Robust MPC of constrained discrete-time nonlinear systems based on ...

Robust MPC of constrained discrete-time nonlinear systems based on uncertain evolution sets: application to a CSTR model ´ D. Lim´on Marruedo† , J.M. Bravo, T. Alamo and E.F. Camacho

1

†

Departamento de Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla Escuela Superior de Ingenieros, Camino de los Descubrimientos s/n 41092. Sevilla Telephone: +34 954487357, Fax: +34 954487340, email: [email protected]

Abstract A robust MPC for constrained discrete-time nonlinear system with additive uncertainties is presented. The MPC is based on the so called uncertain evolution sets, which are the sets containing the predicted evolution of the uncertain system under any admissible uncertainty. These sets are incorporated in the MPC formulation in order to achieve robust stability. By choosing as terminal constraint a robust positively invariant set, a dual-mode controller with guaranteed robust stability is obtained. Stability is guaranteed despite suboptimality of the computed solution. This controller is applied to a highly nonlinear system: a simulation model of a CSTR for an exothermic irreversible reaction. The uncertain evolution sets are computed on-line by using interval arithmetic. This procedure obtains quite good results with a similar computational effort than the nominal prediction.

1 Introduction The Moving Horizon or Model Predictive Control has become a preferred control strategy in academia and in the process industry. The reasons of this success include the constraints handling and the optimal criteria in the computation of the control action. Underlying control theoretic problems on linear MPC [1] and on nonlinear MPC are well studied. An interesting survey of nonlinear MPC can be found in [8] where an standard formulation of the MPC is established and sufficient conditions to guarantee asymptotic stability are given. When uncertainties are present, they must be taken into account in the computation of the control law in order to get robust stability. Some authors have tackled this problem. In [9] a dual-mode receding horizon controller is proposed and robustness under decaying 1 The authors acknowledge MCYT-Spain (contract QUI990663-C02-01) for funding this work.

additive uncertainties is achieved by a proper choice of the terminal region. In [6] a robust MPC strategy based on H∞ cost function is presented and in [11] a closed-loop min-max technique is shown. In this paper, a robust MPC for constrained discretetime nonlinear system with additive uncertainties is presented. It is based on the uncertain evolution sets. These are the sets which contain the predicted evolution of the uncertain system under any admissible uncertainty. Since the nonlinearity of the model makes these sets difficult to be accurately obtained, conditions are established to compute them by using approximate procedures. These sets are incorporated in the MPC formulation in order to consider the uncertainties in the optimal input computation. By choosing as terminal constraint a robust positively invariant set, a dual-mode controller is proposed. If the initial state is such that the optimization problem is feasible, then robust stability is guaranteed. Hence, the uncertain closed-loop system reaches the terminal region in a finite number of steps, and the local controller keeps the system in the terminal region. This controller does not require optimality of the solution to guarantee stability. Furthermore, the commutation to the local controller may be avoided by a modified terminal constraint. Interval arithmetic is used for the computation of the uncertain evolution set. This procedure is very useful for the on-line implementation of the proposed controller, since the computational effort is similar to the nominal prediction. Furthermore, quite good results are obtained since the method provides local approximations to the uncertain set. This controller is applied to a highly nonlinear system: a simulation model of a CSTR for an exothermic reaction. The coolant temperature is the input which is constrained. Simulation results of the proposed MPC are shown.

The paper is organized as follows: In section 2, some preliminary results are established, uncertain evolution sets are presented and some basic results in interval arithmetic are given. In section 3, the Robust MPC strategy is presented, and in the following section, closed loop stability is proved. The application of the proposed controller to a CSTR is shown in section 5, and finally some conclusions are given.

2 Preliminary results 2.1 System description Consider an uncertain nonlinear discrete-time system given by xk+1 = f (xk , uk ) + wk (1) n

m

where xk ∈ IR is the state of the system and uk ∈ IR is the control vector at sample time k. The system is subject to constraints on the state xk ∈ X and on the control action uk ∈ U , where X is a closed set and U a compact set, both of them containing the origin. The vector wk ∈ IRn is the uncertainty which is modeled as additive. The only assumption on wk is that it is bounded in a compact set W that contains the origin. wk ∈ W

(2)

Notice that the additive uncertainty can model perturbed systems and a wide class of model mismatches taking into account that these ones might depend on the state of the system, since xk+1 = f˜(xk , uk ) = f (xk , uk ) + ∆f (xk , uk ) | {z } wk

where wk ∈ W forall xk ∈ X, uk ∈ U . The only assumption on W is that it is bounded. This kind of model uncertainty has been used in previous papers about robustness in MPC, as in [9] and [7]. The model given by x ˆk+1 = f (xk , uk ) denotes the nominal model of the system. For the real state of the system at sample time k, xk and for a given sequence of control inputs denoted uF (k) = {u(k|k), u(k + 1|k), · · · , u(k + N − 1|k)}, the future state of the system at time k + j predicted by using the nominal model is denoted x ˆ(k + j|k). Hence, x ˆ(k + j + 1|k) = f (ˆ x(k + j|k), u(k + j|k)), where x ˆ(k|k) = xk . 2.2 Uncertain evolution sets Since there are mismatches between the real system and the nominal model, the predicted evolution using the nominal model differs from the real evolution of the system. In order to consider this effect in the controller synthesis, it is interesting to compute the region around

the nominal prediction that confines the state of the system under any admissible uncertainty. This idea is the basis of the so-called uncertain evolution sets. Consider that the state of the system at sample time k is xk and a sequence of control inputs uF (k) is applied to the uncertain system. The evolution of the system depends on the uncertainties, that are known to belong to the bounded set W . The uncertain evolution set at sample time k + j is denoted as Xj (xk , uF (k)). This set is the region that confines the evolution of the uncertain system under any admissible uncertainty at sample time k + j. Note that this set depends on xk , on the sequence of inputs from k to k + j − 1, i.e. {u(k|k), · · · , u(k + j − 1|k)} and on the set of uncertainties W . Since the exact computation of these sets is difficult for non-linear systems, approximations to this sets are used. This approximate sets are easily computed using different approaches: based on the Lipschitz continuity of the system or based on interval arithmetic for instance. Interval arithmetic has been used in this paper. Hereafter some definitions and results related to the uncertain evolution sets are presented. Consider sets A and B ⊂ IRn , a vector x ∈ IRn and a function g(x) : IRn → IRn then the following sets are defined: x+A = {x+a, a ∈ A}, g(A) = {g(a), a ∈ A}, A + B = {a + b, a ∈ A, b ∈ B}, and A ∼ B = {c ∈ IRn : c + B ⊆ A}. Definition 1 Consider a system given by (1), consider that the state at sample time k is xk and consider a sequence of control inputs uF (k) then the uncertain evolution set at sample time k + j is Xj (xk , uF (k)) = f (Xj−1 (xk , uF (k)), u(k +j −1|k))+W (3) where X1 (xk , uF (k)) = f (xk , u(k|k)) + W . Note that Xj (xk , uF (k)) is the set that contains the uncertain evolution of all the states of Xj−1 (xk , uF (k)), that is Xj (xk , uF (k)) =

[

f (x, u(k + j − 1|k)) + W

x∈Xj−1

Due to the nonlinear nature of the model, these sets are very difficult to be computed and thus, they are not useful from a practical point of view. In order to reduce the complexity of the computation, these sets can be substituted by approximated approaches. These approaches are based on conservative procedures with less computational burden. The aim of these procedures is

to compute an approximation of f (A, u) for a given set A ⊆ IRn and a given control action u. In the sequel ψ(A, u) denotes the approximate set of f (A, u). The procedure to compute approximate sets must satisfy the following conditions:

2.3 Interval arithmetic In order to implement the computation of the uncertain evolution sets, it is necessary to find procedures that satisfies assumption 1. A procedure based on interval arithmetic is used in this paper.

Assumption 1 Let A be a set A ⊆ IRn and let u be a control action. Consider a procedure ψ(·, u) : A ⊂ IRn → ψ(A, u) ⊂ IRn , then ψ(·, ·) can be used to compute uncertain evolution sets if it satisfies the following conditions

Interval mathematics is a generalization of real mathematics in which intervals numbers replace real numbers, interval arithmetic replaces real arithmetic, and interval analysis replaces real analysis [10]. An interval number X = [a, b] is the set of real numbers such that {x : a ≤ x ≤ b}. If the dimension of the variable is n, then it is extended to interval vectors, where each component is an interval variable. Note that an interval vector X is a set in IRn . The set of real compact intervals [a, b], a, b ∈ IR is denoted by I, and the sets of interval vectors in IRn is denoted by I n .

• Inclusion condition: f (A, u) ⊆ ψ(A, u). • Monotonic condition: Let B a set such that B ⊆ A, then ψ(B, u) ⊆ ψ(A, u). Based on this procedure it is possible to compute a sequence of sets which are outer approximations to the exact sets of the uncertain evolution of the system. These approximate sets are approximate uncertain evolution sets. With slight abuse of notation, these approximate sets are merely called uncertain evolution sets in this paper. Definition 2 (Uncertain evolution set) Consider a system given by (1), consider a procedure ψ(·, ·) that satisfies assumption 1 for the system, consider the state at sample time k is xk and a sequence of control inputs uF (k) then the uncertain evolution set at sample time k + j is Xˆj (xk , uF (k)) = ψ(Xˆj−1 (xk , uF (k)), u(k+j −1|k))+W (4) where Xˆ1 (xk , uF ) = f (xk , u(k|k)) + W . Due to the inclusion condition, the uncertain evolution set Xˆj (xk , uF (k)) contains all the states where the uncertain system may be at sample time k + j from xk and applying the sequence of inputs uF (k). That is Xj (xk , uF (k)) ⊆ Xˆj (xk , uF (k))

(5)

Another interesting property of these sets is stated in the following lemma. Lemma 1 Consider that the state of the system at time k is xk , consider a sequence of control inputs uF (k), consider the uncertain evolution set Xˆj (xk , uF (k)). Consider xk+1 the state where the system evolves at k+1 when at k the input u(k|k) is applied to the system. Consider a sequence of inputs at k + 1, uF (k + 1), such that u(k + j|k + 1) = u(k + j|k) for all j = 1, · · · , N −1, then the predicted uncertain evolution set at k + 1 satisfies the condition Xˆj−1 (xk+1 , uF (k + 1)) ⊆ Xˆj (xk , uF (k))

(6)

The four basic interval operations [10] are given by [a, b] + [c, d] [a, b] − [c, d] [a, b] × [c, d] [a, b] / [c, d]

= = =

[a + c, b + d] [a − d, b − c] [min(a·c, a·d, b·c, b·d), max(a·c, £ a·d,¤ b·c, b·d)] / [c, d] = [a, b] × d1 , 1c if 0 ∈

(7)

The ranges of the four elementary interval arithmetic operations are exactly the ranges of the corresponding real operations. Extension of the interval arithmetic to include 0 in division can be found in [2]. Consider a function g : IRn → IRm and consider an interval vector X ∈ I n , then the set g(X) denotes the range of g(·) over the interval X. Note that it is not an interval vector in general. Computing the exact range of an arbitrary function g(·) over an interval vector X is a hard problem. However, interval arithmetic can be used to obtain interval bounds of the exact range g(X). Definition 3 (Inclusion function) A function G : IRn → IRm is called an inclusion function for g(·) if g(X) ⊆ G(X) for any X of I n . Definition 4 (Inclusion monotonic function) The inclusion function G(·) is inclusion monotonic if and only if for any X, Y ∈ I such that X ⊆ Y then G(X) ⊆ G(Y ). Definition 5 (Natural interval extension [3]) If g : IRn → IRm is a function computable as an expression, algorithm or computer program involving the four elementary operations interspersed with evaluations of standard functions, then a natural interval extension of g(·) is obtained by replacing each occurrence of each component xi of x by the corresponding interval Xi

of X, by executing all operations according to formulas (7) and by computing exact ranges of standard functions. Theorem 1 [3] Natural interval extensions are inclusion monotonic functions, i.e. for any X ∈ I n , g(X) ⊆ ψ(X) and for any Y ⊆ X, ψ(X) ⊆ ψ(Y ). Let ψ(X, u) be a natural interval extension of the model f (x, u), considering the inputs u as a parameter . Hence, it can be used as inclusion functions. From theorem 1, we get that for any X, Y ∈ I n such that X ⊆ Y and for any u, f (X, u) ⊆ ψ(X, u), and ψ(X, u) ⊆ ψ(Y, u). Therefore this procedure satisfies assumption 1 and it can be used to compute uncertain evolution sets. Note that the set of uncertainties W must be an interval vector, since the set ψ(X, u) + W must be an interval vector in order to compute the following set in (4). This is a mild condition since an interval vector which contains W can be used for the computation of the uncertain evolution sets. The only consequence is that the obtained sets might be overconservative.

3 Robust MPC strategy Model predictive control is a well established optimal control strategy which considers constraints on the states and on the control actions [8]. The control law KM P C (xk ) is obtained solving a constrained optimization problem and applying the optimal control action to the system in a receding horizon manner. Consider the finite horizon MPC optimization problem stated as follows min JN (xk , uF (k)) uF (k)

subject to:

function and a terminal constraint given by the region Ω. Taking into account that the optimal minimizer u∗F (xk ) only depends on xk , and the receding horizon policy, the control law is given by uk = KM P C (xk ) = u∗ (k|k). In absence of uncertainties, this control law asymptotically stabilizes the system under some assumptions on the terminal cost and the terminal region [8]. More∗ over, the optimal cost function JN (xk ) is a Lyapunov function of the closed loop system. The domain of attraction of the controller XN is the set where the optimization problem is feasible. If the system is uncertain, then the stability, and probably, the feasibility of the nominal MPC may be lost. Due to the receding horizon policy, the feedback loop provides some degree of robustness to the closed loop system [4, 12]. In order to achieve robustness, the controller must stabilize the system for all possible realizations of the uncertainty along the prediction horizon. Different robust MPC techniques have been proposed: some of them are based on a nominal prediction as in [9], others are based on H∞ [6] or on the worst possible uncertainty as in min-max MPC [11]. In this paper a robust dual-mode MPC is proposed. It is based on the computation on the uncertain evolution sets shown in section 2.2. These sets allow that all possible realizations of the uncertainty to be considered in the computation of the MPC control law. The system must satisfy the following assumption. Assumption 2 There is a region Ω ⊆ X with an associated control law u = h(x) such that Ω is a robust positively invariant set for the uncertain system i.e. {∀x ∈ Ω, f (x, h(x)) + w ∈ Ω, ∀w ∈ W } and u = h(x) ∈ U for all x ∈ Ω. The optimization problem to be solved at time k is:

x ˆ(k + j|k) ∈ X

∀j = 1, · · · , N

u(k + j|k) ∈ U x ˆ(k + N |k) ∈ Ω

∀j = 0, · · · , N − 1

Robust dual-mode MPC optimization problem (Pkd (xk )) min JN −k (xk , uF (k)) uF (k)

where JN (xk , uF (k))

subject to: =

i=N X−1

L(ˆ x(k + i|k), u(k + i|k))

i=0

+V (ˆ x(k + N |k)) and the vector of decision variables uF (k) = {u(k|k), u(k + 1|k), · · · , u(k + N − 1|k)} denotes the future sequence of control inputs of the system along the prediction horizon N and x ˆ(k + i|k) is the predicted nominal state of the system applying uF (k). Notice that the MPC includes a terminal cost V (·) in the cost

Xˆj (xk , uF (k)) ⊆ X u(k + j|k) ∈ U XˆN −k (xk , uF (k)) ⊆ Ω

∀j = 1, · · · , N − k

(8)

∀j = 0, · · · , N − k − 1(9) (10)

Note that the prediction horizon is reduced at each sample time. Therefore, this optimization problem is only defined for k = 0 to k = N − 1. In next section it is proved that xN ∈ Ω, and, hence, the local control law u = h(x) makes the system to remain in Ω.

The robust dual-mode control law is given by ½ ∗ u (k|k) if xk ∈ /Ω d KM P C (xk ) = h(xk ) if xk ∈ Ω For k ≥ 1 it is possible to find a feasible solution (¯ uF (k)) of the optimization problem in k, based on the optimal solution in k − 1 (u∗F (k − 1)). It is the sequence of N − k inputs, u ¯F (k), such that u ¯(k+j −1|k) = u∗ (k+j −1|k−1)

for

j = 1, ..., N −k

The approach proposed in here is different to the one presented in [9]: the uncertain evolution sets are incorporated in the optimization problem and then, the effect of uncertainty is considered in the solution of the optimization problem. It is not necessary to use a more conservative terminal region and the effect of the uncertainty is considered for all the prediction horizon. Furthermore, the constraints on the states are considered in a simpler way. The procedure used for the computation of the uncertain evolution sets are local, and hence, less conservative than global approaches as, for instance, the ones based on a global Lipschitz constant used in [9]. It is worthy of remark that the proposed controller, as the one proposed in [9], is over-conservative since both of them are based in open-loop predictions of the system. Closed-loop formulation are obtained by posing the MPC problem in terms of a sequence of control laws. These approaches are not conservative but their implementation is quite difficult.

4 Stability analysis Since the uncertainties are merely bounded and they may not be decaying, the origin is not an steady state of the uncertain system. Therefore, it is necessary the next definition of stability: Definition 6 A system is asymptotically ultimately bounded if the system evolves asymptotically to a bounded set, i.e. there exist positive constants b and c such that for every α ∈ (0, c), there is a k ∗ such that for all kx0 k ≤ α then kxk k ≤ b, ∀k > k ∗ . Hence, the aim of an stabilizing controller is to steer the state to a neighborhood of the origin and keep the state evolution in it. This set is a robust positively invariant set for the closed loop system and its size depends on the bound on the uncertainties. Therefore, the system is ultimately bounded to this set. The controller proposed in this paper steers the uncertain system to the terminal region, which is a robust invariant set. Hence, the closed loop system is ultimately bounded.

Theorem 2 (Robust stability) Consider a system given by (1) with additive uncertainties subject to (2) and with constraints on the states xk ∈ X and on the inputs uk ∈ U . Consider a robust invariant set for the system Ω with an associated local controller u = h(x) such that both satisfy assumption 2. Consider that it is available a procedure to compute the uncertain evolud tion sets, then the system controlled by u = KM P C (xk ) is ultimately bounded for all x0 such that the optimization problem P0d (x0 ) is feasible. Stability is guaranteed by the feasibility of the computed control action at each sample time. Hence, optimality is not required and a suboptimal solution of the optimization problem suffices to guarantee stability. Furthermore, the cost function only affects the performance but not the stability of the closed-loop system.

5 Application to a CSTR model As illustrative example of the robust MPC controller proposed the technique is applied to a highly nonlinear system: a continuous stirred tank reactor (CSTR) simulation model. The continuous time model of an CSTR for an exothermic, irreversible reaction A → B with constant liquid volume is given by [5]: d CA dt dT dt

= =

³

´

q E ·CA ·(CAf − CA ) − k0 · exp − V R·T ´ ³ E q ∆H·k0 ·CA + · exp − ·(Tf − T ) − V ρ·Cp R·T U ·A + ·(Tc − T ) V ·ρ·Cp

where CA is the concentration of A in the reactor, T is the reactor temperature and Tc is the temperature of the coolant stream. The parameters of the model are: ρ = 1000 g/l, Cp = 0.239 J/g K, ∆H = −5 × 104 J/mol, E/R = 8750 K, k0 = 7.2 × 1010 min−1 , U ·A = 5 × 104 J/min K. The nominal operating conditions are given by: q = 100 l/min, Tf = 350 K, V = 100 l, o CAf = 1.0 mol/l. The steady state is CA = 0.5 mol/l, o o T = 350 K, Tc = 300 K. The temperature of the coolant is constrained to 280K ≤ Tc ≤ 370. The state o , T − T o ]T , of the system is defined as x = [CA − CA o and the input as u = Tc − Tc . The model is discretized with a sampling period Ts = 0.03 min. The additive uncertainty on the discreteof the system are bounded by w1 ∈ £time model ¤ −3·10−3 , 3·10−3 and w2 ∈ [−0.15, 0.15], where w = [w1 , w2 ]T . The different scales of w1 and w2 are due to the different scales of the corresponding states variables (concentration and temperature). The robust positively invariant set is Ω = {x ∈ IR2 : xT ·P ·x ≤ 0.6226}

for the system controlled by uk = K·xk , where P =

·

54.3945 0.5928

0.5928 0.0351

¸

K = [−91.3362 − 4.7002]

Figure 1: Uncertain evolution sets at x0 .

6 Conclusions In this paper, a robust dual-mode MPC controller for constrained discrete-time nonlinear systems with additive uncertainties is presented. It is based on the addition of the uncertain evolution sets to the MPC optimization problem and considering a robust positively invariant set as terminal region. The main contribution of the proposed MPC is that there is not an explicit uncertainty bound to guarantee robust stability. It is guaranteed under feasibility of the optimization problem. Furthermore, suboptimal solution of the problem also guarantees stability. The paper has also shown that interval arithmetic is an appropriate technique for the on-line computation of the uncertain evolution sets, because of the small computational burden required. Using this technique, the proposed dual-mode controller has been applied to a simulation model of a CSTR. The highly nonlinear model of the system and the bounded inputs make the system difficult to be controlled.

5

0

x2

−5

−10

−15

−20 −0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

x1

Figure 2: Closed loop state portrait. In Fig.1 the solution of P0d (x0 ) for a couple of feasible initial states with a prediction horizon of N = 5 is shown. The uncertain evolution sets are also depicted. It can be seen that for all possible realization of the uncertainties, the system evolution is bounded by the uncertain evolution sets. These sets have been obtained using interval arithmetic. Note that the optimal solution found is such that Xˆ5 ⊆ Ω. In Fig.2 closed loop trajectories of the uncertain system are shown. For all the initial states, the optimization problem is feasible and hence, the system is steered to Ω despite the uncertainties. It is worthy of remark that the system to be controlled is highly nonlinear, and quite difficult to control in some regions due to the bounds on the input. Furthermore, the effect of the uncertainties is critical. This effect and the openloop nature of the MPC controller makes the controller over-conservative. The main contribution is that it is possible to guarantee robust stability if the optimization problem is feasible at the initial state.

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