|xT Z 1 x 1} can be 1 found and that the transformation x = Z 2 x enables the use of Theorem 2. This observation essentially indicates that S can be described by a nite number of constraints if the system (1)-(2) is quadratically stable. Remark 6: Theorem 2 also provides a method to show that if max < 1, the resulting set S is non-empty. It can easily be found that, under the same assumptions of Theorem 1, all states x with x bmin /a will satisfy all constraints of S, which proves the existence of S if max < 1. This also shows that c bmin /a will also hold and that therefore the numerator of (12) will always be negative, leading to the observation that n will always be positive. Although Theorem 2 provides a more practical way to calculate S by reducing the number of inequality constraints to a nite number, the method provided by this theorom can still become computationally intractable, even for relatively small values of n, because of the fact that the number of constraints increases exponentially with n. Therefore a more practical algorithm is provided in Section IV. IV. PRACTICAL APPROACH In this section we rst reformulate P1 into a different but equivalent problem P2, for which we then propose an ef cient algorithm. We rst de ne the max = max i , i c = max x x s.t. x S0 , (11d) with bS0 (i) denoting the i-th element of vector bS0 and assuming max < 1, then the sets S (as in Theorem 1) n and S n i=0 Si , with Si also de ned as in Theorem 1 and n de ned as n= are identical. Proof: We prove that S0 Sk , k > n, which then proves the theorem. Therefore we assume that x0 S S0 and then calculate an upper bound to ASk x0 as a function of k : ASk x0 max max i0 ...ik 1 i0 ...ik 1 ln bmin ln a ln c , ln max (12) AS0 ik 1 . . . i0 x0 (13a) AS0 ik 1 . . . i0 x0 (13b) (13c) a k c. max -operator : (16) The largest element of ASk x0 is therefore bounded above by a k c. The smallest element of bSk is the same as max the smallest element of bS0 and is therefore equal to bmin . Hence if k satis es the following condition, it is guaranteed that all inequalities of Sk are satis ed if x0 S0 : k max which is equivalent with ln bmin ln a ln c k . ln max (15) bmin , ac (14) S = {x| x S, } . S can be interpreted as the set of all previous states for which it is guaranteed that the current state lies inside S. This now enables us to formulate a necessary and suf cient condition for positive robust invariance for a set. Lemma 1: A set S is a robust positive invariant set for the system (1) iff S S . (17) Proof: If (17) is satis ed then if x0 S, also x0 S and therefore also x1 S, which proves that (17) is a suf cient condition for robust positive invariance. On the other hand, if there exists a state x (S \ S ) then there / exists such that x S, which proves that (17) is also a necessary condition. Remark 7: Lemma 1 is a generalisation of the invariance condition proven in [3, p. 1010, Theorem 2.2] for the case L = 1, stating that S n is invariant if S n = S n+1 . (18) This can be seen by observing that S n+1 S n Sn+1 = S n S n and by then rewriting (18) as S n S n , which is clearly a special case of (17), which does not impose a speci c structure on S. The inversion of the inequality is necessary since max < 1 and therefore ln max < 0. It is clear that, because k N, (15) is satis ed if k > n, which proves the theorem. Remark 4: Theorem 2 shows that if the closed loop system satis es a certain convergence condition then set S can be constructed with a nite number constraints, of which then guarantees that S is polyhedral. Furthermore, (12) shows 1 that n increases proportional to 1 max for values of max close to 1. Remark 5: For systems with dynamics such that the eigenvalues of the different i lie strictly within the unit circle, but where max 1, an appropriate state transformation can also enable the use of Theorem 2 to calculate n. It 806 Lemma 1 enables us to reformulate problem P1 into the following problem. Problem 2 (P2): Given a system (1)-(2) satisfying (4) and given the constraints (3), nd matrices AS and bS such that the set S = {x Rnx |AS x bS } satis es S S {x Rnx |AS x bS }, Cx Y, x S, (19a) (19b) F [ 0.3 0.1] [ 0.5 0.3] nc 118 44 Alg. 1a T (sec.) 31.6 5.5 nc 30 14 Alg. 1b T (sec.) 53.9 8.8 nc 30 14 Alg. 1c T (sec.) 14.4 5.9 TABLE I NUMBER OF CONSTRAINTS AND CALCULATION TIMES FOR THE INVARIANT SETS DEPICTED IN FIGURES 1 AND 2 FOR THREE DIFFERENT VARIANTS OF ALGORITHM 1 (1A : ALGORITHM 1 WITHOUT 1B : ALGORITHM 1A WITH : ADDITIONAL GARBAGE COLLECTION, with AS [AS 1 ; . . . ; AS L ] and bS [bS ; . . . ; bS ]. In the rest of this paper we refer to this problem as P2. We can now formulate an algorithm for solving P2 that starts with the set S0 and then iteratively adds constraints from S1 , S2 , . . . in order to satisfy (17). Algorithm 1: Given a linear system (1)-(2) satisfying (4) and given the constraints (3). 1) Set the initial values for AS and bS to AS := Ay C bS := by . (20) 2) Initialize the index i := 1. 3) Perform the following steps iteratively while i is not strictly larger than the number of rows in AS : a) Select row i from AS and bS : a = (AS )(i,:) , b = (bS )(i,:) . (21) ADDITIONAL GARBAGE COLLECTION AFTER TERMINATION, 1C ALGORITHM 1B WITH ADDITIONAL GARBAGE COLLECTION AFTER EVERY 10 ITERATIONS). it is clear that this property will also still hold for the rst i 1 rows. Hence, after termination of the algorithm and due to the termination condition in step 3), this property will hold for all the rows of AS and bS , which is identical to satisfaction of (19a) and concludes the proof. Remark 8: Correctness of Algorithm 1 can also be proven in alternative ways, for example by considering the operator S = S S and then showing that the algorithm converges to S = S, but this is left up to the reader. Lemma 3 (Convergence): Under the same conditions as Theorem 2, Algorithm 1 will terminate in a nite number of iterations. Proof: Theorem 2 states that S from Theorem 1 and n Sn i=0 Si are identical and therefore, by virtue of Lemma 1, S also satis es (17). Since step 3b) of Algorithm 1 only adds constraints also found in ASi , bSi , i = 1, . . . , and in the same order as they are found in these matrices for increasing i, Algorithm 1 will never add any constraints from ASi , bSi , i = n+1, . . . , and therefore the maximum number of rows in AS as constructed by Algorithm 1 is bounded by a nite number, namely the number of constraints in ASi , bSi , i = 0, . . . , n. Since i is incremented in each iteration, Algorithm 1 must therefore also reach the termination condition of step 3) in a nite number of iterations, which proves the lemma. Remark 9: After termination of Algorithm 1 it is advised to check whether any of the constraints in AS , bS are redundant, meaning that they can be removed without increasing the size of S. This can occur when constraints added in later iterations modify S in such a way that previously added constraints become irrelevant. A way of checking the redundance of a constraint is by solving an LP similar to (22). This process of garbage collection can also be incorporated in the algorithm itself in order to speed up the solution of (22) in each iteration. This does not invalidate the arguments used in the proofs of Lemma s 2 and 3 since S itself is not modi ed by the removal of the redundant constraints. V. EXAMPLE In this section, a numerical example is presented in order to show the validity of the theory and the effectiveness of the b) Check whether adding any of the constraints a i x b, i = 1, . . . , L to AS , bS would decrease the size of S, by solving the following LP for i = 1, . . . , L : ci = max a i x b x (22a) (22b) s.t. AS x bS For each i = 1, . . . , L, if ci > 0, then add the constraint a i x b to AS , bS as follows : AS := c) Increment i : i := i + 1. (24) AS a i , bS := bS b . (23) We now prove correctness and convergence of Algorithm 1. Lemma 2 (Correctness): If Algorithm 1 terminates in a nite number of iterations then the resulting matrices AS , bS are a valid solution to P2. Proof: From the initialization step 1) and the fact that the algorithm only adds constraints and never removes constraints, it is clear that the resulting set S will satisfy (19b). Satisfaction of (19a) after termination of the algorithm follows directly from the observation that after step 3b) row i of AS and bS (denoted with a and b) satis es the property {x|AS x bS } {x|a i x b, i = 1, . . . , L}. Since constraints are only added to AS , bS and never removed, 807 (5, I) (5, 2 ) (5, 2 ) 2 (5, 3 ) 2 (5, 4 ) 2 (5, 5 ) 2 (5, 5 1 ) 2 (5, 5 2 ) 21 (5, 5 3 ) 21 (5, 5 4 ) 21 (5, 5 5 ) 21 (5, 6 ) 2 (5, 6 1 ) 2 (5, 6 2 ) 21 (5, 6 3 ) 21 (6, 5 1 ) 2 (6, 5 2 ) 21 (6, 5 3 ) 21 (6, 5 4 ) 21 (6, 5 5 ) 21 (6, I) (6, 2 ) (6, 2 ) 2 (6, 3 ) 2 (6, 4 ) 2 (6, 5 ) 2 6 4 2 0 2 4 6 4 2 0 2 4 (6, 6 ) 2 (6, 6 1 ) 2 (6, 6 2 ) 21 (6, 6 3 ) 21 Fig. 1. Invariant polyhedral and ellipsoidal invariant set for the closed loop system formed by (25) and the feedback law u = [ 0.3 0.1]x. Left : Structure of the constraints de ning the invariant set. The notation (i, M ) denotes the constraint aT M x bi , with aT , bi denoting the i-th rows of i i AY and bY respectively. Right : Shape of the invariant set. 50 state trajectories starting from the leftmost vertex of the polyhedral invariant set are depicted in dotted lines. algorithm presented in the previous section. We consider a linear uncertain system representing a double integrator with an uncertainty polytope de ned by the following two vertices : A1 = A2 = 1 0.1 01 1 0.2 01 , , B1 = B2 = 0 1 0 1.5 , . (25a) (25b) seems to be the best overall method in terms of calculation time. A signi cant decrease in calculation time is obtained for the set depicted in Figure 1, while only a small penalty in calculation time is observed for the set depicted in Figure 2. The mentioned calculation times are obtained on a P42GHz PC using Matlab 6.5. VI. CONCLUSION In this paper the construction of polyhedral robust positive invariant sets for linear systems with polytopic model uncertainty subject to linear constraints is explored. A theoretical approach is initially pursued after which a new invariance condition is proposed leading to a new ef cient algorithm for the construction of the invariant set. The resulting set is shown to consist of a nite number of constraints if the system is quadratically stable and to be the maximal admissable set for the system. The resulting sets are shown to be larger than ellipsoidal invariant sets, especially if the invariant set can be represented with a small number of constraints. Additionally the elimination of redundant constraints (garbage collection) in the set description during and after the construction of the invariant set is shown to signi cantly improve the computation speed. Another advantage that is worth mentioning is the fact that polyhedral invariant set can easily deal with non-symmetric constraints, whereas ellipsoidal sets can only deal with suc constraints in a very conservative way. It is important to note that quadratic Lyapunov functions as in [5] can still be used within the polyhedrons constructed in this paper. Due to the garbage collection and due to the fact that the algorithm is based on a more general invariance condition, it is also expected to have a lower computation time for The system is subject to state and input constraints [ 10 10]T xk [10 10]T and 1 uk 1, k = 0, . . . , . Figures 1 and 2 depict polyhedral invariant sets computed with Algorithm 1. Redundant constraints are not depicted. A comparison with the largest ellipsoidal invariant set is also made indicating a signi cantly larger area for the polyhedral invariant set, especially when a high feedback gain is used. To verify the invariance of the polyhedral sets, 50 trajectories are calculated, with the initial state situated at the leftmost vertex of the invariant set and with the system matrices [A(k) B(k)] randomly chosen from [A1 B1 ] and [A2 B2 ] at each time instant. Both gures con rm that the polyhedral sets are indeed positively invariant. The tree structures depicted in the gures indicate that not all possible predictions have to be included in the invariant sets. The maximum tree depths indicate that respectively 11step and 7-step ahead predictions are needed to construct the invariant sets. However, constructing S 11 = 11 Si and i=0 S 7 = 7 Si (cfr. Theorem 1) would take 6(212 1) = i=0 24570 and 6(28 1) = 1530 constraints, out of which only respectively 30 and 14 constraints are considered to be nonredundant according to Algorithm 1, as can be seen in the gures. Table I shows calculation times for three variants of Algorithm 1. Garbage collection during and after the algorithm 808 (5, I) (5, 2 ) (5, 2 ) 2 (5, 3 ) 2 (5, 4 ) 2 (5, 4 1 ) 2 6 5 4 (6, I) (6, 2 ) (6, 2 ) 2 (6, 3 ) 2 (5, 5 ) 2 (6, 3 1 ) 2 3 2 1 0 1 2 4 3 2 1 0 1 2 (5, 3 1 ) 2 (5, 4 1 2 ) 2 Fig. 2. Invariant polyhedral and ellipsoidal invariant set for the closed loop system formed by (25) and the feedback law u = [ 0.5 0.3]x. Input constraints were changed into 0.4 u 1. Left : Structure of the constraints de ning the invariant set. The notation (i, M ) denotes the constraint aT M x bi , with aT , bi denoting the i-th rows of AY and bY respectively. Right : Shape of the invariant set. 50 state trajectories starting from the i i leftmost vertex of the polyhedral invariant set are depicted in dotted lines. systems without uncertainty (L = 1) compared to the algorithm described in [3, p. 1011, Algorithm 3.2]. VII. FUTURE WORK The results presented in this paper can be seen as an enabling technology for several future applications. One possible future research direction is the use of robust invariant polyhedral sets in Modelbased Predictive Control, where invariant sets are generally used as a terminal state constraint in order to ensure stability and avoid infeasibilities in future time steps. Polyhedral invariant sets have the advantage of being larger that ellipsoidal invariant sets, can be non-symmetrical and that they can be imposed on a terminal state by means of linear inequality constraints instead of quadratic constraints. Another possible research direction is the further reduction of the number of constraints at the cost of the volume of the invariant set. Other interesting future research directions are the construction of Lyapunov functions induced by polyhedral robust invariant sets or robust controller synthesis based on polyhedral invariant sets, similar to [5] where ellipsoidal invariant sets are used. Finally, inclusion of robustness with respect to disturbance inputs is also an interesting future research area. VIII. ACKNOWLEDGMENTS Research supported by Research Council KULeuven: GOA-Me sto 666, several PhD/postdoc & fellow grants; Flemish Government: FWO: PhD/postdoc grants, projects, G.0240.99 (multilinear algebra), G.0407.02 (support vector machines), G.0197.02 (power islands), G.0141.03 (Identi cation and cryptography), G.0491.03 (control for intensive care glycemia), G.0120.03 (QIT), research communities (ICCoS, ANMMM); AWI: Bil. Int. Collaboration Hungary/ Poland; IWT: PhD Grants, Soft4s (softsensors), Belgian Federal Government: DWTC (IUAP IV-02 (1996-2001) and IUAP V-22 (2002-2006)), PODO-II (CP/40: TMS and Sustainibility); EU: CAGE; ERNSI; Eureka 2063-IMPACT; Eureka 2419-FliTE; Contract Research/agreements: Data4s, Electrabel, Elia, LMS, IPCOS, VIB. Bert Pluymers is a research assistant with the I.W.T. (Flemish Institute for Scienti c and Technological Research in Industry) at the Katholieke Universiteit Leuven. Dr. Johan Suykens is an associate professor at the Katholieke Universiteit Leuven, Belgium. Dr. Bart De Moor is a full professor at the Katholieke Universiteit Leuven, Belgium. REFERENCES [1] F. Blanchini. Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions. IEEE Transactions on Automatic Control, 39:428 433, 1994. [2] F. Blanchini. Set invariance in control. Automatica, 35:1747 1767, 1999. [3] E. G. Gilbert and Tan. K. T. Linear systems with state and control constraints : The theory and application of maximal output admissable sets. IEEE Transactions on Automatic Control, 36(9):1008 1020, 1991. [4] E. Kerrigan. Robust Constraint Satisfaction: Invariant Sets and Predictive Control. PhD thesis, Cambridge, 2000. [5] M. V. Kothare, V. Balakrishnan, and M. Morari. Robust constrained model predictive control using linear matrix inequalities. Automatica, 32:1361 1379, 1996. [6] Y. I. Lee and B. Kouvaritakis. Robust receding horizon predictive control for systems with uncertain dynamics and input saturation. Automatica, 36:1497 1504, 2000. [7] W.-J. Mao. Robust stabilization of uncertain time-varying discrete systems and comments on an improved approach for constrained robust model predictive control . Automatica, 39:1109 1112, 2003. 809

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: