In this section we analyze exponential controllability with respect to the stage cost (4.3) according to Definition 3.3 in order to estimate the minimal stabilizing horizon length depending on the overshoot C and the decay rate δ in (3.6).
One promising candidate for an exponentially stabilizing control sequence in (3.6) is the constant control ¯u. In this case, the second term in the stage cost (4.3) vanishes and the left-hand side of (3.6), which is given by `(ρ(k),u), can be calculated explicitly¯ 2 thanks to (4.2):
1Since the controlu(1) only influences the subsequent states, which are not included in the objective function, choosing u(1) = ¯uis always the best option when minimizingJ2.
2In this chapter, we rely on the explicit solution formula. The more general case, which is independent of such formulas, is provided in Lemma 5.5.
40 Chapter 4. Stabilizing MPC – Space-independent control
Figure 4.1: A sample desired PDF ¯ρ(x) (dotted blue) and three initial PDFs ˚ρ(x) for α = 1 (solid orange),α <1 (dashed green) and α >1 (dot-dashed red).
On the right-hand side of (3.6) we have Cδkmin
u `(˚ρ, u) =Cδk`(˚ρ,u).¯
Thus, to prove the exponential controllability property (3.6), we show that
`(ρ(t),u)¯ ≤Ce−κt`(ρ(0),u)¯ (4.7) in continuous time for some κ > 0, C ≥ 1 and define δ := e−κTs, where Ts is the MPC sampling time, to arrive at (3.6). The constant C is the overshoot bound from Definition 3.3, δ is the decay rate. Since we can ignore the constant factor √θ/2√
2πς2
in (4.6), this is equivalent to proving
Wα(t)≤Ce−κtWα(0), (4.8)
where
Wα(t) := 1 + 1
pζ(t) − 2√
2 exp(−η(t))
pζ(t) + 1 . (4.9)
Hence, in the following we show that (4.8) holds, and moreover with C = 1, as we then get stability for the shortest meaningful horizon length N = 2.
Before that, however, we give an interpretation of the above parameters α and β.
These depend on the model parameters θ and ς2 as well as on the initial PDF, which is characterized by (˚µ,˚σ2). The value of β indicates the distance between the initial mean ˚µ and the mean of the target equilibrium PDF ¯ρ. Similarly, the former parameter,α, relates the initial variance ˚σ2 to that of the target equilibrium PDF ¯ρ. Ifα= 1, the variance does not change in time since ˚σ2 =ς2/(2θ) in (4.2). Forα <1, the variance of the distribution is increasing in time since ˚σ2 < ς2/(2θ). Analogously, it shrinks in time if α > 1. All cases are illustrated in Figure 4.1. We recall that we cannot control the variance, only the mean, see (4.2).
In order to conclude stability of the MPC closed loop from the exponential control-lability condition (3.6), an exponentially stabilizing control needs to exist for the initial state ˚z =zF(n) = ρ(tn,·) in every MPC iteration. Hence, the value of αmay change from
one step to the next, i.e., αn+1 6= αn, where αn denotes the value of α in the n-th MPC iteration. It is important to note, however, that for space-independent control the sign of αn−1 does not change with n. This is due to the monotone convergence ofαn to 1 that we get from reformulating the change in the variance in (4.2),
˚σ2n+1 = ς2 2θ +
˚σn2− ς2 2θ
e−2θTs, to
αn+1 = 1 + (αn−1)e−2θTs. (4.10) In order to prove (4.8) we now consider the three cases α = 1, α < 1, and α > 1 separately.
The case α= 1:
In this case, the shape of the PDF stays the same since the space-independent control can only move the PDF as a whole. We have
W1(t) = 2−2e−βe−2θt/2 (4.11)
and we can prove the following proposition.
Proposition 4.1. For W1(t), inequality (4.8) holds withC = 1 and κ= 2θe−β/2. Proof. We show W10(t)≤ −κW1(t) to conclude our assertion. To this end, consider
W10(t) +κW1(t) = −4θ β
2e−2θte−βe−2θt/2−e−β/2+e−β/2e−βe−2θt/2
=−4θ
e−βe−2θt/2 β
2e−2θt+e−β/2
−e−β/2
=−4θ e−βτ˜ h
βτ˜ +e−β˜i
−e−β˜ ,
where ˜β := β/2 ≥ 0 and τ :=e−2θt ∈ ]0,1]. For arbitrary but fixed ˜β we define the C∞ function
h1(τ) :=e−βτ˜ ( ˜βτ+e−β˜)−e−β˜.
It can easily be shown that h1(0) = 0 andh1(1)≥0. By calculating h01(τ), one can show that h1(τ) is monotonously increasing on ]0, τ∗[, with τ∗ := (1−e−β˜)/β˜being the unique root of h01(τ), and monotonously decreasing on ]τ∗,1]. Therefore, h1(τ) ≥ 0 on ]0,1], which concludes the proof.
Since C = 1, the MPC closed loop is asymptotically stable even for the shortest possible horizon N.
42 Chapter 4. Stabilizing MPC – Space-independent control since for W1(0)6= 0, which we can assume w.l.o.g., we then use Proposition 4.1 to get
Wα(t)≤ Wα(0)
W1(0)W1(t)≤ Wα(0)
W1(0)e−κtW1(0) =e−κtWα(0) for κ as in Proposition 4.1.
Obviously, h2(0) = 0 and limt→∞h2(t) = 0. Analogously to the proof of Proposi-tion 4.1, one can show there exists at most one root t∗ ∈[0,∞[ of h02(t) and that h2(t) is monotonously decreasing on [0, t∗[ (or [0,∞[ in case there is no root ofh02(t) in [0,∞[) and monotonously increasing on ]t∗,∞[. Hence, (4.12) holds and we have shown the following.
Proposition 4.2. For α <1, Wα(t) satisfies (4.8) with C and κ from Proposition 4.1.
The case α >1:
If α > 1, the shrinking variance of the distribution may lead to increasing stage costs at the beginning, i.e., Wα0(t) > 0 for t ∈ [0, t∗[ and some t∗ >0. This occurs, for instance, forθ = ˚µ=ς = 1, ˚σ = 100, and control ¯u= 2000, cf. Figure 4.2. It is due to theL2 norm used in the stage cost (4.3). Obviously, condition (4.8) does not hold for C = 1.
To circumvent this issue, we can add (time-dependent) control-independent terms to Wα(t). One possibility is to add Just as Wα(t) from (4.9) stemmed from the stage cost ` from (4.3), there exists a stage cost ˜`that, for u≡u, yields ˜¯ Wα(t) in the one-dimensional case. A short calculation reveals that the terms added to Wα(t) can be formulated in terms of ρ and ¯ρ: Note that ˜`yields the same optimal control sequence as`from (4.3) and thus Theorem 3.4 can be applied to ˜`. This is because the added terms to`do not depend on the control u.
At first glance, this might seem counterintuitive, as the PDF ρ of course does depend on u. TheL2(R) norm of ρ, however, does not. At second glance the reason is clear: All the control u can do is shift the PDFρto the left or to the right. But moving a function does not change its L2 norm on R.
Figure 4.2: Wα(t) (solid red), ˜Wα(t) (dotted blue), and ˜Wα(0)e−κt (dashed green) with κ from Proposition 4.3 for (θ,˚µ, ς2,˚σ2,u) = (1,¯ 0,1,25,200) (left) and for (θ,˚µ, ς2,˚σ2,u) =¯ (1,0,16,1/10000,1/4) (right), giving (α, β) = (25/4,5000) and (α, β) = (4/625,2), respectively.
Proposition 4.3. ForW˜α(t)withα >1, inequality (4.8)holds withC= 1andκ=θe−β/2. Proof. To prove that inequality (4.8) holds for ˜Wα(t) we can equivalently consider ˆWα(t) from (4.13). Since α >1 we may drop the absolute value in ˆWα(t). Then we can rewrite
To keep notation brief, we introduce the variables
τ :=e−2θt ∈]0,1], ω :=ζ(t) + 1 >2, κ˜:= κ
44 Chapter 4. Stabilizing MPC – Space-independent control
Second, the derivative h05(β) is given by h05(β) = −τ
To summarize, in all three cases we can apply Theorem 3.4 in order to conclude asymptotic stability of the MPC closed loop for the shortest possible horizon N = 2.
Remark 4.4. (a) Figure 4.2 suggests that—at least in some cases—a much better decay rate can be obtained. However, this is irrelevant if C = 1 and the goal is to show asymptotic stability of the MPC closed loop for N = 2.
(b) It is possible to employ W˜α(t) for all three cases of α. For α = 1, W˜1(t) coincides withW1(t). Forα <1, the proof is structurally similar to the one of Proposition 4.3.
Figure 4.2 (right) depicts exemplarily the case of α <1.
(c) As we will see more clearly in Chapter 5, in the case of Gaussian PDFs we can replace the PDF ρ in the stage costs ` and `, cf.˜ (4.3) and (4.14), by its mean µ and its covariance matrix Σ. In the one-dimensional case Σ corresponds to the variance σ2. Figure 4.3 depicts the terms penalizing the state in both stage costs—
i.e., `(ρ,u)¯ and `(ρ,˜ u)—in terms of¯ (µ,Σ), where the desired PDF ρ¯is a Gaussian PDF with (¯µ,Σ) = (0,¯ 1). The exact formulas can be obtained from Lemma 5.5 and the proof thereof.