- Email: [email protected]
Is suboptimal nonlinear MPC inherently robust?*
Proceedings of the 18th World Congress The International Federation of Automatic Control Milano (Italy) August 28 - September 2, 2011
Is suboptimal nonlinear MPC inherently robust? ? Gabriele Pannocchia ∗ James B. Rawlings ∗∗ Stephen J. Wright ∗∗∗ ∗ Dept.
of Chem. Engineering (DICCISM), Univ. of Pisa, Pisa, Italy (e-mail: [email protected]). ∗∗ Chemical and Biological Engineering Dept., Univ. of Wisconsin, Madison, WI, USA (email: [email protected]) ∗∗∗ Computer Science Dept., Univ. of Wisconsin, Madison, WI, USA (e-mail: [email protected]) Abstract: We study the inherent robust stability properties of nonlinear discrete-time systems controlled by suboptimal model predictive control (MPC). The unique requirement to suboptimal MPC is that it does not increase the cost function with respect to a well-defined warm start, and it is therefore implementable in general nonconvex problems for which no suboptimality margins can be enforced. Because the suboptimal control law is a set-valued map, the closed-loop system is described by a difference inclusion. Under conventional assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium. If, in addition, a continuity assumption of the feasible input set holds, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. To obtain these results, we show that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system. Moreover, we show that robust recursive feasibility is implied by such (nominal) exponential cost decay. We present an illustrative example to clarify the main ideas and assumptions. Keywords: Robust stability, suboptimal model predictive control, difference inclusions, inherent robustness, Lyapunov functions. 1. INTRODUCTION
asymptotic stability of the equilibrium holds. [See also (Rawlings and Mayne, 2009, Sections 2.8 and 6.1.2).]
Nominal stability of the equilibrium of constrained nonlinear systems in closed loop with model predictive control (MPC) is usually proved by showing that the optimal MPC cost function is a Lyapunov function for the closed-loop system (Mayne et al., 2000). This also requires enforcing, explicitly or not, a suitable constraint on the terminal state at the end of the finite horizon. Different variations on this theme can be considered, as thoroughly discussed in (Rawlings and Mayne, 2009, Ch. 2) and references therein. Thus, for most MPC formulations stability proofs rely on the fact that the optimal control problem is solved exactly. For linear systems subject to polytopic constraints and linear (or quadratic) cost functions, this assumption may be considered valid in practice, thanks to the associated convex LP (or QP) formulation. For nonlinear systems, instead, the resulting optimization problem may not be convex (Boyd and Vandenberghe, 2004), and hence optimal solutions are in general not achievable. If a suboptimal solution is used in closed loop, stability may not hold or may be difficult to prove because the associated suboptimal MPC cost is no longer a function of the current state, as it depends, e.g., on the initial guess, on the number of iterations and algorithm, etc. Hence, the MPC cost cannot serve as a Lyapunov function for the closed-loop system. However, as established by Scokaert et al. (1999), if the optimization provides a feasible, suboptimal solution that only improves the cost from a well chosen warm start (and some other more technical assumptions are satisfied),
When dealing with perturbed systems, in most cases, the perturbations have been directly taken into account in the controller formulation leading to a so-called robust MPC formulations [see e.g. (Bemporad and Morari, 1999; Mayne and Langson, 2001; Pannocchia and Kerrigan, 2005; Rakovic et al., 2006), (Rawlings and Mayne, 2009, Ch. 3) and references therein]. Robust MPC formulations, which are usually numerically trackable only for linear systems, tend to be conservative in order to preserve recursive feasibility for all possible scenarios. Assessment of the inherent robustness, i.e., robust stability analysis of a perturbed system in closed-loop with an MPC that ignores such perturbations, has instead received much less attention (De Nicolao et al., 1996; Scokaert et al., 1997) as pointed out by Teel and coworkers (Grimm et al., 2004). In particular, Grimm et al. (2004) presented examples of nonlinear systems controlled by MPC in which the asymptotic stability of the equilibrium is destroyed by arbitrarily small perturbations. Assuming that the optimization problem remains feasible at all times, they proved that existence of a continuous Lyapunov function for the closed-loop system implies robustness to sufficiently small perturbations. Grimm et al. (2007) presented conditions to ensure recursive feasibility, using a constraint tightening approach (Lim´on Marruedo et al., 2002). Grimm et al. (2004) also show that, for linear systems with a quadratic cost, the optimal MPC cost function is a continuous Lyapunov function for the closed-loop system, because the optimal state-feedback law is continuous, hence achieving inherent robustness. A suboptimal MPC law, instead, is not necessarily continuous, even for linear systems, and hence in-
? This work was supported by National Science Foundation (Grant CTS0456694) and TWCCC members. Corresponding author: G. Pannocchia.
978-3-902661-93-7/11/$20.00 © 2011 IFAC
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18th IFAC World Congress (IFAC'11) Milano (Italy) August 28 - September 2, 2011
herent robustness cannot be established by the same argument. Inherent robustness of suboptimal MPC was first addressed in (Lazar and Heemels, 2009), by showing input-to-state stability of the equilibrium of the closed-loop system provided that a suboptimality margin is satisfied by the suboptimal MPC algorithm. For nonlinear systems, however, the resulting optimization problem is usually nonconvex and therefore such margins cannot be ensured. The objective of this paper is to present novel results about the inherent robust stability of (a class of) nonlinear systems controlled by (a general and implementable) suboptimal MPC. By treating the closed-loop system as a difference inclusion (Kellett and Teel, 2004), we show that such robustness is achieved with respect to sufficiently small but arbitrary perturbations, without requiring a priori recursive feasibility. As corollary to these results, we note that optimal MPC is a particular suboptimal MPC, and thus the inherent robust stability properties are consequently established also for the optimal controller. Notation. The symbols I≥0 and R≥0 denote the sets of nonnegative integers and reals, respectively. The symbol I0:N−1 denotes the set {0, 1, . . . , N − 1}. The symbol | · | denotes the Euclidean norm and B denotes the closed ball of radius 1 centered at the origin. 2. PRELIMINARIES
Assumption 1. The functions f : Rn × Rm → Rn , ` : Rn × Rm → R≥0 and V f : Rn → R≥0 are continuous, f (0, 0) = 0, `(0, 0) = 0, and V f (0) = 0. Assumption 2. The sets X and X f , X f ⊆ X, are closed and contain the origin in their interiors, and U is compact and contains the origin. Assumption 3. For any x ∈ X f there exists u ∈ U such that f (x, u) ∈ X f and V f ( f (x, u)) + `(x, u) ≤ V f (x). Assumption 4. There exist positive constants a, a01 , a02 , a f and r¯, such that the cost function satisfies the inequalities `(x, u) ≥ a01 |(x, u)|a for all (x, u) ∈ X × U VN (x, u) ≤ a02 |(x, u)|a for all (x, u) ∈ r¯B V f (x) ≤ a f |x|a for all x ∈ X We observe that in Assumption 2 we allow the origin to be on the boundary of the (deviation) input space U. This case often happens in industrial applications where an upper (typically economic) optimization layer pushes the operating equilibrium towards the boundaries of the (absolute) input space. Assumption 3 implies that X f is control invariant and that X f ⊂ XN . Let u0 (x) be the optimal solution of PN (x) and let κN (x) = u0 (0; x) denote its first component, written as a function of x. The evolution of the nominal system (1) in closed loop with optimal MPC can be written as x+ = f (x, κN (x)) (2)
2.1 MPC problem: assumptions and optimal solution
2.2 Suboptimal MPC solutions
We consider discrete-time systems in the form: x+ = f (x, u), (1) n in which x ∈ R and u ∈ Rm are the state and the input at a given time, while x+ ∈ Rn is the successor state. Both state and input are subject to constraints: x(k) ∈ X , u(k) ∈ U for all k ∈ I≥0 . Given an integer N (referred to as the finite horizon), and an input sequence u of length N, u = {u(0), u(1), . . . , u(N − 1)}, let φ (k; x, u) denote the solution of (1) at time k for a given initial state x(0) = x. For any state x ∈ Rn and input sequence u ∈ UN , we define the following cost function:
Problem PN (x) is, in general, a nonconvex nonlinear program, and practical solvers cannot ensure achievement of the global minimum. Thus, we consider using any (unspecified) suboptimal algorithm having the following properties. Let u ∈ UN (x) denote the (suboptimal) control sequence for the initial state x, and let u˜ denote a warm start for the successor initial state x+ = f (x, u(0; x)), obtained from (x, u) by u˜ = {u(1; x), u(2; x), . . . , u(N − 1; x), u+ }, in which u+ ∈ U is any input that satisfies the invariance conditions of Assumption 3 for x = φ (N; x, u). We observe that the warm start satisfies u˜ ∈ UN (x+ ). Then, the suboptimal input sequence for any given x+ ∈ XN is defined as any u+ ∈ UN that satisfies: u+ ∈ UN (x+ ) (3a) + + + ˜ VN (x , u ) ≤ VN (x , u) (3b) VN (x+ , u+ ) ≤ V f (x+ ) when x+ ∈ rB, (3c) in which r is a positive scalar sufficiently small that rB ⊆ X f . Notice that constraint (3c) is required to hold only if x+ ∈ rB, and by Lemma 12, it implies that |u+ | → 0 as |x+ | → 0. Notice that condition (3b) ensures that the computed suboptimal cost is no larger than that of the warm start. Proposition 5. Any u0 (x+ ), optimal solution to PN (x+ ), satisfies conditions (3a)–(3b) for all x+ ∈ XN . Moreover, if x+ ∈ X f condition (3c) is satisfied by u0 (x+ ).
N−1
VN (x, u) =
∑ `(φ (k; x, u), u(k)) +V f (φ (N; x, u)). k=0
We then define the set of admissible initial states and associated control sequences: ZN = {(x, u) | u(k) ∈ U, φ (k; x, u) ∈ X for all k ∈ I0:N−1 , and φ (N; x, u) ∈ X f }, in which X f ⊆ X is the set of admissible states at the end of the finite horizon. Consequently, we can define the set of admissible initial states as: XN = {x ∈ Rn | ∃u ∈ UN such that (x, u) ∈ ZN } and for each x ∈ XN , the corresponding set of admissible input sequences is defined as: UN (x) = {u | (x, u) ∈ ZN }. Finally, we consider the following finite horizon optimal control problem: PN (x) : min VN (x, u) s.t. u ∈ UN (x). u
We consider the following assumptions throughout the paper.
Proof. See (Pannocchia et al., 2011). 2 Corollary 6. For any x+ ∈ XN , there exists a u+ satisfying all conditions (3). We now observe that u+ is a set-valued map of the state x+ , and so is the associated first component u(0; x+ ). If we, again,
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denote the latter map as κN (·), we can write the evolution of the closed-loop system as the following difference inclusion: x+ ∈ F(x) = { f (x, u) | u ∈ κN (x)}. (4) Proposition 7. κN (0) = {0} and F(0) = {0}. Proof. See (Pannocchia et al., 2011). 2 2.3 Difference inclusions: definitions and results Given a difference inclusion z+ ∈ H(z), H(0) = {0}, we denote by ψ(k; z) = z(k) a solution at time k ∈ I≥0 starting from the initial state z(0) = z. Definition 8. (Exponential stability). The origin of the difference inclusion z+ ∈ H(z) is exponentially stable (ES) on Z , 0 ∈ Z , if there exist scalars b > 0 and 0 < λ < 1, such that for any z ∈ Z , all solutions ψ(k; z) satisfy: ψ(k; z) ∈ Z , |ψ(k; z)| ≤ bλ k |z| for all k ∈ I≥0 . Definition 9. (Exponential Lyapunov function). A function V is an exponential Lyapunov function on the set Z for the difference inclusion z+ ∈ H(z) if there exist positive scalars a, a1 , a2 , a3 such that z ∈ Z implies that: a1 |z|a ≤ V (z) ≤ a2 |z|a , max V (z+ ) ≤ V (z) − a3 |z|a . z+ ∈H(z)
Proposition 10. If V is an exponential Lyapunov function on the set Z for the difference inclusion z+ ∈ H(z), there exists 0 < γ < 1 such that: max V (z+ ) ≤ γV (z).
Proof. From Lemma 13 we have that VN (z) is an exponential Lyapunov function for (5) in any given compact subset of Zr . Let V¯ be an arbitrary positive scalar, and consider the set S = {(x, u) ∈ Zr | VN (x, u) ≤ V¯ }. We observe that S ⊆ Zr is compact and is invariant for (5). By Lemma 11, these facts prove that the origin of the extended system (5) is ES on S , i.e., there exist scalars b0 > 0 and 0 < λ < 1, such that for any z ∈ S we can write: ψ(k; z) ∈ S and |ψ(k; z)| ≤ b0 λ k |z| for all k ∈ I≥0 in which ψ(k; z) = z(k) is a solution of (5) at time k for a given initial extended state z(0) = z. We define C = {x ∈ XN | ∃u ∈ UN (x) s.t. (x, u) ∈ S } and note that C ⊆ XN and that C is compact because it is the projection onto Rn of the compact set S . Thus for any x ∈ C and its associated suboptimal input sequence u such that z = (x, u) ∈ S , we denote with φ (k; x) the state component of ψ(k; z), i.e., a solution of the nonextended system (4), and for all k ∈ I≥0 we can write: φ (k; x) ∈ C and |φ (k; x)| ≤ |ψ(k; z)| ≤ b0 λ k |z| ≤ bλ k |x|, in which b = b0 (1 + c), because from Lemma 12 it follows that |z| ≤ |x| + |u| ≤ (1 + c)|x|. This concludes the proof because it states that the origin of the closed-loop system (4) is ES on C , and V¯ can be chosen large enough for C to contain any given compact subset of XN . 2 Corollary 15. Under Assumptions 1, 2, 3, and 4, if XN is compact, the origin of the closed-loop system (4) is ES on XN .
z+ ∈H(z)
4. ROBUST EXPONENTIAL STABILITY
Proof. See (Pannocchia et al., 2011). 2 Lemma 11. If the set Z , 0 ∈ Z , is positively invariant for the difference inclusion z+ ∈ H(z), H(0) = {0}, and there exists an exponential Lyapunov function V on Z the origin is ES on Z . Proof. See (Pannocchia et al., 2011). 2 3. NOMINAL EXPONENTIAL STABILITY 3.1 Extended state and supporting results We define the extended state z = (x, u), and we observe that it evolves according to the following difference inclusion: z+ ∈ H(z) = {(x+ , u+ ) | x+ = f (x, u(0; x)), u+ ∈ G(z)}, (5) in which (note that both x+ = f (x, u(0; x)) and u˜ depend on z): ˜ G(z) = {u+ | u+ ∈ UN (x+ ), VN (x+ , u+ ) ≤ VN (x+ , u), + + + + and VN (x , u ) ≤ V f (x ) if x ∈ rB}. We also define the following set (notice that rB ⊆ X f ): Zr = {(x, u) ∈ ZN | VN (x, u) ≤ V f (x) if x ∈ rB}. Lemma 12. There exists a positive constant c such that |u| ≤ c|x| for any (x, u) ∈ Zr . Proof. See (Pannocchia et al., 2011). 2 Lemma 13. VN (z) is an exponential Lyapunov function for the extended closed-loop system (5) in any compact subset of Zr . Proof. See (Pannocchia et al., 2011). 2 3.2 Main results Theorem 14. (ES). Under Assumptions 1, 2, 3, and 4, the origin of the closed-loop system (4) is ES on (arbitrarily large) compact subsets of XN .
4.1 Perturbed system and robust stability definitions In order to assess the inherent robustness, we consider the closed-loop evolution of the perturbed system: x+ ∈ Fed (x) = { f (x, u) + d | u ∈ κN (x + e)}, (6) in which d ∈ Rn is an unknown process disturbance and e ∈ Rn represents an unknown state measurement/estimate error. Notice that the control sequence u is computed as a suboptimal solution of PN (xm ), with xm = x + e. Hence, u is based on the nominal system (1) evolution, for the initial measured state. We denote with φed (k; x) = x(k) a solution to (6) for the initial state x(0) = x and given disturbance and measurement error sequences {d(k)}, {e(k)}. We now present the definition of robust exponential stability (RES), close to that of robust asymptotic stability (RAS) given in (Grimm et al., 2004). Definition 16. (RES). The origin of the closed-loop system (6) is robustly exponentially stable (RES) on int(XN ) if there exist scalars b > 0 and 0 < λ < 1 such that for all compact sets C ⊂ XN , with 0 ∈ int(C ), the following property holds: Given any ε > 0, there exists δ > 0 such that for all sequences {d(k)} and {e(k)} with x(0) = x ∈ C satisfying: max |d(k)| ≤ δ , max |e(k)| ≤ δ , k≥0
k≥0
xm (k) = x(k) + e(k) ∈ XN , x(k) ∈ XN , for all k ∈ I≥0 , it follows that |φed (k; x)| ≤ bλ k |x| + ε, for all k ∈ I≥0 . (8) We remark that in RES (or RAS) the robust stability condition (8) is presented for those (if any) initial states, disturbance and measurement error sequences that a-priori ensure feasibility of the perturbed closed-loop trajectories. The next definition instead requires that feasibility is satisfied at all times for all
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sufficiently small disturbance and measurement error sequences and all initial states in a given compact subset of int(XN ) . Definition 17. (SRES). The origin of the closed-loop system (6) is strongly robustly exponentially stable (SRES) on a compact set C ⊂ XN , 0 ∈ int(C ), if there exist scalars b > 0 and 0 < λ < 1 such that the following property holds: Given any ε > 0, there exists δ > 0 such that for all sequences {d(k)} and {e(k)} satisfying |d(k)| ≤ δ and |e(k)| ≤ δ for all k ∈ I≥0 , and all x ∈ C , we have that xm (k) = x(k) + e(k) ∈ XN , x(k) ∈ XN , for all k ∈ I≥0 , (9a)
u
ZN
Sρ
S
UN
u0(x)
x
ρ
Cρ
|φed (k; x)| ≤ bλ k |x| + ε, for all k ∈ I≥0 . (9b)
XN
4.2 Recursive feasibility in perturbed system evolution
Fig. 1. Sketch of the main sets involved in SRES. While the warm start u˜ is feasible for the predicted suc˜ ∈ ZN , it may cessor state x˜+ = f (xm , u(0; xm )), i.e., (x˜+ , u) + = not be feasible for the measured successor state, i.e., xm f (x, u(0; xm )) + d + e+ . Notice that the true successor state, which is unknown in general, is x+ = f (x, u(0; xm )) + d. If + , u) ˜ ∈ (xm / ZN , the right-hand side of the cost inequality (3b) is not meaningful. In such cases, we need to modify the warm +, u ˜ + p) ∈ ZN , and we consider start with a term p such that (xm the following additional assumption. Assumption 18. For any x, x0 ∈ XN and u ∈ UN (x), there exists u0 ∈ UN (x0 ) such that |u − u0 | ≤ σ (|x − x0 |) for some K function σ (·). We remark that Assumption 18 has been shown to hold, e.g., for linear systems subject to polytopic constraints on (x, u), and for nonlinear systems without state (or mixed) constraints. Among various options for finding p, we consider the following feasibility problem (notice that x˜+ is known): + + Find p s.t. u˜ + p ∈ UN (xm ) and |p| ≤ σ (|xm − x˜+ |), (10) + ), it immediately follows that We observe that if u˜ ∈ UN (xm p = 0 satisfies the feasibility problem (10), and hence Assumption 18 is unnecessary. ˜ ∈ ZN Proposition 19. Under Assumption 18, for any (x˜+ , u) + ∈ X , the set of solutions to (10) is nonempty. and xm N Proof. The result follows directly from Assumption 18 by + ). 2 noticing that u˜ ∈ UN (x˜+ ) and u˜ + p ∈ UN (xm + ∈ X , we Given any p satisfying (10), and for any given xm N replace conditions (3) with the following: + u+ ∈ UN (xm ) (11a) + + + (11b) VN (xm , u ) ≤ VN (xm , u˜ + p) + + + + when xm ∈ rB. (11c) VN (xm , u ) ≤ V f (xm )
In the perturbed case, the extended state is z = (x, u), where u is a suboptimal solution to PN (xm ) where xm = x + e is the measured state. The extended system evolves as follows: z+ ∈ Hed (z) = {(x+ , u+ ) | x+ = f (x, u(0; xm )) + d, u+ ∈ Ged (z)}, (12) + = x+ + e+ and u ˜ + p depend on z): in which (note that both xm + + + + Ged (z) = {u+ | u+ ∈ UN (xm ), VN (xm , u ) ≤ VN (xm , u˜ + p), + + + + VN (xm , u ) ≤ V f (xm ) if xm ∈ rB}
4.3 Main results We define zm = (xm , u) = (x + e, u) = z + (e, 0), and we observe that zm ∈ Zr . The following supporting result is fundamental. Lemma 20. For every µ > 0, there exists a δ > 0 such that, for all (zm , e, d, e+ ) ∈ Zr ×δ B×δ B×δ B, z = zm −(e, 0), such that + ∈ X , and some γ, 0 < γ < 1, we have: xm N max VN (z+ ) ≤ max{γVN (z), µ}. z+ ∈Hed (z)
Proof. See (Pannocchia et al., 2011). 2 We now characterize the compact sets over which SRES is guaranteed to hold. Consider a scalar V¯ > 0 such that the set: S = {z ∈ Rn × UN | VN (z) ≤ V¯ } satisfies S ⊆ ZN , i.e., S is a sublevel set of Rn × UN fully contained in ZN . Thus, by definition, for any z = (x, u) ∈ S , it follows that x ∈ XN . Next, given a scalar ρ > 0 and any zm ∈ Zr , we define the following measure and associated set: ρ VN (zm ) = max VN (z) s.t. z = zm − (e, 0), (13) e∈ρB
Sρ = {zm ∈ Zr | VN (zm ) ≤ V¯ }, (14) in which we assume that ρ is small enough that Sρ is nonempty. Finally, we define the following compact set: Cρ = {x ∈ Rn | x = xm − e, e ∈ ρB, ∃u s.t. (xm , u) ∈ Sρ }, (15) and observe that 0 ∈ int(Cρ ) ⊂ XN for ρ sufficiently small. These sets are depicted in Figure 1. The main SRES result of this paper is as follows. Theorem 21. (SRES of suboptimal MPC). Under Assumptions 1, 2, 3, 4, and 18, the origin of the perturbed closed-loop system (6) is SRES on Cρ . ρ
Proof. (Robust feasibility) Suppose that x ∈ Cρ and let z = (x, u) be the corresponding augmented state where u is a suboptimal sequence computed for the measured state xm = x + e, e ∈ ρB. We recall that VN (z) ≤ V¯ , i.e., z ∈ S and that ˜ Since VN (·) is zm ∈ Sρ ⊆ Zr . Moreover, we define z˜+ = (x˜+ , u). an exponential Lyapunov function for the nominal system and ¯ N (zm ) for some zm ∈ Zr , Proposition 10 gives that VN (˜z+ ) ≤ γV ˜ ≤ γ¯V¯ < V¯ . 0 < γ¯ < 1. Because zm ∈ Sρ it follows that VN (x˜+ , u) + = f (x , u(0; x )) − f (x, u(0; x )) − d − Recalling that: x˜+ − xm m m m e+ , it follows from continuity of f that there exists a δ¯1 > 0 such + , u) + ∈ X for all (z , e, d, e+ ) ∈ ˜ < V¯ and thus xm that VN (xm N m ¯ ¯ ¯ Sρ × δ1 × δ1 × δ1 . Hence, the initialization step (10) is well
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¯ V¯ . From continuity of defined. Define any 0 < µ < (1 − γ) + |), we can choose VN and f , and because |p| ≤ σ (|x˜+ − xm +, u ˜+ δ¯2 > 0 such that the following condition holds: VN (xm + + ¯ V¯ ≤ V¯ . This proves ˜ + µ < VN (x˜ , u) ˜ + (1 − γ) p) ≤ VN (x˜ , u) + ˜ + p) < V ¯ . From continuity of VN it that VN (z+ m ) ≤ VN (xm , u also follows that we can choose ρ > 0 sufficiently small that ρ + ¯ ¯ ¯ VN (z+ m ) ≤ V . Taking δ = min{ρ, δ1 , δ2 } it follows that zm ∈ Sρ + for all (zm , e, d, e ) ∈ Sρ × δ B × δ B × δ B. This implies: x(k) ∈ Cρ ⊆ XN for all k ∈ I≥0 , and also that xm (k) ∈ XN for all k ∈ I≥0 . Hence, (9a) holds. (Robust stability) We denote by ψed (k; z) a solution of the perturbed difference inclusion (12) at time k ∈ I≥0 starting from the initial state z(0) = z and given disturbance and measurement error sequences {d(k)}, {e(k)}. As established in the proof of Proposition 7 we have that there exists a scalar a1 > 0 such that a1 |z|a ≤ VN (z) for any z ∈ Cρ × UN ⊆ X × UN . Moreover, by Lemma 13 there exists a scalar a2 > 0 such that VN (z) ≤ a2 |z|a for any z ∈ Cρ × UN . From Lemma 20, by induction, we write: a1 |ψed (k; z)|a ≤ VN (ψed (k; z)) ≤ k
k
4
2 1 x2
max{γ VN (z), µ} ≤ max{γ a2 |z| , µ},
-1 -2 -3 -4 -4
¯ k |z|, ε¯ } |φed (k; x)| ≤ |ψed (k; z)| ≤ max{bλ ¯ k |z| + ε¯ ≤ bλ k |x| + ε, ≤ bλ ¯ + c) and ε = ε¯ + bcδ ¯ , completing the proof with b = b(1 because 0 < λ < 1. 2 Corollary 22. (RES of suboptimal MPC). Under Assumptions 1, 2, 3, 4, and 18, the origin of the perturbed closed-loop system (6) is RES on int(XN ). Proof. This result follows immediately because robust feasibility is assumed in RES. Thus, for any compact set C ⊂ int XN , the second part (Robust stability) of the proof of Theorem 21 can be readily applied to obtain for all x ∈ C : for all k ∈ I≥0 .
We observe that while RES holds in int(XN ), SRES is guaranteed to hold in a compact subset Cρ ⊂ int(XN ), 0 ∈ int(Cρ ). However, SRES may hold even in larger subsets of int(XN ). 4.4 Further comments on robust stability under optimal MPC By Proposition 5, it follows that all robust stability results that we proved for suboptimal MPC, readily apply to optimal MPC even for cases in which there exist more than one optimal solution point, i.e., when the optimal control u0 (·) is a setvalued map. Moreover, for optimal MPC, the definition (13) ρ of VN (and hence of Sρ ) can be modified as follows: ρ
e∈ρB
5. APPLICATION EXAMPLE As an example, we consider the following system:
-3
-2
-1
0
1
2
3
4
x1
Fig. 2. Approximate feasibility sets of the three controllers. x1+ = x1 + u x2+ = bx2 + u3 ,
C1. No state constraints are enforced, X = R2 , and the terminal constraint set is the origin, X f = {0}. C2. No state constraints are enforced, X = R2 , and the terminal constraint set is X f = {x ∈ R2 | V f (x) ≤ α} with V f (x) = x0 Px, α > 0 and P later defined. C3. State constraints are enforced, X = [−2, 2]2 , and the terminal constraint set is the same of C2.
¯ k |z|, (µ/a1 )1/a } ≤ max{bλ ¯ k |z|, ε¯ }, |ψed (k; z)| ≤ max{bλ in which λ = γ 1/a , b¯ = (a2 /a1 )1/a and ε¯ = (µ/a1 )1/a . Finally, from Lemma 12 recalling that |u| ≤ c|xm | ≤ c|x| + cδ and that φed (k; x) represents the state component of ψed (k; z), for all x ∈ Cρ we write:
VN (xm , u0 (xm )) = max VN (xm − e, u0 (xm )).
0
with b = 0.9. The control horizon is N = 3, the input set is U = [−1, 1], and the stage cost function is given by `(x, u) = |x|2 + u2 . Three different nonlinear MPC formulations are considered.
a
which implies
|φed (k; x)| ≤ bλ k |x| + ε
C1 C2 C3
3
(16)
We remark that controller C1 does not satisfy Assumption 2 because X f does not contain the origin in its interior. In the definition of controllers C2 and C3, we note that the linearization of of system at the origin can be written as: 1 0 1 x+ = Ax + Bu with A = , B= , 0 b 0 and we observe that the pair (A, B) is stabilizable. Therefore, we follow the procedure described in (Rawlings and Mayne, 2009, Par. 2.5.3.2). We define a linear control law κ f (x) = Kx = [−1, 0]x and observe that such control law which is stabilizing for the linearized system. Let QK = I + K 0 K, AK = A + BK, and solve the following Lyapunov equation: A0K PAK + 2QK = P, 0 obtaining P = 0 10.53 ; we notice that P is positive definite. Consequently, we define the terminal cost V f (x) = x0 Px, while the terminal constraint set is given by X f = {x ∈ R2 | V f (x) ≤ α}, in which α > 0. It can be shown (Rawlings and Mayne, 2009, Par. 2.5.3.2) that there exists α > 0 such that Assumption 3 holds for for u = Kx = [−1, 0]x In particular, it can be verified that α = 1.1 is one such value. Furthermore, it can be verified that Assumptions 1 and 2 hold. Similarly, the three conditions of Assumption 4 hold for a = 2. We report in Fig. 2 the approximate feasibility sets, XN , for the three controllers. As expected, we notice that the feasibility set XN for C2 contains both the feasibility sets for C1 and C3. We report in Fig. 3 the first component of UN (x), with x = 0.5 [cos(θ ), sin(θ )], as a function of θ for the controllers C1 and C2 (the plot for C3 is not reported as it is identical to that of C2). For C1, we can notice that Assumption 18 does not hold at the point indicated by the arrow, whereas no such points can be noticed for C2.
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this specialization will be discussed in a separate publication (Pannocchia et al., 2010).
First component of UN (x)
1
ACKNOWLEDGMENTS
0.5
We thank Prof. A.R. Teel for providing an updated and detailed proof of Proposition 8 in (Grimm et al., 2004).
Assumption 18 does not hold 0
REFERENCES -0.5
-1 -4
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Fig. 3. First component of UN (x) as a function of the initial state x = 0.5 [cos(θ ), sin(θ )]. C1 (top), C2 (bottom). 6. CONCLUSIONS Attaining global solutions of MPC problems for nonlinear system is, in general, not practical, but when a suboptimal solution is implemented in closed loop many stability issues arise. Such issues were first addressed by Scokaert et al. (1999) who proved nominal asymptotic stability of the origin of the closed-loop system in a neighborhood of the origin. The requirement to suboptimal MPC is that it does not increase the cost function with respect to a well-defined warm start sequence, and it is therefore of general applicability and implementation. In this paper, under similar assumptions, we established nominal exponential stability of the origin of the closed-loop system in arbitrarily large compact subsets of the feasible set. Furthermore, we established inherent exponential robust stability of the origin of the closed-loop system with respect to additive process disturbances and measurement errors, in the spirit of the ideas developed by Teel and coworkers (Grimm et al., 2004, 2007). In particular, we proved robust recursive feasibility in an appropriate compact subset of the nominal region of attraction for all sufficiently small, but arbitrary, perturbations. The main requirement is a local continuity assumption of the input feasibility set, which holds, e.g., for linear systems or nonlinear systems in absence of state constraints. We observe that no specific modification, e.g. constraint tightening (Lim´on Marruedo et al., 2002; Grimm et al., 2007), of the MPC problem was used to preserve robust feasibility. All robustness results established for suboptimal MPC apply to optimal MPC as well, and this represents itself an improvement in stability analysis of optimal nonlinear MPC systems. These results can be specialized further to suboptimal MPC of linear systems, and
Bemporad, A. and Morari, M. (1999). Robust model predictive control: A survey. In A. Garulli, A. Tesi, and A. Vicino (eds.), Robustness in Identification and Control, number 245 in Lecture Notes in Control and Information Sciences, 207– 226. Springer-Verlag. Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press, Cambridge (UK). De Nicolao, G., Magni, L., and Scattolini, R. (1996). On the robustness of receding-horizon control with terminal constraints. IEEE Trans. Auto. Contr., 41(3), 451 –453. Grimm, G., Messina, M.J., Tuna, S.E., and Teel, A.R. (2004). Examples when nonlinear model predictive control is nonrobust. Automatica, 40, 1729–1738. Grimm, G., Messina, M.J., Tuna, S.E., and Teel, A.R. (2007). Nominally robust model predictive control with state constraints. IEEE Trans. Auto. Contr., 52(10), 1856–1870. Kellett, C.M. and Teel, A.R. (2004). Smooth Lyapunov functions and robustness of stability for difference inclusions. Syst. Contr. Lett., 52, 395–405. Lazar, M. and Heemels, W. (2009). Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions. Automatica, 45(1), 180–185. ´ Lim´on Marruedo, D., Alamo, T., and Camacho, E.F. (2002). Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive disturbances. In Proceedings of the 41st IEEE Conference on Decision and Control, 4619–4624. Las Vegas, Nevada. Mayne, D.Q. and Langson, W. (2001). Robustifying model predictive control of constrained linear systems. Electronics Letters, 37, 1422–1423. Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O.M. (2000). Constrained model predictive control: stability and optimality. Automatica, 36, 789–814. Pannocchia, G. and Kerrigan, E.C. (2005). Offset-free receding horizon control of constrained linear systems. AIChE J., 51, 3134–3146. Pannocchia, G., Rawlings, J.B., and Wright, S.J. (2011). Addendum to the paper “Is suboptimal nonlinear MPC inherently robust?”. Technical Report 2011–02, TWCCC. Available at http://twccc.che.wisc.edu. Pannocchia, G., Wright, S.J., and Rawlings, J.B. (2010). Robust stability of suboptimal linear MPC based on partial enumeration. Submitted to J. Proc. Contr. Rakovic, S., Kerrigan, E., Mayne, D., and Lygeros, J. (2006). Reachability analysis of discrete-time systems with disturbances. IEEE Trans. Auto. Contr., 51(4), 546–561. Rawlings, J.B. and Mayne, D.Q. (2009). Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, WI. Scokaert, P.O.M., Mayne, D.Q., and Rawlings, J.B. (1999). Suboptimal model predictive control (feasibility implies stability). IEEE Trans. Auto. Contr., 44, 648–654. Scokaert, P.O.M., Rawlings, J.B., and Meadows, E.S. (1997). Discrete-time stability with perturbations: Application to model predictive control. Automatica, 33, 463–470.
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Is suboptimal nonlinear MPC inherently robust? ? Gabriele Pannocchia ∗ James B. Rawlings ∗∗ Stephen J. Wright ∗∗∗ ∗ Dept.
of Chem. Engineering (DICCISM), Univ. of Pisa, Pisa, Italy (e-mail: [email protected]). ∗∗ Chemical and Biological Engineering Dept., Univ. of Wisconsin, Madison, WI, USA (email: [email protected]) ∗∗∗ Computer Science Dept., Univ. of Wisconsin, Madison, WI, USA (e-mail: [email protected]) Abstract: We study the inherent robust stability properties of nonlinear discrete-time systems controlled by suboptimal model predictive control (MPC). The unique requirement to suboptimal MPC is that it does not increase the cost function with respect to a well-defined warm start, and it is therefore implementable in general nonconvex problems for which no suboptimality margins can be enforced. Because the suboptimal control law is a set-valued map, the closed-loop system is described by a difference inclusion. Under conventional assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium. If, in addition, a continuity assumption of the feasible input set holds, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. To obtain these results, we show that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system. Moreover, we show that robust recursive feasibility is implied by such (nominal) exponential cost decay. We present an illustrative example to clarify the main ideas and assumptions. Keywords: Robust stability, suboptimal model predictive control, difference inclusions, inherent robustness, Lyapunov functions. 1. INTRODUCTION
asymptotic stability of the equilibrium holds. [See also (Rawlings and Mayne, 2009, Sections 2.8 and 6.1.2).]
Nominal stability of the equilibrium of constrained nonlinear systems in closed loop with model predictive control (MPC) is usually proved by showing that the optimal MPC cost function is a Lyapunov function for the closed-loop system (Mayne et al., 2000). This also requires enforcing, explicitly or not, a suitable constraint on the terminal state at the end of the finite horizon. Different variations on this theme can be considered, as thoroughly discussed in (Rawlings and Mayne, 2009, Ch. 2) and references therein. Thus, for most MPC formulations stability proofs rely on the fact that the optimal control problem is solved exactly. For linear systems subject to polytopic constraints and linear (or quadratic) cost functions, this assumption may be considered valid in practice, thanks to the associated convex LP (or QP) formulation. For nonlinear systems, instead, the resulting optimization problem may not be convex (Boyd and Vandenberghe, 2004), and hence optimal solutions are in general not achievable. If a suboptimal solution is used in closed loop, stability may not hold or may be difficult to prove because the associated suboptimal MPC cost is no longer a function of the current state, as it depends, e.g., on the initial guess, on the number of iterations and algorithm, etc. Hence, the MPC cost cannot serve as a Lyapunov function for the closed-loop system. However, as established by Scokaert et al. (1999), if the optimization provides a feasible, suboptimal solution that only improves the cost from a well chosen warm start (and some other more technical assumptions are satisfied),
When dealing with perturbed systems, in most cases, the perturbations have been directly taken into account in the controller formulation leading to a so-called robust MPC formulations [see e.g. (Bemporad and Morari, 1999; Mayne and Langson, 2001; Pannocchia and Kerrigan, 2005; Rakovic et al., 2006), (Rawlings and Mayne, 2009, Ch. 3) and references therein]. Robust MPC formulations, which are usually numerically trackable only for linear systems, tend to be conservative in order to preserve recursive feasibility for all possible scenarios. Assessment of the inherent robustness, i.e., robust stability analysis of a perturbed system in closed-loop with an MPC that ignores such perturbations, has instead received much less attention (De Nicolao et al., 1996; Scokaert et al., 1997) as pointed out by Teel and coworkers (Grimm et al., 2004). In particular, Grimm et al. (2004) presented examples of nonlinear systems controlled by MPC in which the asymptotic stability of the equilibrium is destroyed by arbitrarily small perturbations. Assuming that the optimization problem remains feasible at all times, they proved that existence of a continuous Lyapunov function for the closed-loop system implies robustness to sufficiently small perturbations. Grimm et al. (2007) presented conditions to ensure recursive feasibility, using a constraint tightening approach (Lim´on Marruedo et al., 2002). Grimm et al. (2004) also show that, for linear systems with a quadratic cost, the optimal MPC cost function is a continuous Lyapunov function for the closed-loop system, because the optimal state-feedback law is continuous, hence achieving inherent robustness. A suboptimal MPC law, instead, is not necessarily continuous, even for linear systems, and hence in-
? This work was supported by National Science Foundation (Grant CTS0456694) and TWCCC members. Corresponding author: G. Pannocchia.
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herent robustness cannot be established by the same argument. Inherent robustness of suboptimal MPC was first addressed in (Lazar and Heemels, 2009), by showing input-to-state stability of the equilibrium of the closed-loop system provided that a suboptimality margin is satisfied by the suboptimal MPC algorithm. For nonlinear systems, however, the resulting optimization problem is usually nonconvex and therefore such margins cannot be ensured. The objective of this paper is to present novel results about the inherent robust stability of (a class of) nonlinear systems controlled by (a general and implementable) suboptimal MPC. By treating the closed-loop system as a difference inclusion (Kellett and Teel, 2004), we show that such robustness is achieved with respect to sufficiently small but arbitrary perturbations, without requiring a priori recursive feasibility. As corollary to these results, we note that optimal MPC is a particular suboptimal MPC, and thus the inherent robust stability properties are consequently established also for the optimal controller. Notation. The symbols I≥0 and R≥0 denote the sets of nonnegative integers and reals, respectively. The symbol I0:N−1 denotes the set {0, 1, . . . , N − 1}. The symbol | · | denotes the Euclidean norm and B denotes the closed ball of radius 1 centered at the origin. 2. PRELIMINARIES
Assumption 1. The functions f : Rn × Rm → Rn , ` : Rn × Rm → R≥0 and V f : Rn → R≥0 are continuous, f (0, 0) = 0, `(0, 0) = 0, and V f (0) = 0. Assumption 2. The sets X and X f , X f ⊆ X, are closed and contain the origin in their interiors, and U is compact and contains the origin. Assumption 3. For any x ∈ X f there exists u ∈ U such that f (x, u) ∈ X f and V f ( f (x, u)) + `(x, u) ≤ V f (x). Assumption 4. There exist positive constants a, a01 , a02 , a f and r¯, such that the cost function satisfies the inequalities `(x, u) ≥ a01 |(x, u)|a for all (x, u) ∈ X × U VN (x, u) ≤ a02 |(x, u)|a for all (x, u) ∈ r¯B V f (x) ≤ a f |x|a for all x ∈ X We observe that in Assumption 2 we allow the origin to be on the boundary of the (deviation) input space U. This case often happens in industrial applications where an upper (typically economic) optimization layer pushes the operating equilibrium towards the boundaries of the (absolute) input space. Assumption 3 implies that X f is control invariant and that X f ⊂ XN . Let u0 (x) be the optimal solution of PN (x) and let κN (x) = u0 (0; x) denote its first component, written as a function of x. The evolution of the nominal system (1) in closed loop with optimal MPC can be written as x+ = f (x, κN (x)) (2)
2.1 MPC problem: assumptions and optimal solution
2.2 Suboptimal MPC solutions
We consider discrete-time systems in the form: x+ = f (x, u), (1) n in which x ∈ R and u ∈ Rm are the state and the input at a given time, while x+ ∈ Rn is the successor state. Both state and input are subject to constraints: x(k) ∈ X , u(k) ∈ U for all k ∈ I≥0 . Given an integer N (referred to as the finite horizon), and an input sequence u of length N, u = {u(0), u(1), . . . , u(N − 1)}, let φ (k; x, u) denote the solution of (1) at time k for a given initial state x(0) = x. For any state x ∈ Rn and input sequence u ∈ UN , we define the following cost function:
Problem PN (x) is, in general, a nonconvex nonlinear program, and practical solvers cannot ensure achievement of the global minimum. Thus, we consider using any (unspecified) suboptimal algorithm having the following properties. Let u ∈ UN (x) denote the (suboptimal) control sequence for the initial state x, and let u˜ denote a warm start for the successor initial state x+ = f (x, u(0; x)), obtained from (x, u) by u˜ = {u(1; x), u(2; x), . . . , u(N − 1; x), u+ }, in which u+ ∈ U is any input that satisfies the invariance conditions of Assumption 3 for x = φ (N; x, u). We observe that the warm start satisfies u˜ ∈ UN (x+ ). Then, the suboptimal input sequence for any given x+ ∈ XN is defined as any u+ ∈ UN that satisfies: u+ ∈ UN (x+ ) (3a) + + + ˜ VN (x , u ) ≤ VN (x , u) (3b) VN (x+ , u+ ) ≤ V f (x+ ) when x+ ∈ rB, (3c) in which r is a positive scalar sufficiently small that rB ⊆ X f . Notice that constraint (3c) is required to hold only if x+ ∈ rB, and by Lemma 12, it implies that |u+ | → 0 as |x+ | → 0. Notice that condition (3b) ensures that the computed suboptimal cost is no larger than that of the warm start. Proposition 5. Any u0 (x+ ), optimal solution to PN (x+ ), satisfies conditions (3a)–(3b) for all x+ ∈ XN . Moreover, if x+ ∈ X f condition (3c) is satisfied by u0 (x+ ).
N−1
VN (x, u) =
∑ `(φ (k; x, u), u(k)) +V f (φ (N; x, u)). k=0
We then define the set of admissible initial states and associated control sequences: ZN = {(x, u) | u(k) ∈ U, φ (k; x, u) ∈ X for all k ∈ I0:N−1 , and φ (N; x, u) ∈ X f }, in which X f ⊆ X is the set of admissible states at the end of the finite horizon. Consequently, we can define the set of admissible initial states as: XN = {x ∈ Rn | ∃u ∈ UN such that (x, u) ∈ ZN } and for each x ∈ XN , the corresponding set of admissible input sequences is defined as: UN (x) = {u | (x, u) ∈ ZN }. Finally, we consider the following finite horizon optimal control problem: PN (x) : min VN (x, u) s.t. u ∈ UN (x). u
We consider the following assumptions throughout the paper.
Proof. See (Pannocchia et al., 2011). 2 Corollary 6. For any x+ ∈ XN , there exists a u+ satisfying all conditions (3). We now observe that u+ is a set-valued map of the state x+ , and so is the associated first component u(0; x+ ). If we, again,
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denote the latter map as κN (·), we can write the evolution of the closed-loop system as the following difference inclusion: x+ ∈ F(x) = { f (x, u) | u ∈ κN (x)}. (4) Proposition 7. κN (0) = {0} and F(0) = {0}. Proof. See (Pannocchia et al., 2011). 2 2.3 Difference inclusions: definitions and results Given a difference inclusion z+ ∈ H(z), H(0) = {0}, we denote by ψ(k; z) = z(k) a solution at time k ∈ I≥0 starting from the initial state z(0) = z. Definition 8. (Exponential stability). The origin of the difference inclusion z+ ∈ H(z) is exponentially stable (ES) on Z , 0 ∈ Z , if there exist scalars b > 0 and 0 < λ < 1, such that for any z ∈ Z , all solutions ψ(k; z) satisfy: ψ(k; z) ∈ Z , |ψ(k; z)| ≤ bλ k |z| for all k ∈ I≥0 . Definition 9. (Exponential Lyapunov function). A function V is an exponential Lyapunov function on the set Z for the difference inclusion z+ ∈ H(z) if there exist positive scalars a, a1 , a2 , a3 such that z ∈ Z implies that: a1 |z|a ≤ V (z) ≤ a2 |z|a , max V (z+ ) ≤ V (z) − a3 |z|a . z+ ∈H(z)
Proposition 10. If V is an exponential Lyapunov function on the set Z for the difference inclusion z+ ∈ H(z), there exists 0 < γ < 1 such that: max V (z+ ) ≤ γV (z).
Proof. From Lemma 13 we have that VN (z) is an exponential Lyapunov function for (5) in any given compact subset of Zr . Let V¯ be an arbitrary positive scalar, and consider the set S = {(x, u) ∈ Zr | VN (x, u) ≤ V¯ }. We observe that S ⊆ Zr is compact and is invariant for (5). By Lemma 11, these facts prove that the origin of the extended system (5) is ES on S , i.e., there exist scalars b0 > 0 and 0 < λ < 1, such that for any z ∈ S we can write: ψ(k; z) ∈ S and |ψ(k; z)| ≤ b0 λ k |z| for all k ∈ I≥0 in which ψ(k; z) = z(k) is a solution of (5) at time k for a given initial extended state z(0) = z. We define C = {x ∈ XN | ∃u ∈ UN (x) s.t. (x, u) ∈ S } and note that C ⊆ XN and that C is compact because it is the projection onto Rn of the compact set S . Thus for any x ∈ C and its associated suboptimal input sequence u such that z = (x, u) ∈ S , we denote with φ (k; x) the state component of ψ(k; z), i.e., a solution of the nonextended system (4), and for all k ∈ I≥0 we can write: φ (k; x) ∈ C and |φ (k; x)| ≤ |ψ(k; z)| ≤ b0 λ k |z| ≤ bλ k |x|, in which b = b0 (1 + c), because from Lemma 12 it follows that |z| ≤ |x| + |u| ≤ (1 + c)|x|. This concludes the proof because it states that the origin of the closed-loop system (4) is ES on C , and V¯ can be chosen large enough for C to contain any given compact subset of XN . 2 Corollary 15. Under Assumptions 1, 2, 3, and 4, if XN is compact, the origin of the closed-loop system (4) is ES on XN .
z+ ∈H(z)
4. ROBUST EXPONENTIAL STABILITY
Proof. See (Pannocchia et al., 2011). 2 Lemma 11. If the set Z , 0 ∈ Z , is positively invariant for the difference inclusion z+ ∈ H(z), H(0) = {0}, and there exists an exponential Lyapunov function V on Z the origin is ES on Z . Proof. See (Pannocchia et al., 2011). 2 3. NOMINAL EXPONENTIAL STABILITY 3.1 Extended state and supporting results We define the extended state z = (x, u), and we observe that it evolves according to the following difference inclusion: z+ ∈ H(z) = {(x+ , u+ ) | x+ = f (x, u(0; x)), u+ ∈ G(z)}, (5) in which (note that both x+ = f (x, u(0; x)) and u˜ depend on z): ˜ G(z) = {u+ | u+ ∈ UN (x+ ), VN (x+ , u+ ) ≤ VN (x+ , u), + + + + and VN (x , u ) ≤ V f (x ) if x ∈ rB}. We also define the following set (notice that rB ⊆ X f ): Zr = {(x, u) ∈ ZN | VN (x, u) ≤ V f (x) if x ∈ rB}. Lemma 12. There exists a positive constant c such that |u| ≤ c|x| for any (x, u) ∈ Zr . Proof. See (Pannocchia et al., 2011). 2 Lemma 13. VN (z) is an exponential Lyapunov function for the extended closed-loop system (5) in any compact subset of Zr . Proof. See (Pannocchia et al., 2011). 2 3.2 Main results Theorem 14. (ES). Under Assumptions 1, 2, 3, and 4, the origin of the closed-loop system (4) is ES on (arbitrarily large) compact subsets of XN .
4.1 Perturbed system and robust stability definitions In order to assess the inherent robustness, we consider the closed-loop evolution of the perturbed system: x+ ∈ Fed (x) = { f (x, u) + d | u ∈ κN (x + e)}, (6) in which d ∈ Rn is an unknown process disturbance and e ∈ Rn represents an unknown state measurement/estimate error. Notice that the control sequence u is computed as a suboptimal solution of PN (xm ), with xm = x + e. Hence, u is based on the nominal system (1) evolution, for the initial measured state. We denote with φed (k; x) = x(k) a solution to (6) for the initial state x(0) = x and given disturbance and measurement error sequences {d(k)}, {e(k)}. We now present the definition of robust exponential stability (RES), close to that of robust asymptotic stability (RAS) given in (Grimm et al., 2004). Definition 16. (RES). The origin of the closed-loop system (6) is robustly exponentially stable (RES) on int(XN ) if there exist scalars b > 0 and 0 < λ < 1 such that for all compact sets C ⊂ XN , with 0 ∈ int(C ), the following property holds: Given any ε > 0, there exists δ > 0 such that for all sequences {d(k)} and {e(k)} with x(0) = x ∈ C satisfying: max |d(k)| ≤ δ , max |e(k)| ≤ δ , k≥0
k≥0
xm (k) = x(k) + e(k) ∈ XN , x(k) ∈ XN , for all k ∈ I≥0 , it follows that |φed (k; x)| ≤ bλ k |x| + ε, for all k ∈ I≥0 . (8) We remark that in RES (or RAS) the robust stability condition (8) is presented for those (if any) initial states, disturbance and measurement error sequences that a-priori ensure feasibility of the perturbed closed-loop trajectories. The next definition instead requires that feasibility is satisfied at all times for all
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sufficiently small disturbance and measurement error sequences and all initial states in a given compact subset of int(XN ) . Definition 17. (SRES). The origin of the closed-loop system (6) is strongly robustly exponentially stable (SRES) on a compact set C ⊂ XN , 0 ∈ int(C ), if there exist scalars b > 0 and 0 < λ < 1 such that the following property holds: Given any ε > 0, there exists δ > 0 such that for all sequences {d(k)} and {e(k)} satisfying |d(k)| ≤ δ and |e(k)| ≤ δ for all k ∈ I≥0 , and all x ∈ C , we have that xm (k) = x(k) + e(k) ∈ XN , x(k) ∈ XN , for all k ∈ I≥0 , (9a)
u
ZN
Sρ
S
UN
u0(x)
x
ρ
Cρ
|φed (k; x)| ≤ bλ k |x| + ε, for all k ∈ I≥0 . (9b)
XN
4.2 Recursive feasibility in perturbed system evolution
Fig. 1. Sketch of the main sets involved in SRES. While the warm start u˜ is feasible for the predicted suc˜ ∈ ZN , it may cessor state x˜+ = f (xm , u(0; xm )), i.e., (x˜+ , u) + = not be feasible for the measured successor state, i.e., xm f (x, u(0; xm )) + d + e+ . Notice that the true successor state, which is unknown in general, is x+ = f (x, u(0; xm )) + d. If + , u) ˜ ∈ (xm / ZN , the right-hand side of the cost inequality (3b) is not meaningful. In such cases, we need to modify the warm +, u ˜ + p) ∈ ZN , and we consider start with a term p such that (xm the following additional assumption. Assumption 18. For any x, x0 ∈ XN and u ∈ UN (x), there exists u0 ∈ UN (x0 ) such that |u − u0 | ≤ σ (|x − x0 |) for some K function σ (·). We remark that Assumption 18 has been shown to hold, e.g., for linear systems subject to polytopic constraints on (x, u), and for nonlinear systems without state (or mixed) constraints. Among various options for finding p, we consider the following feasibility problem (notice that x˜+ is known): + + Find p s.t. u˜ + p ∈ UN (xm ) and |p| ≤ σ (|xm − x˜+ |), (10) + ), it immediately follows that We observe that if u˜ ∈ UN (xm p = 0 satisfies the feasibility problem (10), and hence Assumption 18 is unnecessary. ˜ ∈ ZN Proposition 19. Under Assumption 18, for any (x˜+ , u) + ∈ X , the set of solutions to (10) is nonempty. and xm N Proof. The result follows directly from Assumption 18 by + ). 2 noticing that u˜ ∈ UN (x˜+ ) and u˜ + p ∈ UN (xm + ∈ X , we Given any p satisfying (10), and for any given xm N replace conditions (3) with the following: + u+ ∈ UN (xm ) (11a) + + + (11b) VN (xm , u ) ≤ VN (xm , u˜ + p) + + + + when xm ∈ rB. (11c) VN (xm , u ) ≤ V f (xm )
In the perturbed case, the extended state is z = (x, u), where u is a suboptimal solution to PN (xm ) where xm = x + e is the measured state. The extended system evolves as follows: z+ ∈ Hed (z) = {(x+ , u+ ) | x+ = f (x, u(0; xm )) + d, u+ ∈ Ged (z)}, (12) + = x+ + e+ and u ˜ + p depend on z): in which (note that both xm + + + + Ged (z) = {u+ | u+ ∈ UN (xm ), VN (xm , u ) ≤ VN (xm , u˜ + p), + + + + VN (xm , u ) ≤ V f (xm ) if xm ∈ rB}
4.3 Main results We define zm = (xm , u) = (x + e, u) = z + (e, 0), and we observe that zm ∈ Zr . The following supporting result is fundamental. Lemma 20. For every µ > 0, there exists a δ > 0 such that, for all (zm , e, d, e+ ) ∈ Zr ×δ B×δ B×δ B, z = zm −(e, 0), such that + ∈ X , and some γ, 0 < γ < 1, we have: xm N max VN (z+ ) ≤ max{γVN (z), µ}. z+ ∈Hed (z)
Proof. See (Pannocchia et al., 2011). 2 We now characterize the compact sets over which SRES is guaranteed to hold. Consider a scalar V¯ > 0 such that the set: S = {z ∈ Rn × UN | VN (z) ≤ V¯ } satisfies S ⊆ ZN , i.e., S is a sublevel set of Rn × UN fully contained in ZN . Thus, by definition, for any z = (x, u) ∈ S , it follows that x ∈ XN . Next, given a scalar ρ > 0 and any zm ∈ Zr , we define the following measure and associated set: ρ VN (zm ) = max VN (z) s.t. z = zm − (e, 0), (13) e∈ρB
Sρ = {zm ∈ Zr | VN (zm ) ≤ V¯ }, (14) in which we assume that ρ is small enough that Sρ is nonempty. Finally, we define the following compact set: Cρ = {x ∈ Rn | x = xm − e, e ∈ ρB, ∃u s.t. (xm , u) ∈ Sρ }, (15) and observe that 0 ∈ int(Cρ ) ⊂ XN for ρ sufficiently small. These sets are depicted in Figure 1. The main SRES result of this paper is as follows. Theorem 21. (SRES of suboptimal MPC). Under Assumptions 1, 2, 3, 4, and 18, the origin of the perturbed closed-loop system (6) is SRES on Cρ . ρ
Proof. (Robust feasibility) Suppose that x ∈ Cρ and let z = (x, u) be the corresponding augmented state where u is a suboptimal sequence computed for the measured state xm = x + e, e ∈ ρB. We recall that VN (z) ≤ V¯ , i.e., z ∈ S and that ˜ Since VN (·) is zm ∈ Sρ ⊆ Zr . Moreover, we define z˜+ = (x˜+ , u). an exponential Lyapunov function for the nominal system and ¯ N (zm ) for some zm ∈ Zr , Proposition 10 gives that VN (˜z+ ) ≤ γV ˜ ≤ γ¯V¯ < V¯ . 0 < γ¯ < 1. Because zm ∈ Sρ it follows that VN (x˜+ , u) + = f (x , u(0; x )) − f (x, u(0; x )) − d − Recalling that: x˜+ − xm m m m e+ , it follows from continuity of f that there exists a δ¯1 > 0 such + , u) + ∈ X for all (z , e, d, e+ ) ∈ ˜ < V¯ and thus xm that VN (xm N m ¯ ¯ ¯ Sρ × δ1 × δ1 × δ1 . Hence, the initialization step (10) is well
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¯ V¯ . From continuity of defined. Define any 0 < µ < (1 − γ) + |), we can choose VN and f , and because |p| ≤ σ (|x˜+ − xm +, u ˜+ δ¯2 > 0 such that the following condition holds: VN (xm + + ¯ V¯ ≤ V¯ . This proves ˜ + µ < VN (x˜ , u) ˜ + (1 − γ) p) ≤ VN (x˜ , u) + ˜ + p) < V ¯ . From continuity of VN it that VN (z+ m ) ≤ VN (xm , u also follows that we can choose ρ > 0 sufficiently small that ρ + ¯ ¯ ¯ VN (z+ m ) ≤ V . Taking δ = min{ρ, δ1 , δ2 } it follows that zm ∈ Sρ + for all (zm , e, d, e ) ∈ Sρ × δ B × δ B × δ B. This implies: x(k) ∈ Cρ ⊆ XN for all k ∈ I≥0 , and also that xm (k) ∈ XN for all k ∈ I≥0 . Hence, (9a) holds. (Robust stability) We denote by ψed (k; z) a solution of the perturbed difference inclusion (12) at time k ∈ I≥0 starting from the initial state z(0) = z and given disturbance and measurement error sequences {d(k)}, {e(k)}. As established in the proof of Proposition 7 we have that there exists a scalar a1 > 0 such that a1 |z|a ≤ VN (z) for any z ∈ Cρ × UN ⊆ X × UN . Moreover, by Lemma 13 there exists a scalar a2 > 0 such that VN (z) ≤ a2 |z|a for any z ∈ Cρ × UN . From Lemma 20, by induction, we write: a1 |ψed (k; z)|a ≤ VN (ψed (k; z)) ≤ k
k
4
2 1 x2
max{γ VN (z), µ} ≤ max{γ a2 |z| , µ},
-1 -2 -3 -4 -4
¯ k |z|, ε¯ } |φed (k; x)| ≤ |ψed (k; z)| ≤ max{bλ ¯ k |z| + ε¯ ≤ bλ k |x| + ε, ≤ bλ ¯ + c) and ε = ε¯ + bcδ ¯ , completing the proof with b = b(1 because 0 < λ < 1. 2 Corollary 22. (RES of suboptimal MPC). Under Assumptions 1, 2, 3, 4, and 18, the origin of the perturbed closed-loop system (6) is RES on int(XN ). Proof. This result follows immediately because robust feasibility is assumed in RES. Thus, for any compact set C ⊂ int XN , the second part (Robust stability) of the proof of Theorem 21 can be readily applied to obtain for all x ∈ C : for all k ∈ I≥0 .
We observe that while RES holds in int(XN ), SRES is guaranteed to hold in a compact subset Cρ ⊂ int(XN ), 0 ∈ int(Cρ ). However, SRES may hold even in larger subsets of int(XN ). 4.4 Further comments on robust stability under optimal MPC By Proposition 5, it follows that all robust stability results that we proved for suboptimal MPC, readily apply to optimal MPC even for cases in which there exist more than one optimal solution point, i.e., when the optimal control u0 (·) is a setvalued map. Moreover, for optimal MPC, the definition (13) ρ of VN (and hence of Sρ ) can be modified as follows: ρ
e∈ρB
5. APPLICATION EXAMPLE As an example, we consider the following system:
-3
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-1
0
1
2
3
4
x1
Fig. 2. Approximate feasibility sets of the three controllers. x1+ = x1 + u x2+ = bx2 + u3 ,
C1. No state constraints are enforced, X = R2 , and the terminal constraint set is the origin, X f = {0}. C2. No state constraints are enforced, X = R2 , and the terminal constraint set is X f = {x ∈ R2 | V f (x) ≤ α} with V f (x) = x0 Px, α > 0 and P later defined. C3. State constraints are enforced, X = [−2, 2]2 , and the terminal constraint set is the same of C2.
¯ k |z|, (µ/a1 )1/a } ≤ max{bλ ¯ k |z|, ε¯ }, |ψed (k; z)| ≤ max{bλ in which λ = γ 1/a , b¯ = (a2 /a1 )1/a and ε¯ = (µ/a1 )1/a . Finally, from Lemma 12 recalling that |u| ≤ c|xm | ≤ c|x| + cδ and that φed (k; x) represents the state component of ψed (k; z), for all x ∈ Cρ we write:
VN (xm , u0 (xm )) = max VN (xm − e, u0 (xm )).
0
with b = 0.9. The control horizon is N = 3, the input set is U = [−1, 1], and the stage cost function is given by `(x, u) = |x|2 + u2 . Three different nonlinear MPC formulations are considered.
a
which implies
|φed (k; x)| ≤ bλ k |x| + ε
C1 C2 C3
3
(16)
We remark that controller C1 does not satisfy Assumption 2 because X f does not contain the origin in its interior. In the definition of controllers C2 and C3, we note that the linearization of of system at the origin can be written as: 1 0 1 x+ = Ax + Bu with A = , B= , 0 b 0 and we observe that the pair (A, B) is stabilizable. Therefore, we follow the procedure described in (Rawlings and Mayne, 2009, Par. 2.5.3.2). We define a linear control law κ f (x) = Kx = [−1, 0]x and observe that such control law which is stabilizing for the linearized system. Let QK = I + K 0 K, AK = A + BK, and solve the following Lyapunov equation: A0K PAK + 2QK = P, 0 obtaining P = 0 10.53 ; we notice that P is positive definite. Consequently, we define the terminal cost V f (x) = x0 Px, while the terminal constraint set is given by X f = {x ∈ R2 | V f (x) ≤ α}, in which α > 0. It can be shown (Rawlings and Mayne, 2009, Par. 2.5.3.2) that there exists α > 0 such that Assumption 3 holds for for u = Kx = [−1, 0]x In particular, it can be verified that α = 1.1 is one such value. Furthermore, it can be verified that Assumptions 1 and 2 hold. Similarly, the three conditions of Assumption 4 hold for a = 2. We report in Fig. 2 the approximate feasibility sets, XN , for the three controllers. As expected, we notice that the feasibility set XN for C2 contains both the feasibility sets for C1 and C3. We report in Fig. 3 the first component of UN (x), with x = 0.5 [cos(θ ), sin(θ )], as a function of θ for the controllers C1 and C2 (the plot for C3 is not reported as it is identical to that of C2). For C1, we can notice that Assumption 18 does not hold at the point indicated by the arrow, whereas no such points can be noticed for C2.
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this specialization will be discussed in a separate publication (Pannocchia et al., 2010).
First component of UN (x)
1
ACKNOWLEDGMENTS
0.5
We thank Prof. A.R. Teel for providing an updated and detailed proof of Proposition 8 in (Grimm et al., 2004).
Assumption 18 does not hold 0
REFERENCES -0.5
-1 -4
-3
-2
-1
0
1
2
3
4
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2
3
4
θ /π
First component of UN (x)
1
0.5
0
-0.5
-1 -4
-3
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-1
0 θ /π
Fig. 3. First component of UN (x) as a function of the initial state x = 0.5 [cos(θ ), sin(θ )]. C1 (top), C2 (bottom). 6. CONCLUSIONS Attaining global solutions of MPC problems for nonlinear system is, in general, not practical, but when a suboptimal solution is implemented in closed loop many stability issues arise. Such issues were first addressed by Scokaert et al. (1999) who proved nominal asymptotic stability of the origin of the closed-loop system in a neighborhood of the origin. The requirement to suboptimal MPC is that it does not increase the cost function with respect to a well-defined warm start sequence, and it is therefore of general applicability and implementation. In this paper, under similar assumptions, we established nominal exponential stability of the origin of the closed-loop system in arbitrarily large compact subsets of the feasible set. Furthermore, we established inherent exponential robust stability of the origin of the closed-loop system with respect to additive process disturbances and measurement errors, in the spirit of the ideas developed by Teel and coworkers (Grimm et al., 2004, 2007). In particular, we proved robust recursive feasibility in an appropriate compact subset of the nominal region of attraction for all sufficiently small, but arbitrary, perturbations. The main requirement is a local continuity assumption of the input feasibility set, which holds, e.g., for linear systems or nonlinear systems in absence of state constraints. We observe that no specific modification, e.g. constraint tightening (Lim´on Marruedo et al., 2002; Grimm et al., 2007), of the MPC problem was used to preserve robust feasibility. All robustness results established for suboptimal MPC apply to optimal MPC as well, and this represents itself an improvement in stability analysis of optimal nonlinear MPC systems. These results can be specialized further to suboptimal MPC of linear systems, and
Bemporad, A. and Morari, M. (1999). Robust model predictive control: A survey. In A. Garulli, A. Tesi, and A. Vicino (eds.), Robustness in Identification and Control, number 245 in Lecture Notes in Control and Information Sciences, 207– 226. Springer-Verlag. Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press, Cambridge (UK). De Nicolao, G., Magni, L., and Scattolini, R. (1996). On the robustness of receding-horizon control with terminal constraints. IEEE Trans. Auto. Contr., 41(3), 451 –453. Grimm, G., Messina, M.J., Tuna, S.E., and Teel, A.R. (2004). Examples when nonlinear model predictive control is nonrobust. Automatica, 40, 1729–1738. Grimm, G., Messina, M.J., Tuna, S.E., and Teel, A.R. (2007). Nominally robust model predictive control with state constraints. IEEE Trans. Auto. Contr., 52(10), 1856–1870. Kellett, C.M. and Teel, A.R. (2004). Smooth Lyapunov functions and robustness of stability for difference inclusions. Syst. Contr. Lett., 52, 395–405. Lazar, M. and Heemels, W. (2009). Predictive control of hybrid systems: Input-to-state stability results for sub-optimal solutions. Automatica, 45(1), 180–185. ´ Lim´on Marruedo, D., Alamo, T., and Camacho, E.F. (2002). Input-to-state stable MPC for constrained discrete-time nonlinear systems with bounded additive disturbances. In Proceedings of the 41st IEEE Conference on Decision and Control, 4619–4624. Las Vegas, Nevada. Mayne, D.Q. and Langson, W. (2001). Robustifying model predictive control of constrained linear systems. Electronics Letters, 37, 1422–1423. Mayne, D.Q., Rawlings, J.B., Rao, C.V., and Scokaert, P.O.M. (2000). Constrained model predictive control: stability and optimality. Automatica, 36, 789–814. Pannocchia, G. and Kerrigan, E.C. (2005). Offset-free receding horizon control of constrained linear systems. AIChE J., 51, 3134–3146. Pannocchia, G., Rawlings, J.B., and Wright, S.J. (2011). Addendum to the paper “Is suboptimal nonlinear MPC inherently robust?”. Technical Report 2011–02, TWCCC. Available at http://twccc.che.wisc.edu. Pannocchia, G., Wright, S.J., and Rawlings, J.B. (2010). Robust stability of suboptimal linear MPC based on partial enumeration. Submitted to J. Proc. Contr. Rakovic, S., Kerrigan, E., Mayne, D., and Lygeros, J. (2006). Reachability analysis of discrete-time systems with disturbances. IEEE Trans. Auto. Contr., 51(4), 546–561. Rawlings, J.B. and Mayne, D.Q. (2009). Model Predictive Control: Theory and Design. Nob Hill Publishing, Madison, WI. Scokaert, P.O.M., Mayne, D.Q., and Rawlings, J.B. (1999). Suboptimal model predictive control (feasibility implies stability). IEEE Trans. Auto. Contr., 44, 648–654. Scokaert, P.O.M., Rawlings, J.B., and Meadows, E.S. (1997). Discrete-time stability with perturbations: Application to model predictive control. Automatica, 33, 463–470.
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