AY oulaParameter Approach to Robust Constrained Linear Model

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					        A Youla Parameter Approach to Robust Constrained Linear Model
                             Predictive Control
   Qifeng Cheng, Basil Kouvaritakis∗ , Mark Cannon                                   J. Anthony Rossiter
  University of Oxford, Department of Engineering Science, University of Sheffield, Department of Automatic Control
                   Oxford, OX1 3PJ, UK                             and Systems Eng., Sheffield, S1 3JD, UK



   Abstract— Previous work (Kouvaritakis et al., 1992) proposed    this control law for all initial states in the terminal set).
the introduction of a Youla parameter into the receding               As well as providing a means of systematically accounting
horizon control strategy to enhance the degree of robustness       for constraints on states and inputs, MPC provides a feedback
of the closed loop system without affecting the optimality of
nominal performance. The idea is clearly appealing and has         mechanism through its receding horizon implementation
been investigated further by a number of researchers. Yet          which affords a certain degree of robustness to uncertainty
to date the application of the idea has only been considered       due to model mismatch. A convenient mathematical descrip-
for the case of unconstrained systems, which is surprising         tion of such mismatch has the form of polytopic parametric
given that the presence of constraints is main justification        uncertainty. To retain the guarantee of closed loop stability
for the use of MPC for linear systems. It is the purpose of
the present paper to combine recent results on invariance,         it is necessary to ensure that constraints are satisfied by the
feasibility and an augmented and autonomous formulation of         predicted behaviour over the entire prediction horizon (in
the predicted dynamics to derive an effective robust constrained   modes 1 and 2) and propagating the effects of uncertainty
MPC algorithm that takes full advantage of the introduction        over a future horizon poses some very significant compu-
of a suitable Youla parameter. The efficacy of the proposed         tational difficulties. To avoid these [2] removed mode 1 but
algorithm is illustrated by means of a numerical example.
                                                                   allowed for the online adjustment of the terminal control law.
                                                                   Accordingly [2] ensured that the terminal set was invariant
                     I. I NTRODUCTION
                                                                   and feasible over the entire uncertainty class and minimized
   Model Predictive Control (MPC) is a well established            an upper bound (over the uncertainty class) on the cost-
control strategy that uses a receding horizon strategy to          to-go (in mode 2). Despite the elegance of the solution
derive effective approximations to a closed loop constrained       proposed, the restriction to a fixed state feedback law over
optimal control problem. Instead of solving a closed loop          the entire prediction horizon implies that performance can be
problem which is in essence an infinite dimensional problem         conservative while online computational load is quite high.
that cannot be solved online, MPC uses the measured infor-            Later work [3], [4] produced less conservative results by
mation at each instant of time to predict the future behaviour     re-introducing mode 1 but at a considerable extra compu-
of the controlled system for a given future trajectory of          tational load. An alternative [5] which not only preserved
control moves. This trajectory is then optimized in order          mode 1 but also led to a significant reduction in compu-
to minimize a measure, J, of predicted behaviour; J here           tational cost was based on an augmented autonomous state
will be taken to be the usual linear-quadratic (LQ) cost. To       space formulation of the predicted dynamics. According to
reduce computational complexity, the future input trajectory       this, the degrees of freedom (DOF) where reformulated as
comprises only a finite number of free moves, say the first          perturbations on the unconstrained LQ optimal control law
Nc , the remainder being those prescribed by a pre-specified        and were introduced as additional states of an augmented
state feedback control law, referred to as the terminal control    state vector. In this framework it was possible to derive
law, which is normally taken to be the unconstrained LQ            the prediction dynamics as an autonomous system thereby
optimal. Thus MPC effectively adopts a dual mode prediction        converting the dual mode prediction problem into a single
paradigm: mode 1 consisting of the first Nc time instants, and      mode problem and thus obviating the need to propagate the
mode 2 of the remainder. MPC implements only the first              effects of uncertainty over the prediction horizon.
element of the computed optimal future control sequence,              In addition, given that the degrees of freedom are taken to
and repeats the whole procedure at the next time instant.          be perturbations on the unconstrained optimal law in [5], it
   To ensure a guarantee of closed loop stability [1], MPC         can be shown that the optimization of nominal performance
deploys: (i) a terminal penalty term in the cost index J, which    is equivalent to the minimization of a weighted norm of the
is normally taken to be a quadratic function of the terminal       vector of degrees of freedom, f . This defines a cost which
state (i.e. the state at the end of mode 1), and is equal to       is quadratic in f , and, if this is combined with an ellipsoidal
the cost-to-go, namely the cost incurred in mode 2; (ii) a         terminal set which involves constraints that are quadratic in
terminal inequality constraint that restricts the terminal state   f , then the optimal solution can be obtained at a fraction of
to lie in a terminal set which, under the terminal control law,    the cost required by the usual MPC schemes, even those that
is invariant and feasible (i.e. constraints are satisfied under     use polytopic terminal sets enabling a solution through the
use of quadratic programming (QP) with warm starting.               linear-quadratic (LQ) problem with cost index
   Just as the worst case strategy adopted by [2] can be overly                               ∞
pessimistic (at least with respect to nominal behaviour), so                            J=            2
                                                                                                    (yk + Ru2 )
                                                                                                            k                         (5)
the approach of [5] can be overly optimistic (at least with                                   k=0
respect to worst case behaviour), especially so if the worst
                                                                    for the nominal model parameters, say (A(0) , B (0) , C). Then
case and nominal performance differ significantly in closed
                                                                    the prediction dynamics are given by the autonomous state
loop behaviour. Reducing this difference provides an ideal
                                                                    space model
remedy to this problem and an observation made in [6] is                                                                  
key to this. It noted in [6] that in the unconstrained case, the                                                     c0
                                                                                                  x
online optimization of MPC resulted in a fixed term feedback                zk+1 = Ψ(i) zk , zk = k , fk =  .  (6)
                                                                                                                  . 
                                                                                                   fk                 .
control scheme which could be endowed with extra degrees
                                                                                                                   cN −1
of freedom through the introduction of a Youla Parameter.
The interesting observation made there was that this Youla                   A(i) −B (i) K   B (i) eT
                                                                                                    1 , M = 0N −1,1               IN −1
                                                                    Ψ(i) =
Parameter can be used to improve the robustness of the MPC                         0           M               0                 01,N −1
feedback scheme without affecting the optimality of nominal         where e1 is the first column of the N × N identity matrix
performance. Thus the minimization of the RH∞ -norm of              IN . A convenient way to ensure the feasibility of predictions
an appropriate sensitivity transfer function would lead to          is through the stability constraint:
optimal robustness with respect to additive uncertainty to the
                                                                                              T
system transfer function; robustness to additive disturbances                                zk P zk ≤ 1                              (7)
could be optimized instead by minimizing the RH∞ -norm of           where P is a positive definite matrix (denoted P              0)
a different transfer function. This idea of achieving optimal       satisfying the robust invariance and feasibility conditions:
performance first and subsequently optimizing performance
gained favoured with many researchers in subsequent years                      Ψ(i) T P Ψ(i) P, i = 1, . . . p                       (8a)
(e.g. [7], [8], [9], [10]). Surprisingly however, in all this                   ˆ        ˆ        ˆ
                                                                                KP −1 K T ≤ u2 , K = −K
                                                                                             ¯                            eT .       (8b)
                                                                                                                           1
work, consideration was given only to the unconstrained
case, which somehow seems to defeat the purpose of using            It can be shown that the minimization of the cost (5) for the
                                                                                                                               T
MPC; after all, in the absence of constraints, MPC can be           nominal model is equivalent to the minimization of fk fk ,
implemented as a fixed term controller. It is the purpose of         and on the basis of this an MPC algorithm with guaranteed
the present paper to close this gap, through the combined           robust stability/feasibility can be devised [5]:
use of the augmented autonomous formulation of [5], with               Algorithm 1 (Robust efficient MPC): At k = 0, 1, . . .:
                                                                                      ∗
its attendant use of robust invariance and feasibility and the        1. Compute fk = arg minf f 2 subject to (7)
                                                                                                          2
                                                                                                            ∗
use of the Youla Parameter as recommended in [6].                     2. implement uk = −Kxk + eT fk .  1
                                                                    The online optimization in step 1 involves a quadratic cost
II. ROBUST MPC BASED ON AUGMENTED AUTONOMOUS                        and is subject to quadratic constraints. This can be solved
                    PREDICTION MODEL                                very efficiently (at a fraction of the computational cost of QP-
  Consider a SISO discrete time linear system with model:           based MPC algorithms) by using a Newton-Raphson proce-
                                                                    dure with guaranteed quadratic convergence rate to determine
              xk+1 = Axk + Buk            yk = Cxk            (1)   the unique negative root of a 2N th order polynomial.
where x ∈ Rn , u, y ∈ R, subject to polytopic uncertainty                 III. T HE NEED FOR THE YOULA PARAMETER
defined by
                             p                                         Despite its significant advantages in terms of computa-
                 (A, B) =         ηi (A(i) , B (i) )          (2)   tional efficiency, Algorithm 1 optimizes the performance of
                            i=1                                     the nominal system, which could differ considerably from
                                                                    that obtained for other models within the uncertain class
and subject to the constraint
                                                                    of (2). The nominal performance objective of Algorithm 1
                        −¯ ≤ u ≤ u.
                         u       ¯                            (3)   would be better justified if it were possible to desensitize
                                                                    performance under model uncertainty. From the perspec-
Although this constraint applies to the plant input, it is          tive of frequency response desensitization, the Q parameter
equivalent to a general constraint on both the states and the       provides an effective solution and was introduced in the
DOF if the closed loop paradigm is employed. According to           context of MPC in the following way [6]. In the absence of
this, the DOF are not taken to be the future predicted inputs,      constraints, the MPC control law can be implemented using
but instead are perturbations on a state feedback control law:      the fixed term feedback scheme of Figure 1, where r denotes
                  −Kxk+i + ci         i = 0, 1, . . . N − 1         a reference set point signal and where the polynomials M (z),
        uk+i =                                                (4)   N (z) are solutions of
                  −Kxk+i              i = N, N + 1, . . .
                                                                               a(0) (z)M (z) + b(0) (z)N (z) = popt (z)              (9a)
with Nc denoting the prediction horizon, where K will be                                                 (0)        (0)
taken to be the optimal feedback gain for the unconstrained                     popt (z) = det[zI − A          +B         K].        (9b)
                Fig. 1.   The MPC feedback loop


                                                                          Fig. 2.   The MPC feedback loop with the Q parameter
  However (9a) is a Diophantine equation which admits a
whole class of solutions:
                                                                     Since (11) is affine in the parameter Q, it is amenable
M (z) = Y (z) − b(0) (z)Q(z), N (z) = X(z) + a(0) (z)Q(z)
                                                                  to the usual RH∞ treatment (including that of [6] which is
                                                         (10)
                                                                  based on the application of Lawson weighted least squares
where for convenience X, Y , and Q can be chosen to
                                                                  algorithm [12]). Furthermore S is known to give a measure of
be polynomials. Under these circumstances, the transfer
                                                                  the reduction of sensitivity to uncertainty in going from the
functions from r to y and u are b(z)/popt (z), a(z)/popt (z),
                                                                  open to the closed loop system, but that reduction concerns
respectively and as such are not affected by the introduction
                                                                  the frequency response sensitivity. Thus, choosing Q in
of the free variable Q. The idea is therefore to use Q to
                                                                  this manner will optimize the worst case (over the entire
optimize some other property without compromising the
                                                                  frequency range) frequency response sensitivity reduction.
nominal optimality of the feedback system of Figure 1. For
                                                                  Clearly, if desired, S can be scaled by a suitable weighting
example one could choose the Q that minimizes the RH∞ -
                                                                  factor that allows information about the frequency response
norm of an appropriate transfer function in order to optimize
                                                                  of the open loop sensitivity to be taken into account. This
robustness properties in terms of either uncertainty in the
                                                                  procedure will result in a Q which desensitizes the closed
model, or additive disturbances. In the case of the former
                                                                  loop to additive uncertainty, but may well not be optimal in
objective, the work of Caratheodory and Fejer [11] provides
                                                                  the context of desensitizing the cost of (5). Our simulation
a justification for the use of Lawson’s weighted least squares
                                                                  results show that the solutions for Q based on the well known
algorithm [12] for the derivation of optimal polynomials Q
                                                                  RH∞ optimal solutions produced results which were very
of any chosen order so as to maximize the tolerance of the
                                                                  close to optimal, even in the context of the cost of (5).
system of Figure 1 to parametric uncertainty in the open loop
                                                                     The analysis above applies to the unconstrained case only
system transfer function b(z)/a(z). This method of obtaining
                                                                  and will not in general hold true for all reference signals
robustness over and above the optimality of MPC was first
                                                                  r. However when constraints are present, as described in
introduced in [6], and was subsequently explored further by
                                                                  Section 2, to avoid infeasibility it is necessary to perturb the
many researchers (e.g. [7], [8], [9], [10]). Yet remarkably
                                                                  optimal control law, and this was achieved in (4) through the
none of the results in this area address constraint handling,
                                                                  use of the closed loop paradigm. This paradigm can also be
which is the main justification for using MPC.
                                                                  applied to the system of Figure 1 through the introduction
                                                                  of the perturbation signal c depicted in Figure 2. Then,
 IV. R EDUCING SENSITIVITY THROUGH THE USE OF THE
                                                                  provided the MPC strategy is able to ensure the feasibility of
                 YOULA PARAMETER
                                                                  predictions at all time instants, the feedback loop of Figure
   The current paper redresses this omission using the aug-       2 will always operate within its linear region and therefore
mented autonomous MPC formulation described in Section            the desensitization achieved through Q will remain valid,
2. We first reformulate the DOF as perturbations on the            throughout the receding horizon of the MPC strategy. The
optimal feedback control law depicted in Figure 1. This leads     following section shows how the strategy of Section 2 may
to the feedback loop of Figure 2, where the perturbation          be used to provide the required guarantee of feasibility.
signal c is introduced in order to force the input u to respect
the constraints of (3), but otherwise should be kept as small        V. T HE USE OF THE YOULA PARAMETER IN ROBUST
as possible in the interests of optimality. This strategy is                        CONSTRAINED MPC
optimal for the nominal model only, but its approximate
                                                                     As explained above, the scheme of Figure 2 is the input-
validity can be extended to a wider class of models (centred
                                                                  output version of the closed loop paradigm of (4). It therefore
on the nominal) provided that the system of Figure 2 is
                                                                  follows that, if it were possible to derive a state space de-
desensitized to the effects of model uncertainty. An indirect,
                                                                  scription for Figure 2, then it would be possible to apply the
but convenient way to achieve this desensitization is to
                                                                  augmented autonomous formulation of Section 2 to derive
minimize the RH∞ -norm of the sensitivity transfer function
                                                                  an MPC algorithm with the robust stability and feasibility
                       X(z) + a(0) Q(z) b(0)
                                                  −1              guarantees of Algorithm 1, but which would also benefit from
          S(z) = 1 +                                              the desensitization effects of Q.
                       Y (z) − b(0) Q(z) a(0)
                                                          (11)       A simple procedure for the transformation of the input-
                   (0)
                  a Y (z) − a(0) b(0) Q(z)                        output setting of Figure 2 to a state space model is the
                =
                          popt (z)                                following. First, in the interests of parsimony (i.e. in retaining
a low model order), it is recommended that the optimal             new augmented autonomous dynamics of (15)). As is the
polynomial Q(z) is approximated by a transfer function,            case for the predictions of (6), the minimization of the
NQ (z)/DQ (z). Such an approximation is likely to result in        nominal predicted cost of (5) can be likewise proved to
                                                                                                           T
an increase in the norm of S, and hence a trade-off has to         be equivalent to the minimization of fk fk . The procedure
be struck between this effect and the desire to keep model         differs however in one respect, since, unlike the model of
complexity to a minimum. Substituting NQ (z)/DQ (z) in                                          ˆ                 ˆˆ
                                                                   Section 2 (where uk = −Kzk ), here uk = C xk + Dvk ,   ˆ
place of S yields:                                                 where C  ˆ is subject to polytopic uncertainty with vertices
                                                                    ˆ
                                                                   C (i) = ( 0T CD θ(i) )T . This means that the conditions
                         DQ (z)a(z)
             u(z) =                       v(z)                                                       ˆ
                                                                   to be satisfied by the parameter, P , defining the augmented
                                ˆ
                    ˆ (z)a(z) + N (z)b(z)
                    M
                                                           (12)    ellipsoid should be modified to the analogous invariance and
                         DQ (z)b(z)
             y(z) =                       v(z)                     feasibility conditions:
                                ˆ
                    ˆ (z)a(z) + N (z)b(z)
                    M
                                                                     ˆ      ˆˆ
                                                                     Ψ(i) T P Ψ(i)   ˆ ˆ ˆ ˆ
                                                                                     P , K (i) P −1 K (i) T ≤ u2 , i = 1, . . . p
                                                                                                              ¯
where v(z) = r(z) + c(z) and
                                                                                                                               (16)
           ˆ
           M (z) = Y (z)DQ (z) − b(0) (z)NQ (z)                       The overall MPC control strategy is summarized by the
           ˆ
           N (z) = X(z)DQ (z) − a(0) (z)NQ (z).                    following algorithm.
                                                                      Algorithm 2 (Robust MPC with Youla parameter):
The observability canonical form for (12) can be written as:       Offline:
 ˆ      ˆx    ˆ          ˆ ˆ          ˆˆ     ˆ
 xk+1 = Aˆk + Bvk , yk = Cy xk , uk = C xk + Dvk                                  ˆ                    ˆ
                                                                     a. Compute P = arg minP det(P ) subject to (16).
                                                                                                ˆ
      0 I      0                                                     b. Compute the Q which minimizes the RH∞ -norm of the
  ˆ
  A=        ˆ
          , B=                   ˆ
                  , d = CM CN θ, D = 1 (13)
      −dT      1         ˆ  ˆ                                           (weighted) sensitivity function S of (11).
                                                                                                                                    T
     ˆT             ˆ
    Cy = CD 0T θ, C T = 0T CD θ − d                                Online, at k = 0, 1, . . ., for xk = yk yk−1 · · · yk−ν+1 :
                                                                                                   ˆ
                                                                                   ∗
                                                                                                                   ˆT ˆ ˆ
                                                                     1. Compute fk = arg minf f 2 subject to zk P zk ≤ 1.
                                                                                                        2
where θ is the column vector of coefficients (in order of                                     ˆ ˆ
                                                                                               (0)     ˆ
                                                                     2. Implement uk = C xk + Dvk ,
ascending powers of z) of a(z), b(z) (with the coefficients                      ˆ
                               ˆ ˆ                                      where C (0) = ( 0T CD θ(0) )T .
of a(z) appearing first), Cy , C are the row vectors of the
                                                                                                                        ˆ
                                                                      Remark 1: Note that the offline optimization of P in step
coefficients of DQ a(z), DQ (z)b(z), respectively taken in
                                ˆ   ˆ
ascending order, CM , CN , CD , CD are the convolution ma-         a can be formulated as a convex LMI problem (e.g. see
                       ˆ     ˆ
                                           ˆ          ˆ            [13]). Furthermore the online optimization in step 1 can be
trices that relate θ to the coefficients of M (z)a(z), N (z)a(z),
                                                                   performed using the efficient NRMPC optimization of [5].
DQ (z)a(z), DQ (z)b(z), respectively. In this formulation, the
states are given by:                                                  In common with Algorithm 1, Algorithm 2 enjoys the
                                                                   benefit of an extremely efficient online optimization and
                                T
 xk = yk yk−1 · · · yk−ν+1
 ˆ                                  , ν = na +nb +nQ +1 (14)       has the guarantee of closed loop stability and feasibility
                                                                   (provided that the online optimization is feasible at initial
where na , nb , nQ denote the degrees of a(z), b(z), and
                                                                   time). However in addition to Algorithm 1, the desirability of
DQ (z). The model (13) caters for a tracking problem where
                                                                   achieving optimality for the nominal predictions is enhanced
r is assumed to be a constant setpoint. The more general
                                                                   on account of the use of the Q parameter.
case requires knowledge of the dynamics that generate any
particular reference signal r(t). With r a constant, it is                              VI. N UMERICAL E XAMPLE
                                                       ˆ
straightforward to compute the steady state value for x and
                                                                     This section illustrates the efficacy of Algorithm 2 by
re-write the model of (13) in terms of a regulation problem
                                                                   means of a numerical example. Consider a system with an
by considering deviations away from the steady state. This is
                                                                   input-output description given by
assumed to have already been done and correspondingly we
take the input v to be c (the deviation of v from the steady                           a(z) b(z) = 1 z z 2 1 z θ                   (17)
state).
   On account of the linear dependence on θ in (13), it is         where θ lies in the convex hull of vertices θ(i) , i = 0, . . . 4:
possible to generate the polytopic uncertainty model for the                                            T
                                                                        θ(0) = 1 0.8 0.3 1 0.6
system of Figure 2 (with r = 0) given the vertices θ(i)
                                                                                                            T
of a corresponding polytopic uncertainty class for the open                  (i)       ± 0 0.16 0.06 0 0        ,       i = 1, 2   (18)
                                                                         θ         =                            T
loop input-output description. The problem then becomes                                ± 0 0 0 0.02 0.012           ,   i = 3, 4
equivalent to that defined in Section 2, after the application
of the closed loop paradigm law of (4). This leads to the          For this example, R = 10−6 results in the optimal closed
augmented autonomous model for the prediction dynamics:            loop polynomial:
                       ˆ
                       x          ˆ
                                  A(i)             ˆ
                                                   BeT                                    popt (z) = 1 + 0.6z −1                   (19)
        ˆ                  ˆ
 zk+1 = Ψ(i) zk , zk = k , Ψ(i) =
 ˆ                ˆ                                  1     (15)
                       fk          0                M
                                                                   and a pair of particular solutions to (9a)
  The perturbation ck can be optimized using an approach
analogous to step 1 of Algorithm 1, but modified for the                Y (z) = 1 + 0.6z −1 ,      X(z) = −0.8 − 0.3z −1 .          (20)
   The 14th order polynomial Q that minimizes the sensi-                Although it may seem paradoxical that the minimax costs
tivity transfer function S of (11) for this system (computed         given Algorithm by 3 are not the best, this can be explained
using Lawson’s weighted least squares algorithm), and its            by the fact that the costs displayed are closed loop costs and
2nd order transfer function approximation are given as:              therefore do not bear a direct relation to the predicted costs
                                                                     that are optimized by any of the algorithms. Furthermore, in
  Q(z) = 0.1 3.514 − 0.651z −1 + 0.105z −2 − 0.176z −3
                                                                     order to make the computation of Algorithm 3 tractable, an
           + 0.227z −4 − 0.175z −5 + 0.09z −6 − 0.027z −7            LTI assumption has been imposed. In summary, Algorithm 2
           − 0.003z −8 + 0.009z −9 − 0.006z −10                      is clearly the best, since it both produces the lowest nominal
           + 0.002z −11 − 0.001z −13 + 0.001z −14                    cost (which, as expected, is the same as that of Algorithm
                                                             (21)    1), while also resulting in smaller cost sensitivity. Moreover,
                                                                     Algorithm 2 achieves this with an online computational
           NQ (z)       0.3505 + 0.0599z −1
                  =                                          (22)    load which is comparable to that of Algorithm 1. Clearly
           DQ (z)   1 + 0.3570z −1 + 0.0229z −2                      there is a certain amount of extra offline computation but
    The degree of optimality of Q is illustrated in Figures          that involves two simple optimization problems which are
3-6 which give the gain of the frequency response of S               known to converge and are not particularly computationally
for the Q of (22) in transfer function form (solid line)             demanding.
and for Q = 0 (dashed line). The solid line plots would
in fact be flat for the optimal Q of (21), but deviate a                                      VII. C ONCLUSIONS
little from flatness for the approximation of (22). Finally              The use of the Youla parameter in the context of MPC
the offline computation of Algorithm 2 is completed by the            has been advocated in a paper as far back as 1992 and since
                                  ˆ
determination of the parameter P of the augmented ellipsoid.         then has been endorsed by various researchers. Yet, to date,
For a control horizon of N = 10 and an input bound                   there has been no attempt to apply this in the presence of
     ¯
of u = 0.3, the maximum volume invariant and feasible                constraints, which is very surprising when it is considered
                            ˆ                       ˆ
ellipsoid is obtained for P with determinant det(P ) = 0.093.        that constraints offer the main justification for the use of
    To illustrate the efficacy of Algorithm 2, Table 1 lists          MPC. This paper proposes a strategy that redresses this
the closed loop costs obtained by Algorithms 1 and 2 for             situation and demonstrates the efficacy of the strategy by
zero initial conditions and a reference setpoint r = 0.36.           means of a numerical example.
The costs are computed for fixed values of θ, namely the                 There remains the issue of how to tune the Youla param-
nominal value θ(0) and each of the vertices θ(i) , i = 1, . . . 4.   eter. In this paper, the aim is to desensitize the closed loop
To assess the desirability of the nominal objective employed         system to additive disturbances. However this desensitization
by these strategies, the table also includes the corresponding       is performed for the sake of convenience (given the body of
costs for a worst case algorithm, referred to as Algorithm 3.        available work) in terms of the frequency response. As a
A truly worst case algorithm would entail online computation         result, there is no guarantee that the resulting Q parameter
which is intractably computationally intensive, since it would       is optimal for the purposes of the robust MPC framework
have to assume an LTV model over the prediction horizon              proposed in the paper. This particular point was tested
at each time instant. To make the computational demand               numerically using the example of section 6, and it was
of Algorithm 3 a little more comparable with those of                found that although some improvement to the results of the
Algorithms 1 and 2, though in essence still a great deal             proposed algorithm could be brought about by the further
more intensive, the assumption is made at each instant that          tuning of the Youla parameter, such improvements were very
the dynamics are LTI over the prediction horizon, even               modest indeed. It is nevertheless left as a topic for further
though at each time instant, the algorithm implements the            research to devise efficient algorithms for the choice of Q
first element of the sequence that optimizes the predicted            which address directly the needs of the robust constrained
cost for whichever vertex gives the greatest cost.                   MPC algorithm proposed in the paper.
    As expected, the desensitization of the closed loop system
to additive uncertainty has the significant effect of reducing                                     R EFERENCES
(almost halving) the cost sensitivity (measured as a percent-
                                                                      [1] D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Con-
age of nominal cost) with the maximum value for Algorithm                 strained model predictive control: Stability and optimality. Automatica,
2 being more than 4% smaller than that of Algorithm 1 but                 36:789–814, 2000.
also 3.6% smaller than that of Algorithm 3. In fact for this          [2] M.V. Kothare, V. Balakrishnan, and M. Morari. Robust constrained
                                                                          model predictive control using linear matrix inequalities. Automatica,
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offer, in that not only is the nominal cost worse than what can       [3] J. Schuurmans and J.A. Rossiter. Robust predictive control using
be obtained by either Algorithm 1 or 2, but the worst case                tight sets of predicted states. Control Theory and Applications, IEE
                                                                          Proceedings -, 147(1):13–18, 2000.
cost is also only marginally better than that of Algorithm 2,         [4] J.W. Lee, W.H. Kwon, and J.H. Choi. On stability of constrained
and in fact worse than that of Algorithm 1. This is the case              receding horizon control with finite terminal weighting matrix. Auto-
even though the computational demand of Algorithm 3 is                    matica, 34(12):1607–1612, 1998.
                                                                      [5] B. Kouvaritakis, J.A. Rossiter, and J. Schuurmans. Efficient robust pre-
considerably higher than that required by either Algorithm 1              dictive control. Automatic Control, IEEE Transactions on, 45(8):1545–
or Algorithm 2.                                                           1549, 2000.
                                                                θ(0)          θ(1)         θ(2)        θ(3)                             θ(4)          Max cost sensitivity
                                          Algorithm 1           0.1296        0.1402       0.1319      0.1445                           0.1303        11.5%
                                          Algorithm 2           0.1296        0.1378       0.1280      0.1385                           0.1271        6.9%
                                          Algorithm 3           0.1321        0.1357       0.1252      0.1453                           0.1342        10.0%

                                                                                              TABLE I
                                                                                     C LOSED LOOP COST VALUES




                         0.165                                                                                                 0.3


                               0.16                                                                                           0.25


                         0.155




                                                                                                           | S [exp(jωT)] |
                                                                                                                               0.2
     | S [exp(j ωT)] |




                               0.15
                                                                                                                              0.15

                         0.145
                                                                                                                               0.1

                               0.14

                                                                                                                              0.05
                                                                                                                                  0       5      10     15     20     25   30   35   40
                         0.135                                                                                                                            Frequency, ω
                              0           5    10    15     20      25   30   35     40
                                                       Frequency, ω

                                                                                                                              Fig. 5.    Sensitivity improvements due to Q – vertex 3.
                               Fig. 3.   Sensitivity improvements due to Q – vertex 1.

                                                                                                                               0.2
                               0.24
                                                                                                                              0.18
                               0.22

                                                                                                                              0.16
                                                                                                           | S [exp(jωT)] |




                                0.2
            | S [exp(jωT)] |




                               0.18                                                                                           0.14


                               0.16                                                                                           0.12

                               0.14
                                                                                                                               0.1

                               0.12
                                                                                                                              0.08
                                                                                                                                  0       5      10     15     20     25   30   35   40
                                0.1                                                                                                                       Frequency, ω
                                   0      5    10    15     20     25    30   35     40
                                                       Frequency, ω
                                                                                                                              Fig. 6.    Sensitivity improvements due to Q – vertex 4.
                               Fig. 4.   Sensitivity improvements due to Q – vertex 2.

                                                                                                                   e                e ¨
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