Min-max Control of Constrained Uncertain Discrete-Time Linear Systems
Alberto Bemporad†, Francesco Borrelli‡, Manfred Morari‡
Abstract— For discrete-time uncertain linear systems with constraints on inputs and states, we develop an approach to determine state feedback controllers based on a min-max control formulation. Robustness is achieved against additive norm-bounded input disturbances and/or polyhedral para-metric uncertainties in the state-space matrices. We show that the finite-horizon robust optimal control law is a con-tinuous piecewise affine function of the state vector and can be calculated by solving a sequence of multiparametric lin-ear programs. When the optimal control law is implemented in a receding horizon scheme, only a piecewise affine func-tion needs to be evaluated on line at each time step. The technique computes the robust optimal feedback controller for a rather general class of systems with modest compu-tational effort without needing to resort to gridding of the state space.
I. Introduction
A control system is robust when stability is preserved and the performance specifications are met for a specified range of model variations and a class of noise signals (un-certainty range). Although a rich theory has been devel-oped for the robust control of linear systems, very little is known about the robust control of linear systems with constraints. This type of problem has been addressed in the context of constrained optimal control, and, in par-ticular, in the context of robust receding horizon control (RHC) and robust model predictive control (MPC), see e.g. [1, 2]. A typical robust RHC/MPC strategy consists of solving a min-max problem to optimize robust perfor-mance (the minimum over the control input of the maxi-mum over the disturbance) while enforcing input and state constraints for all possible disturbances. Min-max robust RHC was originally proposed by Witsenhausen [3]. In the context of robust MPC, the problem was tackled by Campo and Morari [4], and further developed in [5] for SISO FIR plants. Kothare et al. [6] optimize robust per-formance for polytopic/multi-model and linear fractional uncertainty, Scokaert and Mayne [7] for additive distur-bances, and Lee and Yu [8] for linear time-varying and time-invariant state-space models depending on a vector of parametersθ∈Θ, where Θ is either an ellipsoid or a poly-hedron. In all cases, the resulting min-max problems turn out to be computationally demanding, a serious drawback for on-line receding horizon implementation.
In this paper we show how state feedback solutions to min-max robust constrained control problems based on a linear performance index can be computed off-line for sys-tems affected by additive norm-bounded exogenous dis-turbances and/or polyhedral parametric uncertainty. We show that the resulting optimal state feedback control law is piecewise affine so that the on-line computation involves
†Dip. Ingegneria dell’Informazione, Universit`a di Siena, Via Roma
56, 53100 Siena, Italy. Email: [email protected].
‡ Automatic Control Laboratory, ETH - ETL I 26, 8092 Zurich,
Switzerland. Email: borrelli,[email protected].
a simple function evaluation. Earlier results have appeared in [9], and very recently [10] presented various approaches to characterize the solution of the open loop min-max prob-lem with a quadratic objective function. The approach of this paper relies on multiparametric solvers, and follows the ideas proposed earlier in [11–13] for the optimal con-trol of linear systems and hybrid systems without uncer-tainty. More details on multiparametric programming can be found in [14,15] for linear programs, in [16] for nonlinear programs, and in [11, 17, 18] for quadratic programs.
II. Problem Statement
Consider the following discrete-time linear uncertain sys-tem
x(t+ 1) =A(w(t))x(t) +B(w(t))u(t) +Ev(t) (1) subject to the constraints
F x(t) +Gu(t)≤f, (2) where x(t) ∈Rn and u(t)∈ Rnu are the state and input vector, respectively. Vectors v(t) ∈ Rnv and w(t) ∈ Rnw are unknown exogenous disturbances and parametric un-certainties, respectively, and we assume that only bounds onv(t) and w(t) are known, namely thatv(t)∈ V, where
V ⊂ Rnv is a given polytope containing the origin, V =
{v : Lv ≤ }, and that w(t) ∈ W = {w : M w ≤ m}, whereWis a polytope inRnw. We also assume thatA(w),
B(w) are affine functions ofw, A(w) = A0+nw
i=1Aiwi, B(w) = B0+nw
i=1Biwi, a rather general time-domain
description of uncertainty, which includes uncertain FIR models [4]. A typical example is a polytopic uncertainty set given as the convex hull ofnwmatrices (cf. [6]), namely
W ={w: −wi ≤0, nw i=1w i ≤ 1, −nw i=1w i ≤ − 1}, A0= 0,B0= 0.
The following min-max control problem will be referred to as Open-Loop Constrained Robust Optimal Control (OL-CROC) problem: JN∗(x0)u min 0,...,uN−1J(x0, U) (3) subj. to F xk+Guk≤f xk+1=A(wk)xk+B(wk)uk+Evk xN ∈ Xf k= 0, . . . , N−1 ∀vk∈ V, wk ∈ W ∀k= 0, . . . , N−1 (4) J(x0, U) max v0, . . . , vN−1 w0, . . . , wN−1 N−1 k=0 Qxkp+Rukp +P xNp (5) subj. to xk+1=A(wk)xk+B(wk)uk+Evk vk ∈ V wk∈ W, k= 0, . . . , N−1 (6) where xk denotes the state vector at time k, obtained by
the input sequenceU {u0, . . . , uN−1}and the sequences V {v0, . . . , vN−1}, W {w0, . . . , wN−1}; p= 1 or p=
+∞,x∞andx1are the standard∞-norm and 1-norm inRn(i.e.,x∞= maxj=1,...,n|xj|andx1=|x1|+. . .+
|xn|, where xj is the j-th component ofx),Q ∈Rn×n, R
∈ Rnu×nu are non-singular matrices, P ∈Rm×n, and the constraintxN ∈ Xf forces the final statexN to belong to
the polyhedral set
Xf {
x∈Rn: FNx≤fN}. (7)
The choice of Xf is typically dictated by stability and
feasibility requirements when (3)–(6) is implemented in a receding horizon fashion. Receding horizon implementa-tions will be discussed in Section IV. Problem (5)–(6) looks for the worst valueJ(x0, U) of the performance index and the corresponding worst sequences V, W as a function of
x0 andU, while problem (3)–(4) minimizes the worst per-formance subject to the constraint that the input sequence must be feasible for all possible disturbance realizations. In other words, worst-case performance is minimized un-der constraint fulfillment against all possible realizations ofV,W.
In the sequel, we denote by U∗ = {u∗0, . . . , u∗N−1} the
optimal solution to (3)–(6), where u∗j : R
n → Rnu, j = 0, . . . , N −1, and by X0 the set of initial states x0 for which (3)–(6) is feasible.
The min-max formulation (3)–(6) is based on an open-loop prediction, in contrast to the closed-loop prediction schemes of [6–8, 19, 20]. In [19] uk = F xk + ¯uk, where F is a fixed linear feedback law, and ¯uk are new degrees
of freedom optimized on line. In [6] uk = F xk, and F
is optimized on line via linear matrix inequalities. In [20]
uk=F xk+ ¯uk, where ¯uk andF are optimized on line (for
implementation, F is restricted to belong to a finite set of LQR gains). In [7, 8] the optimization is over general feedback laws.
The benefits of closed-loop prediction can be understood by viewing the optimal control problem as a dynamic game between the disturbance and the input. In open-loop pre-diction the whole disturbance sequence plays first, then the input sequence is left with the duty of counteracting the worst disturbance realization. By letting the whole distur-bance sequence play first, the effect of the uncertainty may grow over the prediction horizon and may easily lead to infeasibility of the min problem (3)–(4). On the contrary, in closed-loop prediction schemes the disturbance and the input play one move at a time, which makes the effect of the disturbance more easily mitigable [20].
Without imposing any predefined structure on the closed-loop controller, we define the following Closed-Loop Constrained Robust Optimal Control (CL-CROC)
prob-lem [3, 8, 21, 22]: Jj∗(xj) min uj Jj(xj, uj) (8) subj. to F xj+Guj ≤f A(wj)xj+B(wj)uj+Evj∈ Xj+1 ∀vj∈ V, wj∈ W (9) Jj(xj, uj) max vj∈V, wj∈W Qxjp+Rujp+ Jj∗+1(A(wj)xj+B(wj)uj+Evj) (10) forj= 0, . . . , N−1 and with boundary conditions
JN∗(xN) = P xNp (11)
XN
=Xf, (12)
where Xj denotes the set of statesxfor which (8)–(10) is
feasible,
Xj ={x∈Rn| ∃u, F x+Gu≤f, andA(w)x
+B(w)u+Ev∈ Xj+1,∀v∈ V, w∈ W}. (13)
The reason for including constraints (9) in the minimiza-tion problem and not in the maximizaminimiza-tion problem is that in (10)vj is free to act regardless of the state constraints.
On the other hand, the inputujhas the duty of keeping the
state within the constraints (9) for all possible disturbance realizations.
We will consider different ways of solving OL-CROC and CL-CROC problems in the following sections. First we will briefly review other algorithms that were proposed in the literature.
For models affected by additive norm-bounded distur-bances and parametric uncertainties on the impulse re-sponse coefficients, Campo and Morari [4] show how to solve the OL-CROC problem via linear programming. The idea can be summarized as follows. First, the minimization of the objective function (3) is replaced by the minimiza-tion of an upper-boundµon the objective function subject to the constraint that µ is indeed an upper bound for all sequencesV ={v0, . . . , vN−1} ∈ V × V ×. . .× V (although µis an upper bound, at the optimum it coincides with the optimal value of the original problem). Then, by exploiting the convexity of the objective function (3) with respect to
V, such a continuum of constraints is replaced by a finite number, namely one for each vertex of the setV×V×. . .×V. As a result, for a given value of the initial statex(0), the OL-CROC problem is recast as a linear program (LP).
A solution to the CL-CROC problem was given in [7] us-ing a similar convexity and vertex enumeration argument. The idea there is to augment the number of free inputs by allowing one free sequenceUi for each vertexi of the set
V ×V ×. . .×V, i.e. N·NN
V free control moves, whereNVis
the number of vertices of the setV. By using a causality ar-gument, the number of such free control moves is decreased to (NN
V −1)/(NV−1). Again, using the minimization of
an upper-bound for all the vertices ofV × V ×. . .× V, the problem is recast as a finite dimensional convex optimiza-tion problem, which in the case of ∞-norms or 1-norms,
can be handled via linear programming as in [4] (see [23] for details). By reducing the number of degrees of freedom in the choice of the optimal input moves, other suboptimal CL-CROC strategies have been proposed, e.g., in [6,19,20]. III. State Feedback Solution to CROC Problems In Section II we have reviewed different approaches to compute numerically the optimal input sequence solving the CROC problems for a given value of the initial state
x0. Here we want to find astate feedbacksolution to CROC problems, namely a function u∗k :Rn → Rnu (and an ex-plicit representation of it) mapping the statexk to its
cor-responding optimal inputu∗k,∀k= 0, . . . , N−1.
For a very general parameterization of the uncertainty description, in [8] the authors propose to solve CL-CROC in state feedback form via dynamic programming by dis-cretizing the state-space. Therefore the technique is lim-ited to simple low-dimensional prediction models. In this paper we aim at finding the exact solution to CROC prob-lems via multiparametric programming [11, 14, 15, 24], and in addition, for the CL-CROC problem, by using dynamic programming.
For the problems defined above the task of determining the sequence of optimal control actions can be expressed as a mathematical program with the initial state as a fixed parameter. To determine the optimal state feedback law we consider the initial state as a parameter which can vary over a specified domain. The resulting problem is referred to as a multi-parametric mathematical program. In the following we will first define and analyze various multi-parametric mathematical programs. Then we will show how they can be used to solve the different robust control problems. Finally we will demonstrate the effectiveness of these tools on some numerical examples from the literature. A. Preliminaries on Multiparametric Programming
Consider the multi-parametric program
J∗(x) = minz gz
subj. to Cz≤c+Sx, (14)
where z ∈ Rnz is the optimization vector, x∈ Rn is the vector of parameters, and g ∈ Rnz, C ∈ Rnc×nz, c ∈ Rn
c, S ∈ Rnc×n are constant matrices. We refer to (14) as a (right-hand-side) multi-parametric linear program (mp-LP) [14, 15].
For a given polyhedral set X ⊆ Rn of parameters, solving (14) amounts to determining the set Xf ⊆ X of
parameters for which (14) is feasible, the value function
J∗:Xf→R, and the optimizer function1 z∗:Xf →Rnz.
Theorem 1: Consider the mp-LP (14). Then, the setXf
is a convex polyhedral set, the optimizerz∗:Rn→Rnz is a continuous2 and piecewise affine function3 ofx, and the
1In case of multiple solutions, we definez∗(x) as one of the
opti-mizers [15].
2In case the optimizer is not unique, a continuous optimizer function z∗(x) can always be chosen, see [15, Remark 4] for details.
3We recall that, given a polyhedral set X ⊆ Rn1, a continuous function h : X → Rn2 is piecewise affine (PWA)if there exists a
optimizer functionJ∗:Xf →Ris a convex and continuous
piecewise affine function ofx.
Proof: See [14]. 2
The following lemma deals with the special case of a multi-parametric program where the cost function is a con-vex function ofzandx.
Lemma 1: Let J : Rnz ×Rn → R be a convex piece-wise affine function of (z, x). Then the multi-parametric optimization problem
J∗(x) minz J(z, x)
subj. to Cz≤c+Sx. (15)
is an mp-LP.
Proof: AsJ is a convex piecewise affine function, it fol-lows thatJ(z, x) = maxi=1,...,s{Liz+Hix+Ki}[25]. Then,
it is easy to show that (15) is equivalent to the following mp-LP: minz,εεsubject toCz≤c+Sx,Liz+Hix+Ki≤ε,
i= 1, . . . , s. 2
Lemma 2: Let f : Rnz×Rn×Rnd →R and g: Rnz ×
Rn×Rnd → Rng be functions of (z, x, d) convex in d for each (z, x). Assume that the variable d belongs to the polyhedron D with vertices{d¯i}Ni=1D. Then, the min-max
multi-parametric problem
J∗(x) = minz maxd∈Df(z, x, d)
subj. to g(z, x, d) ≤0∀d∈ D (16)
is equivalent to the multi-parametric optimization problem
J∗(x) = minµ,z µ
subj. to µ≥f(z, x,d¯i), i= 1, . . . , ND g(z, x,d¯i)≤0, i= 1, . . . , ND.
(17) Proof: Easily follows by the fact that the maximum of a convex function over a convex set is attained at an extreme point of the set, cf. also [7]. 2 Corollary 1: If f is also convex and piecewise affine in (z, x), i.e. f(z, x, d) = maxi=1,...,s {Li(d)z+Hi(d)x+ Ki(d)} and g is linear in (z, x) for all d∈ D, g(z, x, d) = Kg(d) +Lg(d)x+Hg(d)z (withKg(·),Lg(·), Hg(·),Li(·), Hi(·),Ki(·),i= 1, . . . , s, convex functions), then the
min-max multi-parametric problem (16) is equivalent to the mp-LP problem J∗(x) = minµ,z µ subj. to µ≥Kj( ¯di) +Lj( ¯di)z+Hj( ¯di)x, ∀i= 1, . . . , ND, ∀j= 1, . . . , s Lg( ¯di)x+Hg( ¯di)z≤ −Kg( ¯di), ∀i= 1, . . . , ND. (18) Remark 1: In caseg(z, x, d) =g1(z, x) +g2(d), the sec-ond constraint in (17) can be replaced by g1(z, x) ≤ −¯g, where ¯g¯g1, . . . ,g¯ng is a vector whosei-th component is ¯ gi= max d∈D g i 2(d), (19)
partition ofXinto convex polyhedraX1,. . . ,XN, andh(x) =Hix+ki
and gi
2(d) denotes thei-th component ofg2(d). Similarly, if f(z, x, d) =f1(z, x) +f2(d), the first constraint in (17) can be replaced byµ≥f1(z, x) + ¯f, where
¯
fi= max
d∈D f i
2(d). (20)
Clearly, this has the advantage of reducing the number of constraints in the multiparametric program (17) from
NDng to ng for the second constraint and from NDs to s for the first constraint. Note that (19)–(20) does not requiref2(·),g2(·),Dto be convex. 2 In the following subsections we propose an approach based on multiparametric linear programming to obtain solutions to CROC problems in state feedback form. B. CL-CROC
Theorem 2: By solving N mp-LPs, the solution of CL-CROC is obtained in state feedback piecewise affine form
u∗k(xk) = Fikxk+gik, if
xk ∈ Xik {x: Tikx≤Sik}, i= 1, . . . , sk
(21) for allxk ∈ Xk, whereXk =∪is=1k Xik is the set of statesxk
for which (8)–(10) is feasible with j=k.
Proof: Consider the first step j =N −1 of dynamic programming applied to the CL-CROC problem (8)–(10)
JN∗−1(xN−1) min uN−1JN−1(xN−1, uN−1) (22a) subj. to F xN−1+GuN−1≤f A(wN−1)xN−1+B(wN−1)uN−1+ EvN−1∈ Xf ∀vN−1∈ V, wN−1∈ W (22b) JN−1(xN−1, uN−1) max vN−1∈V, wN−1∈W QxN−1p+ RuN−1p+P(A(wN−1)xN−1+B(wN−1)uN−1 +EvN−1)p . (22c)
The cost function in the maximization problem (22c) is piecewise affine and convex with respect to the opti-mization vector vN−1, wN−1 and the parameters uN−1, xN−1. Moreover, the constraints in the minimization
problem (22b) are linear in (uN−1, xN−1) for all vectors vN−1, wN−1. Therefore, by Lemma 2 and Corollary 1, JN∗−1(xN−1), uN∗−1(xN−1) andXN−1 are computable via
the mp-LP4: JN∗−1(xN−1) min µ,uN−1µ (23a) subj. toµ≥ QxN−1p+RuN−1p +P(A( ¯wh)xN−1+B( ¯wh)uN−1+Ev¯i)p (23b) F xN−1+GuN−1≤f (23c) A( ¯wh)xN−1+B( ¯wh)uN−1+E¯vi∈ XN(23d) ∀i= 1, . . . , NV, ∀h= 1, . . . , NW.
4In case p = ∞ (23a), (23b) can be rewritten as:
minµ1,µ2,µ3,uN−1 µ1 +µ2+µ3, subject to µ1 ≥ ±PixN, ∀i = 1,2, . . . , m,µ2≥ ±QixN−1, ∀i= 1,2, . . . , n,µ3 ≥ ±RiuN−1, ∀i= 1,2, . . . , nu, wherei denotes the i-th row. The case p = 1 can be treated similarly.
where {v¯i}Ni=1V and {w¯h}Nh=1W are the vertices of the
distur-bance sets V and W, respectively. By Theorem 1, JN∗−1
is a convex and piecewise affine function ofxN−1, the
cor-responding optimizer u∗N−1 is piecewise affine and
contin-uous, and the feasible set XN−1 is a convex polyhedron.
Therefore, the convexity and linearity arguments still hold for j = N −2, . . . ,0 and the procedure can be iterated backwards in timej, proving the theorem. 2 Remark 2: Letna andnb be the number of inequalities
in (23b) and (23d), respectively, for anyiandh. In case of additive disturbances only (w(t)≡0) the total number of constraints in (23b) and (23d) for alliandhcan be reduced from (na+nb)NVNW tona+nbas shown in Remark 1. 2
The following corollary is an immediate consequence of the continuity properties of the mp-LP recalled in Theo-rem 1, and of TheoTheo-rem 2:
Corollary 2: The piecewise affine solution u∗k : Rn →
Rnu to the CL-CROC problem is a continuous function of
xk,∀k= 0, . . . , N−1.
C. OL-CROC
Theorem 3: The solution U∗ : X0 → RN nu to OL-CROC with parametric uncertainties in theBmatrix only (A(w) ≡ A), is a piecewise affine function of x0 ∈ X0, where X0 is the set of initial states for which a solution to (3)–(6) exists. It can be found by solving an mp-LP.
Proof: Since xk = Akx0 + k−1
k=0A i
[B(w)uk−1−i + Evk−1−i] is a linear function of the disturbancesW,V for
any givenU andx0, the cost function in the maximization problem (5) is convex and piecewise affine with respect to the optimization vectorsV, W and the parametersU,x0. The constraints in (4) are linear in U and x0, for any V
andW. Therefore, by Lemma 2 and Corollary 1, problem (3)–(6) can be solved by solving an mp-LP through the enumeration of all the vertices of the setsV × V ×. . .× V
andW × W ×. . .× W. 2
We remark that Theorem 3 covers a rather broad class of uncertainty descriptions, including uncertainty on the coefficients of the impulse and step response [4]. In case of OL-CROC with additive disturbances only (w(t)≡0) the number of constraints in (4) can be reduced as explained in Remark 1.
The following is a corollary of the continuity properties of mp-LP recalled in Theorem 1 and of Theorem 3:
Corollary 3: The piecewise affine solution U∗ : X0 →
RN nuto the OL-CROC problem with additive disturbances and uncertainty in the B matrix only (A(w) ≡ A) is a continuous function ofx0.
IV. Robust Receding Horizon Control A robust receding horizon controller (RHC) for sys-tem (1) which enforces the constraints (2) at each time
t in spite of additive and parametric uncertainties can be obtained immediately by setting
-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 x2 x1
(a) Nominal Optimal Control
-10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 x2 x1 (b) OL-CROC -10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 x2 x1 (c) CL-CROC
Fig. 1. Polyhedral partition of the state-space corresponding to the explicit solution of nominal optimal control, OL-CROC and CL-CROC
where u∗0(x0) :Rn → Rnu is the piecewise affine solution to the OL-CROC or CL-CROC problems developed in the previous sections. In this way we obtain a state feedback strategy defined at all time steps t = 0,1, . . ., from the associated finite time CROC problem.
For stability and feasibility of the closed-loop sys-tem (1), (24) we refer the reader to previously published results on robust RHC, see e.g. [1, 2, 26], although general stability criteria for the case of parametric uncertainties, to the best of our knowledge, are not available.
When the optimal control law is implemented in a mov-ing horizon scheme, the on-line computation consists of a simple function evaluation. However, when the number of constraints involved in the optimization problem increases, the number of regions associated with the piecewise affine control map may increase exponentially. In [27,28] efficient algorithms for the on-line evaluation of the explicit optimal control law were presented, where efficiency is in terms of storage and computational complexity.
V. Examples
In [9] we compared the state feedback solutions to nom-inal RHC [12], open-loop robust RHC, and closed-loop ro-bust RHC for the example considered in [7], using infinity norms instead of quadratic norms in the objective function. For closed-loop robust RHC, the off-line computation time in Matlab 5.3 on a Pentium III 800 was about 1.3 s by using Theorem 2 (mp-LP). Below we consider another example. Example 1: Consider the problem of robustly regu-lating to the origin the system x(t + 1) = [1 10 1]x(t) + [01]u(t) + [1 00 1]v(t). We consider the performance mea-sure P xN∞+
N−1
k=0(Qxk∞+|Ruk|), where N = 4, P = Q = [1 10 1], R = 1.8, and U = {u0, . . . , u3}, sub-ject to the input constraints −3 ≤ uk ≤ 3, k = 0, . . . ,3,
and the state constraints −10 ≤ xk ≤ 10, k = 0, . . . ,4.
The two-dimensional disturbance v is restricted to the set
V ={v: v∞≤1.5}.
We compare the control law (24) for the nominal case, OL-CROC, and CL-CROC. In all cases the closed-loop sys-tem is simulated from the initial statex(0) = [−8,0] with two different disturbances profiles shown in Figure 2.
Nominal case. We ignore the disturbancev(t), and solve the resulting multiparametric linear program by using the approach of [12]. The piecewise affine state feedback con-trol law is computed in 23 s, and the corresponding poly-hedral partition (defined over 12 regions) is depicted in Figure 1(a) (for lack of space, we do not report here the different affine gains for each region). Figures 3(a)-3(b) re-port the corresponding evolutions of the state vector. Note that the second disturbance profile leads to infeasibility at step 3.
OL-CROC. The min-max problem is formulated as in (3)–(6) and solved off-line in 582 s. The resulting poly-hedral partition (defined over 24 regions) is depicted in Figure 1(b). In Figures 3(c)- 3(d) the closed-loop system responses are shown.
CL-CROC. The min-max problem is formulated as in (8)–(10) and solved in 53 s using the approach of Theo-rem 2. The resulting polyhedral partition (defined over 12 regions) is depicted in Figure 1(c). In Figures 3(e)-3(f) the closed-loop system responses can be seen.
As expected, it is apparent the slightly superior perfor-mance of CL-CROC over OL-CROC (shorter settling time for both disturbance profiles). This is due to less conser-vatism of CL-CROC in choosing the control action.
Remark 3: As shown in [23], the approach of [7] to solve CL-CROC, requires the solution of one mp-LP where the number of constraints is proportional to the numberNVN of extreme points of the setV × V ×. . .× V ⊂RN nv of distur-bance sequences, and the number of optimization variables, as observed earlier, is proportional to (NN
V −1)/(NV−1),
whereNV is the number of vertices ofV. LetnJ∗i andnXi be the number of the affine gains of the cost-to-go function
Ji∗and the number of constraints definingXi, respectively.
The dynamic programming approach of Theorem 2 requires
Nmp-LPs where at stepithe number of optimization vari-ables is nu+ 1 and the number of constraints is equal to
a quantity proportional to (nJi∗+nXi). Simulation
experi-ments have shown thatnJi∗ andnXi do not increase expo-nentially during the recursion i=N−1, . . . ,0 (although, in the worst case, they could). For instance in Example 1, we have at step 0 nJ0∗ = 34 and nX0 = 4 while NV = 4
0 2 4 6 8 10 12 14 16 18 20 -1.5 -1 -0.5 0 Time Disturbanc e Sequence Disturbance Sequence
(a) Disturbance profile #1
Disturbanc e Sequence Disturbance Sequence 0 2 4 6 8 10 12 14 16 18 20 -1.5 -1 -0.5 0 Time (b) Disturbance profile #2
Fig. 2. Disturbances profiles
0 2 4 6 8 10 12 14 16 18 20 -10 -5 0 5 Time States
(a) nominal RHC, disturbance profile #1
-15 -10 -5 0 5 States 0 2 4 6 8 10 12 14 16 18 20 Time
(b) nominal RHC, disturbance profile #2
0 2 4 6 8 10 12 14 16 18 20 -10 -5 0 5 Time States
(c) Open-Loop Robust RHC, disturbance profile #1 -15 -10 -5 0 5 StatesStates 0 2 4 6 8 10 12 14 16 18 20 Time
(d) Open-Loop Robust RHC, disturbance profile #2 0 2 4 6 8 10 12 14 16 18 20 -10 -5 0 5 Time States
(e) Closed-Loop Robust RHC, disturbance profile #1 0 2 4 6 8 10 12 14 16 18 20 -15 -10 -5 0 5 Time States
(f) Closed-Loop Robust RHC, disturbance profile #2
Fig. 3. Closed loop simulations of nominal optimal control, open-loop robust RHC and closed-loop robust RHC
and NN
V = 256. As the complexity of an mp-LP depends
mostly (in general combinatorially) on the number of con-straints, one can expect that the approach presented here is numerically more efficient than the approach of [7, 23]. On the other hand, it is also true that the latter approach could benefit from the elimination of redundant inequal-ities before solving the mp-LP (how many inequalinequal-ities is quite difficult to quantify a priori).
2 We remark that the off-line computational time of CL-CROC is about ten times smaller than the one of OL-CROC, where the vertex enumeration would lead to a
prob-lem with 12288 constraints, reduced to 52 by applying Re-mark 1, and further reduced to 38 after removing redun-dant inequalities in the extended space of variables and parameters. We finally remark that by enlarging the dis-turbancev to the set ˜V ={v: v∞≤2}the OL-RRHC problem becomes infeasible for all the initial states, while the CL-RRHC problem is still feasible for a certain set of
initial states. 2
Example 2: We consider here the problem of robustly regulating to the origin the active suspension system [29]
x(t + 1) = 0.809 0.009 0 0 −36.93 0.80 0 0 0.191 −0.009 1 0.01 0 0 0 1 x(t) + 0.0005 0.0935 −0.005 −0.0100 u(t) +
−0.009
0.191 −0.0006
0
v(t) where the input disturbancev(t) represents the vertical ground velocity of the road profile andu(t) the vertical acceleration. We solved the CL-CROC (8)–(10) with N = 4, P =Q= diag{5000,0.1,400,0.1},Xf =R4,
and R = 1.8, with input constraints −5 ≤ u ≤ 5, and the state constraints
−0.02 −∞ −0.05 −∞ ≤ x ≤ 0.02 +∞ 0.05 +∞ . The dis-turbance v is restricted to the set −0.4 ≤ v ≤ 0.4. The problem was solved in less then 5 minutes for the subset
X = x∈R4| −0.02 −1 −0.05 −0.5 ≤x≤ 0.02 1 0.50 0.5
of states, and the resulting piecewise-affine robust optimal control law is
de-fined over 390 polyhedral regions. 2
VI. Conclusions
This paper has shown how to find state feedback solu-tions to constrained robust optimal control problems based on min-max optimization, for both open-loop and closed-loop formulations. The resulting robust optimal control law is piecewise affine. Such a characterization is especially useful in those applications of robust receding horizon con-trol where on-line min-max constrained optimization may be computationally prohibitive. In fact, our technique al-lows the design of robust optimal feedback controllers with modest computational effort for a rather general class of systems.
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