Disturbance–driven Model Predictive Control by

means of Youla–Kucera parameter switching with an

application to drainage canal control

Franco Blanchini,

Dipartimento di Matematica

e Informatica,

Universita di Udine,

33100 Udine, Italy

blanchini@uniud.it

Felice Andrea Pellegrino,

Dipartimento di Ingegneria Industriale

e dell’Informazione,

Universita di Trieste,

via Alfonso Valerio 10,

34127 Trieste, Italy

fapellegrino@units.it

Stefano Miani,

Dipartimento di Ingegneria Elettrica

e Gestionale,

Universita di Udine,

33100 Udine, Italy

miani.stefano@uniud.it

Abstract

Motivated by applications such as drainage canal control, where the disturbance actingon a system can be predicted based on weather forecast, we propose an MPC techniqueconsisting in the on–line tuning of a Youla-Kucera parameter. Essentially, the optimal controlsequence is supplied to the system by properly setting the coefficients of the Youla-Kuceraparameter taken as an FIR, whose length depends on that of the optimal trajectory. Stabilityis guaranteed for any arbitrary choice of the time-varying coefficients that are adapted tothe current forecast of disturbance. The approach is illustrated by numerical simulations ofa drainage canal control system.

Index terms– Model–Predictive Control, Youla–Kucera parametrization, switching sys-tems, relatively optimal control.

1 Introduction

Quite often the control of real plants requires dealing simultaneously with constraints, stabilityand robustness. To this purpose, Model Predictive Control (MPC) (sometimes referred to asReceding–Horizon control) is commonly accepted as one of the most effective techniques. TheMPC idea is relatively old [1] and exhaustive surveys are available since 20 years [2]. Still theresearch in this field is extremely active. Undoubtedly, one of the main reasons is that, due to thecurrent computer technology, implementing the efficient and sophisticated algorithms resultingfrom the MPC approach is today possible. Excellent books which describe several aspects ofthe technology are available [3, 4, 5, 6, 7, 8, 9].

The essential idea of MPC is to perform the trajectory optimization on–line, based on theknowledge of the current system output (or state) and the available information about thesystem and the external input, and to apply to the system only the first portion (typically thefirst element) of the optimal control sequence. Such a sequence is then re–computed based onnew information coming from new measurements on the system. It is apparent (and well–known)that the basic questions are concerned with the overall stability of the scheme, with constraintsfulfillment and, possibly, optimality achievement.

As far as the constraints fulfillment is concerned, quite effective tools are provided by thetheory of invariant sets [10]. In the past, several applications of invariance to assure (robust)constraints satisfaction have been proposed [11, 12, 13, 14]. Stability can be faced by meansof Lyapunov theory. The basic idea is that, at least under suitable conditions, the cost–to–gofunction adopted in the optimization problem turns out to be a Lyapunov function (see [15] and[16] for details). Efficient computational techniques are based on the so–called multi–parametricprogramming [17, 18].

A crucial point in the application of MPC techniques, especially in the presence of constraints,is the fact that they seem well fitted for state–feedback problems. However, several results foroutput feedback problems have been proposed [19, 20, 21, 22] mainly based on properly designedobservers.

1.1 Motivations

In some situations, such as drainage canal control [23, 24], it is possible to predict the disturbanceacting on the system and indeed, the availability of predictions is of fundamental importancefor the control to be effective. In the mentioned case, for example, when the forecasted inflow(i.e. the disturbance) due to rainfall is large, the water level must be lowered with respect toits steady–state value, in order to provide a buffer for the incoming water. Usually, disturbanceforecasts are changing in time, becoming more and more accurate as the forecasting horizonshortens. In this paper we deal with control systems where predictions of future disturbance areavailable and continuously refined based on the available information.

We propose to manage such situations by means of a Youla–Kucera (Y–K) parameter com-puted to provide the optimal control action given the information currently available and adaptedon–line according to the refinement of the disturbance forecasts. Precisely, we borrow ideas fromthe observer–based Youla–Kucera parametrization [25, 26], from the switching control theory[27] (see also [28]), and from the relatively optimal control theory [29, 30], to assure good per-formance as well as optimality under nominal conditions.

The well-known Y–K parametrization [31, 32] allows to express the family of the linearstabilizing controllers of a given linear plant as an affine function of a linear, stable system called

2

the Y-K parameter. By its nature, the Y–K parametrization is suitable for enforcing furtherproperties to a stable control system by the choice of a proper parameter. The use of Y–Kparametrization is not new in the context of Model Predictive Control: in [33] it was introducedin the context of unconstrained linear systems for robustness purposes. In [34] that approach hasbeen extended to constrained MPC. In [35] such a parametrization is used to enforce constraintsatisfaction by adding (in a feedback fashion) an extra term to a fixed stabilizing control action.The Y–K parameter is computed on–line by solving a constrained optimization problem whosedecision variables are the coefficients of a linear combination of fixed Finite Impulse Response(FIR) systems. Robust stability of the scheme has been investigated in [36, 37], where l1–normuncertainties and more general unstructured uncertainties, respectively, are considered. Theproblem of enforcing robust stability for a nominally stable control system, with respect tounstructured and polytopic uncertainties has been faced in [38, 39], where the appropriate Y-Kparameter is the solution of a convex optimization problem.

1.2 Outline of the paper

The scheme proposed in the present paper is depicted in Fig. 1. It resembles that of [35] butit is worth mentioning an important difference: here, in contrast to [35], the decision variablesare the optimal control actions. In other words, a (constrained) optimization problem providesa sequence of control actions that is optimal with respect to a given criterion. Based on thatcontrol sequence, the new Y–K parameter is computed in such a way that the whole controlaction turns out to be the optimal one when the parameter switches to this new value. Moreover,the proposed scheme is strictly related to the Relatively Optimal Control (ROC) [29, 30], sinceit is a purely feedback scheme that results in optimal response to an impulse, while guaranteeingthe stability of the overall control system. In addition, due to the time–varying nature of thecontrol, on–line adjustments are possible; from this point of view, the proposed scheme can bethought of as a time–varying ROC. The contribution of the paper is summarized in the followingpoints.

• The controller is structured as an observed–based Y–K parametrization in which arbitraryadaptation of the FIR Y–K parameter does not compromise stability.

• It is shown how to adapt on–line such a parameter as soon as more information about thedisturbances becomes available, in order to improve the system performance.

• A suitable initialization of the Y-K states allows for a pre–action which, in the occurrence,may be quite efficient in contrasting the disturbance effect.

The reminder of the paper is organized as follows. In Section 2 the state–space Youla–Kucera parametrization is reviewed. Section 3 describes the proposed control scheme, focusingon impulse response optimization and on–line parameter adaptation. Section 4 is dedicated tothe motivating application (drainage canal control). Simulation results are reported in Section5. Finally, Section 6 is dedicated to the conclusions and to further developments.

2 A review of Youla–Kucera parametrization

Consider the discrete–time linear plant

x(k + 1) = Ax(k) +Bu(k) +Ed(k) (1)

y(k) = Cx(k) (2)

3

where x(k) ∈ IRn, y(k) ∈ IRp is the input, u(k) ∈ IRm is the control and d(k) ∈ IRq is adisturbance. We assume that the system is reachable and observable.

A fundamental result in control theory, due to [31] and [32] is the parametrization of all sta-bilizing compensators. We describe here the state–space version described in [25, 26]. Considerany arbitrary state–observer–based compensator

z(k + 1) = [A+ LC]z(k) +Bu(k)− Ly(k) (3)

e(k) = y(k)− Cz(k) (4)

u(k) = Jz(k) + v(k) (5)

where matrices L and J are chosen in such a way that A+BJ and A+LC are Schur matrices,where v(k) is an auxiliary input, while e(k) is the estimation error. Then the following resultshold.

Proposition 1 [25] [26] Consider any proper, asymptotically stable linear system with transferfunction Q(z). Then the compensator given by (3)–(5) along with

v(z) = Q(z)e(z) (6)

stabilizes the system (1)–(2). Conversely, given any stabilizing compensator

u(z) = K(z)y(z) (7)

there exists a stable Q(z) such that the compensator (3)–(5) along with (6) has K(z) as transferfunction.

The following result due to [27], will be adopted in the sequel and concerns the switchingcompensators.

Proposition 2 [27] Given a family of compensators Ki(z), i = 1, 2, . . . , NC , we can realizethem in such a way that the overall system is stable under arbitrary switching. A realization canbe achieved by taking the Youla–Kucera parameters Qi(z) corresponding to these compensatorsand realizing them in such a way that they share a common quadratic Lyapunov function.

This argument can be pushed a little further by noticing that we can actually replace (6), byany nonlinear operator which is Lyapunov stable.

Proposition 3 Let the auxiliary input v(k) in (5) be generated by the nonlinear differenceoperators

w(k + 1) = fi(w(k), e(k)) (8)

v(k) = hi(w(k), e(k)) (9)

with fi and hi continuous, satisfying fi(0, 0) = 0, hi(0, 0) = 0. Assume that the state equationsshare a common positive–definite radially unbounded Lyapunov function V (x) such that

∆Vi(w, e).= V (fi(w, e)) − V (x) ≤ λV (w) +Oi(e, w)

such thatOi(w, e) → 0, as e → 0

then the overall system is asymptotically stable.

4

Proof The proof is simple. For d = 0, recasting the system in the error variable r = x − z, sothat e = Cr we get, after simple computations

x(k + 1) = [A+BJ ]x(k)−BJr(k) +Bhi(w(k), e(k))

r(k + 1) = [A+ LC]r(k)

w(k + 1) = fi(w(k), e(k))

Since A + LC is Schur, r(k) converges to zero. In turn e(k) converges to zero and thenhi(w(k), e(k)) and, finally, x(k) converge to zero by the assumed stability of A + BJ . Theessential message is that we can put any operator suggested by the fantasy, provided it is stable,to get a stabilizing compensator.

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ν

A CB 0

yu

OBSERVER

ESTIM. STATE FEEDBACK

PLANT

YOULA−KUCERA FIR

z = [A + L C] z − L y + B u

+

J

z

C

+

Ju

e

Figure 1: The compensator with a Youla–Kucera FIR parameter

In particular we have the following Lemma which is interesting for the sequel.

Lemma 2.1 Consider the FIR linear operator v = Q(e, k) defined as follows (see Fig. 1)

v(k) = Q0(k)e(k) +Q1(k)e(k − 1) +Q2(k)e(k − 2) + . . .+QN (k)e(k −N) (10)

with arbitrary time–varying and uniformly bounded parameters, ‖Qi(k)‖ ≤ µ for some µ, thenthe corresponding closed–loop system is stable.

Proof The operator admits the stable state–space representation with matrices

AQ =

0 0 . . . 0I 0 . . . 00 I . . . 00 . . . I 0

, BQ =

I000

5

and output matrix

CQ(k) =[

Q0(k) Q1(k) . . . QN (k)]

.

The basic idea is apparent from the previous lemma and based on the following two points.

• Stability is assured for any choice of the parameters Qi(k);

• The degrees of freedom in the choice of the parameters Qi(k) can be exploited to improvethe system performance.

3 MPC control via Youla–Kucera parameter switching

3.1 Impulse response optimization

In this section we show how to optimize the transient with guaranteed stability. We work underan assumption which will be reconsidered later.

Assumption 1 Matrix CE is left invertible (so p ≥ q).

This assumption basically means that the system has relative degree one and implies that it hasat least as many measured outputs as disturbance inputs.

On the basis of Assumption 1 it is possible to detect the initial state when a disturbanceimpulse occurs. Indeed, assuming by convenience that the (unknown) impulse d(−1) occurs attime −1 and that the system state x(−1) is null, it is possible to detect the initial state asfollows1

x(0) = Ed(−1) = E(CE)⊥ CEd(−1)︸ ︷︷ ︸

y(0)

= E(CE)⊥y(0)

where (CE)⊥ is a left inverse of CE.

Given any horizon length N , one can compute the “optimal” input sequence uopt(k) for thedetected initial state x(0). The optimization can be carried out because we are dealing withan impulse disturbance, say d(k) = 0 for k ≥ 0. Since z(0) is available, one can compute the“observer produced” control component Jz(k), the estimated output sequence y(k) hence theerror sequence e(k) = y(k)− yopt(k). Finally, the sequence v(k) = uopt(k)− Jz(k) is computed.

Clearly, this sequence should not be injected open–loop for well known reasons. The essentialidea is to synthesize, on line, a Youla–Kucera parameter, which will provide the optimal sequenceanyway.

This can be achieved, on–line, by the following procedure.

Procedure 3.1

1. Given y(0) compute the “guessed” initial state x(0) = E(CE)⊥y(0).

2. Compute the “optimal” sequence uopt(0), uopt(1), . . . , uopt(N), correspondingly.

3. Compute the corresponding output sequence yopt(0), yopt(1), . . . , yopt(N).

1If the state is non–null but known, a similar condition is easily derived.

4. Compute z(k) and e(k) = yopt(k) −Cz(k).

5. Denote by uJ(k) = Jz(k) and if e(0) 6= 0 compute the FIR coefficients as follows

uopt(0)− uJ(0) = Q0e(0)uopt(1)− uJ(1) = Q0e(1) +Q1e(0)uopt(2)− uJ(2) = Q0e(2) +Q1e(1) +Q2e(0)+

:uopt(N)− uJ(N) = Q0e(N) +Q1e(N − 1) + . . . +QNe(0)

(11)

Assuming e(0) 6= 0, is equivalent to say that a non-trivial event occurs. As long as the systemstarts from 0 initial conditions this property is assured for non–zero impulses.

The previous equations are solvable and, in the general case (when p > 1), under determined.Indeed, by transposing and rewriting (11) in matrix form

uTopt(0)− uTJ (0)

uTopt(1)− uTJ (1)...

uTopt(N)− uTJ (N)

︸ ︷︷ ︸

∆u

=

eT (0) 0 0 0e(1)T eT (0) 0 0

...e(N)T eT (N − 1) · · · eT (0)

︸ ︷︷ ︸

Φ

QT (0)QT (1)

...QT (N)

︸ ︷︷ ︸

Q

(12)

it is straightforward to check that the matrix Φ is full row rank whenever e(0) 6= 0 and thus theparameters Qi can be determined, for example, by least squares as

Q = ΦT(

ΦΦT)−1

∆u.

We formalize the results in a proposition.

Proposition 4 The impulse response of the system with the Youla-Kucera parameters computedaccording to procedure 3.1 is optimal with respect to the criterion adopted to synthesize the controluopt. The closed–loop system is linear and stable.

Remark 3.1 The set of equations in (11) to be solved to derive the parameters Qi is in trian-gular form and thus it can be solved efficiently.

3.2 Optimization for the response to a given disturbance profile

The proposed technique is quite general, flexible and suitable for many applications. Considersystem (1)–(2) and assume, for the moment being, that an “expected” disturbance sequencedexp(k) is exactly known, and that x(0) = 0 (a lucky and quite unrealistic case). Then, givenany optimality criterion with constraints and an horizon N , it is possible to choose an optimalsequence uopt(k) which produces an optimal state sequence xopt(k).

xopt(k + 1) = Axopt(k) +Buopt(k) + Edexp(k)

yopt(k) = Cxopt(k)

Accordingly, the observer transient (for given z(0)) is

xopt(k + 1) = [A+ LC]z(k) +Buopt(k)

uopt(k) = Jz(k)

e(k) = Cz(k)− yopt(k).

7

By means of the above equations it is possible to synthesize the Youla–Kucera FIR parametersin order to produce the exact sequence uopt(0), . . . uopt(N). Thus the nominal transient is theoptimal one and the stability of the closed–loop system is assured (relative optimality).

Exact knowledge of the disturbance is seldom available. Then, more realistically, we can“forecast” or “hypothesize” some disturbance patterns. For instance one may assume that thedisturbance is a step of unknown amplitude and unknown endurance (a truncated step). Insuch cases, the considered scheme works quite well. Indeed one can “set” the Y–K parameter assoon as the step occurs, in order to optimize the transient once the true amplitude is available.Moreover, if the disturbance stops, then one can immediately reset the Y–K parameter. Itis apparent that, if the state observer–based control (namely with Qi = 0, ∀i = 0, . . . , N) isthe optimal quadratic one, a natural choice is to set the Y–K parameters to 0 as soon as thedisturbance is over.

Generally speaking, the provided set-up may be used as an hybrid scheme as follows. TheY–K parameter is set to the “optimal one” in normal conditions and “selected via optimization”in case of abnormal events producing system transients as we will see next.

Remark 3.2 The advantage of the proposed scheme with respect to the simpler feedforwardinjection is that the compensation through Y–K parameter is activated only when the eventoccurs. In the impulse case the activation is triggered by the impulse, since d(0) 6= 0 produces anon–zero error e(0). On the other hand, as it will be explained with more detail in Subsection5.2, by properly initializing the state of the FIR system, it is possible to anticipate predictedabnormal events with optimal control actions before they occur.

4 Drainage Canal Control

A drainage canal system is composed of storage and conveyance canals along with structuressuch as gates and pumps for manipulating the flow in the canals. It is designed to collect waterfrom an area and drain it out to a river or to the sea. Efficient drainage canal systems areimportant in areas with considerable amounts of precipitation. Controlling the canal systemmeans maintaining the water levels within an allowed range in “normal” conditions, such asfor regular storm events, and counteracting properly the heavy storm events, that cause largeinflows and can result in over-topping of dikes and inundations. A heavy storm event can befaced by lowering the water levels in advance, in order to create an extra storage for the largepredicted inflow. In deciding the control policy, different constraints have to be taken intoaccount: physical constraints, such as the capacity of the structures and transport canals andperformance constraints such as the minimum allowed water level. Maintaining the water levelclose to the target level while minimizing the control action and preventing inundations dueto large inflow are the (conflicting) objectives. Normally, the control actions are taken basedon the predicted inflow, obtained from weather forecasts by means of a hydrological model ofthe terrain [40]. As shown below, the predicted inflow can be considered as a prediction of thedisturbance that is going to act on the system.

4.1 Model of the drainage system

We follow [23], where the drainage canal system is modelled as a single large reservoir (see Figure4), whose level h(k) obeys the following difference equation:

h(k + 1) = h(k) −Tc

As

Qc(k − 1) +Tc

As

Qd(k), (13)

8

h(k)

Mean Sea Level (MSL)

Qc(k − 1)Qd(k)

Figure 2: Model of the drainage canal system as a single reservoir

where As [m2] is the average storage area, Tc [s] the control time step, Qc [m

3/s] the manipulatedoutflow, Qd [m3/s] the inflow (disturbance) and k the time step index. Notice that a one–stepdelay appears between the control action and the water level change. Such a model, called forobvious reasons Integrator Delay Model, although extremely simple, has proven to be satisfactoryfor low–land drainage canal systems, where the canals are wide and well interconnected [23]. Astate–space realization of (13) is given by the following second order discrete–time linear system:

x(k + 1) =

[

1 10 0

]

x(k)−

[

0Tc/As

]

u(k) +

[

Tc/As

0

]

d(k) (14)

y(k) =[

1 0]

x(k) (15)

where the control action u(k) is the controlled outflow Qc(k) and the disturbance d(k) is theinflow due to rainfall Qd(k). The output y(k) of the system is the water level in the reservoir.The model is completed by a water level set–point y, maximum and minimum allowed levelsymax and ymin, maximum and minimum control flows umin and umax.

4.2 Control of the drainage system

The control problem can be formulated as follows: based on the water level measurements andon the disturbance predictions, keep the water level as close as possible to the set–point whileminimizing the control action2 and satisfying the constraints. A review of the various controlstrategies that have been proposed for facing the above control problem is well beyond thescope of this paper; the interested reader is referred to [40] and the references therein. It isclear, however, that by its nature the problem is suitable for MPC as far as a proper objectivefunction is chosen in order to balance the two conflicting objectives of tracking a constantreference and minimizing the control action. Indeed, in [23] the problem is faced in an MPCfashion by formulating a constrained quadratic programming problem solved at each time step.Here we implement the scheme of Fig. 1, thus proposing the following strategy: a basic observer–based output feedback controller is synthesized, capable of controlling the system in normalcircumstances. In the simulations below, such a controller, defined by the estimated state gain Jand by the observer gain L, is chosen as a linear quadratic optimal regulator. The purpose of the

2Minimizing the control action, in particular its variations u(k)−u(k− 1), is important for many reasons suchas cost, stress on the structures, and waves that propagate through the canals [40].

9

0 1 2 3 4 5 6 7 8 9−0.05

0

0.05

0.1

0.15

time[h]

wal

ter

leve

l [m

]

0 1 2 3 4 5 6 7 8 9−20

0

20

40

60

80

time[h]

cont

rol f

low

[m3 /s

]

Figure 3: Closed–loop trajectories of water level and control action for an impulse occurringat time t = 1h. The basic observer–based controller (null FIR coefficients) is stabilizing butresults in a weak control action (dash line). On the contrary, by setting the FIR coefficients Qk

according to Table 1 , the optimal (constrained) impulse response is achieved in a closed–loopfashion.

basic controller is that of operating in normal circumstances, when the physical and performanceconstraints are far from being active. In those circumstances, the coefficients Qk of the Y–Kparameter are set to zero. The disturbance forecasts are continuously monitored, in such a waythat, as soon as a large inflow due to a heavy storm event is predicted, an optimal (constrained)control sequence is computed by solving a constrained optimization problem. Then, the FIRcoefficients are set accordingly, to “correct” the basic controller output via the terms v(k) thusobtaining the desired sequence. Such a sequence is obtained in a feedback fashion and, thanksto the properties of the Y–K parametrization, the whole control system remains stable.

10

5 Simulation results

In the following we apply the proposed control strategy to the system (14)-(15) where Tc = 900 sand As = 7, 3× 106 m2. The state, input and disturbance matrices are then:

A =

[

1 10 0

]

, B =

[

0−1.23 · 10−4

]

, C =[

1 0]

, E =

[

1.23 · 10−4

0

]

. (16)

The set–point and the maximum and minimum allowed levels are taken, respectively, as y =−0, 40 mMSL, ymax = −0, 30 mMSL and ymin = −0, 55 mMSL. The constraints on the controlaction are umin = 0 m3/s and umax = 75 m3/s. According to the data reported in [23], weassume an inflow of 3 m3/s in absence of rainfall. As a consequence the equilibrium input isu = 3 m3/s and the constraints can be rewritten as −3 m3/s ≤ ∆u ≤ 72 m3/s. Similarly, thewater level constraints become −0.15 m ≤ ∆y ≤ 0.1 m. For easiness of notation, from now onthe variables have to be intended as the deviations from the equilibrium values.

In the following, the same stabilizing observer–based controller will be used for the differentsimulations. It corresponds to

J =[

−99.4 −99.4]

, L =

[

0.7320

]

. (17)

Such a choice, obtained by solving an LQG control problem for the system (14)-(15) , results inσ(A−BJ) = {0.9877, 0} and σ(A− LC) = {0.2679, 0}.

5.1 Optimal impulse response

As a first example, we show how to obtain optimal impulse response by means of a proper Y–Kparameter. We consider a disturbance impulse d(−1) such that the water level is driven fromzero to y(0) = ymax = 0.1 m (from then on, the disturbance is assumed null). As soon as thedisturbance affects the output (i.e. at time k = 0) the current state is easily estimated to be

x0 = E(CE)⊥y(0) =

[

Tc/As

0

]([

1 0][

Tc/As

0

])⊥

y(0) =

[

0.10

]

.

Then, the optimal input sequence is computed by solving the following constrained (quadratic)optimization problem.

minu(·)

Z =N−1∑

k=0

xT (k)Qx(k) +R (u(k) − u(k − 1))2 (18)

s.t. (19)

x(k + 1) = Ax(k) +Bu(k) + Ed(k) (20)

y(k) = Cx(k) (21)

−3 ≤ u(k) ≤ 72 (22)

−0.15 ≤ y(k) ≤ 0.1 (23)

x(0) = x0 (24)

x(N) = 0. (25)

A prediction horizon of 3.5 hours is achieved by taking N = 14 time steps while the state andthe input costs are taken respectively as

Q =

[

25 00 0

]

, R = 5 · 10−2. (26)

11

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13uopt(k) 31.7 54.3 67.7 72 72 72 72 72 72 72 67.6 54.2 31.6 0yopt(k) 0.1 0.1 0.0961 0.0894 0.0811 0.0722 0.0633 0.0544 0.0456 0.0367 0.0278 0.0189 0.0106 0.0039Qk 317 389 468 482 477 483 490 496 503 509 472 355 170 -83.7

Table 1: Optimal control sequence uopt(k), optimal output sequence yopt(k) and FIR coefficientsQk for the optimal impulse response problem.

k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Qk 3310 -1120 321 -134 131 26.9 189 -264 300 -78.5 96.2 -3.95 175 580 351 -92.5

wk+1(0) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.26 0.124

k 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30Qk 255 -11.6 147 -43.2 -35 -34.8 34.1 -86.1 -6.05 -48.2 88.7 40.9 -47.9 -64.9 -61.1

wk+1(0) -0.308 -1.73 -0.726 2.1 -4.16 0.932 2.36 -1.72 -4.74 12.9 -13.8 5.76 -1.88 7.84 -38.9

Table 2: Coefficients Qk and components of w(0), the initial state of the FIR system for theoptimal disturbance response problem.

Notice that the equality terminal constraint (25) is not a requirement of the proposed method-ology: indeed, a less restrictive inequality constraint could be employed as well. The solutionuopt(k), k = 0, . . . , 13 of the above problem is reported in Table 1, along with the correspondingoptimal output sequence yopt(k). The expected observer trajectory z(k) and output estima-tion error trajectory e(k) are easily computed thanks to (3)-(4) by setting u(k) = uopt(k) andy(k) = yopt(k). Finally, the triangular system (11) is solved, given e(0) = 0.1 and uJ(k) = Jz(k),leading to the FIR coefficients Qk reported in Table 1. Now, by setting the FIR coefficients tothese values, the FIR system driven by the estimation error e(k) generates the proper correctionterms v(k) to the output of the basic observer–based controller while guaranteeing stability. InFig. 3, the optimal closed–loop trajectories obtained with the proposed scheme are reported(solid line) compared to the trajectories obtained by the observer–based controller alone. Theimpulse occurs at time t = 1h. Notice that the basic controller results in a weak control actionthat does not exploit the maximum available control flow (72 m3/s). On the contrary, the opti-mal constrained control action, that drives the level to zero in 3.5h, is obtained by means of theproposed scheme.

5.2 Optimal response to a given disturbance profile

A more realistic situation is the following: based on weather forecasts and hydrological models,a large inflow due to a heavy storm event is predicted. Such an inflow is too large to be facedby the basic controller, because it requires lowering the level before the beginning of the actualinflow, in order to get more space for the incoming water. The proposed scheme can be easilyapplied to this situation by means of a proper initialization of the FIR system. We assume thatat time t = 0 the predicted disturbance profile is that reported in top of Fig. 4. Then, we takeN = 48 (i.e. we consider a 12 hours horizon) and solve the optimization problem (18)-(25). Theresulting optimal control action is depicted in Fig. 4 (bottom, solid line). It is apparent that sucha control action anticipates the predicted disturbance, occurring a time t = 4h. However, as itis clear from the scheme of Fig. 1, the input to the FIR system is null whenever the estimationerror e = y − Cz is zero. The predicted trajectory of the estimation error corresponding tothe optimal control action is reported in Fig. 5. Since the FIR is driven by e, from k = 0 tok = 16, the correction term v(k) must be obtained by a proper initialization of the FIR state. Inparticular, the FIR is taken of order 48− 17 = 31 and synthesized based on the last 31 steps ofthe optimal trajectory (in other words, the FIR is computed as if the optimal trajectory began

12

0 2 4 6 8 10 12−0.1

0

0.1

0.2

0.3

time[h]

wat

er le

vel [

m]

0 2 4 6 8 10 12−20

0

20

40

60

80

time[h]

cont

rol f

low

[m3 /s

]

0 2 4 6 8 10 120

50

100

150

200

250

time[h]

dist

urba

nce

[m3 /s

]

Figure 4: Disturbance profile (top), water level (middle) and control action (bottom) obtainedwith the basic observer–based controller (dash line) and with the proposed scheme (solid line).Thanks to the initialization of the state of the Y–K parameter, the predicted inflow can beanticipated by a suitable (optimal) control action, although the input to the Y–K parameter isnull until the disturbance acts on the system.

as soon as e 6= 0). Given the FIR coefficients and being the pair AQ, CQ observable3, a suitableinitial state w(0) of the FIR system can be computed, in order to obtain the first correction termsv(k), k = 0, . . . , 16 and, ultimately, the first part of the optimal trajectory. The FIR coefficientsand the components of w(0) are reported in Table 2. The 17 non–null components of w(0) areresponsible of the first part of the optimal control action. In Fig. 4 (middle and bottom), theclosed–loop water level trajectories and control trajectories obtained by the proposed schemeand by the basic observer–based controller are reported, respectively in solid line and dash line.

3More precisely, we rely on the observability of the subsystem composed of the last 17 components of the FIRstate.

13

0 5 10 15 20 25 30 35 40 45 50

0

0.01

0.02

0.03

k

e(k)

Figure 5: Trajectory of the estimation error e = y−Cz, i.e. the input of the FIR system, underthe control action that is optimal for the disturbance profile of Fig. 4 (top). The estimationerror is null until the disturbance occurs, at time step k = 17.

5.3 Time–varying disturbance predictions

Finally, we consider the following scenario: at time t = 0h a certain disturbance profile ispredicted, as shown in Fig. 6 (top, dash–dot line). Accordingly, an optimal control sequenceis computed and the FIR coefficients are chosen along with a proper initialization, as describedin the previous subsection. The optimal control sequence is reported in Fig. 6 (bottom, dash–dot line). Then, at time t = 1.5h, the disturbance prediction is updated: a larger inflow is nowexpected, as shown in Fig. 6 (top, dash line). In order to face the new situation, another optimalcontrol sequence is computed (by solving the same optimization problem, with updated initialconditions and disturbance predictions) and the controller is simply switched to the new oneby changing the FIR coefficients with proper initialization. The new optimal control sequenceis reported in Fig. 6 (bottom, dash line). At time t = 3h the disturbance predictions changeagain: an even larger inflow is now expected (Fig. 6, top, solid line) hence the Y–K parameteris updated accordingly. The most recently predicted disturbance is the one that actually affectsthe system. The overall closed–loop control action and the water level trajectory are reported(solid line) respectively in bottom and middle of Fig. 6.

6 Conclusions and discussion

In this paper we have discussed an MPC scheme based on Youla–Kucera parametrization, alongthe lines proposed in previous literature. We have seen that the MPC scheme can be im-plemented by setting on–line the coefficients of a FIR Youla–Kucera parameter based on thecurrent information about the system conditions to improve performance. The adaptation ofthese parameters can be done persistently with guaranteed stability. The scheme can be appliedto many problems and appears particularly useful when the information about the disturbanceis updated on–line. A possible drawback is sensitivity, in the case of large FIR, since smallvariations of the measurements can lead to strong variations of the coefficients, hence possiblehigh actuator exploitation. The scheme could be extended to more general classes of systemssuch as that of switching plants. In this case, the techniques suggested in [28] are likely tobe appropriate. In particular the “events” for which a compensations through Youla–Kuceraparameter can be quite effective can be the system switchings (which often cause transients tonew equilibria). The observer–based scheme proposed there can be successfully combined withthe present technique.

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0 5 10 15−0.05

0

0.05

0.1

0.15

time[h]

wat

er le

vel [

m]

0 5 10 150

20

40

60

80

time[h]

cont

rol f

low

[m3 /s

]

0 5 10 150

50

100

150

200

time[h]

dist

urba

nce

[m3 /s

]

Figure 6: Simulation results obtained by changing the Y–K parameter when predicted distur-bance changes. Top: increasing inflows are predicted at time t = 0h, t = 1.5h and t = 3h.Middle: the overall closed–loop water level trajectory. Bottom: the overall control action (solidline) and the optimal trajectories corresponding to the first and second disturbance predictions(dash–dot line and dash line).

References

[1] A. Propoi, “Use of linear programming methods for synthesizing sampled data automaticsystems,” Automation and Remote Control, vol. 24, pp. 837–844, 1963.

[2] C. Garcia, D. Prett, and M. Morari, “Model predictive control theory and practice—asurvey,” Automatica, vol. 25, no. 3, pp. 335–348, 1989.

[3] J. Rossiter, Model-Based Predictive Control. CRC-Press, 2003.

[4] W. Kwon and S. Han, Receding Horizon Control. London: Springer, 2005.

15

[5] G. C. Goodwin, M. Seron, and J. De Dona, Constrained control and estimation: an opti-misation approach. Springer, London ; New York, 2004.

[6] E. Camacho and C. Bordons, Model predictive control, ser. Advanced textbooks in controland signal processing. Springer, 2004.

[7] M. Alamir, Stabilization of Nonlinear Systems Using Receding Horizon Technique, ser. Lec-ture Notes in Control and Information Sciences, 339. Berlin: Springer, 2006.

[8] J. Rawlings and D. Mayne, Model Predictive Control Theory and Design. Nob Hill Pub.,2009.

[9] L. Grune and J. Pannek, Nonlinear Model Predictive Control: Theory and Algorithms,1st ed., ser. Communications and Control Engineering. Springer, 2011.

[10] F. Blanchini and S. Miani, Set-Theoretic Methods in Control, 1st ed. Birkhauser Basel,2007.

[11] M. Sznaier and M. J. Damborg, “Control of linear systems with state and control inequalityconstraint,” in Proceedings of the 26th Conference on Decision and Control, Los Angeles,CA, 1987, pp. 761–762.

[12] D. W. Clarke and C. Mohtadi, “Properties of generalized predictive control,” Automatica,vol. 25, no. 6, p. 859875, 1989.

[13] F. Blanchini, “Control synthesis of discrete time systems with control and state bounds inthe presence of disturbance,” J. Optim. Theory and Appl., vol. 65, no. 1, pp. 29–40, 1991.

[14] D. Q. Mayne, M. Seron, and S. Rakovic, “Robust model predictive control of constrainedlinear systems with bounded disturbances,” AUT, vol. 41, pp. 291–224, 2005.

[15] D. Q. Mayne, “Control of constrained dynamic systems,” Europ. J. Contr., vol. 7, pp.87–99, 2001.

[16] D. Q. Mayne, J. Rawlings, C. Rao, and P. Scokaert, “Constrained model predictive control:stability and optimality,” AUT, vol. 36, pp. 789–814, 2000.

[17] P. Tøndel, T. Johansen, and A. Bemporad, “An algorithm for multi-parametric quadraticprogramming and explicit MPC solutions,” Automatica, vol. 39, no. 3, pp. 489–497, 2003.

[18] C. Jones, M. Baric, and M. Morari, “Multiparametric linear programming with applicationsto control,” European Journal of Control, vol. 13, no. 2-3, pp. 152–170, Mar. 2007.

[19] D. W. Yoon, Tae-Woong; Clarke, “Observer design in receding-horizon predictive control,”Int. J. Control, vol. 61, no. 1, p. 171191, 1995.

[20] D. Q. Mayne, S. V. Rakovic, R. Findeisen, and F. Allgower, “Robust output feedback modelpredictive control of constrained linear systems,” Automatica, vol. 42, no. 7, pp. 1217–1222,2006.

[21] B. Kouvaritakis, W. Wang, and Y. I. Lee, “Observers in nonlinear model-based predictivecontrol,” IJRNC, vol. 10, no. 10, pp. 749–761, 2000.

[22] A. Bemporad and A. Garulli, “Output-feedback predictive control of constrained linearsystems via set-membership state estimation,” Int. J. of Control, vol. 73, no. 8, pp. 655–665, 2000.

16

[23] P. van Overloop, S. Weijs, and S. Dijkstra, “Multiple Model Predictive Control on a drainagecanal system,” Control Engineering Practice, vol. 16, no. 5, pp. 531–540, May 2008.

[24] P. van Overloop, R. Negenborn, D. Schwanenberg, and B. De Schutter, “Towards integratingwater prediction and control technology,” in Networking, Sensing and Control (ICNSC),2011 IEEE International Conference on, no. April. Delft: IEEE, 2011, pp. 11–13.

[25] R. Sanchez-Pena and M. Sznaier, Robust systems, Theory and Applications. New York:Wiley, 1998.

[26] K. Zhou, J. C. Doyle, and J. D. Glover, Robust and Optimal Control. Engelwood Cliffs,NJ: Prentice Hall, 1996.

[27] P. Hespanha and A. S. Morse, “Switching between stabilizing controllers,” Automatica,vol. 38, pp. 1905 – 1917, 2002.

[28] F. Blanchini, S. Miani, and F. Mesquine, “A separation principle for linear switching systemsand parametrization of all stabilizing controllers,” IEEE Trans. on Autom. Control, vol. 54,no. 2, pp. 279–292, 2009.

[29] F. Blanchini and F. A. Pellegrino, “Relatively Optimal Control and Its Linear Implemen-tation,” IEEE Transactions on Automatic Control, vol. 48, no. 12, pp. 2151–2162, 2003.

[30] ——, “Relatively Optimal Control: A Static Piecewise-Affine Solution,” SIAM Journal onControl and Optimization, vol. 46, no. 2, p. 585, 2007.

[31] D. C. Youla, H. A. Jabr, and J. J. Bongiorno, “Modern Wiener-Hopf Design of OptimalControllers - Part II : The Multivariable Case,” IEEE Transactions on Automatic Control,vol. 21, no. 3, pp. 319–338, 1976.

[32] V. Kucera, Analysis and Design of Discrete Linear Control Systems. Engelwood Cliffs,NJ: Prentice Hall, 1991.

[33] B. Kouvaritakis, J. Rossiter, and A. Chang, “Stable generalised predictive control: analgorithm with guaranteed stability,” IEE Proceedings D Control Theory and Applications,vol. 139, no. 4, p. 349, 1992.

[34] Q. Cheng, B. Kouvaritakis, M. Cannon, and J. A. Rossiter, “A Youla parameter approach torobust constrained linear model predictive control,” Proceedings of the 48h IEEE Conferenceon Decision and Control (CDC), pp. 2771–2776, Dec. 2009.

[35] T. van den Boom and T. de Vries, “Constrained Predictive Control using a Time-varyingYoula Parameter: A State Space Solution,” in European Control Conference, Rome, 1995,pp. 3235–3240.

[36] R. Van den Boom, T.J.J. , De Vries, “Robust predictive control using a time-varying Youlaparameter,” International Journal of Applied Mathematics and Computer Science, vol. 9,no. 1, pp. 101–128, 1999.

[37] S. C. Thomsen, H. Niemann, and N. K. Poulsen, “Robust stability in constrained predictivecontrol through the Youla parameterisations,” International Journal of Control, vol. 84,no. 4, pp. 653–664, 2011.

[38] C. Stoica, P. Rodriguez-Ayerbe, and D. Dumur, “Off-Line Method for Improving Robust-ness of Model Predictive Control Laws,” Journal of Control Engineering and Applied In-formatics, vol. 9, no. 3, 4, pp. 76–83, 2007.

17

[39] C. Stoica, P. Ayerbe, and D. Dumur, “Off-line Robustification of Model Predictive Controlfor Uncertain Multivariable Systems,” in 17th IFAC World Congress on Automatic Control,no. 1, 2008, pp. 7832–7837.

[40] P. van Overloop, Model predictive control on open water systems. Amsterdam: Ios Press,2006.

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