Improved Stability Conditions for Unconstrained Nonlinear Model ...
51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Improved Stability Conditions for Unconstrained Nonlinear Model Predictive Control by using Additional Weighting Terms Marcus Reble, Daniel E. Quevedo, and Frank Allg¨ower Abstract— In this work, we present two unconstrained MPC schemes using additional weighting terms which allow to obtain improved stability conditions. First, we consider unconstrained MPC with general terminal cost functions. If the terminal cost is not a control Lyapunov function, but satisfies a relaxed condition, then our results yield improved estimates for a stabilizing prediction horizon. Furthermore, our analysis also allows to recover two well-known results as special cases: if the terminal cost function is chosen as zero, we recover previous conditions on the length of the prediction horizon such that stability is guaranteed; and if the terminal cost is a control Lyapunov function conform to the stage cost, stability follows independently of the length of the prediction horizon. Second, we propose to use an exponential weighting on the stage cost in order to improve the stability properties of the closed-loop. This also allows to consider local controllability assumptions in combination with a suitable terminal constraints and thereby gives a connection to the classical MPC approaches using terminal constraints.
I. I NTRODUCTION Model predictive control (MPC) is a modern control method which has received much attention in academic research as well as in practical applications in particular due to its ability to handle hard constraints and to take performance criteria directly into account [1]. However, MPC with a finite prediction horizon does not guarantee stability in general, which was also demonstrated on a practical example in [2]. In order to overcome this problem, one of the following elements, or a combination thereof, is usually used: introducing additional constraints in the finite horizon optimal control problem (in particular terminal constraints), using control Lyapunov functions (CLF) as terminal cost and/or a prediction horizon chosen sufficiently large based on a suitably defined controllability assumption, see also Figure 1. MPC schemes using terminal constraints and a terminal cost are the most well-studied approaches [3–5]. In [6, 7], the terminal constraint is omitted but it is nevertheless guaranteed to be satisfied for a defined set of initial states. The approach presented in [1, Section 2.6.1] replaces the terminal constraint by a sufficiently large terminal cost using Marcus Reble and Frank Allg¨ower are with the Institute for Systems Theory and Automatic Control (IST), University of Stuttgart, Germany. {reble,allgower}@ist.uni-stuttgart.de. Research supported by the German Research Foundation (DFG) within the Priority Programme 1305 “Control Theory of Digitally Networked Dynamical Systems” and within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. Daniel E. Quevedo is with the School of Electrical Engineering & Computer Science, The University of Newcastle, Australia. [email protected]. Research supported by Australian Research Council’s Discovery Projects funding scheme (project number DP0988601).
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a controllability assumption, which requires an upper bound of the optimal cost of an associated problem with terminal equality constraint. Using a similar controllability assumption, but a significantly different analysis, so-called unconstrained MPC schemes [8–12], i.e., schemes without terminal constraints, guarantee stability by providing an explicit bound on the minimal stabilizing prediction horizon. The first goal of our work is to give a connection between the unconstrained MPC schemes [8–12] and the classical schemes using terminal constraints and cost terms [3–5]. A first result in this respect has been reported in [13] by considering a generalized integral cost. In the present work, we provide further results with a different point of view. Secondly, many examples exhibit a significant difference between the actual minimal stabilizing prediction horizon observed in practice and the one provided by theoretical results. The second goal of our work is to overcome this conservatism. For a similar goal, an algorithmic approach has been presented in [14], which takes advantage of the symmetry properties of the stability condition with respect to the sampling time and reuses optimal input trajectories calculated at previous sampling instants in order to guarantee stability for a shorter prediction horizon. In Section III, we consider unconstrained MPC with a general positive semi-definite terminal cost function. In particular, we investigate the use of assumptions from both MPC schemes in contrast to using a generalized assumption as in [13]. This result is related to the discrete-time results in [15], which takes properties of the terminal cost function into account in order to improve the stability conditions obtained in [8]. However, we obtain better estimates for the stabilizing prediction horizon and our analysis allows to recover recent results for unconstrained MPC without terminal cost. In contrast, setting the terminal cost equal to zero in the result of [15] does not allow to guarantee asymptotic stability even for an arbitrarily large prediction horizon. The combination proposed in our result enables us to relax the assumptions from both MPC schemes and provides a different point of view. Namely, we show that the combination of a relaxed condition on the terminal cost, which does not need to be a control Lyapunov function, and a controllability assumption with a shorter prediction horizon provides guaranteed stability. Our analysis recovers previous well-known results as special cases and, hence, allows to bridge the gap between the approaches, which have been mostly considered separately in literature so far, see Figure 1. In Section IV, we propose to use unconstrained MPC
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Local CLF and Constraints [3–5]
Terminal Constraints [16, 17]
s.t.
Stability in MPC
Terminal Cost (Global CLF) [18]
Fig. 1.
u ¯(t0 ) ∈ U ,
Generalized Integral Cost [13] Exponential Weighting (Section V)
ti
+ E(¯ x(ti + T ; ti )) .
F (x, u) ≥ F (x, 0) ≥ αF (|x|) and E(x) ≥ 0 .
with an additional exponential weighting on the stage cost in order to reduce the prediction horizon necessary for existing stability guarantees. A result similar to [13] is derived in Section V, which replaces the generalized integral cost by the exponential weighting introduced in the preceding section. Both of these results allow to consider local controllability assumptions in combination with terminal constraints in contrast to the unconstrained MPC schemes [8–12], for which such a modification is not possible. Hence, these schemes bridge another, yet similar, gap between previous approaches, see Figure 1.
uMPC (t) = u∗ (t; ti ) ,
III. U NCONSTRAINED MPC WITH G ENERAL T ERMINAL C OST F UNCTION
u ¯(·)
(5a)
E (t; ti ) = E(x (t; ti )) ,
(5b)
∗
for t ∈ [ti , ti + T ] and ti ∈ {0, δ}. From this definition, it directly follows that Z ti +T ∗ JT (x(ti )) = F ∗ (t0 ; ti )dt0 + E ∗ (ti + T ; ti ) . (6) ti
We will use the following simple general result on asymptotic stability to derive our main results, see also [9, Proposition 2.4] for related results in discrete-time. Lemma 1 (General Condition for Asymptotic Stability) Suppose the following holds for all x ∈ Rn Z δ JT∗ (x(δ)) − JT∗ (x(0)) ≤ −α F ∗ (t0 ; 0)dt0 , (7) 0
In this section, we derive sufficient stability conditions for unconstrained MPC with a general positive semi-definite terminal cost function. The open-loop finite horizon optimal control problem at sampling instant ti given measured state x(ti ) is formulated as minimize JT (x(ti ), u ¯)
(4)
F ∗ (t; ti ) = F (x∗ (t; ti ), u∗ (t; ti )) , ∗
with state x(t) ∈ Rn , initial condition x0 ∈ Rn , and control input u(t) ∈ U ⊆ Rm . The function f is continuously differentiable and the constraint set U ⊂ Rm is compact and contains the origin in its interior. Throughout this work, we assume that System (1) has a unique solution for all t ∈ R≥0 for any initial condition x0 ∈ Rn and any piecewise- and right-continuous input function u(·) : R≥0 → U. Without loss of generality, we assume x = 0 to be an equilibrium of System (1) for u = 0, i.e., f (0, 0) = 0. The problem of interest is to stabilize the equilibrium x = 0 via continuoustime model predictive control with periodic sampling.
ti ≤ t < ti + δ .
Without loss of generality, we consider the two consecutive sampling instants t0 = 0 and t1 = δ in the following. Since System (1) is time-invariant, all results hold analogously for any other two consecutive sampling instants ti and ti+1 . With slight abuse of notation, we use the following abbreviations
We consider nonlinear continuous-time systems (1)
(3)
We assume that the optimal control which minimizes (2) is given by u∗ (t0 ; ti ) with associated predicted state trajectory x∗ (t0 ; ti ), t0 ∈ [ti , ti + T ]. The optimal cost is denoted by JT∗ (x(ti )). For a given sampling time δ with 0 < δ ≤ T , the control input to the system is defined by the optimal solution of Problem (2) at sampling instants ti = i δ , i ∈ N0 , in the usual continuous-time receding horizon fashion
II. P ROBLEM S ETUP x(0) = x0
(2d)
x ¯(t0 ; ti ) is the predicted trajectory starting from initial condition x ¯(ti ; ti ) = x(ti ) and driven by u ¯(t0 ) for t0 ∈ [ti , ti + T ]. n m The stage cost F : R × R → R≥0 and the terminal cost function E : Rn → R≥0 are continuous, F (0, 0) = E(0) = 0 and there is a class K∞ function1 αF : R≥0 → R≥0 such that for all x ∈ Rn , u ∈ Rm
Schematic overview on stability conditions in MPC.
x(t) ˙ = f (x(t), u(t)) ,
(2c)
for all t0 ∈ [ti , ti + T ] with Z ti +T JT (x(ti ), u ¯) = F (¯ x(t0 ; ti ), u ¯(t0 )) dt0
Controllability Assumption [8–12]
Unconstrained MPC with Terminal Cost (Section III)
x ¯˙ (t0 ; ti ) = f (¯ x(t0 ; ti ), u ¯(t0 )) , x ¯(ti ; ti ) = x(ti ) , (2b)
(2a)
in which α > 0. Then, the closed-loop using MPC is asymptotically stable. Proof: Similar to the result in [3], it can be shown that JT∗ (x) is continuous in x in a neighborhood of the origin. Condition (7) directly implies that JT∗ is non-increasing along 1 A function α : R ≥0 → R≥0 is said to belong to class K∞ if it is continuous, strictly increasing, α(0) = 0 and α(s) → ∞ as s → ∞.
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trajectories of the closed-loop and stability directly follows. R∞ Asymptotic stability follows from F (x(t0 ), u(t0 ))dt0 ≤ 0
JT∗ (x(0)), continuity of f , compactness of U, the lower bound (3), and Barbalat’s Lemma. In contrast to the unconstrained MPC schemes without terminal cost, see, e.g., [9, 10, 12], JT∗ is not necessarily monotonically increasing in T and, hence, α does not give a suboptimality estimate of the closed-loop compared to the infinite horizon optimal controller. Our main results, namely Theorems 6, 8, and 9, are based on the above lemma. In order to derive α in (7), we use the following two controllability assumptions. Assumption 1 (Controllability Assumption based on F ) For all T 0 ∈ R≥0 and x0 ∈ Rn , there exists a piece-wise continuous input trajectory u ˆ(·; 0) with u ˆ(t; 0) ∈ U for all t ∈ [0, T 0 ] and JT∗ 0 (x0 ) ≤ JT 0 (x0 , u b) ≤ B(T 0 )F (x0 , 0), in which B : R≥0 → R≥0 is a continuous and bounded function. Assumption 2 (Controllability Assumption based on E) For all x0 ∈ Rn , there exists a piece-wise continuous input trajectory u ˆ(·; 0) with u ˆ(t; 0) ∈ U for all t ∈ [0, δ] and δ Z Γ F (¯ x(t0 ; 0), u ˆ(t0 ; 0))dt0 + E(¯ x(δ; 0)) ≤ E(x0 )
Lemma 3 (Calculation of Ξ) Suppose that Controllability Assumption 1 is satisfied. Then, JT∗ (x(δ))
ZT ≤Ξ
F ∗ (t0 ; 0)dt0 ,
(9)
δ
in which
1 Ξ
= 1 − exp −
RT δ
! 1 B(T +δ−t∗ )
dt
∗
.
Proof: The proof is similar to the proof of Lemma 7 in [12] for unconstrained MPC without terminal cost. Lemma 4 (Calculation of γ) Suppose that the Controllability Assumption 1 is satisfied. Then, ZT
F (t ; 0)dt + E (T ; 0) ≤ γ ∗
0
0
∗
Zδ
F ∗ (t0 ; 0) dt0 ,
(10)
0
δ
in which
1 γ
= exp
Rδ 0
! 1 B(T −t∗ )
dt
∗
− 1.
Proof: The proof mirrors the proof of Lemma 8 in [12] for unconstrained MPC without terminal cost.
0
in which Γ ∈ [0, 1]. Assumption 1 is a standard assumption in unconstrained MPC [8–12]. However, note that Assumption 1 depends explicitly on the terminal cost due to the definition of JT . Assumption 2 is always satisfied for Γ = 0. On the other hand, if no terminal cost term is considered, i.e., E(x) = 0 as in [9–12], Assumption 2 can only be satisfied for Γ = 0. For 0 < Γ < 1, Assumption 2 can be interpreted as the terminal cost E(x) being “similar” to a CLF, however in a significantly weaker sense as shown in the following proposition. Proposition 2 (CLF-like Condition for Assumption 2) Suppose that there exists ν ∈ R≥0 such that for all x ∈ Rn , there exists u ∈ U such that ˙ E(x) ≤ −F (x, u) + ν E(x) . Then, Assumption 2 is satisfied for Γ = e
can show the following intermediate results using the two assumptions above.
−νδ
(8)
.
Proof: The result follows directly from integrating (8) from 0 to δ and taking F (x, u) ≥ 0 into account. For ν = 0, we obtain Γ = 1 and consequently Assumption 2 corresponds to E being an F -conform control Lyapunov function (CLF), which is a common assumption in MPC in order to guarantee stability of the closed-loop, see, e.g., [3– 5, 18]. For ν > 0, this assumption is relaxed and additional arguments are required in order to guarantee stability. In this work, a prediction horizon chosen “sufficiently large” is considered with explicit conditions on the prediction horizon given in Theorem 6 based on Assumption 1. To this end, we
Lemma 5 (Direct Consequence of Assumption 2) Suppose that Controllability Assumption 2 is satisfied. Then, Γ JT∗ (x(δ))
ZT ≤Γ
F ∗ (t0 ; 0)dt0 + E ∗ (T ; 0) .
(11)
δ
Proof: The proof follows directly from optimality of JT∗ (x(δ)) and using Assumption 2 for x0 = x∗ (T ; 0), which provides an upper bound on the cost on the interval [T, T +δ]. We are now able to state our main result on stability of unconstrained MPC with general positive semi-definite terminal cost functions. Theorem 6 (Stability of Unconstrained MPC) Suppose that the Controllability Assumptions 1 and 2 are satisfied for System (1) and consider Ξ α=1−γ −1 , (12) 1 + (Ξ − 1)Γ with γ and Ξ defined in Lemmata 3 and 4, respectively. If α > 0, then the closed-loop is asymptotically stable. Proof: Multiplying Inequality (9) in Lemma 3 with 1 − Γ and multiplying Inequality (11) in Lemma 5 with Ξ, and adding the two resulting inequalities yields T Z Ξ F ∗ (t0 ; 0)dt0 + E ∗ (T ; 0) . JT∗ (x(δ)) ≤ 1 + (Ξ − 1)Γ
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δ
Following as in the proof of [12, Theorem 9] and using (6) and Lemma 4, we obtain that (7) holds and the result follows from Lemma 1. Note that α > 0 and thus asymptotic stability can always be guaranteed for a prediction horizon chosen large enough because Ξ → 1 and α → 1 for T → ∞. For Γ = 0, for which Assumption 2 is trivially satisfied, we recover the stability condition from [12, Theorem 9] and consequently the results for unconstrained MPC without terminal cost. These results have been shown to be the “best possible” stability conditions based only on the Controllability Assumption 1, see [12, Theorem 10]. Here, “best possible” refers to the largest α for which (7) in Lemma 1 holds. Loosely speaking, this can be seen because Fb∗ , defined in the proofs of Lemmata 3 and 4, satisfies all conditions implied by Controllability Assumption 1 and yields α = 1−γ(Ξ−1) for (7). Hence, one can only guarantee (7) for a larger α by taking into account additional information, e.g., about the system or the optimal cost function. If Assumption 1 is satisfied for Γ = 1, for instance if E is a global F -conform control Lyapunov function, then asymptotic stability is guaranteed independently of the Controllability Assumption 1. Thus, we recover the previous stability result using global control Lyapunov functions. Summarizing, Theorem 6 allows in some sense to bridge the gap between the stability results for MPC schemes using control Lyapunov functions as terminal cost and the more recently developed MPC schemes based on controllability assumptions. However, the case of Assumption 2 only being satisfied in a terminal region around the origin has not been treated in this work because an additional terminal constraint makes the verification of Assumption 1 significantly harder. This issue will be subject to future research. IV. U NCONSTRAINED MPC WITH E XPONENTIAL W EIGHTING In this section, we consider a similar MPC setup as in the previous section with the following two modifications. First, in the open-loop finite horizon optimal control problem (2) we replace the cost functional JT (x(ti ), u ¯) in (2d) by JT,µ (x(ti ), u ¯) =
tZ i +T
β(t0 − ti ) F (¯ x(t0 ; ti ), u ¯(t0 )) dt0 ,
ti
(13) in which β(t) = eµ t for some constant µ ≥ 0 is an exponential weighting on the stage cost and no terminal cost terms are considered. Second, we restrict ourselves to the following exponential controllability assumption, which is a special case of Assumption 1 and a standard assumption in unconstrained MPC [9–12]. Assumption 3 (Exponential Controllability) For all x0 ∈ Rn , there exists a piece-wise continuous input trajectory u ˆ(·; 0) with u ˆ(t; 0) ∈ U for all t ∈ R≥0 and corresponding state trajectory x ¯uˆ such that F (¯ xuˆ (t; 0), u ˆ(t; 0)) ≤ C e−λ t F (x0 , 0) , ∀t ∈ R≥0
with overshoot constant C ≥ 1 and decay rate λ > µ ≥ 0. Following the analysis of [12] and Section III, we consider the two consecutive sampling instants t0 = 0 and t1 = δ and we will use the abbreviation F ∗ (t; ti ) introduced in (5) and ∗ Lemma 1. The optimal cost is denoted by JT,µ (x(ti )) and analogous to (6), we have ∗ JT,µ (x(ti ))
tZ i +T
=
β(t0 − ti )F ∗ (t0 ; ti )dt0 .
(14)
ti
Lemma 7 (Unconstrained MPC with Exp. Weighting) Suppose that the Exponential Controllability Assumption 3 is satisfied. Then, ZT
∗ JT,µ (x(δ))
≤Ξ
β(t0 )F ∗ (t0 ; 0)dt0 ,
(15a)
β(t0 )F ∗ (t0 ; 0) dt0 ,
(15b)
δ
ZT
β(t0 )F ∗ (t0 ; 0)dt0 ≤ γ
Zδ 0
δ
in which (λ−µ) δ 1 e −1 C e−µδ , =1− Ξ e(λ−µ) T − 1 C1 1 1 − e−(λ−µ) T = − 1. γ e−(λ−µ) δ − e−(λ−µ) T
(16a) (16b)
Proof: Due to (14) and the definition of β(t), we have ∗ JT,µ (x(δ))
=e
−µδ
δ+T Z
β(t0 )F ∗ (t0 ; δ)dt0 .
δ
The Exponential Controllability Assumption 3 and optimality ∗ of JT,µ (x(δ)) then imply that ∗ eµδ JT,µ (x(δ)) ≤
t∗
Z
β(t0 )F ∗ (t0 ; 0)dt0
δ
+ B(T + δ − t∗ )β(t∗ )F ∗ (t∗ ; 0) C in which B(T ) = λ−µ (1 − e−(λ−µ) T ) holds for all t∗ ∈ [δ, T ]. Noting the similarities allows to follow the proof of Lemma 3 in order to prove (15a) and (16a). The proof of (15b) and (16b) follows exactly the lines of the proof of Lemma 4 when setting E ∗ (T ; 0) = 0 and replacing F ∗ (t0 ; 0), Fb∗ (t) by β(t0 )F ∗ (t0 ; 0), β(t0 )Fb∗ (t), respectively, in all expressions. Comparing these results to [12, Section 4.1], or equivalently by substitution of B(T ) = Cλ (1 − e−λ T ) in the results of Lemmata 3 and 4, we note two differences. First, λ is replaced by λ − µ in all expressions. Second, the expression for Ξ is multiplied by an additional term e−µδ . This second term is directly caused by shifting the prediction horizon when using an exponential weighting on the stage cost and beneficial for achieving α > 0, and hence for stability guarantees.
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α
1 0.95
10
0.9
T∗
15
0.85
5
0.8
0
0.75 0
−5 4 2
T
0
Fig. 2. α from (17) in Theorem 8 as a function of the prediction horizon T and the exponential weighting parameter µ. The red line depicts the stability limit α = 0.
Theorem 8 (Stability of MPC with Exp. Weighting) Suppose that the Controllability Assumption 3 is satisfied for System (1) and consider α = 1 − γ (Ξ − 1) ,
(17)
with γ and Ξ given by (16). If α > 0, then the closed-loop is asymptotically stable. Proof: The proof follows closely the proof of Theorem 6. By noting that Ξ > 1 and with (14),(15), we obtain ∗ JT,µ (x(δ))
−
∗ JT,µ (x(0))
Zδ ≤ −α
3
4
Fig. 3. Shortest prediction horizon T ∗ which guarantees α > 0 and therefore closed-loop stability.
µ
0
2 µ
4
2
1
β(t0 )F ∗ (t0 ; 0)dt0 .
0
For α > 0, asymptotic stability follows directly from ∗ Lemma 1, which also holds when replacing JT∗ by JT,µ . In order to briefly illustrate the effect of the newly introduced exponential weighting, we consider the following case: C = 2.5, λ = 4, and δ = 0.15. Figure 2 shows the values for α given by Theorem 8 as a function of the prediction horizon T and the parameter of the exponential weighting µ. In Figure 3, the smallest prediction horizon is shown for which asymptotic stability is guaranteed by virtue of Theorem 8, i.e., T ∗ = inf T s.t. α > 0. In this example, we see the T ∈R≥0
benefits of the additional exponential weighting with respect to the minimal prediction horizon with guaranteed stability. However, we also note that T ∗ does not necessarily decrease monotonically in µ. Unfortunately, the opposite can also be the case, i.e., the use of exponential weighting might require longer prediction horizons for satisfaction of the sufficient stability conditions established in the present work. V. MPC WITH E XPONENTIAL W EIGHTING AND T ERMINAL C ONSTRAINTS Besides guaranteeing asymptotic stability for a possibly shorter prediction horizon, the exponential weighting introduced in Section IV has another advantage: if the exponential controllability assumption is only satisfied locally in
a (terminal) region Ω around the origin and appropriately defined terminal constraints are added to the finite horizon optimal control problem, we are still able to show stability of the closed-loop. Note that this is not possible for the MPC schemes proposed in [12] and Section III unless Γ = 1, see also the remark after Theorem 9. However, a similar result was presented in [13] by considering a generalized integral cost. In the present work, the exponential weighting provides a similar additional weighting term as the generalized integral cost in [13]. In this section, we consider a terminal region Ω ⊂ Rn which is compact and contains the origin in its interior. Additionally, we modify Assumption 3 as follows. Assumption 4 (Local Exponential Controllability) For all x0 ∈ Ω, there exists a piece-wise continuous input trajectory u ˆ(·; 0) with u ˆ(t; 0) ∈ U for all t ∈ R≥0 such that the corresponding state trajectory x ¯uˆ satisfies F (¯ xuˆ (t; 0), u ˆ(t; 0)) ≤ C e−λ t F (x0 , 0) and x ¯uˆ (t; 0) ∈ Ω for all t ∈ R≥0 with overshoot constant C ≥ 1 and decay rate λ > µ > 0. There are two main differences compared to Assumption 3. On the one hand, we now only require the exponential controllability assumption to be satisfied locally in a (terminal) region Ω around the origin. On the other hand, we additionally assume that this region is controlled invariant. We next replace (2) by the following finite horizon optimal control problem. minimize JT,µ,τ (x(ti ), u ¯) u ¯(·)
s.t. x ¯˙ (t0 ; ti ) = f (¯ x(t0 ; ti ), u ¯(t0 )), x ¯(ti ; ti ) = x(ti ),
(18a)
(18b)
0
u ¯(t ) ∈ U ,
(18c)
x ¯(t ; ti ) ∈ Ω ,
(18d)
00
for all t0 ∈ [ti , ti + T ] and for all t00 ∈ [ti + T − τ, ti + T ], in which τ ∈ (0, T − δ] is a constant design parameter and
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with tZ i +T
JT,µ,τ (x(ti ), u ¯) =
β(t0 − ti ) F (¯ x(t0 ; ti ), u ¯(t0 )) dt0 .
ti
Note that we do not only impose the terminal constraint on the “last” state, i.e., x ¯(ti +T ; ti ) ∈ Ω, but actually require all states after a certain time ti +T −τ to lie within Ω, which is similar to the generalized terminal constraint considered in [13]. Using this MPC setup, we have the following result. Theorem 9 (MPC with Local Exp. Weighting) Suppose that the Controllability Assumption 4 is satisfied for System (1) and assume that Ξ ≤ 1 for Ξ defined by (λ−µ) δ 1 e −1 C e−µδ . (19) =1− Ξ e(λ−µ) τ − 1 Then, the closed-loop is asymptotically stable. Proof: Similar to (15a) and (16a), we can show that ∗ JT,µ,τ (x(δ))
ZT ≤ max{Ξ, 1} ·
β(t0 )F ∗ (t0 ; 0)dt0 .
(20)
δ
For Ξ ≤ 1, asymptotic stability is guaranteed by Lemma 1, ∗ . which also holds when replacing JT∗ by JT,µ,τ The stability guarantee relies on the fact that µ > 0. Using only a local controllability assumption does not allow to conclude (10) or (15b). Hence, in order to show that α > 0 in (7), we require Ξ ≤ 1 in (9) or (15a), respectively. However, this is not possible for any finite horizon unless an additional weighting is employed. The result in this section can be regarded as in-between those established in [3–5] for MPC schemes using a terminal constraint and a terminal cost, and the stability results for unconstrained MPC schemes based on a controllability assumption [8–12]. The advantages of this in-between scheme can be summarized as follows: in contrast to [3–5], no control Lyapunov function is required. In contrast to [8–12], the controllability assumption does not have to be satisfied globally, but is only required locally in a region around the origin. A possible drawback of the current approach is that the region Ω has to be reachable in finite time and an additional terminal constraint is added to the optimization problem. The latter does not only confine the terminal state at the end of the prediction horizon, but all predicted states in some interval of length τ . VI. C ONCLUSIONS In this work, we presented two unconstrained MPC schemes using additional weighting terms which allow to obtain improved stability conditions. We have shown that stability properties can be improved by applying suitable weighting terms either on the terminal state or on the stage cost along the whole prediction horizon. A particular nice feature of our first result is that it bridges the gap between MPC schemes using control Lyapunov functions as terminal
cost and the more recently developed MPC schemes based on controllability assumptions. Our second result allows to consider a local controllability assumption in combination with a generalized terminal constraint in order to guarantee stability. Both results underpin that, although only the first part of the optimal open-loop input trajectory calculated at each sampling instant is actually applied in MPC, the last part of the prediction horizon plays a crucial role with respect to stability. R EFERENCES [1] J. B. Rawlings and D. Q. Mayne, Model Predictive Control: Theory and Design. Madison, WI, USA: Nob Hill Publishing, 2009. [2] T. Raff, S. Huber, Z. K. Nagy, and F. Allg¨ower, “Nonlinear model predictive control of a four tank system: An experimental stability study,” in Proc. IEEE Conf. Control Applications, Munich, Germany, 2006, pp. 237–242. [3] H. Chen and F. Allg¨ower, “A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability,” Automatica, vol. 34, no. 10, pp. 1205–1218, 1998. [4] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: stability and optimality,” Automatica, vol. 26, no. 6, pp. 789–814, 2000. [5] F. A. C. C. Fontes, “A general framework to design stabilizing nonlinear model predictive controllers,” Syst. Contr. Lett., vol. 42, no. 2, pp. 127–143, 2001. [6] B. Hu and A. Linnemann, “Toward infinite-horizon optimality in nonlinear model predictive control,” IEEE Trans. Autom. Control, vol. 47, no. 4, pp. 679–682, 2002. [7] D. Limon, T. Alamo, F. Salas, and E. F. Camacho, “On the stability of constrained MPC without terminal constraint,” IEEE Trans. Autom. Control, vol. 51, no. 5, pp. 832–836, 2006. [8] G. Grimm, M. J. Messina, S. E. Tuna, and A. R. Teel, “Model predictive control: for want of a local control Lyapunov function, all is not lost,” IEEE Trans. Autom. Control, vol. 50, no. 5, pp. 546–558, 2005. [9] L. Gr¨une, “Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems,” SIAM Journal on Control and Optimization, vol. 48, no. 2, pp. 1206–1228, 2009. [10] L. Gr¨une, J. Pannek, M. Seehafer, and K. Worthmann, “Analysis of unconstrained nonlinear MPC schemes with time-varying control horizon,” SIAM Journal on Control and Optimization, vol. 48, no. 8, pp. 4938–4962, 2010. [11] L. Gr¨une and J. Pannek, Nonlinear Model Predictive Control: Theory and Algorithms. London: Springer, 2011. [12] M. Reble and F. Allg¨ower, “Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems,” Automatica, vol. 48, no. 8, pp. 1812–1817, 2012. [13] M. Reble, D. E. Quevedo, and F. Allg¨ower, “A unifying framework for stability in MPC using a generalized integral terminal cost,” in Proc. Amer. Contr. Conf., Montr´eal, Canada, 2012, pp. 1211–1216. [14] J. Pannek and K. Worthmann, “Reducing the prediction horizon in NMPC: An algorithm based approach,” in Proc. 18th IFAC World Congress, Milano, Italy, 2011, pp. 7969–7974. [15] S. E. Tuna, M. J. Messina, and A. R. Teel, “Shorter horizons for model predictive control,” in Proc. Amer. Contr. Conf., Minneapolis, MN, USA, 2006, pp. 863–868. [16] S. S. Keerthi and E. G. Gilbert, “Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations,” Journal of Optimization Theory and Applications, vol. 57, no. 2, pp. 265–293, 1988. [17] D. Q. Mayne and H. Michalska, “Receding horizon control of nonlinear systems,” IEEE Trans. Autom. Control, vol. 35, no. 7, pp. 814–824, 1990. [18] A. Jadbabaie, J. Yu, and J. Hauser, “Unconstrained receding-horizon control of nonlinear systems,” IEEE Trans. Autom. Control, vol. 46, no. 5, pp. 776–783, 2001.
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Improved Stability Conditions for Unconstrained Nonlinear Model Predictive Control by using Additional Weighting Terms Marcus Reble, Daniel E. Quevedo, and Frank Allg¨ower Abstract— In this work, we present two unconstrained MPC schemes using additional weighting terms which allow to obtain improved stability conditions. First, we consider unconstrained MPC with general terminal cost functions. If the terminal cost is not a control Lyapunov function, but satisfies a relaxed condition, then our results yield improved estimates for a stabilizing prediction horizon. Furthermore, our analysis also allows to recover two well-known results as special cases: if the terminal cost function is chosen as zero, we recover previous conditions on the length of the prediction horizon such that stability is guaranteed; and if the terminal cost is a control Lyapunov function conform to the stage cost, stability follows independently of the length of the prediction horizon. Second, we propose to use an exponential weighting on the stage cost in order to improve the stability properties of the closed-loop. This also allows to consider local controllability assumptions in combination with a suitable terminal constraints and thereby gives a connection to the classical MPC approaches using terminal constraints.
I. I NTRODUCTION Model predictive control (MPC) is a modern control method which has received much attention in academic research as well as in practical applications in particular due to its ability to handle hard constraints and to take performance criteria directly into account [1]. However, MPC with a finite prediction horizon does not guarantee stability in general, which was also demonstrated on a practical example in [2]. In order to overcome this problem, one of the following elements, or a combination thereof, is usually used: introducing additional constraints in the finite horizon optimal control problem (in particular terminal constraints), using control Lyapunov functions (CLF) as terminal cost and/or a prediction horizon chosen sufficiently large based on a suitably defined controllability assumption, see also Figure 1. MPC schemes using terminal constraints and a terminal cost are the most well-studied approaches [3–5]. In [6, 7], the terminal constraint is omitted but it is nevertheless guaranteed to be satisfied for a defined set of initial states. The approach presented in [1, Section 2.6.1] replaces the terminal constraint by a sufficiently large terminal cost using Marcus Reble and Frank Allg¨ower are with the Institute for Systems Theory and Automatic Control (IST), University of Stuttgart, Germany. {reble,allgower}@ist.uni-stuttgart.de. Research supported by the German Research Foundation (DFG) within the Priority Programme 1305 “Control Theory of Digitally Networked Dynamical Systems” and within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart. Daniel E. Quevedo is with the School of Electrical Engineering & Computer Science, The University of Newcastle, Australia. [email protected]. Research supported by Australian Research Council’s Discovery Projects funding scheme (project number DP0988601).
978-1-4673-2064-1/12/$31.00 ©2012 IEEE
a controllability assumption, which requires an upper bound of the optimal cost of an associated problem with terminal equality constraint. Using a similar controllability assumption, but a significantly different analysis, so-called unconstrained MPC schemes [8–12], i.e., schemes without terminal constraints, guarantee stability by providing an explicit bound on the minimal stabilizing prediction horizon. The first goal of our work is to give a connection between the unconstrained MPC schemes [8–12] and the classical schemes using terminal constraints and cost terms [3–5]. A first result in this respect has been reported in [13] by considering a generalized integral cost. In the present work, we provide further results with a different point of view. Secondly, many examples exhibit a significant difference between the actual minimal stabilizing prediction horizon observed in practice and the one provided by theoretical results. The second goal of our work is to overcome this conservatism. For a similar goal, an algorithmic approach has been presented in [14], which takes advantage of the symmetry properties of the stability condition with respect to the sampling time and reuses optimal input trajectories calculated at previous sampling instants in order to guarantee stability for a shorter prediction horizon. In Section III, we consider unconstrained MPC with a general positive semi-definite terminal cost function. In particular, we investigate the use of assumptions from both MPC schemes in contrast to using a generalized assumption as in [13]. This result is related to the discrete-time results in [15], which takes properties of the terminal cost function into account in order to improve the stability conditions obtained in [8]. However, we obtain better estimates for the stabilizing prediction horizon and our analysis allows to recover recent results for unconstrained MPC without terminal cost. In contrast, setting the terminal cost equal to zero in the result of [15] does not allow to guarantee asymptotic stability even for an arbitrarily large prediction horizon. The combination proposed in our result enables us to relax the assumptions from both MPC schemes and provides a different point of view. Namely, we show that the combination of a relaxed condition on the terminal cost, which does not need to be a control Lyapunov function, and a controllability assumption with a shorter prediction horizon provides guaranteed stability. Our analysis recovers previous well-known results as special cases and, hence, allows to bridge the gap between the approaches, which have been mostly considered separately in literature so far, see Figure 1. In Section IV, we propose to use unconstrained MPC
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Local CLF and Constraints [3–5]
Terminal Constraints [16, 17]
s.t.
Stability in MPC
Terminal Cost (Global CLF) [18]
Fig. 1.
u ¯(t0 ) ∈ U ,
Generalized Integral Cost [13] Exponential Weighting (Section V)
ti
+ E(¯ x(ti + T ; ti )) .
F (x, u) ≥ F (x, 0) ≥ αF (|x|) and E(x) ≥ 0 .
with an additional exponential weighting on the stage cost in order to reduce the prediction horizon necessary for existing stability guarantees. A result similar to [13] is derived in Section V, which replaces the generalized integral cost by the exponential weighting introduced in the preceding section. Both of these results allow to consider local controllability assumptions in combination with terminal constraints in contrast to the unconstrained MPC schemes [8–12], for which such a modification is not possible. Hence, these schemes bridge another, yet similar, gap between previous approaches, see Figure 1.
uMPC (t) = u∗ (t; ti ) ,
III. U NCONSTRAINED MPC WITH G ENERAL T ERMINAL C OST F UNCTION
u ¯(·)
(5a)
E (t; ti ) = E(x (t; ti )) ,
(5b)
∗
for t ∈ [ti , ti + T ] and ti ∈ {0, δ}. From this definition, it directly follows that Z ti +T ∗ JT (x(ti )) = F ∗ (t0 ; ti )dt0 + E ∗ (ti + T ; ti ) . (6) ti
We will use the following simple general result on asymptotic stability to derive our main results, see also [9, Proposition 2.4] for related results in discrete-time. Lemma 1 (General Condition for Asymptotic Stability) Suppose the following holds for all x ∈ Rn Z δ JT∗ (x(δ)) − JT∗ (x(0)) ≤ −α F ∗ (t0 ; 0)dt0 , (7) 0
In this section, we derive sufficient stability conditions for unconstrained MPC with a general positive semi-definite terminal cost function. The open-loop finite horizon optimal control problem at sampling instant ti given measured state x(ti ) is formulated as minimize JT (x(ti ), u ¯)
(4)
F ∗ (t; ti ) = F (x∗ (t; ti ), u∗ (t; ti )) , ∗
with state x(t) ∈ Rn , initial condition x0 ∈ Rn , and control input u(t) ∈ U ⊆ Rm . The function f is continuously differentiable and the constraint set U ⊂ Rm is compact and contains the origin in its interior. Throughout this work, we assume that System (1) has a unique solution for all t ∈ R≥0 for any initial condition x0 ∈ Rn and any piecewise- and right-continuous input function u(·) : R≥0 → U. Without loss of generality, we assume x = 0 to be an equilibrium of System (1) for u = 0, i.e., f (0, 0) = 0. The problem of interest is to stabilize the equilibrium x = 0 via continuoustime model predictive control with periodic sampling.
ti ≤ t < ti + δ .
Without loss of generality, we consider the two consecutive sampling instants t0 = 0 and t1 = δ in the following. Since System (1) is time-invariant, all results hold analogously for any other two consecutive sampling instants ti and ti+1 . With slight abuse of notation, we use the following abbreviations
We consider nonlinear continuous-time systems (1)
(3)
We assume that the optimal control which minimizes (2) is given by u∗ (t0 ; ti ) with associated predicted state trajectory x∗ (t0 ; ti ), t0 ∈ [ti , ti + T ]. The optimal cost is denoted by JT∗ (x(ti )). For a given sampling time δ with 0 < δ ≤ T , the control input to the system is defined by the optimal solution of Problem (2) at sampling instants ti = i δ , i ∈ N0 , in the usual continuous-time receding horizon fashion
II. P ROBLEM S ETUP x(0) = x0
(2d)
x ¯(t0 ; ti ) is the predicted trajectory starting from initial condition x ¯(ti ; ti ) = x(ti ) and driven by u ¯(t0 ) for t0 ∈ [ti , ti + T ]. n m The stage cost F : R × R → R≥0 and the terminal cost function E : Rn → R≥0 are continuous, F (0, 0) = E(0) = 0 and there is a class K∞ function1 αF : R≥0 → R≥0 such that for all x ∈ Rn , u ∈ Rm
Schematic overview on stability conditions in MPC.
x(t) ˙ = f (x(t), u(t)) ,
(2c)
for all t0 ∈ [ti , ti + T ] with Z ti +T JT (x(ti ), u ¯) = F (¯ x(t0 ; ti ), u ¯(t0 )) dt0
Controllability Assumption [8–12]
Unconstrained MPC with Terminal Cost (Section III)
x ¯˙ (t0 ; ti ) = f (¯ x(t0 ; ti ), u ¯(t0 )) , x ¯(ti ; ti ) = x(ti ) , (2b)
(2a)
in which α > 0. Then, the closed-loop using MPC is asymptotically stable. Proof: Similar to the result in [3], it can be shown that JT∗ (x) is continuous in x in a neighborhood of the origin. Condition (7) directly implies that JT∗ is non-increasing along 1 A function α : R ≥0 → R≥0 is said to belong to class K∞ if it is continuous, strictly increasing, α(0) = 0 and α(s) → ∞ as s → ∞.
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trajectories of the closed-loop and stability directly follows. R∞ Asymptotic stability follows from F (x(t0 ), u(t0 ))dt0 ≤ 0
JT∗ (x(0)), continuity of f , compactness of U, the lower bound (3), and Barbalat’s Lemma. In contrast to the unconstrained MPC schemes without terminal cost, see, e.g., [9, 10, 12], JT∗ is not necessarily monotonically increasing in T and, hence, α does not give a suboptimality estimate of the closed-loop compared to the infinite horizon optimal controller. Our main results, namely Theorems 6, 8, and 9, are based on the above lemma. In order to derive α in (7), we use the following two controllability assumptions. Assumption 1 (Controllability Assumption based on F ) For all T 0 ∈ R≥0 and x0 ∈ Rn , there exists a piece-wise continuous input trajectory u ˆ(·; 0) with u ˆ(t; 0) ∈ U for all t ∈ [0, T 0 ] and JT∗ 0 (x0 ) ≤ JT 0 (x0 , u b) ≤ B(T 0 )F (x0 , 0), in which B : R≥0 → R≥0 is a continuous and bounded function. Assumption 2 (Controllability Assumption based on E) For all x0 ∈ Rn , there exists a piece-wise continuous input trajectory u ˆ(·; 0) with u ˆ(t; 0) ∈ U for all t ∈ [0, δ] and δ Z Γ F (¯ x(t0 ; 0), u ˆ(t0 ; 0))dt0 + E(¯ x(δ; 0)) ≤ E(x0 )
Lemma 3 (Calculation of Ξ) Suppose that Controllability Assumption 1 is satisfied. Then, JT∗ (x(δ))
ZT ≤Ξ
F ∗ (t0 ; 0)dt0 ,
(9)
δ
in which
1 Ξ
= 1 − exp −
RT δ
! 1 B(T +δ−t∗ )
dt
∗
.
Proof: The proof is similar to the proof of Lemma 7 in [12] for unconstrained MPC without terminal cost. Lemma 4 (Calculation of γ) Suppose that the Controllability Assumption 1 is satisfied. Then, ZT
F (t ; 0)dt + E (T ; 0) ≤ γ ∗
0
0
∗
Zδ
F ∗ (t0 ; 0) dt0 ,
(10)
0
δ
in which
1 γ
= exp
Rδ 0
! 1 B(T −t∗ )
dt
∗
− 1.
Proof: The proof mirrors the proof of Lemma 8 in [12] for unconstrained MPC without terminal cost.
0
in which Γ ∈ [0, 1]. Assumption 1 is a standard assumption in unconstrained MPC [8–12]. However, note that Assumption 1 depends explicitly on the terminal cost due to the definition of JT . Assumption 2 is always satisfied for Γ = 0. On the other hand, if no terminal cost term is considered, i.e., E(x) = 0 as in [9–12], Assumption 2 can only be satisfied for Γ = 0. For 0 < Γ < 1, Assumption 2 can be interpreted as the terminal cost E(x) being “similar” to a CLF, however in a significantly weaker sense as shown in the following proposition. Proposition 2 (CLF-like Condition for Assumption 2) Suppose that there exists ν ∈ R≥0 such that for all x ∈ Rn , there exists u ∈ U such that ˙ E(x) ≤ −F (x, u) + ν E(x) . Then, Assumption 2 is satisfied for Γ = e
can show the following intermediate results using the two assumptions above.
−νδ
(8)
.
Proof: The result follows directly from integrating (8) from 0 to δ and taking F (x, u) ≥ 0 into account. For ν = 0, we obtain Γ = 1 and consequently Assumption 2 corresponds to E being an F -conform control Lyapunov function (CLF), which is a common assumption in MPC in order to guarantee stability of the closed-loop, see, e.g., [3– 5, 18]. For ν > 0, this assumption is relaxed and additional arguments are required in order to guarantee stability. In this work, a prediction horizon chosen “sufficiently large” is considered with explicit conditions on the prediction horizon given in Theorem 6 based on Assumption 1. To this end, we
Lemma 5 (Direct Consequence of Assumption 2) Suppose that Controllability Assumption 2 is satisfied. Then, Γ JT∗ (x(δ))
ZT ≤Γ
F ∗ (t0 ; 0)dt0 + E ∗ (T ; 0) .
(11)
δ
Proof: The proof follows directly from optimality of JT∗ (x(δ)) and using Assumption 2 for x0 = x∗ (T ; 0), which provides an upper bound on the cost on the interval [T, T +δ]. We are now able to state our main result on stability of unconstrained MPC with general positive semi-definite terminal cost functions. Theorem 6 (Stability of Unconstrained MPC) Suppose that the Controllability Assumptions 1 and 2 are satisfied for System (1) and consider Ξ α=1−γ −1 , (12) 1 + (Ξ − 1)Γ with γ and Ξ defined in Lemmata 3 and 4, respectively. If α > 0, then the closed-loop is asymptotically stable. Proof: Multiplying Inequality (9) in Lemma 3 with 1 − Γ and multiplying Inequality (11) in Lemma 5 with Ξ, and adding the two resulting inequalities yields T Z Ξ F ∗ (t0 ; 0)dt0 + E ∗ (T ; 0) . JT∗ (x(δ)) ≤ 1 + (Ξ − 1)Γ
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δ
Following as in the proof of [12, Theorem 9] and using (6) and Lemma 4, we obtain that (7) holds and the result follows from Lemma 1. Note that α > 0 and thus asymptotic stability can always be guaranteed for a prediction horizon chosen large enough because Ξ → 1 and α → 1 for T → ∞. For Γ = 0, for which Assumption 2 is trivially satisfied, we recover the stability condition from [12, Theorem 9] and consequently the results for unconstrained MPC without terminal cost. These results have been shown to be the “best possible” stability conditions based only on the Controllability Assumption 1, see [12, Theorem 10]. Here, “best possible” refers to the largest α for which (7) in Lemma 1 holds. Loosely speaking, this can be seen because Fb∗ , defined in the proofs of Lemmata 3 and 4, satisfies all conditions implied by Controllability Assumption 1 and yields α = 1−γ(Ξ−1) for (7). Hence, one can only guarantee (7) for a larger α by taking into account additional information, e.g., about the system or the optimal cost function. If Assumption 1 is satisfied for Γ = 1, for instance if E is a global F -conform control Lyapunov function, then asymptotic stability is guaranteed independently of the Controllability Assumption 1. Thus, we recover the previous stability result using global control Lyapunov functions. Summarizing, Theorem 6 allows in some sense to bridge the gap between the stability results for MPC schemes using control Lyapunov functions as terminal cost and the more recently developed MPC schemes based on controllability assumptions. However, the case of Assumption 2 only being satisfied in a terminal region around the origin has not been treated in this work because an additional terminal constraint makes the verification of Assumption 1 significantly harder. This issue will be subject to future research. IV. U NCONSTRAINED MPC WITH E XPONENTIAL W EIGHTING In this section, we consider a similar MPC setup as in the previous section with the following two modifications. First, in the open-loop finite horizon optimal control problem (2) we replace the cost functional JT (x(ti ), u ¯) in (2d) by JT,µ (x(ti ), u ¯) =
tZ i +T
β(t0 − ti ) F (¯ x(t0 ; ti ), u ¯(t0 )) dt0 ,
ti
(13) in which β(t) = eµ t for some constant µ ≥ 0 is an exponential weighting on the stage cost and no terminal cost terms are considered. Second, we restrict ourselves to the following exponential controllability assumption, which is a special case of Assumption 1 and a standard assumption in unconstrained MPC [9–12]. Assumption 3 (Exponential Controllability) For all x0 ∈ Rn , there exists a piece-wise continuous input trajectory u ˆ(·; 0) with u ˆ(t; 0) ∈ U for all t ∈ R≥0 and corresponding state trajectory x ¯uˆ such that F (¯ xuˆ (t; 0), u ˆ(t; 0)) ≤ C e−λ t F (x0 , 0) , ∀t ∈ R≥0
with overshoot constant C ≥ 1 and decay rate λ > µ ≥ 0. Following the analysis of [12] and Section III, we consider the two consecutive sampling instants t0 = 0 and t1 = δ and we will use the abbreviation F ∗ (t; ti ) introduced in (5) and ∗ Lemma 1. The optimal cost is denoted by JT,µ (x(ti )) and analogous to (6), we have ∗ JT,µ (x(ti ))
tZ i +T
=
β(t0 − ti )F ∗ (t0 ; ti )dt0 .
(14)
ti
Lemma 7 (Unconstrained MPC with Exp. Weighting) Suppose that the Exponential Controllability Assumption 3 is satisfied. Then, ZT
∗ JT,µ (x(δ))
≤Ξ
β(t0 )F ∗ (t0 ; 0)dt0 ,
(15a)
β(t0 )F ∗ (t0 ; 0) dt0 ,
(15b)
δ
ZT
β(t0 )F ∗ (t0 ; 0)dt0 ≤ γ
Zδ 0
δ
in which (λ−µ) δ 1 e −1 C e−µδ , =1− Ξ e(λ−µ) T − 1 C1 1 1 − e−(λ−µ) T = − 1. γ e−(λ−µ) δ − e−(λ−µ) T
(16a) (16b)
Proof: Due to (14) and the definition of β(t), we have ∗ JT,µ (x(δ))
=e
−µδ
δ+T Z
β(t0 )F ∗ (t0 ; δ)dt0 .
δ
The Exponential Controllability Assumption 3 and optimality ∗ of JT,µ (x(δ)) then imply that ∗ eµδ JT,µ (x(δ)) ≤
t∗
Z
β(t0 )F ∗ (t0 ; 0)dt0
δ
+ B(T + δ − t∗ )β(t∗ )F ∗ (t∗ ; 0) C in which B(T ) = λ−µ (1 − e−(λ−µ) T ) holds for all t∗ ∈ [δ, T ]. Noting the similarities allows to follow the proof of Lemma 3 in order to prove (15a) and (16a). The proof of (15b) and (16b) follows exactly the lines of the proof of Lemma 4 when setting E ∗ (T ; 0) = 0 and replacing F ∗ (t0 ; 0), Fb∗ (t) by β(t0 )F ∗ (t0 ; 0), β(t0 )Fb∗ (t), respectively, in all expressions. Comparing these results to [12, Section 4.1], or equivalently by substitution of B(T ) = Cλ (1 − e−λ T ) in the results of Lemmata 3 and 4, we note two differences. First, λ is replaced by λ − µ in all expressions. Second, the expression for Ξ is multiplied by an additional term e−µδ . This second term is directly caused by shifting the prediction horizon when using an exponential weighting on the stage cost and beneficial for achieving α > 0, and hence for stability guarantees.
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α
1 0.95
10
0.9
T∗
15
0.85
5
0.8
0
0.75 0
−5 4 2
T
0
Fig. 2. α from (17) in Theorem 8 as a function of the prediction horizon T and the exponential weighting parameter µ. The red line depicts the stability limit α = 0.
Theorem 8 (Stability of MPC with Exp. Weighting) Suppose that the Controllability Assumption 3 is satisfied for System (1) and consider α = 1 − γ (Ξ − 1) ,
(17)
with γ and Ξ given by (16). If α > 0, then the closed-loop is asymptotically stable. Proof: The proof follows closely the proof of Theorem 6. By noting that Ξ > 1 and with (14),(15), we obtain ∗ JT,µ (x(δ))
−
∗ JT,µ (x(0))
Zδ ≤ −α
3
4
Fig. 3. Shortest prediction horizon T ∗ which guarantees α > 0 and therefore closed-loop stability.
µ
0
2 µ
4
2
1
β(t0 )F ∗ (t0 ; 0)dt0 .
0
For α > 0, asymptotic stability follows directly from ∗ Lemma 1, which also holds when replacing JT∗ by JT,µ . In order to briefly illustrate the effect of the newly introduced exponential weighting, we consider the following case: C = 2.5, λ = 4, and δ = 0.15. Figure 2 shows the values for α given by Theorem 8 as a function of the prediction horizon T and the parameter of the exponential weighting µ. In Figure 3, the smallest prediction horizon is shown for which asymptotic stability is guaranteed by virtue of Theorem 8, i.e., T ∗ = inf T s.t. α > 0. In this example, we see the T ∈R≥0
benefits of the additional exponential weighting with respect to the minimal prediction horizon with guaranteed stability. However, we also note that T ∗ does not necessarily decrease monotonically in µ. Unfortunately, the opposite can also be the case, i.e., the use of exponential weighting might require longer prediction horizons for satisfaction of the sufficient stability conditions established in the present work. V. MPC WITH E XPONENTIAL W EIGHTING AND T ERMINAL C ONSTRAINTS Besides guaranteeing asymptotic stability for a possibly shorter prediction horizon, the exponential weighting introduced in Section IV has another advantage: if the exponential controllability assumption is only satisfied locally in
a (terminal) region Ω around the origin and appropriately defined terminal constraints are added to the finite horizon optimal control problem, we are still able to show stability of the closed-loop. Note that this is not possible for the MPC schemes proposed in [12] and Section III unless Γ = 1, see also the remark after Theorem 9. However, a similar result was presented in [13] by considering a generalized integral cost. In the present work, the exponential weighting provides a similar additional weighting term as the generalized integral cost in [13]. In this section, we consider a terminal region Ω ⊂ Rn which is compact and contains the origin in its interior. Additionally, we modify Assumption 3 as follows. Assumption 4 (Local Exponential Controllability) For all x0 ∈ Ω, there exists a piece-wise continuous input trajectory u ˆ(·; 0) with u ˆ(t; 0) ∈ U for all t ∈ R≥0 such that the corresponding state trajectory x ¯uˆ satisfies F (¯ xuˆ (t; 0), u ˆ(t; 0)) ≤ C e−λ t F (x0 , 0) and x ¯uˆ (t; 0) ∈ Ω for all t ∈ R≥0 with overshoot constant C ≥ 1 and decay rate λ > µ > 0. There are two main differences compared to Assumption 3. On the one hand, we now only require the exponential controllability assumption to be satisfied locally in a (terminal) region Ω around the origin. On the other hand, we additionally assume that this region is controlled invariant. We next replace (2) by the following finite horizon optimal control problem. minimize JT,µ,τ (x(ti ), u ¯) u ¯(·)
s.t. x ¯˙ (t0 ; ti ) = f (¯ x(t0 ; ti ), u ¯(t0 )), x ¯(ti ; ti ) = x(ti ),
(18a)
(18b)
0
u ¯(t ) ∈ U ,
(18c)
x ¯(t ; ti ) ∈ Ω ,
(18d)
00
for all t0 ∈ [ti , ti + T ] and for all t00 ∈ [ti + T − τ, ti + T ], in which τ ∈ (0, T − δ] is a constant design parameter and
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with tZ i +T
JT,µ,τ (x(ti ), u ¯) =
β(t0 − ti ) F (¯ x(t0 ; ti ), u ¯(t0 )) dt0 .
ti
Note that we do not only impose the terminal constraint on the “last” state, i.e., x ¯(ti +T ; ti ) ∈ Ω, but actually require all states after a certain time ti +T −τ to lie within Ω, which is similar to the generalized terminal constraint considered in [13]. Using this MPC setup, we have the following result. Theorem 9 (MPC with Local Exp. Weighting) Suppose that the Controllability Assumption 4 is satisfied for System (1) and assume that Ξ ≤ 1 for Ξ defined by (λ−µ) δ 1 e −1 C e−µδ . (19) =1− Ξ e(λ−µ) τ − 1 Then, the closed-loop is asymptotically stable. Proof: Similar to (15a) and (16a), we can show that ∗ JT,µ,τ (x(δ))
ZT ≤ max{Ξ, 1} ·
β(t0 )F ∗ (t0 ; 0)dt0 .
(20)
δ
For Ξ ≤ 1, asymptotic stability is guaranteed by Lemma 1, ∗ . which also holds when replacing JT∗ by JT,µ,τ The stability guarantee relies on the fact that µ > 0. Using only a local controllability assumption does not allow to conclude (10) or (15b). Hence, in order to show that α > 0 in (7), we require Ξ ≤ 1 in (9) or (15a), respectively. However, this is not possible for any finite horizon unless an additional weighting is employed. The result in this section can be regarded as in-between those established in [3–5] for MPC schemes using a terminal constraint and a terminal cost, and the stability results for unconstrained MPC schemes based on a controllability assumption [8–12]. The advantages of this in-between scheme can be summarized as follows: in contrast to [3–5], no control Lyapunov function is required. In contrast to [8–12], the controllability assumption does not have to be satisfied globally, but is only required locally in a region around the origin. A possible drawback of the current approach is that the region Ω has to be reachable in finite time and an additional terminal constraint is added to the optimization problem. The latter does not only confine the terminal state at the end of the prediction horizon, but all predicted states in some interval of length τ . VI. C ONCLUSIONS In this work, we presented two unconstrained MPC schemes using additional weighting terms which allow to obtain improved stability conditions. We have shown that stability properties can be improved by applying suitable weighting terms either on the terminal state or on the stage cost along the whole prediction horizon. A particular nice feature of our first result is that it bridges the gap between MPC schemes using control Lyapunov functions as terminal
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