Model Predictive Control of Continuous-Time Nonlinear Systems With ...

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Model Predictive Control of Continuous-Time Nonlinear Systems With Piecewise Constant Control Lalo Magni and Riccardo Scattolini

Abstract—A new model predictive control (MPC) algorithm for nonlinear systems is presented. The plant under control, the state and control constraints, and the performance index to be minimized are described in continuous time, while the manipulated variables are allowed to change at fixed and uniformly distributed sampling times. In so doing, the optimization is performed with respect to sequences, as in discrete-time nonlinear MPC, but the continuous-time evolution of the system is considered as in continuous-time nonlinear MPC. Index Terms—Nonlinear control, nonlinear model predictive control (MPC), sampled systems.

I. INTRODUCTION

T

HE extraordinary industrial success of model predictive control (MPC) techniques based on linear plant models, see, e.g., the survey paper [20], motivates the development of MPC algorithms of nonlinear systems. Nowadays, there are many theoretical results, see [17] and [12], as well as industrial applications, see [21], which witness that MPC for nonlinear systems is going to have a diffusion and popularity similar to the one achieved by MPC algorithms for linear systems. MPC methods for nonlinear systems are developed by assuming that the plant under control is either described by a continuous-time model, see [3], [9], [10], [15], [16], and [18], or by a discrete-time one, see [5], [11], and [13]. A continuous-time representation is much more natural, since the plant model is usually derived by resorting to first principles equations, but it results in a more difficult development of the MPC control law, which in principle calls for the solution of a functional optimization problem. As a matter of fact, the performance index to be minimized is defined in a continuous-time setting and the overall optimization procedure is assumed to be continuously repeated after any vanishingly small sampling time, which often turns out to be a computationally intractable task. On the contrary, MPC algorithms based on a discrete-time system representation are computationally simpler, but require the discretization of the model equations, so that they rely from the very beginning on an approximate system representation. Moreover, the performance index to be minimized as well as the state constraints Manuscript received February 27, 2002; revised May 7, 2003 and January 30, 2004. Recommended by Associate Editor A. Bemporad. This work was supported in part by the MURST Project ”New techniques for identification and adaptive control of industrial systems.” L. Magni is with the Dipartimento di Informatica e Sistemistica, Universita’ di Pavia, 27100 Pavia, Italy (e-mail: [email protected]). R. Scattolini is with the Dipartimento di Elettronica e Informazione, Politecnico di Milan, 20133 Milan, Italy (e-mail: [email protected]) Digital Object Identifier 10.1109/TAC.2004.829595

only consider the system behavior in the sampling instants, so ignoring the intersample behavior, which in some cases could be significant in the evaluation of the control performance. In this paper, a different approach is taken, which accounts fully for the hybrid nature of sampled data control systems. The plant under control, the state and control constraints and the performance index to be minimized are described in continuous-time, while the manipulated variables are allowed to change at fixed and uniformly distributed sampling times. In so doing, one has to deal with the optimization with respect to sequences, as in discrete-time nonlinear MPC, while taking into account the continuous-time evolution of the system. The stability properties of the algorithm are established by including in the cost function a penalty and a terminal constraint on the state at the end of the prediction horizon, according to well-known results, see [17]. However, the proof of stability is not trivial due to the sampled-data nature of the problem and the use of piece-wise constant signals. A similar approach was already taken in [7], where the sampling mechanism was explicitly considered, for the solution of a regulation problem, but a functional optimization problem had to be solved iteratively. Piecewise constant signals have also been used in [1] and [8] for the design of reference governors. Preliminary results concerning the hybrid approach considered in this paper are reported in [14]. All the proofs are gathered in an Appendix to improve readability. II. PROBLEM STATEMENT AND PRELIMINARY RESULTS , denotes the EuIn the paper, for any vector clidean norm in , , where is an arbitrary Hermitian matrix, denotes the weighted norm. For any and denote the largest Hermitian matrix , and the smallest real part of the eigenvalues of the matrix , stands for the induced 2-norm of . respectively and denotes the closed ball of radius defined with the weighted , . norm , i.e., Consider a plant described by the nonlinear continuous-time dynamic system (1) where

is the state, is the input, and is a function of its arguments. The state and control variables are restricted to fulfill the following constraints:

0018-9286/04$20.00 © 2004 IEEE

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MAGNI AND SCATTOLINI: MODEL PREDICTIVE CONTROL OF CONTINUOUS-TIME NONLINEAR SYSTEMS

where and are compact subsets of and , respectively, both containing the origin as an interior point. The movefor a conment of (1) from the initial time and initial state trol signal is denoted by . , Given a suitable sampling period , and letting nonnegative integer, be the sampling instants, the goal is to determine a “sampled” feedback control law for the computation of a piecewise constant control signal which exponentially stabilizes the origin of the associated closed-loop system. For the solution of this regulation problem, the following preliminary assumption is introduced. Assumption 1: Letting

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be called feasible hereafter. Besides, one can also wish to find a control law (3) in order to enlarge the maximal output admissible set guaranteed by a given feasible control law and to optimize a given performance index. Let us now suppose that a feasible control law (3) satisfying the following assumption is known. is a function Assumption 2: The feasible control law with Lipschitz constant . For this control law, an associated sampled output admissible set can be computed as follows. First, define the linearization of (1) at the origin (5) Then, introduce the discretization of (5) given by (6)

the pair is stabilizable. Given a generic sampled feedback control law

with (3)

with , the description of the hold mechanism implicit in (3) calls for a state augmentation. Letting , the closed-loop system (1)–(3) is

(4) and its movement from the initial time and initial state is denoted by

With reference to the closed-loop system (4), define the following sets. Definition 1: A sampled output admissible set associated such that for all , to (4) is a set , , , , . In is a state invariant set, associated to the other words, closed-loop system (4), defined at the sampling instants and such that: i) the state and control constraints (2) are satisfied in all the future continuous-time instants, and ii) the regulation problem is asymptotically solved. The (unique) maximal is defined as the union of sampled output admissible set all sampled output admissible sets. Definition 2: An output admissible set associated to (4) such that for all , is a set , where is the closest sampling time in , , . In the future, other words, is a set, defined at any continuous-time instant , of states of the closed-loop system (4) such that: i) the state of (1) at the closest sampling time in the future , and ii) the state and control constraints (2) belongs to are satisfied in all the future continuous-time instants. The is defined as (unique) maximal output admissible set the union of all output admissible sets. The regulation problem can now be formally stated as the problem of finding a sampled control law (3) such that its maximal output admissible set is non empty. Such a control law will

Finally, let

In view of Assumption 2, it is then easy to show that the of the linearized closed-loop matrix discrete-time system (6) is Hurwitz and the following result holds. be a feasible control law, suppose that Lemma 1: Let Assumptions 1 and 2 are satisfied and consider a positive–defsuch that inite matrix and two real positive scalars and . Define by the unique symmetric positive–definite solution of the following Lyapunov equation: (7) where

and (8) Then, there exist two constants specifying a neighborhood

and of the origin of the form (9)

such that i) ii)

,

,

;

(10) iii)

is a sampled output admissible set for (4);

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iv)

is a positive–definite function decreasing in the sampling times along the trajectory of (4). Remark 1: An obvious way to determine a feasible sampled control law is to choose a suitable , to consider the linearization of (1) around the origin and the sampled linear model described by (6) and to synthesize, with any standard linear control synthesis technique, a linear control law (11) such that is discrete-time Hurwitz. Provided that defined in (6) is discrete-time Hurwitz, an even simpler choice is to set equal to zero in (11). However, the maximal output admissible set associated to these linear sampled control laws is in general quite small due to their local nature and their performance are likely to be improved by taking into account the nonlinearity of the system under control.

where the terminal penalty

is selected as

The minimization of (13) must be performed under the following constraints: ; i) the state dynamics (1) with with given by (12); ii) the constraints (2), iii) the terminal state constraint . According to the receding horizon approach, the state-feedback MPC control law is derived by solving FHOCP at every sampling time instant , and applying the constant control , where is the signal . In so doing, first column of the optimal sequence one implicitly defines the sampled state-feedback control law (14)

III. SAMPLED MPC CONTROL LAW , Let us suppose that a feasible sampled control law henceforth called the auxiliary control law, is known together with an associated sampled output admissible set and the Lyapunov function both given in Lemma 1. It is now shown how MPC allows one to extend the maximal output admissible set of and to improve the control performance by minimizing a cost function suitably chosen by the designer. and the control seTo this end, given the sampling time quence

with , define the Finite Horizon piece-wise constant control signal

(12) where and . the signal in the Moreover, denote by . interval For (1), the MPC control problem here considered is based on the solution of the following. Finite Horizon Optimal Control Problem (FHOCP) Given the sampling time , the control horizon , the pre, two positive–definite matrices diction horizon , and , a feasible auxiliary control law satisfying Assumpgiven in Lemma 1 tion 2, the matrix and the region with and , at every sampling , the perfortime instant minimize, with respect to mance index

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In order to establish the properties of the control law (14), first let

be the movement of (4) with . Then, define the following sets. be the set of states Definition 3: Let of system (1) at the sampling times such that there exists a for FHOCP. feasible control sequence Definition 4: Let be the set of states such that for all , , where is the closest sampling time in the future, , , . The main stability results of the proposed MPC algorithm can now be stated. Theorem 2: Under Assumptions 1 and 2 i) the origin is an exponentially stable equilibrium point for the closed-loop system formed by (1) and (14) with ; maximal output admissible set , ; ii) iii) , ; such that , iv) there exist a finite . Remark 2: The use of different control ( ) and prediction ( ) horizons is already well known in MPC for linear systems, see the popular GPC algorithm [4]. As discussed in [13], it is even more important in nonlinear MPC, where the need to reduce the size of the optimization problem is of crucial importance for computational reasons. Remark 3: In Remark 1, it has been suggested to derive the auxiliary control law by means of linearization and Lemma 1 provides a systematic tool to derive the terminal penalty and the terminal inequality constraint. However, it is possible to relax Assumptions 1 and 2 and to consider any other feasible

MAGNI AND SCATTOLINI: MODEL PREDICTIVE CONTROL OF CONTINUOUS-TIME NONLINEAR SYSTEMS

piecewise constant control law provided that an associated possuch that itive–definite function

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out to be substantially greater than the one implied by the proposed solution. In fact, the maximum integration time used in the integration of the state (1) can be (and is usually) chosen much smaller than the sampling time required for the solution of the MPC optimization problem. IV. CONCLUSION

and a sampled output admissible set associated to it are known. In particular, this allows one to include in this framework the results of [7], which show how MPC can be used to control nonholonomic systems. On the contrary, in [7] it is required that

As is well know for linear systems too, this condition is not fulfilled by the terminal penalty derived in Lemma 1. Remark 4: According to [5], the terminal weight in (13) can be dropped with the terminal inequality constraint iii) in FHOCP provided that . This can appear to be only of theoretical interest, however it is in practice sufficient to consider such that the settling time of the system (1) is a value of completely included into the prediction horizon. This is what is usually done in many practical implementations of linear and nonlinear MPC. In the FHOCP optimization problem, continuous-time state constraints are considered. It can appear that this approach is only conceptual, because any numerical implementation needs a time discretization and the constraints satisfaction can be checked only in the integration time instants. However, this is not a significant limitation; in fact, letting be the (maximum) integration step used in the optimization phase to simulate the plant (1) with the control signal (12), the following result holds. Theorem 3: Let

if: a) and b) , then . From this result it is clear that one can choose the maximum integration step and a more conservative discrete-time state so as to guarantee continuous-time constraint (defined by state constraint satisfaction. More precisely, given and such , a nonnegative integer , a constant integration that , condition ii) in the FHOCP can be replaced by step

The intrinsic characteristics of sampled data control systems have been explicitly considered in this paper for the development of an MPC algorithm for nonlinear systems. The main peculiarities of the proposed method are related to the piecewise nature of the control signal and to the use of different control and prediction horizons. These two features lead to a tractable optimization problem, where the cost minimization is performed with respect to sequences and the number of future control moves to be selected can be small. Notably, once a stabilizing auxiliary control law is known, the method here proposed can only improve its performance with respect to the adopted cost function. This renders more attractive the approach also when a stabilizing control law is already available. APPENDIX A. Proof of Lemma 1 i) From the smoothness of and and recalling that , , it follows that there exist and , such that , , . Thus, let . , the inii) Letting equality (10) is equivalent to

(15) From (7), it is easy to see that inequality (15) is equivalent to

Moreover, from the definition of The use of a more conservative constraints set has already been proposed for linear systems in [2]. Remark 5: An alternative approach to the solution of the MPC problem consists in reformulating it in a completely discrete-time framework using the sampling time . In this way, also the state constraints (2) should be modified according to the results of Theorem 3, but the conservatism introduced turns

it follows that

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Define , then (16) becomes (17), as shown at the bottom of the page. Define and

, given in (8), as . Then, by the of and there exists definition of , such that the inequality (18) holds [see (9)], which implies that (10) holds as well. Moreover, from (10) it follows that, (19)

Note that and are finite because (17) is satisfied provided that

. Then,

so that

implies . This fact with ( ) means that is a sampled output admissible set associated to (4). Finally, from (19) it iv) is satisfied. follows that

B. Proof of Theorem 2

(18) In fact

is the maximal In view of Definitions 2 and 4, if then sampled output admissible set of (4) with is the maximal output admissible set of (4) with . Note in fact that, from Definition 4 it follows the constraints are not satisfied for that and/or . But in view of cannot belong to any output Definition 2 this means that . admissible set of (4) with First of all, from Lemma 1 it follows that is and are nonempty, then also is the maximal nonempty. Let us now show that sampled output admissible set for (1) and (14). In fact, letting and the associated solution of the FHOCP at time , a feasible solution at for the FHOCP is time

(20) where

Moreover, in view of the zero-order-hold sampling of (5), given in (6), is formed by terms of of order higher than one so as . Similarly, considering the definition that

, is the optimal control signal

computed at time and is the value of the state of the MPC closed-loop system at time . Then, by definition, and, in view of constraints ii) of the FHOCP, (2) are satisfied along the trajec. Finally, tory of (4) with the MPC control law is not defined so that a sample output cannot exist. admissible set larger than Let us now show that the origin is an asymptotically stable equilibrium point for the closed-loop system (1), (14). To this

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end, define that •

, as shown at the top of the page, and note is bounded . Moreover

and, since

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and are positive–definite matrices, if , both and

are bounded. As shown in [19], these facts prove that (21) • At time given by (20) is a (suboptimal) feasible solution for the new FHOCP so that

As for the exponential stability property, it easily follows by analyzing the linearized closed-loop system, see [6]. The proof of ii)–iv) can be derived as in [13, Th. 6]. C. Proof of Theorem 3 First, note that exists in view of the smoothness of and , and the compactness of , because . Finally, suppose that there exist some times such that and call the first time such that . Then, , and then

(22) where

and this contradicts the assumption that there exists some times such that . REFERENCES

From Lemma 1 and the conditions , , introduced in the formulation of the FHOCP, it follows that

(23) In conclusion, using (21) when (23) when , ,

,

, and

[1] A. Bemporad, “Reference governor for constrained nonlinear systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 415–419, Mar. 1998. [2] L. Berardi, E. De Santis, M. D. Di Benedetto, and G. Pola, “Controlled safe sets for continuous time linear systems,” in Proc. Eur. Control Conf., J. L. M. de Carvalho, F. A. C. C. Fontes, and M. D. R. De Pinho, Eds., Porto, Portugal, 2001, pp. 803–808. [3] H. Chen and F. Allgöwer, “A quasiinfinite horizon nonlinear model predictive control scheme with guaranteed stability,” Automatica, vol. 34, pp. 1205–1217, 1998. [4] D. W. Clarke, C. Mothadi, and P. S. Tuffs, “Generalized predictive control-Parts I and II,” Automatica, vol. 23, pp. 137–160, 1987. [5] G. De Nicolao, L. Magni, and R. Scattolini, “Stabilizing receding-horizon control of nonlinear time-varying systems,” IEEE Trans. Automat. Contr., vol. 43, pp. 1030–1036, June 1998. , “Stability and robustness of nonlinear receding-horizon control,” [6] in Nonlinear Model Predictive Control, F. Allgöwer and A. Zheng, Eds. Boston, MA: Birkhäuser, 2000, Progress in Systems and Control Theory, pp. 3–22. [7] F. A. C. C. Fontes, “A general framework to design stabilizing nonlinear model predictive controllers,” Syst. Control Lett., vol. 42, pp. 127–143, 2001. [8] E. G. Gilbert and I. V. Kolmanovsky, “Set-point control of nonlinear systems with state and control constraints: A Lyapunov-function, reference-governor approach,” presented at the 38th Conf. Decision Control, Phoenix, AZ, 1999. [9] A. Jadbabaie and J. Hauser, “Unconstrained receding-horizon control of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 46, pp. 776–783, May 2001.

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[10] A. Jadbabaie, J. Primbs, and J. Hauser, “Unconstrained receding horizon control with no terminal cost,” presented at the Amer. Control Conf., Arlington, VA, June 25–27, 2001. [11] S. S. Keerthi and E. G. Gilbert, “Optimal, infinite-horizon feedback laws for a general class of constrained discrete-time systems,” J. Optim. Theor. Appl., vol. 57, pp. 265–293, 1988. [12] L. Magni, “Editorial of the special issue on control of nonlinear systems with model predictive control,” Int. J. Robust Nonlinear Control, vol. 13, pp. 189–190, 2003. [13] L. Magni, G. De Nicolao, L. Magnani, and R. Scattolini, “A stabilizing model-based predictive control for nonlinear systems,” Automatica, vol. 37, pp. 1351–1362, 2001. [14] L. Magni, R. Scattolini, and K. J. Åström, “Global stabilization of the inverted pendulum using model predictive control,” presented at the 15th IFAC World Congr., Barcelona, Spain, 2002. [15] L. Magni and R. Sepulchre, “Stability margins of nonlinear receding horizon control via inverse optimality,” Syst. Control Lett., vol. 32, pp. 241–245, 1997. [16] D. Q. Mayne and H. Michalska, “Receding horizon control of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 35, pp. 814–824, Sept. 1990. [17] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, vol. 36, pp. 789–814, 2000. [18] H. Michalska and D. Q. Mayne, “Robust receding horizon control of constrained nonlinear systems,” IEEE Trans Automat. Contr., vol. 38, pp. 1623–1633, Oct. 1993. [19] H. Michalska and R. B. Vinter, “Nonlinear stabilization using discontinuous moving-horizon control,” IMA J. Math Control Inform., vol. 11, pp. 321–340, 1994. [20] S. J. Qin and T. A. Badgwell, “An overview of industrial model predictive control technology,” in Proc. 5th Int. Conf. Chemical Process Control, J. C. Kantor, C. E. Garcia, and B. Carnahan, Eds., 1996, pp. 232–256. [21] , “An overview of nonlinear model predictive control applications,” in Nonlinear Model Predictive Control, F. Allgower and A. Zheng, Eds. Berlin, Germany: Birkhäuser, 2000, pp. 369–392.

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Lalo Magni was born in Bormio (SO), Italy, in 1971. He received the Laurea degree (summa cum laude) in computer engineering from the University of Pavia, Pavia, Italy, in 1994 and the Ph.D. degree in electronic and computer engineering in 1998. Currently, he is an Assistant Professor with the University of Pavia. From October 1996 to February 1997, and in March 1998, he was with CESAME, Universitè Catholique de Louvain, Louvain La Neuve, Belgium. From October to November 1997, he was at the University of Twente, Twente, The Netherlands, 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 nonlinear control, predictive control, receding-horizon control, robust control and process control. His research is testified by about 25 papers published in the main international journals of the field. Dr. Magni was Guest Editor for the Special Issue ”Control of Nonlinear Systems with Model Predictive Control” in the International Journal of Robust and Nonlinear Control. He also serves as an Associate Editor for the IEEE TRANSACTIONS ON AUTOMATIC CONTROL.

Riccardo Scattolini was born in Milan, Italy, in 1956. He received the Laurea degree in electrical engineering from the Politecnico di Milan, Milan, Italy, in 1979. He is currently a Professor of Process Control with the Politecnico di Milan. During the academic year 1984/1985, he was with the Department of Engineering Science, Oxford University, Oxford, U.K. He also spent one year working in industry on the simulation and control of chemical plants. His current research interests include predictive and robust control theory, with applications to automotive engine control and power plants control. Dr. Scattolini was awarded the Heaviside Premium of the Institution of Electrical Engineers (IEE) in 1991.