This is a self-archive to the following paper.

This is a self-archive to the following paper. S. Hanba, Robust Nonlinear Model Predictive Control With Variable Bloc Length, IEEE Transactions on Automatic Control, Vol. 54, No. 7, pp. 68 622, 2009 doi: 0.09/TAC.2009.207965 To use this material, see the Copyright Notice below. Copyright Notice 2009 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective wors for resale or redistribution to servers or lists, or to reuse any copyrighted component of this wor in other wors must be obtained from the IEEE. The published version may be found in http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=552965&isnumber=559545

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED Robust nonlinear model predictive control with variable bloc length Shigeru Hanba, Member, IEEE Abstract Robust model predictive controllers for discretetime nonlinear systems are proposed in this paper. The algorithms first search for an open-loop controller with some bloc length, and then try to improve it in a closed-loop fashion by solving minimax problems on-line. It is proved that the controllers are capable of maing a subsequence of the state converge into a target set in the presence of bounded disturbances. I. INTRODUCTION AND PROBLEM STATEMENT The importance of constructing a robust model predictive controller for a nonlinear system is widely recognized [7], and extensive researches have been carried out [2] [4], [6], [8] [0]. However, there is a common drawbac in existing methods that many restrictive conditions on the stage cost, terminal constraints, and feasibility are required. This impairs the benefit of model predictive control (MPC) of being readily applicable to a variety of systems. The objective of this paper is to remedy this situation and construct robust model predictive controllers under relatively mild conditions. Consider a discrete-time nonlinear system of the form x + = f(x, u, w ), x R n, u R n u, w R n w, () where x is the state, u is the control input, and w is the disturbance at the time instant. We assume that f is C. The design objective is to drive the state of the system into a neighborhood of a target point x target. We intend to employ dual-mode MPC, and we assume that once the state is inside the neighborhood, the model predictive controller is turned off and some other controller (not discussed in this paper) is used. Typical applications the author has in mind are i) stabilization and ii) reaching control to a (controlled) attractors. For the former, x target is generally an equilibrium, whereas for the latter, x target is in a neighborhood of the attractor and not always an equilibrium; however, for the simplicity of notation, we let x target = 0. In what follows, u [i,j] denotes the sequence (u i,..., u j ). We identify u [i,i] with u i, and u [i,j] with the empty sequence if j < i. The sequence obtained by joining (u i,..., u j ) and (u j+,..., u l ) is denoted by u [i,j] : u [j+,l]. Similar notation is used for w i. The solution of () at time N with inputs u [0,N ] and disturbances w [0,N ] initialized at x 0 is denoted by φ(n, u [0,N ], w [0,N ], x 0 ); if the disturbances are identically zero, the solution is denoted by φ(n, u [0,N ], 0, x 0 ). For a vector, denotes any norm; for a matrix, it denotes S. Hanba is with the Department of Electrical and Electronics Engineering, University of the Ryuyus,, Senbaru, Nishihara, Naagamu-gun, Oinawa, 903-023, Japan (email: sh@gargoyle.eee.u-ryuyu.ac.jp) the induced norm. The symbols B(x, σ x ) and B(x, σ x ) denote the open and closed balls centered at x with radius σ x. For a sequence u [i,j], B (u [i,j], σ u ) denotes the set {v [i,j] : {i,..., j}, v B(u, σ u )}. The symbol 0 either denotes the scalar zero, or a zero vector of any dimension. We assume that the disturbances are bounded, that is, for some σ w > 0 and for all, w σ w. II. VARIABLE BLOCK-LENGTH CONTROLLERS We assume a particular ind of controllability to the target point (the origin) as follows. Assumption For any x, there exist a N N and a sequence u [0,N ] that satisfy: i) φ(n, u [0,N ], 0, x) = 0, and ii) ran ( φ/ u [0,N ] ) = n at (u [0,N ], 0, x), where φ/ u [0,N ] denotes (( φ/ u 0 ) T,..., ( φ/ u N ) T ) T. The first condition of Assumption is the reachability and the second condition implies that the reachable set contains a non-empty open set. Assumption does not say anything about the bound of N or the norm of u [0,N ] ; however, the following lemma asserts that they are in fact bounded if the initial condition is on a compact set. Lemma 2 For any compact set Ω, there exist a N N and a continuous function ρ u on Ω that have the following properties: ( x Ω)( N N)( u [0,N ] B (0, ρ u (x)))(φ(n, u [0,N ], 0, x) = 0). Proof. For a x 0 Ω, choose some N x0 and u 0 [0,N x0 ] that satisfy Assumption. Let p be a vector consisting of elements of u [0,Nx0 ] that satisfy ran ( φ/ p)(n x0, u 0 [0,N x0 ], 0, x 0) = n, and q be the vector consisting of remaining elements. The vectors corresponding to u 0 [0,N x0 ] are denoted by p 0 and q 0. By omitting unnecessary arguments, we rewrite φ(n x0, u [0,Nx0 ], 0, x) = ψ(x, q, p). Choose some ε v > 0 and ε p > 0. Define Ψ(v, p) = (v T, (ψ(x 0 + v, q 0, p)) T ) T, where v R n. Because f is C and the Jacobian of Ψ is nonsingular at (0, p 0 ), for sufficiently small ε v and ε p, Ψ is a homeomorphism on B(0, ε v ) B(p 0, ε p ). Because ψ(x 0, q 0, p 0 ) = 0, for sufficiently small ε v ε v, the set {(v T, x T ) T : v ε v, x = 0} is contained in the image of Ψ. This implies that ( x B(x 0, ε v))( p x

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED 2 B(p 0, ε p ))(ψ(x, q 0, p x ) = 0). Let the sequence corresponding to p x and q 0 be u x [0,N x0 ], κ 0 = max j {0,...,Nx0 } u 0 j, and κ x0 = κ 0 + ε p. Our conclusion at this stage is as follows: ( x B(x 0, ε v))( u x [0,N x0 ] B (0, κ x0 )) (φ(n x0, u x [0,N x0 ], 0, x) = 0). Choose an open set V that contains B(x 0, ε v). By Urysohn s lemma, there is a continuous function ζ x0 : R n [0, κ x0 +ε p ] that is identically κ x0 + ε p on B(x 0, ε v) and identically zero on the complement of V. Consider the set of triples {(N x, ζ x, B(x, ε x )) : x Ω}, where N x, ζ x, and B(x, ε x ) are constructed for each x by the procedure described above. Because {B(x, ε x ) : x Ω} covers Ω and Ω is compact, there is a finite subcollection {B(x j, ε xj ) : x j Ω, j {,..., M}} for some M that also covers Ω. Let N = max j {,...,M} N xj and ρ u (x) = sup j {,...,M} ζ xj (x). They satisfy the desired property. Remar 3 The bound ρ u (x) obtained in Lemma 2 is of theoretical nature. It is difficult to evaluate it effectively, and there is no assurance that it is compatible to the physical limitations of engineering systems. Generally, solving the nonlinear equation φ(n, u [0,N ], 0, x) = 0 numerically requires infinitely many iterations. Thus, we replace it with the problem of finding u [0,N ] (an open-loop control sequence) which maes φ(n, u [0,N ], 0, x) α x for some 0 < α < and N N, whose existence is assured by Lemma 2. Because our objective is to construct a robust controller, we analyze the effect of w [0,N ] to φ(n, u [0,N ], 0, x). Temporally, let us denote U = (u T 0,..., u T N )T, W = (w0 T,..., wn T )T, W = ( N =0 w i 2 ) /2, and F N (U, W, x) = φ(n, u [0,N ], w [0,N ], x). Note that F N is C. Let θ n [0, ] R n, and define ( F Jw N N / W )(U, θ W, x) (U, W, θ, x) =.... (2) ( Fn N / W )(U, θ n W, x) Then, from the intermediate function theorem, for some θ, F N (U, W, x) F N (U, 0, x) = J N w (U, W, θ, x)w. (3) Choose a σ x > 0. Because the set Ω of Lemma 2 is arbitrary, one may assume that B(0, σ x ) Ω. Let σ u = max x B(0,σx ) ρ u(x), σ Jw (N) = max x B(0,σ x ), U B (0,σ u ), W B (0,σ w ), θ Qn [0,] J N w (U, W, θ, x), and σ Jw = max {j=,...,n} σ Jw (N). Then, by (3), for all x B(0, σ x ), F N (U, W, x) F N (U, 0, x) σ Jw Nσ w. (4) By using above results, we construct a piecewise openloop controller, which is similar to the bloc MPC of Sun et al. [4]. It consists of open-loop blocs with length N j. Choose some α (0, ). For x 0 B(0, σ x ), let u 0 [0,N 0 ] be a sequence of inputs that satisfy φ(n 0, u 0 [0,N ], 0, x 0) α x 0 with N 0 N. Let U 0 be the vector corresponding to u 0 [0,N 0 ]. Then, from (4), we obtain x N0 α x 0 + σ Jw Nσ w. (5) Let M j = j i=0 N i (the empty sum implies zero). Inductively, assume that x Mj B(0, σ x ). Then, it is possible to find N j N and U j that satisfy F Nj (U j, 0, x Mj ) α x Mj. Therefore, for all j, x Mj α j x 0 + (( α j )/( α))σ Jw Nσ w, provided that they are well defined. Therefore, in order for all x Mj to be in B(0, σ x ), it is sufficient that x 0 + σ J w Nσw σ x, (6) α and the above discussion shows that, if (6) is satisfied, the subsequence (x Mj ) j N converge into the set B(0, σ Jw Nσ w /( α)). (7) Remar 4 It should be noted that (6) imposes a restriction on not only the initial condition but also the disturbances, that is, σ w ( α)σ x /( Nσ Jw ) (8) If (8) is not satisfied, no initial condition is permissible. For linear systems, this is not restrictive because σ Jw is a constant and σ x may be arbitrarily large. For nonlinear systems, σ Jw depends on σ x ; hence, the restriction (8) may become more conservative by maing σ x larger. However, the convergence is semi-global in some sense, in that, an arbitrarily large initial condition is permitted by maing σ x larger at the expense of smaller permissible disturbances. Our new model predictive controllers are the closed-loop counterparts of aforementioned piecewise open loop controller. The endpoint of each bloc is fixed and the open-loop control sequence inside the bloc is improved by solving minimax problems on-line. We first consider the terminal condition only. Algorithm ) Choose a constant α satisfying 0 < α <, and let j = 0 and M j = 0. 2) The control terminates if x Mj is inside the target set. Otherwise, find N j ( N j N) and u 0 [0,N j ] B (0, σ u ) that give φ(n j, u 0 [0,N j ], 0, x M j ) α x Mj. (9) Let = 0 and proceed to step 3. 3) The control terminates if x Mj+ is inside the target set. If = N j, let M j+ = M j + N j, We define the target set to be a neighborhood of the target point x target specified by the designer. The controller is supposed to switch to the auxiliary control law after the state reaches into the target set according to the dualmode policy, but the auxiliary control law itself is not dealt with in this paper.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED 3 j = j +, and return to step 2. Otherwise, solve min u [,N j ] B (0,σ u) max w [,N j ] B (0,σ w) φ(n j, u [,N j ], w [,N j ], x M j +). Let the solution be u [,N j ] and w[,n j ]. Apply u to () to obtain x Mj ++; let = +, and return to the beginning of this step. In order to analyze Algorithm, we require a simple technical lemma. Lemma 5 For a scalar function h(x, x 2, y, y 2 ), let (x b, x b 2, y b, y b 2) be a solution of min max h(x, x 2, y, y 2 ). x,x 2 y,y 2 For x b 2 and some y c 2, let (x a, y a ) be a solution of min max h(x, x b x y 2, y, y2). c Then, h(x a, x b 2, y a, y c 2) h(x b, x b 2, y b, y b 2). Proof. ( y )( y 2 )(h(x b, x b 2, y, y 2 ) h(x b, x b 2, y, b y2)) b and h(x a, x b 2, y, a y2) c h(x b, x b 2, y, a y2). c Proposition 6 Assume that (6) is satisfied. Then, by using Algorithm, the subsequence (x Mj ) j N converges into the set (7). Proof. From the construction of step 2 of Algorithm and (4), φ(n 0, u 0 [0,N 0 ], w0 [0,N 0 ], x 0) φ(n 0, u 0 [0,N 0 ], w0 [0,N 0 ], x 0) α x 0 + σ Jw Nσw. By Lemma 5, for all, φ(n 0, u + [+,N 0 ], w+ [+,N 0 ], x +) = φ(n 0, u act φ(n 0, u [,N 0 ], w,[n 0 ], x ), : u + [+,N 0 ], wact : w + [+,N 0 ], x ) where u act and w act are the actual control input and disturbance applied to the plant at the time instant (note that u act = u ). Moreover, x N 0 φ(, u N 0 N, 0 wn 0 N, x 0 N 0 ) because w N 0 N 0 is the worst disturbance. Therefore, x M = x N0 satisfies (5). Inductively, assume that x Mj α j x 0 +( j i=0 αi )σ Jw Nσw. Then, x Mj B(0, σ x ), and arguments similar to the above give x Mj+ α j+ x 0 + ( j i=0 αi )σ Jw Nσw, hence x Mj+ B(0, σ x ). Therefore, Algorithm is executable for all j and the subsequence (x Mj ) j N converges into the set B(0, σ Jw Nσw /( α)). If the target set contains the set (7) and is open, Algorithm terminates after finitely many iterations. Remar 7 The difficult step of Algorithm is to find a N j and a u [0,Nj ]. Computationally, this step is performed by minimizing φ(n j, u 0 [0,N j ], 0, x M j ) with increasing N j until φ(n j, u 0 [0,N j ], 0, x M j ) α x Mj is satisfied (true minimum is unnecessary). The solution can be found relatively easily if one chooses α. Next, we construct a variant of Algorithm that taes the stage cost into account. In the j-th bloc with length N j, consider the cost functions J j = N j i=0 l(i, φ(i +, u act [0, ] : u [,i], w act [0, ] : w [,i], x M j ), u i ), (0) for {0,..., N j }, in which u act [0, ] and wact [0, ] denote the sequences of control inputs and disturbances, respectively, that have already been applied to the system at the time instant M j +. In the summation of (0), if i <, u act [0, ] and w[0, ] act are interpreted as u act [0,i] and w[0,i] act, respectively; u i is interpreted as an element of u act [0, ] for i < and an element of u [,...,N j ] for i. It should be noted that, for evaluating (0), w[0,i ] act (i ) are in fact unnecessary, because x Mj +i = φ(i, u act [0,i ], wact [0,i ], x M j ) is nown. Algorithm 2 ) Choose a constant α satisfying 0 < α <, and let j = 0 and M j = 0. 2) The control terminates if x Mj is inside the target set. Otherwise, find a N j N and a u 0 [0,N j ] B (0, σ u ) that satisfy (9). Let = 0, J j = max φ(n j, u 0 [0,N w 0 j ], [0,N j ] B (0,σ w ) w 0 [0,N j ], x M j ), () and proceed to step 3. 3) The control terminates if x Mj + is inside the target set. If = N j, let M j+ = M j + N j, j = j +, and return to step 2. Otherwise, solve min u [,N j ] B (0,σ u ) max w [,N j ] B (0,σ w ) J j subject to max φ(n j, u [,N, j ] w [,N j ] B (0,σ w) w [,N j ], x M j +) J j. (2) Let the solution be (u [,N j ], w [,N j ] ). Apply u to () to obtain x Mj++, let = +, and return to the beginning of this step. Because of (2), Algorithm 2 is executable for all j, and the same convergence property as Proposition 6 is obtained. Let the optimal value of Jj be Jj. Then, by Lemma 5 (with a slight modification), it follows that Jj 0 Jj. Remar 8 Our stage cost has nothing to do with the convergence and may be designed rather freely. Time-varying or even discontinuous stage costs are permitted. A barrier-lie function may be used in order to mildly constrain the state inside a set. Exactly constraining the state in a specified set would require an extended function that taes the value + outside the set, but we do not consider such a function here, because introducing such a function causes a problem on feasibility. If it is desirable, instead of (0), a standard receding horizon type stage cost may be used together with the constraint (2).

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED 4 In this case, the constraint is not terminal, but the analysis of the convergence is still unaffected. Thus far, we have fixed the bloc length N j after determining N j and u [0,Nj ] that satisfy (9). However, a shorter bloc length is sometimes preferable. A variant of Algorithm that reduces the bloc length is as follows. Algorithm 3 ) Choose a constant α satisfying 0 < α <, and let j = 0 and M j = 0. 2) The control terminates if x Mj is inside the target set. Otherwise, find a N j N and a u 0 [0,N j ] B (0, σ u ) that satisfy (9). Let = 0, obtain J j by (), let Jj = J j, and proceed to step 3. 3) The control terminates if x Mj + is inside the target set. If = N j, let M j+ = M j + N j, j = j +, and return to step 2. Otherwise, for m = 0,..., N j, solve min max u [,+m] B (0,σ u ) w [,+m] B (0,σ w ) φ(m +, u [,+m], w [,+m], x M j +). Let the solution be (u,m [,+m], w,m [,+m]), and the corresponding cost be J,m. If J,m J j for some m, let Jj = J,m, N j = + m +, apply u,m to () to obtain x Mj++, let = +, and return to the beginning of this step. For m = N j, J,N J j j is always satisfied. Hence, Algorithm 3 is executable. Its convergence property is the same as that of Algorithm. Remar 9 Our algorithms, which we have been describing as a robust counterparts of the bloc MPC [4] thus far, may also be interpreted as a non-linear analogue of the tube MPC [8], [2], [3] with more freedom on the selection of the horizon. Our decrease condition (9) may be interpreted as a variant of the constraint tightening which is frequently used in existing MPC algorithms; however, our framewor has the important feature that, under Assumption, the feasibility is automatically assured by Lemma 2. III. NUMERICAL EXAMPLES In this section, we apply our algorithms to a slightly modified version of the system with zero-robustness given by Grimm et al [2] and a chained system []. Example Consider the two-dimensional system ( ) ( x ()( u x + = ) + w x u ), (3) where u [0, ] is the control input and w [0, ] is the disturbance. The symbol x (i) denotes the i-th component of x. For this system, we assume x = x() + x(2). In fact, the function f(x) = (x ()( u ), x u ) T is not C under this norm; nevertheless, our algorithms function effectively. We apply all our algorithms to (3). For Algorithm 2, we use the cost function l(i, φ(i+, ), u i ) = φ(i+, ) + u i (the symbol denotes an omission of arguments). Let x 0 = (a, b) T with a > 0, b > 0, and consider the first bloc. Because u 0 = and u = 0 gives x 2 = 0 if w 0 = w = 0, we begin with N 0 = 2 together with the reference controller (u 0 0, u 0 ) = (, 0). Let = 0 in the minor loop of each algorithm. Note that φ(, u 0, w 0, x 0 ) = (a( u 0 ) + w 0, (a + b)u 0 + w 0 ) T and ( ) (a( u0 ) + w φ(2, u [0,], w [0,], x 0 ) = 0 )( u ) + w. (a + bu 0 + 2w 0 )u + w (4) For Algorithm, let and h 2 (u [0,], w [0,] ) = φ(2, u [0,], w [0,], x 0 ) h 2 (u 0, u ) = max w [0,] h 2 (u [0,], w [0,] ). Then, h 2 taes the maximum at w 0 = w = ; therefore, h 2 (u 0, u ) = 3 + a au 0 + u + (a + b)u 0 u. (5) The minimum of (5) is on the border of [0, ] [0, ] because h/ u = + (a + b)u 0 = 0 implies u 0 < 0. By evaluating h on the border, the minimum of 3 is obtained at u 0 = and u = 0. Hence, u 0 [0,] = (, 0) and u 0 0 = is applied to (3). For Algorithm 2, J0 = 3 gives the constraint on the inputs, that is, a au 0 + u + (a + b)u 0 u 0, which has the unique solution u 0 [0,] = (, 0). Therefore, u0 0 = is applied to (3). Algorithm 3 is slightly different. For all u 0, φ(, u 0, w 0, x 0 ) taes the maximum h (u 0 ) = a + bu 0 + 2 at w 0 =, which in turn taes the minimum a + 2 at u 0 = 0. Hence, if a, u 0 0 = 0 is applied to (3) and the first bloc terminates; otherwise, the behavior is the same as Algorithm. Next, let = (this step does not exist for Algorithm with a ). For Algorithm, we must solve min u max w x ()+ x (2)u +2w, whose solution is u = 0 and w =. Thus, u = 0 is applied to (3) and the first bloc terminates. For Algorithm 2, the constraint becomes w 0 + (a + b + w 0 )u. (6) By performing some calculations and omitting constant terms, we obtain the problem min u max w 2w +(+a+b+w 0 )u subject to (6), which has the solution u = 0 and w =. Thus, u = 0 is applied to (3) and the first bloc terminates. For Algorithm 3 with a, the result is the same as that of Algorithm. The analyses of succeeding blocs are the same. For all j, N j = 2, and except for Algorithm 3, a two-length control sequence (u Mj, u Mj +) = (, 0) is obtained. For Algorithm 3, a one-length control sequence (u Mj ) = (0) may obtained when x j (). The results of applying each algorithm to (3) with the initial condition (a, b) = (0, 0) are shown in Fig., in which the horizontal axis is the time and the vertical axis is the norm of the state at each time instant. For all algorithms, an identical randomly generated disturbance sequence whose components are uniformly distributed in the interval [0, ] is applied. From Fig., it can be observed that all algorithms successfully made the state bounded without causing any instability.

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED 5 x 20 5 0 5 Algorithm and Algorithm 2 Algorithm 3 0 0 5 0 5 20 time Fig.. Results of applying each algorithm to the system with zerorobustness. Example 2 Consider the following four-dimensional chained system []: x t () u t () d x t (2) dt x t (3) = u t (2) x t (2)u t (), (7) x t (4) x t (3)u t () where x t = (x t (), x t (2), x t (3), x t (4)) T is the state and u t = (u t (), u t (2)) T is the input at the time instant t. By assuming that u t is constant during the sampling period T and integrating, we obtain the following discrete-time system: x + () x () x + (2) x + (3) = x (2) x (3) x + (4) x (4) + T u () T u (2) T u ()x (2) + T 2 2 u ()u (2) T u ()x (3) + T 2 2 u2 ()x (2) + T 3 6 u2 ()u (2). (8) In the following, we give the results of applying Algorithm 2, a minimax MPC [9], [0], bloc MPC [4], and a conventional MPC [7] to (8) with T =. To see the robustness, randomly generated disturbance vectors were added to the state at each time instant. For all algorithms, the cost functions of the form (0) were used, with l(i, φ(i+, ), u i ) = φ(i+, ) 2 + u i 2, i = 0,..., N j. (9) Note that the bloc MPC and the conventional MPC assumes zero disturbance in the prediction, whereas Algorithm 2 and the minimax MPC assumes the worst-case disturbances. For Algorithm 2, N j was initialized with 4, whereas for other algorithms, N j was fixed at 4. No terminal constraints were used for the minimax MPC, the bloc MPC and the conventional MPC. The solutions to the minimax problems were obtained approximately by solving the maximization and minimization problems alternatively. Constrained problems were converted to unconstrained ones by the penalty function method. To solve the optimization problems, Powell s NEWUOA [] pacage (Scilab version by Guilbert [5]) was used. All parameters of NEWUOA were tuned so that the slowest algorithm is executable within one second (rhobeg = 0; maxfun 200 and rhoend=0 for the maximization; maxfun= 400 and rhoend=0 2 for the minimization; see the documentation of [5] for the meanings of these parameters). For Algorithm 2, different values of α were used for blocs of different lengths: for blocs with length, 2, and 3, α = 0 4, whereas for those with length greater than 3, α = 0. The reason of this possibly unnatural selection is that it is difficult to obtain a good minimizer in limited CPU time for higher dimensional blocs. The experiments were executed on an IBM-compatible PC (Core2 Quad CPU of 2.40GHz and 2GB of memory) with FreeBSD-7.0-STABLE, Scilab-4..2, and Atlas-3.8.0. The CPU time was measured by Scilab s timer() function. The state was initialized with (0, 0, 0, 0) T, and each algorithm was applied over,000 steps of time. The results are shown in Table I, where E[ ] denotes the expectation (excluding the initial condition of the state) and τ denotes the CPU time required in calculating the control input. In (a), the range of the disturbances was [, ] (uniform distribution), whereas in (b), the range was [0, ], and the corresponding minimax problems were solved within this range. In this example, all algorithms successfully stabilized the system. As far as E[ x ] is considered, for (a), the conventional MPC outperformed other algorithms, and Algorithm 2 was next to it, whereas for (b), the minimax MPC outperformed other algorithms, and Algorithm 2 was next to it. Bloc MPC required the least average CPU time (about 0% of the minimax MPC), but the robustness was relatively poor against disturbances with non-zero expectations. Overall, the behavior of Algorithm 2 and the minimax MPC was similar, but the average CPU time of Algorithm 2 was about 50% 60% of that of the minimax MPC. Algorithm 2 and the minimax MPC required larger inputs (about 50% 290%) of other types. The distribution of the bloc lengths of Algorithm 2 is shown in (c). For both (a) and (b), the most frequently used bloc length was 3. Blocs with length greater than five were never used. TABLE I COMPARISON OF SEVERAL MPC ALGORITHMS FOR THE CHAINED SYSTEM. (a) Disturbance with range [, ]. E[τ ] E[ x ] E[ u ] Algorithm 2 0.27234 2.59966.8554 Minimax 0.206945 2.752 2.33358 Bloc 0.0235859 2.74776 0.797282 Conventional 0.0576406.8268 0.837802 (b) Disturbance with range [0, ]. E[τ ] E[ x ] E[ u ] Algorithm 2 0.4047 2.3328.3735 Minimax 0.20764 2.20833.62888 Bloc 0.026787 4.0366.09776 Conventional 0.0689844 2.5057.3862 (c) The distribution of the bloc lengths of Algorithm 2. Bloc length 2 3 4 5 Case (a) 0 49 97 84 55 Case (b) 0 63 53 68 29

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, SUBMITTED 6 The performance degradation the conventional and bloc MPC in Table I (b) is due to the bias of the disturbance, and good performance may be recovered by using a disturbance observer. IV. CONCLUSION In this paper, we have constructed robust bloc model predictive controllers under relatively mild conditions on controllability to the target point. The strategy of these methods is to improve the open-loop controller in a closed-loop fashion by solving minimax problems on-line. If the target point is an equilibrium and the linear approximation of the system is controllable at the point, a dual-mode scheme that combines one of the proposed methods with linear state feedbac around the equilibrium provides a semi-global practical stabilizing controller. To the best of the author s nowledge, this is a rare example of a nonlinear closed-loop stabilizing controller being obtained in a constructive manner merely from the conditions on controllability. ACKNOWLEDGMENTS The author would lie to express his gratitude toward the anonymous referees for their careful reading and helpful and constructive comments. REFERENCES [] A. M. Bloch, Nonholonomic Mechanics and Control, Springer, 2003. [2] G. Grimm, M. J. Messina, S. E. Tuna and A. R. Teel, Examples when nonlinear model predictive control is nonrobust, Automatica, Volume 40, Issue 0, pp. 729 738, 2004. [3] G. Grimm, M. J. Messina, S. E. Tuna and A. R. Teel, Model predictive control: for want of a local control Lyapunov function, all is not lost, IEEE Transactions on Automatic Control, Volume 50, Issue 5, pp. 546 558, 2005. [4] G. Grimm, M. J. Messina, S. E. Tuna and A. R. Teel, Nominally robust model predictive control with state constraints, IEEE Transactions on Automatic Control, Volume 52, Issue 0, pp. 856 870, 2007. [5] M. Guilbert, SCILAB-NEWUOA Interface, Available online at:www.inrialpes.fr, accessed 8 July 2008. [6] L. Magni, D. M. Raimondo and R. Scattolini, Regional input-to-state stability for nonlinear model predictive control, IEEE Transactions on Automatic Control, Volume 5, Issue 9, pp. 548 553, 2006. [7] D. Q. Mayne, J. B. Rawlings, C. V. Rao and P. O. M. Scoaert, Constrained model predictive control: stability and optimality, Automatica, Volume 36, Issue 6, pp. 789 84, 2000. [8] W. Langson, I. Chryssochoos, S. V. Raovi and D. Q. Mayne, Robust model predictive control using tubes, Automatica, Volume 40, Issue, pp. 25 33, 2004. [9] M. Lazar, D. M. de la Peña, W. P. M. H. Heemels and T. Alamo, On input-to-state stability of min-max nonlinear model predictive control, Systems & Control Letters, Volume 57, Issue, pp. 39 48, 2008. [0] D. Limon, T. Alamo, F. Salas and E. F. Camacho, Input to state stability of min-max MPC controllers for nonlinear systems with bounded uncertainties, Automatica, Volume 42, Issue 5, pp. 797 803, 2006. [] M. J. D. Powell, The NEWUOA software for unconstrained optimization without derivatives. 40th Worshop on Large Scale Nonlinear Optimization, Erice, Italy, 2004. [2] S. V. Raovic, E. C. Kerrigan, K. I. Kouramas and D. Q. Mayne, Invariant approximations of the minimal robust positively Invariant set, IEEE Transactions on Automatic Control, Volume 50, Issue 3, pp. 406 40, 2005. [3] S. V. Raovic, E. C. Kerrigan, D. Q. Mayne and J. Lygeros, Reachability analysis of discrete-time systems with disturbances, IEEE Transactions on Automatic Control, Volume 5, Issue 4, pp. 546 56, 2006. [4] J. Sun, I. V. Kolmanovsy, R. Ghaemi and S Chen, A stable bloc model predictive control with variable implementation horizon, Automatica, Volume 43, Issue, pp. 945 953, 2007.