LMI Conditions for Contraction Metrics

Chapter 3: Robust Nonlinear Control and Estimation via Contraction Theory . 50

3.2 LMI Conditions for Contraction Metrics

We design a nonlinear feedback tracking control law parameterized by a matrix- valued function𝑀(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) (or𝑀(𝑥 , 𝑡), see Theorem 3.2) as follows:

𝑢 =𝑢_𝑑−𝐾(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) (𝑥−𝑥_𝑑) (3.10)

=𝑢_𝑑−𝑅(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡)⁻¹𝐵(𝑥 , 𝑡)^⊤𝑀(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) (𝑥−𝑥_𝑑)

where 𝑅(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) ≻ 0 is a weight matrix on the input𝑢 and𝑀(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) ≻ 0 is a positive definite matrix (which satisfies the matrix inequality constraints for a contraction metric, to be given in Theorem 3.1). As discussed in Sec. 3.1.III, the extended linear form of the tracking control (3.10) enables LTV systems-type ap- proaches to Lyapunov function construction, while being general enough to capture the nonlinearity of the underlying dynamics due to Lemma 3.2 [36].

Lemma 3.2. Consider a general feedback controller𝑢defined as𝑢= 𝑘(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) with𝑘(𝑥_𝑑, 𝑥_𝑑, 𝑢_𝑑, 𝑡) =𝑢_𝑑, where𝑘 :R^𝑛×R^𝑛×R^𝑚 ×R≥0↦→ R^𝑚. If𝑘 is piecewise continuously differentiable, then ∃𝐾 : R^𝑛 × R^𝑛 × R^𝑚 × R≥0 ↦→ R^𝑚×𝑛 s.t. 𝑢 = 𝑘(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) =𝑢_𝑑−𝐾(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) (𝑥−𝑥_𝑑).

Proof. Using𝑘(𝑥_𝑑, 𝑥_𝑑, 𝑢_𝑑, 𝑡) =𝑢_𝑑,𝑢can be decomposed as𝑢=𝑢_𝑑+(𝑘(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡)−

𝑘(𝑥_𝑑, 𝑥_𝑑, 𝑢_𝑑, 𝑡)). Since we have𝑘(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) −𝑘(𝑥_𝑑, 𝑥_𝑑, 𝑢_𝑑, 𝑡) =∫1

0 (𝑑 𝑘(𝑐𝑥+ (1− 𝑐)𝑥_𝑑, 𝑥_𝑑, 𝑢_𝑑, 𝑡)/𝑑𝑐)𝑑𝑐, selecting𝐾as

𝐾 =−

∫ 1 0

𝜕 𝑘

𝜕 𝑥

(𝑐𝑥+ (1−𝑐)𝑥_𝑑, 𝑥_𝑑, 𝑢_𝑑, 𝑡)𝑑𝑐 gives the desired relation [36].

Remark 3.4. Lemma 3.2 implies that designing optimal 𝑘 of𝑢 = 𝑘(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) re- duces to designing the optimal gain𝐾(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡)of𝑢=𝑢_𝑑−𝐾(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) (𝑥−𝑥_𝑑). We could also generalize this idea further using the CCM-based differential feedback controller𝛿𝑢= 𝑘(𝑥 , 𝛿𝑥 , 𝑢, 𝑡)[3], [14], [15], [19], [20], [33] (see Theorem 4.6).

Substituting (3.10) into (3.1) and (3.2) yields the following virtual system of a smooth path𝑞(𝜇, 𝑡), parameterized by 𝜇 ∈ [0,1] to have𝑞(0, 𝜇) =𝑥_𝑑 and𝑞(1, 𝑡) =𝑥, for partial contraction in Theorem 2.2:

¤

𝑞(𝜇, 𝑡) =𝜁(𝑞(𝜇, 𝑡), 𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) +𝑑(𝜇, 𝑥 , 𝑡) (3.11) 𝑑𝑞(𝜇, 𝑡) =𝜁(𝑞(𝜇, 𝑡), 𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡)𝑑 𝑡+𝐺(𝜇, 𝑥 , 𝑡)𝑑𝒲(𝑡) (3.12) where𝑑(𝜇, 𝑥 , 𝑡) =𝜇 𝑑_𝑐(𝑥 , 𝑡),𝐺(𝜇, 𝑥 , 𝑡) =𝜇𝐺_𝑐(𝑥 , 𝑡), and𝜁(𝑞, 𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡)is defined as

𝜁 =(𝐴(𝜚, 𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) −𝐵(𝑥 , 𝑡)𝐾(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡)) (𝑞−𝑥_𝑑) + 𝑓(𝑥_𝑑, 𝑡) +𝐵(𝑥_𝑑, 𝑡)𝑢_𝑑 (3.13) where𝐴is the SDC matrix of Lemma 3.1 with(𝑠,𝑠,¯ 𝑢¯) =(𝑥 , 𝑥_𝑑, 𝑢_𝑑). Setting𝜇=1 in (3.11) and (3.12) results in (3.1) and (3.2), respectively, and setting𝜇=0 simply results in (3.3). Consequently, both 𝑞 = 𝑥 and 𝑞 = 𝑥_𝑑 are particular solutions of (3.11) and (3.12). If there is no disturbance acting on the dynamics (3.1) and (3.2), the differential dynamics of (3.11) and (3.12) for𝜕_𝜇𝑞=𝜕 𝑞/𝜕 𝜇is given as

𝜕_𝜇𝑞¤ =(𝐴(𝜚, 𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) −𝐵(𝑥 , 𝑡)𝐾(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡))𝜕_𝜇𝑞 . (3.14) In [12], [13], [16], [17], it is proposed that the contraction conditions of Theorems 2.1 and 2.5 for the closed-loop dynamics (3.11) and (3.12) can be expressed as convex constraints as summarized in Theorem 3.1.

Theorem 3.1. Let 𝛽be defined as 𝛽=0for deterministic systems(3.1)and 𝛽=𝛼_𝑠 = 𝐿_𝑚𝑔¯²

𝑐(𝛼_𝐺 +1/2)

for stochastic systems (3.2), respectively, where 𝑔¯_𝑐 is given in (3.2), 𝐿_𝑚 is the Lipschitz constant of 𝜕 𝑀/𝜕 𝑥_𝑖 for 𝑀 of (3.10), and 𝛼_𝐺 ∈ R^>0 is an arbitrary constant as in Theorem 2.5. Also, let𝑊 = 𝑀(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡)⁻¹(or𝑊 =𝑀(𝑥 , 𝑡)⁻¹, see

Theorem 3.2),𝑊¯ = 𝜈𝑊, and 𝜈 = 𝑚. Then the following three matrix inequalities are equivalent:

𝑀¤ +𝑀

𝜕 𝜁

𝜕 𝑞 + 𝜕 𝜁

𝜕 𝑞

⊤

𝑀 ⪯ −2𝛼 𝑀−𝛽I, ∀𝜇∈ [0,1] (3.15) 𝑀¤ +2 sym(𝑀 𝐴) −2𝑀 𝐵 𝑅⁻¹𝐵^⊤𝑀 ⪯ −2𝛼 𝑀− 𝛽I (3.16)

− ¤𝑊¯ +2 sym(𝐴𝑊¯) −2𝜈 𝐵 𝑅⁻¹𝐵^⊤ ⪯ −2𝛼𝑊¯ − 𝛽 𝜈

¯

𝑊² (3.17)

where 𝜁 is as defined in (3.13). For stochastic systems with 𝛽 = 𝛼_𝑠 > 0, these inequalities are also equivalent to

"

− ¤𝑊¯ +2 sym(𝐴𝑊¯) −2𝜈 𝐵 𝑅⁻¹𝐵^⊤+2𝛼𝑊¯ 𝑊¯

𝑊¯ −^𝜈

𝛽I

#

⪯0. (3.18)

Note that𝜈and𝑊¯ are required for(3.17)and(3.18)and the arguments(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) for each matrix are suppressed for notational simplicity.

Furthermore, under these equivalent contraction conditions, Theorems 2.4 and 2.5 hold for the virtual systems (3.11) and (3.12), respectively. In particular, if 𝑚I⪯ 𝑀 ⪯ 𝑚Iof(2.26)holds, or equivalently

I⪯ 𝑊¯(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) ⪯ 𝜒I (3.19)

holds for 𝜒=𝑚/𝑚, then we have the following bounds:

∥𝑥(𝑡) −𝑥_𝑑(𝑡) ∥ ≤ 𝑉_ℓ(0)

√ 𝑚

𝑒^−𝛼𝑡+ 𝑑¯_𝑐 𝛼

√

𝜒(1−𝑒^−𝛼𝑡) (3.20)

E

∥𝑥(𝑡) −𝑥_𝑑(𝑡) ∥²

≤ E[𝑉_𝑠ℓ(0)]

𝑚

𝑒^−2𝛼𝑡+ 𝐶_𝐶 2𝛼

𝜒 (3.21)

where𝑉_𝑠ℓ =∫^𝑥

𝑥𝑑

𝛿𝑞^⊤𝑀 𝛿𝑞and𝑉_ℓ =∫^𝑥

𝑥𝑑

∥Θ𝛿𝑞∥are as given in Theorem 2.3 with𝑀 = Θ^⊤Θ, the disturbance bounds 𝑑¯_𝑐 and𝑔¯_𝑐 are given in (3.1) and(3.2), respectively, and 𝐶_𝐶 = 𝑔¯²

𝑐(2𝛼_𝐺⁻¹ +1). Note that for stochastic systems, the probability that

∥𝑥−𝑥_𝑑∥is greater than or equal to𝜀 ∈R^>0is given as P[∥𝑥(𝑡) −𝑥_𝑑(𝑡) ∥ ≥𝜀] ≤ 1

𝜀²

E[𝑉_𝑠ℓ(0)]

𝑚

𝑒^−2𝛼𝑡+ 𝐶_𝐶 2𝛼

𝜒

. (3.22)

Proof. Substituting (3.13) into (3.15) gives (3.16). Since 𝜈 > 0 and 𝑊 ≻ 0, multiplying (3.16) by𝜈and then by𝑊from both sides preserves matrix definiteness.

Also, the resultant inequalities are equivalent to the original ones [37, p. 114]. These operations performed on (3.16) yield (3.17). If 𝛽 = 𝛼_𝑠 > 0 for stochastic systems,

applying Schur’s complement lemma [37, p. 7] to (3.17) results in the Linear Matrix Inequality (LMI) constraint (3.18) in terms of ¯𝑊 and𝜈. Therefore, (3.15) – (3.18) are indeed equivalent.

Also, since we have ∥𝜕_𝜇𝑑(𝜇, 𝑥 , 𝑡) ∥ ≤ 𝑑¯_𝑐 for𝑑 in (3.11) and∥𝜕_𝜇𝐺(𝜇, 𝑥 , 𝑡) ∥²

𝐹 ≤ 𝑔¯²

𝑐

for𝐺in (3.12), the virtual systems in (3.11) and (3.12) clearly satisfy the conditions of Theorems 2.4 and 2.5 if it is equipped with (3.15), which is equivalent to (3.16) – (3.18). This implies the exponential bounds (3.20) – (3.22) rewritten using𝜒=𝑚/𝑚, following the proofs of Theorems 2.4 and 2.5.

Because of the control and estimation duality in differential dynamics similar to that of the Kalman filter and Linear Quadratic Regulator (LQR) in LTV systems, we have an analogous robustness result for the contraction theory-based state estimator as to be derived in Sec. 4.2.

Although the conditions (3.15) – (3.18) depend on (𝑥_𝑑, 𝑢_𝑑), we could also use the SDC formulation with respect to a fixed point [12], [13] in Lemma 3.1 to make them independent of the target trajectory as in the following theorem.

Theorem 3.2. Let (𝑥 ,¯ 𝑢¯) be a fixed point selected arbitrarily in R^𝑛 × R^𝑚, e.g., (𝑥 ,¯ 𝑢¯)= (0,0), and let𝐴(𝑥 , 𝑡)be an SDC matrix constructed with(𝑠,𝑠,¯ 𝑢¯)= (𝑥 ,𝑥 ,¯ 𝑢¯) in Lemma 3.1, i.e.,

𝐴(𝜚, 𝑥 , 𝑡) (𝑥−𝑥¯) = 𝑓(𝑥 , 𝑡) +𝐵(𝑥 , 𝑡)𝑢¯− 𝑓(𝑥 , 𝑡¯ ) −𝐵(𝑥 , 𝑡¯ )𝑢 .¯ (3.23) Suppose that the contraction metric of Theorem 3.1 is designed by 𝑀(𝑥 , 𝑡) with 𝐴 of(3.23), independently of the target trajectory (𝑥_𝑑, 𝑢_𝑑), and that the systems(3.1) and(3.2)are controlled by

𝑢 =𝑢_𝑑−𝑅(𝑥 , 𝑡)⁻¹𝐵(𝑥 , 𝑡)^⊤𝑀(𝑥 , 𝑡) (𝑥−𝑥_𝑑) (3.24) with such 𝑀(𝑥 , 𝑡), where 𝑅(𝑥 , 𝑡) ≻ 0 is a weight matrix on 𝑢. If the function 𝜙(𝑥 , 𝑥_𝑑, 𝑢_𝑑, 𝑡) = 𝐴(𝜚, 𝑥 , 𝑡) (𝑥_𝑑−𝑥¯) +𝐵(𝑥 , 𝑡) (𝑢_𝑑−𝑢¯) is Lipschitz in𝑥 with its Lips- chitz constant𝐿¯, then Theorem 3.1 still holds with𝛼of the conditions(3.15)–(3.18) replaced by𝛼+𝐿¯√︁

𝑚/𝑚. The same argument holds for state estimation of Theo- rem 4.3 to be discussed in Sec. 4.2.

Proof. The unperturbed virtual system of (3.1), (3.2), and (3.3) with 𝐴 of (3.23) and𝑢of (3.24) is given as follows:

¤

𝑞 =(𝐴(𝜚, 𝑥 , 𝑡) −𝐵(𝑥 , 𝑡)𝐾(𝑥 , 𝑡)) (𝑞−𝑥_𝑑) +𝐴(𝜚, 𝑞, 𝑡) (𝑥_𝑑−𝑥¯) +𝐵(𝑞, 𝑡) (𝑢_𝑑−𝑢¯)

+ 𝑓(𝑥 , 𝑡¯ ) +𝐵(𝑥 , 𝑡¯ )𝑢¯ (3.25)

where𝐾(𝑥 , 𝑡) = 𝑅(𝑥 , 𝑡)⁻¹𝐵(𝑥 , 𝑡)^⊤𝑀(𝑥 , 𝑡). Following the proof of Theorem 3.1, the computation of𝑉¤, where𝑉 =𝛿𝑞^⊤𝑀(𝑥 , 𝑡)𝛿𝑞, yields an extra term

2𝛿𝑞^⊤𝑀

𝜕 𝜙

𝜕 𝑞

(𝑞, 𝑥_𝑑, 𝑢_𝑑, 𝑡)𝛿𝑞 ≤ 2 ¯𝐿

√︄

𝑚 𝑚

𝛿𝑞^⊤𝑀 𝛿𝑞 (3.26)

due to the Lipschitz condition on 𝜙, where 𝜙(𝑞, 𝑥_𝑑, 𝑢_𝑑, 𝑡) = 𝐴(𝑞, 𝑡) (𝑥_𝑑 − 𝑥¯) + 𝐵(𝑞, 𝑡) (𝑢_𝑑 −𝑢¯). This indeed implies that the system (3.25) is contracting as long as the conditions (3.15) – (3.18) hold with 𝛼 replaced by 𝛼+ 𝐿¯√︁

𝑚/𝑚. The last statement on state estimation follows from the nonlinear control and estimation duality to be discussed in Sec. 4.2.

Remark 3.5. As in [12], we could directly use the extra term2𝛿𝑞^⊤𝑀(𝜕 𝜙/𝜕 𝑞)𝛿𝑞of (3.26)in(3.15)–(3.18)without upper-bounding it, although now the constraints of Theorem 3.2 depend on(𝑥 , 𝑞, 𝑡)instead of(𝑥 , 𝑡). Also, the following two inequalities given in [12] with𝛾¯ =𝜈 𝛾, 𝛾 ∈R≥0:

− ¤𝑊¯ + 𝐴𝑊¯ +𝑊 𝐴¯ ^⊤+𝛾¯I−𝜈 𝐵 𝑅⁻¹𝐵^⊤ ⪯0

"

¯

𝛾I+𝜈 𝐵 𝑅⁻¹𝐵^⊤−𝑊 𝜙¯ ^⊤−𝜙𝑊¯ −2𝛼𝑊¯ 𝑊¯

¯

𝑊 ^𝜈

2𝛼𝑠I

#

⪰ 0. are combined as one LMI(3.18)in Theorems 3.1 and 3.2.

Example 3.4. The inequalities in Theorem 3.1 can be interpreted as in the Riccati inequality inH_∞control. Consider the following system:

¤

𝑥 = 𝐴𝑥+𝐵_𝑢𝑢+𝐵_𝑤𝑤 , 𝑧=𝐶_𝑧𝑥 (3.27)

where 𝐴 ∈ R^𝑛×𝑛, 𝐵_𝑢 ∈ R^𝑛×𝑚, 𝐵_𝑤 ∈ R^𝑛×𝑤, and 𝐶_𝑧 ∈ R^𝑜×𝑛 are constant matrices, 𝑤 ∈ R^𝑤 is an exogenous input, and 𝑧 ∈ R^𝑜 is a system output. As shown in [4]

and [37, p. 109], there exists a state feedback gain𝐾 = 𝑅⁻¹𝐵^⊤

𝑢𝑃 such that theL₂ gain of the closed-loop system(3.27),sup_{∥𝑤∥≠0}∥𝑧∥/∥𝑤∥, is less than or equal to𝛾 if

2 sym(𝑃 𝐴) −2𝑃 𝐵_𝑢𝑅⁻¹𝐵^⊤

𝑢𝑃+ 𝑃 𝐵_𝑤𝐵^⊤

𝑤𝑃 𝛾²

+𝐶^⊤

𝑧𝐶_𝑧 ⪯ 0 (3.28)

has a solution𝑃 ≻ 0, where𝑅 ≻ 0is a constant weight matrix on the input𝑢. If we select 𝐵_𝑤 and𝐶_𝑧 to have 𝐵_𝑤𝐵^⊤

𝑤 ⪰ (𝑃⁻¹)² and𝐶^⊤

𝑧 𝐶_𝑧 ⪰ 2𝛼 𝑃 for some𝛼 > 0, the contraction condition(3.16)in Theorem 3.1 can be satisfied with 𝑀 = 𝑃, 𝐵 = 𝐵_𝑢, and𝛽 =1/𝛾²due to(3.28).

In Sec. 3.3 and Sec. 3.4, we will discuss the relationship to input-output stability theory as in Example 3.4, using the results of Theorem 3.1.