International Journal of Systems Science Vol. 36, No. 8, 20 June 2005, 477–484
On the solution of the tracking problem for non-linear systems with MPC L. MAGNI*{ and R. SCATTOLINIz {Dipartimento di Informatica e Sistemistica, Universita’ di Pavia, via Ferrata 1, 27100 Pavia, Italy {Dipartimento di Elettronica e Informazione, Politecnico di Milano, P.za Leonardo da Vinci 32, 20100 Milano, Italy (Received 27 November 2003; in final form 21 March 2005) This paper presents a Model Predictive Control (MPC) algorithm for non-linear systems which solves the tracking problem for asymptotically constant references. Closed-loop stability of the equilibrium and asymptotic zero-error regulation are guaranteed. The performance of the method is discussed with the classical Continuous Stirred Tank Reactor (CSTR) control application. Keywords: Non-linear systems; Nonlinear Model Predictive Control; Discrete-time systems; Tracking
1. Introduction Nowadays, many Model Predictive Control (MPC) algorithms are available to solve the regulation problem with guaranteed stability for non-linear systems; see Mayne et al. (2000). However, for a practical use of non-linear MPC, some issues are still to be faced. Among them, the fundamental problem of tracking constant exogenous reference signals using output feedback stabilizing regulators is still largely open. In this context, an input–output approach was used in De Nicolao et al. (1997), while in Magni et al. (2001) a state-space algorithm for tracking exogenous signals (not necessarily constant) and the asymptotic rejection of disturbances generated by a properly defined exosystem was developed. Although this method solves the tracking and disturbance rejection problems, its applicability can be limited by the computational load. For example, in the case of piecewise constant references, it needs to compute the equilibrium corresponding to the required constant output. In Findeisen et al. (2000), the pseudolinearization method described in Reboulet and Champetier (1984) is adopted to derive a model predictive control strategy. In so doing, an invertible state variable change and a state feedback law are introduced such that the transformed and controlled closed-loop
*Corresponding author. Email:
[email protected]
system has the same (constant) linearization independent of the constant reference signal. Considering systems for which a pseudolinearization can be obtained, the MPC control law stabilizes all the fixed set points in the considered set point family. However, the computation of the pseudolinearization transformation and of the feedback law may be rather cumbersome. This paper presents a new MPC algorithm for non-linear models which guarantees local stability and asymptotic tracking of constant references. The proposed method builds on the state-space regulation approach described in De Nicolao et al. (1998), and its main peculiarities with respect to other approaches e.g. Magni et al. (2001), are: 1. It does not require the knowledge of the steadystate value of the input and state variables corresponding to the desired equilibrium. In fact, the control parameters, like the terminal penalty and the terminal constraint, are directly implicitly related to the reference signal, and their explicit computation is not required. 2. It allows for the use of pre-programmed references, while in Magni et al. (2001), the reference is viewed as the output of a given exosystem. 3. It takes advantage of the a priori knowledge of a locally stabilizing linear regulator, which can be the one already used to control the plant. This property allows one to easily integrate the MPC algorithm
International Journal of Systems Science ISSN 0020–7721 print/ISSN 1464–5319 online ß 2005 Taylor & Francis Group Ltd http://www.tandf.co.uk/journals DOI: 10.1080/00207720500139666
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in an already existing structure, made for example by standard PID regulators, with the aim to enhance performance. Alternatively, the linear regulator can be derived by means of empirical design procedures, for example the classical Ziegler–Nichols tuning rules. In this case, the synthesis of the linear regulator does not even need the computation of the linearized plant model. 4. It can be applied also in the output feedback case by resorting to the ‘separation principle’ for non-linear systems presented in Magni et al. (2004). The paper is structured as follows. In section 2, the control problem is stated, while the new MPC method is presented in section 3, together with its stabilizing properties. Finally, in section 4, the performances of the proposed algorithm are tested on a simulated Continuous Stirred Tank Reactor (CSTRÞ control application.
exists, let ~ @ f , A~ ðy Þ ¼ @ 0 0 ðy Þ, uðy Þ ~ @ h : C~ ðy 0 Þ ¼ @u 0 0
~ @ f B~ ðy Þ ¼ @u
0
0
, ðy0 Þ, u ðy0 Þ
ðy Þ, u ðy Þ
Assumption A1: The linear system (A~ ðy0 Þ, B~ ðy 0 Þ, C~ ðy0 ÞÞ is reachable and observable, and does not have transmission zeros equal to one. Theorem 2.1: Consider y 0 2 Rm and the associated equilibrium (ðy 0 Þ, u ðy 0 Þ) of (1) with h~ððy 0 ÞÞ ¼ y 0 . If A1 holds, there exists one and only one pair of functions ðy0 Þ, u ðy0 Þ, continuous in an open neighbourhood Rm of y0 , such that, 8y0 2 , ðy0 Þ ¼ ~ f ððy0 Þ, u ðy0 ÞÞ, y0 ¼ h~ððy0 ÞÞ, ðy0 Þ 2 , u ðy0 Þ 2 U: According to the Internal Model principle, an integrator is preliminarily plugged in front of the system to guarantee the solution of the asymptotic tracking problem. Then, the control variable u(k) is given by
2. Problem formulation Consider a non-linear dynamic system ðk þ 1Þ ¼ f~ððkÞ, uðkÞÞ ð1Þ
yðkÞ ¼ h~ððkÞÞ,
where k is the discrete time index, 2 Rn , u 2 Rm , y 2 Rm are the state, input and output vectors respectively, f~ð, Þ and h~ðÞ are C1 functions of their arguments. m 0 Assume now that, for a0 given0 vector y 2 R , there exists an equilibrium ðy Þ, u ðy Þ such that ðy 0 Þ ¼ f~ððy0 Þ, u ðy 0 ÞÞ
y 0 ¼ h~ððy 0 ÞÞ:
The problem is to design a control algorithm such that for any constant reference signal y0 2 , where is an open neighbourhood of y0 , the resulting closedloop system reaches an asymptotically stable equilibrium point (ðy0 Þ, u ðy0 ÞÞ such that y0 ¼ h~ððy0 ÞÞ: Moreover, the state and control variables are required to fulfil the following constraints: ðkÞ 2 ,
uðkÞ 2 U,
k t,
ð2Þ
where and U are compact subsets of Rn and Rm , containing ðy 0 Þ and u ðy 0 Þ; respectively, as an interior point. Although this appears as a reasonable assumption, the reader should be warned that it rules out the case of constraints that are active at a steady state, which may be a case of industrial interest. Assuming that the equilibrium (ðy0 Þ, u ðy 0 ÞÞ
zðk þ 1Þ ¼ zðkÞ þ uðkÞ uðkÞ ¼ zðkÞ
ð3Þ
so that the overall system (1), (3) is described by xðk þ 1Þ ¼ f ðxðkÞ, uðkÞÞ,
xðtÞ ¼ x^ ,
kt
ð4Þ
yðkÞ ¼ hðxðkÞÞ
ð5Þ
eðkÞ ¼ y0 yðkÞ,
ð6Þ
where x ¼ ½0 z0 0 , x ðy0 Þ ¼ ½ðy0 Þ0 uðy0 Þ0 0 and
~ð, zÞ f f ðx, uÞ ¼ : z þ u For the augmented system (4)–(5), the following wellknown result holds (see De Nicolao et al. 1997, for example, for a complete proof). Lemma 2.2: Consider y0 2 Rm and the associated equilibrium (ðy 0 Þ, jujðy0 Þ) of (1) with h~ððy 0 ÞÞ ¼ y 0 . Then, under A1, for system (4)–(6) there exist an open neighbourhood 0 of y 0 and a Linear Dynamic Regulator (LDR): wðk þ 1Þ ¼ Ar wðkÞ þ Br eðkÞ, uðkÞ ¼ Cr wðkÞ þ Dr eðkÞ,
wðtÞ ¼ w^ ,
k t ð7Þ ð8Þ
479
Solving the tracking problem with w 2 Rr such that 8y0 2 0 , x cl ðy0 Þ ¼ ½xðy0 Þ0 , 00 is a stable equilibrium point of the closed-loop system (4)–(8). Assumption A2: With reference to system (4)–(5), with uðkÞ ¼ 0, if yðkÞ ¼ y0 , 8k t, then xðtÞ ¼ x ðy0 Þ: Remark: Under A1, from standard continuity arguments it follows that there exist finite neighbourhoods of y 0 and of the equilibrium point x ð y0 Þ where A2 is satisfied. In fact, A2 is implied by the observability of the linearization of (4)–(5) around (x ðy0 Þ, 0).
In (9), the terminal penalty is Vf ðx^ , w^ , LDRÞ ¼ wðtÞ0 Rw wðtÞ 1 X eðiÞ0 Q e eðiÞ þ 2eðiÞ0 Q ew wðiÞ ð11Þ þ i¼t
þ wðiÞ0 Q w wðiÞ subject to (4)–(8) with
Q e ¼ ½Qe þ D0r Ru Dr þ B0r Rw Br , Q w ¼ ½Qw þ Cr0 Ru Cr þ A0r Rw Ar Q ew ¼ ½D0 Ru Cr þ B0 Rw Ar : r
3. Non-linear predictive control algorithm In order to enlarge the stability region and improve the performance provided by the LDR (7)–(8), a nonlinear MPC algorithm is now derived by suitably extending the method proposed in De Nicolao et al. (1998) for the regulation problem. To this end, for any y0 2 0 , denote by XðLDR, y0 Þ the set of all states x^ cl ¼ ½x^ 0 w^ 0 0 2 Rn þ m þ r of (4)–(8) which are steered to x cl ð y0 Þ without violating the state and control constraints (2), that is (under A2) n XðLDR, y0 Þ ¼ x^ cl 2 Rn þ m þ r lim eðkÞ ¼ 0 k!1
o subject to ð4Þ–ð8Þ and ð2Þ with xcl ðtÞ ¼ x^ cl :
r
Note that
Q e Q 0ew
Q ew Q w
> 0:
In the sequel, a sequence ½ut, tþN1 , wðt þ NÞ is termed admissible with respect to x^ if, when applied to (4) with xðtÞ ¼ x^ , it satisfies (10) and (2). Assuming that the plant state x is known, the MPC control law is obtained by solving at any time instant the FHOCP and by applying only the first control move uðtÞ ¼ MPC ðx^ Þ, where MPC ðx^ Þ is the first column of the optimal sequence u0t, tþN1 : Correspondingly, the closed-loop system is xMPC ðk þ 1Þ ¼ f ðxMPC ðkÞ, MPC ðxMPC ðkÞÞÞ:
3.1. Finite Horizon Optimal Control Problem (FHOCP) Given the prediction horizon Np , four positive definite matrices Qe , Ru , Qw and Rw of suitable dimensions, a feasible auxiliary regulator (7, 8), the control sequence ut, tþN1 ¼ ½uðtÞ, uðt þ 1Þ, . . . , uðt þ N 1Þ, minimize with respect to ½ut, tþN1 , wðt þ NÞ, the performance index Jðx^ , ½ut, tþN1 , wðt þ NÞ, y0 Þ ¼
tþN1 X
eðiÞ0 Qe eðiÞ þ uðiÞ0 Ru uðiÞ
i¼t
þ Vf ðxðt þ NÞ, wðt þ NÞ, LDRÞ
ð9Þ
subject to (4)–(6), (2) and
xðt þ NÞ 2 XðLDR, y0 Þ: wðt þ NÞ
ð10Þ
ð12Þ
ð13Þ
Denoting by X0 ðN, y0 Þ the set of states x^ for which an admissible solution for the FHOCP exists, the following stability result holds. Theorem 3.1: Consider a constant reference signal y0 2 0 , and suppose that A1–A2 hold. Then X0 ðN, y0 Þ is non-empty and (13) admits the state x ð y0 Þ as an asymptotically stable equilibrium point with region of attraction X0 ðN, y0 Þ: Remark 1: In the proposed approach, (9) must be minimized with respect both to ut, tþN1 and to the state wðt þ NÞ of the LDR at the end of the control horizon. In so doing, the algorithm does not require computation of steady state value of the state at the desired equilibrium. The optimization is performed subject to the constraint that at the end of the control horizon, the LDR is applied, while the constraint (10) is added to guarantee that at the end of the control horizon, this linear regulator is stabilizing for the considered operating condition. However, the LDR is never applied in practice, and the terminal set XðLDR, y0 Þ needs not be computed explicitly off-line; in fact, the terminal
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constraint (10) is satisfied if Vf is finite and the state constraint (2) is satisfied for all the times. In this way an implicit direct relation between y0 and the controller parameters like the terminal region and terminal penalty is obtained.
corresponding output feedback control law enjoys regionally stabilizing properties similar to those stated in Theorem 3.1. 4. Illustrative example
Remark 2: In principle, the terminal penalty (11) should be computed over an infinite horizon. However, in a practical implementation of the method, its computation can be approximated as suggested in De Nicolao et al. (1998), Remark 3. In fact, after a sufficiently long horizon, it can be assumed that the system has been driven to a sufficiently small neighbourhood of the equilibrium, where the cost to go is neligligible.
The Continuous Stirred Tank Reactor (CSTR) for an exothermic, irreversible reaction, A ! B, is described by the following dynamic model based on a component balance for reactant A and an energy balance (see Seborg et al. 1989):
q E _ CA ¼ CAf CA k0 exp CA V RT
H q E _ ð14Þ Tf T k0 exp T¼ CA V Cp RT UA þ ðTc T Þ, VCp
Remark 3: The output admissible set X0 ðN, y0 Þ is not smaller than XðLDR, y0 Þ and the performance in terms of the cost function (9) improved; otherwise, the set of constant references that can be tracked with the NMPC is no larger than that associated with the LDR: Remark 4: In the statement of Theorem 3.1, only constant references y0 have been considered. However, it is easy to extend the problem definition and the proof of Theorem 3.1 to the case of asymptotically constant reference signals. This possibility allows one to use preprogrammed set-points with an asymptotically constant behaviour, as shown in section 4.
where CA is the concentration of A, T is the reactor temperature, and Tc is the temperature of the coolant stream. The constraints are 280 K Tc 380 K, 280 K T 380 K, 0 CA 1: The objective is to control T by manipulating Tc. Letting q ¼ 100 l min1 , CAf ¼ 1 mol l1 , Tf ¼ 350 K, V ¼ 100 l, ¼ 1000 g l1 , Cp ¼ 0:239 JgK1 , H ¼ 5 104 J mol1 , E=R ¼ 8750 K, k0 ¼ 7:21010 min1 , UA ¼ 5 104 J min K1, the constant input Tc ¼ 300 K corresponds to the unstable equilibrium state CA ¼ 0:5 mol l1 , T ¼ 350 K. The open-loop responses for 5 K changes in Tc , reported in figure 1, demonstrate
Remark 5: When the system state is not measurable, one can combine the stabilizing control law with an asymptotic state observer, such as the popular Extended Kalman Filter (EKF ). By resorting to the results in Magni et al. (2004) it is possible to prove that the 0.7
1 0.9
0.5
CA (mol/L)
CA (mol/L)
0.6
0.4 0.3 0.2
0.7 0.6
0.1 0
0.8
0
2
4
6
8
0.5
10
0
2
Time (min) 440
10
8
10
345 340
400
T (K)
T (K)
8
350
420
380
335 330 325
360 340
4 6 Time (min)
320 0
Figure 1.
2
4
6
8
10
315 0
2
4
6
Time (min)
Time (min)
(a)
(b)
Open-loop response for þ5 K (a) and 5 K (b) changes in Tc.
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Solving the tracking problem
steady-state value of the temperature set-point T equal to 365 K and 375 K, respectively. In figures 2 and 3, the non-linear (continuous line) closed-loop responses are reported together with the considered reference signals. For comparison, these transients are compared with those obtained with the CRHPC algorithm (Clarke and Scattolini 1991) (dashed line), an MPC method for linear systems which guarantees closed-loop stability. As the figures clearly show, even with a small control horizon (N ¼ 4), the
CA (mol/L)
0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
380
T (K)
that the reactor exhibits highly non-linear behaviour in this operating regime. The non-linear discrete-time state-space model (1) of system (14) can be obtained by defining the state vector ¼ ½CA T 0 , the manipulated input u ¼ Tc , and the controlled output y ¼ T, and by discretizing equations (14) with sampling period t ¼ 0:05 min (numerical integrations were performed using the function ode23 of Matlab). The MPC algorithm, together with a standard EKF, was used to compute the output feedback non-linear control law. Specifically, an optimization horizon N ¼ 4 was used together with Qe ¼ 10, Qw ¼ Rw ¼ Ru ¼ 1, while the EKF was designed assuming that two mutually independent white noises ’ WGNð0, Q Þ, Q ¼diag(1, 1000, 1000) and
WGNð0, R Þ, R ¼ 10, act on the enlarged state ([CA T z]0 ) and output vector (T ), respectively. The LDR used in the implementation of the MPC algorithm is given by Ar ¼ 0:98, Br ¼ 4, Cr ¼ 5:734, Dr ¼ 12:2,
360
and it is computed with the root-locus technique applied to the model obtained by linearization of the plant around the nominal operating conditions.
Tc (K)
340
350 300 250
4.1. Simulation results
Time (min)
4.1.1. Simulation 1. Starting from the nominal operating conditions, a pre-programmed step reference signal and a reference signal with a truncated ramp-type behaviour were used in order to reach a
Figure 2. Pre-programmed step reference for T from 350 to 365 K: linear CRHPC regulator (dashed), non-linear MPC regulator (continuous), constraints (dotted) and set-point (dashed-dotted).
CA (mol/L)
0.8 0.6 0.4 0.2 0
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
0
0.5
1
1.5
2
2.5
3
T (K)
420 400 380 360
Tc (K)
340
400 200 0
Time (min)
Figure 3. Pre-programmed truncated ramp-type behaviour reference for T from 350 to 375 K: linear CRHPC regulator (dashed), nonlinear MPC regulator (continuous), constraints (dotted), and set-point (dashed-dotted).
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L. Magni and R. Scattolini 8750
0.52
CA (mol/L)
E/R (K)
0.5 0.48
8700
0.46 0.44
8650
0.42 0.4
0
2
4 6 Time (min) (a)
8
10
0.38
0
2
4 6 Time (min) (b)
8
10
0
2
4 6 Time (min)
8
10
Time (min) (c)
(d)
352
305
351.5
300
Tc (K)
T (K)
8600
351 350.5 350
295 290
0
2
4
Figure 4.
6
8
10
285
Closed-loop response for a change in E=R.
control performance is significantly improved by the proposed non-linear MPC method. Notably, the Root Mean Square Error (RMSE), computed in the intervals [0, 3] for the step response are 37 for the linear CRHPC and 16 for the non-linear MPC. Note also that the linear controller violates substantially the input constraint on Tc in the interval [0.5, 1] min. For the truncated-ramp response experiment, the RMSE computed in the intervals [0, 4] is 103 for the linear CRHPC and 2.3 for the nonlinear MPC, respectively. This time, the linear controller heavily violates the state constraint on T as well as the input constraint Tc.
4.1.2. Simulation 2. The robustness of the proposed MPC method was investigated by forcing a ramptype change on the parameter E=R starting from nominal operating conditions, as shown in figure 4(a). Correspondingly, the transients of the state and control variables reported in figure 4(b)–(d) were computed. These results clearly show that the MPC algorithm can reject plant-parameter perturbations without the need for an explicit estimation of the plant parameters.
face of asymptotically constant reference signals. The stability results rely on very mild assumptions on the linearized plant model. The method requires the availability of a (dynamic) output feedback linear regulator stabilizing the ensemble plant þ integrator in the considered operating conditions.
Acknowledgement The authors acknowledge the partial financial support by MURST Project ‘New techniques for identification and adaptive control of industrial system’.
Appendix Proof of Theorem 2.1: Define ’ð, u, y0 Þ ¼ ½ðf~ð, uÞ Þ0 ðh~ðÞ y0 Þ0 0 . By assumption, it follows that ’ððy0 Þ, u ðy 0 Þ, y 0 Þ ¼ 0, and, in view of A1, " @’ð, u, y0 Þ A~ ðy 0 Þ I rank ¼ rank @ð, uÞ ðy0 Þ, uðy0 Þ, y0 C~ ðy0 Þ
B~ ðy 0 Þ
#
0
¼ n þ m: 5. Conclusions
Then, the proof follows from the implicit function theorem.
The novel output-feedback MPC algorithm for nonlinear systems proposed in this paper ensures local closed-loop stability and zero-error regulation in the
Proof of Theorem 3.1: Given a constant reference signal y0 2 0 , we show that 8x 2 X0 ðN, y0 Þ, Vðx^ Þ :¼ Jðx^ , ½u0t, tþN1 , wðt þ NÞ0 , y0 Þ is a Lyapunov function
483
Solving the tracking problem for (13). First, note that in view of Theorem 2.1, A2 and (9), VðxÞ > 0 8x 6¼ x ðy0 Þ 2 X0 ðN, y0 ÞÞ and Vðx ðy0 ÞÞ ¼ 0. Then, the closed-loop system (4)–(8) formed by the non-linear system and the linear dynamic regulator can be rewritten in the following way ðk þ 1Þ ¼ fððkÞ, ðkÞÞ
ðkÞ ¼
xðkÞ
"ðkÞ ¼ hððkÞ, ðkÞ, y0 ðkÞÞ
ð15Þ
ðkÞ ¼ K"ðkÞ
ð16Þ
wðkÞ
"
# ,
ðkÞ ¼ "
fððkÞ, ðkÞÞ ¼
uðkÞ
#
2 ðkÞ
" ,
"ðkÞ ¼
f ðxðkÞ, uðkÞÞ
eðkÞ
#
wðkÞ
# ,
2 ðkÞ " hððkÞ, ðkÞÞ ¼
y0 ðkÞ hðxðkÞÞ
#
wðkÞ "
Dr
Cr
Br
Ar
K¼
# :
Then, the optimization problem can be rewritten as the minimization, with respect to t, tþN1 ¼ ½ðtÞ, ðt þ 1Þ, . . . , ðt þ N 1Þ, of
J ðx^ , t, tþN1 , y0 Þ ¼
tþN1 X
"ðiÞ0 Q"ðiÞ þ ðiÞ0 RðiÞ
i¼t 1 X
þ
tþN1 X
eðiÞ0 Qe eðiÞ þ uðiÞ0 Ru uðiÞ
i¼t
þ 2 ðt þ N 1Þ0 Rw 2 ðt þ N 1Þ 1 X þ eðiÞ0 Q e eðiÞ þ 2eðiÞ0 Q ew wðiÞ þ wðiÞ0 Q w wðiÞ , i¼tþN
with "
to the minimization (subject to (4)) of
"ðiÞ0 Q"ðiÞ þ ðiÞ0 RðiÞ,
i¼tþN
with Q ¼ diagðQe , Qw Þ, R ¼ diagðRu , Rw Þ, subject to the open-loop dynamics (15) and
with Q e , Q w , Q ew given by (12) with uðiÞ given by (7)–(8) 2 ðt þ N 1Þ: In this way, we equivalence of the minimization of We can now show that
and, for i t þ N, where wðt þ NÞ ¼ have proven the J and J :
Vðx^ Þ ¼ Jðx^ , ½u0t, tþN1 , wðt þ NÞ0 , y0 Þ ¼ J x^ , 0t, tþN1 , y0 is a Lyapunov function. Letting 0t, tþN1 be the optimal solution at time t, the sequence ~ tþ1, tþN ¼ ½0tþ1, tþN1 , K"ðt þ NÞ is a feasible (but not necessarily optimal) solution at time t þ 1, in view of the invariance of XðLDR, y0 Þ with respect to the LDR control. Then, Vðf ðx^ , MPC ðx^ ÞÞ J f ðx^ , MPC ðx^ ÞÞ, ~ tþ1, tþN1 , y0 Vðx^ Þ "ðtÞ0 Q"ðtÞ ðtÞ0 RðtÞ which implies Vðf ðx^ , MPC ðx^ ÞÞ Vðx^ Þ < 0, 8"ðtÞ, ðtÞ 6¼ 0: Therefore, we are left to show that there are no perturbed closed-loop trajectories of xMPC ðÞ such that VðxMPC ðÞÞ remains constant. By contradiction this would entail "ðtÞ ¼ ðtÞ ¼ 0, 8t, that is eðtÞ ¼ uðtÞ ¼ 0, 8t, or also y0 hðxMPC ðtÞÞ ¼ 0, 8t: Then, in view of A2, xMPC coincides with the equilibrium x ðy0 Þ and the thesis is proven: Finally, note that X0 ðN, y0 Þ is a positive invariant set. In fact, if x^ 2 X0 ðN, y0 Þ, letting 0t, tþN1 be the optimal solution at time t, the sequence ~ tþ1, tþN ¼ ½0tþ1, tþN1 , K"ðt þ NÞ is a feasible sequence at time t þ 1, so that f ðx^ , MPC ðx^ ÞÞ 2 X0 ðN, y0 Þ.
ðtÞ ¼ ½x^ 0 00 ðiÞ ¼ K"ðiÞ,
for i t þ N:
To see this, first note that for t k t þ N 1, the variables eðkÞ, uðkÞ, and x(k) are not affected by 2 ðÞ: Hence, the choice 2 ðkÞ ¼ 0, for t k t þ N 2 is the one that minimizes the cost function, and only 2 ðt þ N 1Þ ¼ wðt þ NÞ is a real optimization parameter. Then, the minimization of J is equivalent
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Lalo Magni was born in Bormio (SO), Italy, in 1971. He took the Laurea Degree (summa cum laude) in Computer Engineering at the University of Pavia, Italy, in 1994. He received his Ph.D. in Electronic and Computer Engineering in 1998. Currently, he is Associate Professor at the University of Pavia. From October 1996 to February 1997 and in March 1998, he was at CESAME, Universite´ Catholique de Louvain, Louvain La Neuve (Belgium). From October to November 1997 he was at the University of Twente with the System and Control Group in the Faculty of Applied Mathematics. In 2003 he was a plenary speaker at the 2nd IFAC Conference Control Systems Design (CSD’03). His current research interests include non-linear control, predictive control, receding-horizon control, robust control and process control. His research is testified by more than 25 papers published in the main international journals of the field. He has been Guest Editor of the Special Issue Control of Nonlinear Systems with model predictive control in the International Journal of Robust and Nonlinear Control. He serves as an Associate Editor of the IEEE Transactions on Automatic Control. Riccardo Scattolini was born in Milan, Italy, in 1956. In 1979, he received the Laurea degree in Electrical Engineering from the Politecnico di Milano, where he is presently Professor of Process Control. During the academic year 1984/85, he was with the Department of Engineering Science, Oxford University. He also spent one year working in industry on the simulation and control of chemical plants. In 1991, he was awarded the Heaviside Premium of the Institution of Electrical Engineers. His current research interests include predictive and robust control theory, with applications to automotive engine control and power plants control.
