Partial-State Deadzone-Adapted Disturbance Suppression Control For Strict-Feedback Systems

1. Introduction

• Robustness to dynamic uncertainties is harder than robustness to static uncertainties, and partial-state feedback matters when states are unmeasured or system dynamics are neglected.The introduction relates dynamic-uncertainty robustness to partial-state adaptive feedback.Source: §1 · pp. 1–2
• Prior approaches include leakage and dynamic gains with small-gain arguments, universal λ-tracking, and prescribed-performance adaptive control.The introduction also notes later extensions of the Praly-Jiang and prescribed-performance approaches.Source: §1 · p. 2
• Earlier DADS combines nonlinear damping, a single adaptive gain, and a deadzone update to attenuate plant state and prevent gain and state drift.Prior work applied it to partial-state feedback under a matching condition and to systems with neglected infinite-dimensional dynamics.Source: §1 · p. 2
• DADS is extended to strict-feedback systems with and without integrators, attenuating output to an assignable small level despite unknown, time-varying disturbances, parameters, and input coefficients.The uncertainties may have arbitrary and unknown bounds, and the results extend to the infinite-dimensional case.Source: §1 · p. 2
• The paper presents finite- and infinite-dimensional examples, using the finite-dimensional example for a detailed comparison with the controller in.Sections 2 and 3 give finite- and infinite-dimensional results; Section 5 contains proofs and Section 6 concluding remarks.Source: §1 · p. 2

2.1. Results

• Inequalities (2.11) enable a backstepping lemma for partial-state feedback design; the nonnegative constant Λ is important for exact regulation, which is possible when Λ = 0.Source: §2 · p. 6
• The growth and dissipation conditions characterize interconnections that a partial-state controller can compensate using only the measured state components.For the no-integrator case the controller depends only on ξ; the integrator case has a corresponding dependence on (ζ, ξ).Source: §2 · pp. 6, 8
• For systems with integrators, Assumption (B) uses bounds involving the additional state component and a dissipation inequality different from Assumption (A), enabling partial-state backstepping.The known function Q appears on the right-hand side of the dissipation condition; exact regulation is associated with Λ = 0 and specified conditions on γ.Source: §2 · pp. 6–7
• Under Assumption (A), Theorem 1 establishes a partial-state DADS result for strict-feedback systems without integrators, with controller-related functions and a function V positive for nonzero regulated state.The theorem introduces estimates for solutions, but the supplied passages do not include their full form.Source: §2 · p. 8

2. Results for the finite-dimensional case

• As t→∞, the cited asymptotic estimates bound the regulated-output quantity and Lyapunov function V(t,z(t)) by ε in limiting upper value.Source: §2 · p. 8
• Under the stated constant-parameter and vanishing-disturbance conditions, the no-integrator closed loop is UGAOS with output (w, ξ).The theorem’s additional asymptotic alternative allows convergence to zero under these conditions.Source: §2 · p. 9
• With integrators, Theorem 2 gives corresponding asymptotic bounds for the regulated output and Lyapunov function, assuming (B) and that (A, b) is stabilizable.Its estimates also include bounds involving initial conditions and system quantities in (2.27)–(2.28).Source: §2 · pp. 9–10
• Under corresponding conditions, the integrator case is UGAOS with output (w, ζ, ξ), and the theorem does not exclude regulation of the full plant state.The theorem also gives an asymptotic alternative for the regulated states under its stated vanishing-disturbance conditions.Source: §2 · p. 10
• The debt function cannot depend on unknown parameters or disturbances, but its growth is unrestricted through the construction of the Lyapunov function.Theorem 1 allows dependence on dynamic gain, whereas Theorem 2’s debt function depends on (ζ, ξ), reflecting the different backstepping roles of these states.Source: §2 · pp. 11–12

2.3. Formulae for the feedback law

• Theorem 1’s existential feedback law is constructive: its proof yields a backstepping algorithm under assumption (A).The algorithm permits arbitrary positive c, κ, ε, and Γ and β in (0, 1).Source: §2.3 · pp. 12–13
• If b_1(t) ≥ 1 for all t ≥ 0, the authors state that a simpler formula can be used as an alternative to the preceding feedback law.Source: §2.3 · pp. 14–15
• No similar algorithm is given for the DADS feedback law guaranteed under assumption (B), because its proof requires additional technical steps.The algorithm described under assumption (A) is characterized as aggressive; the passage does not provide a corresponding construction for assumption (B).Source: §2.3 · p. 15

Assumptions

• The unmeasured state w in X is a dynamic uncertainty affecting strict-feedback system (2.1); unlike finite-dimensional assumptions, (A′) and (B′) require functions a1, a2, a3, a4 in K∞ satisfying (3.8)–(3.9).Source: §3 · pp. 15–17

No integrators

• In the no-integrator case, closed-loop trajectories satisfy limsup |ξ_1(t)| ≤ ε/2 and limsup V(z(t), ξ(t)) ≤ ε.The theorem assumes (A’) and gives these bounds for every κ, ε, Γ > 0, subject to its stated disturbance and input-coefficient conditions.Source: §3 · pp. 17–18

With integrators

• For the integrator case, Theorem 4 extends the result under (B’) when the pair (A, b) is stabilizable.The theorem states that functions V and k exist for every κ, ε, Γ > 0; the supplied passage does not include its ensuing trajectory estimates.Source: §3 · p. 18, §5 · p. 62

3. Results for the infinite-dimensional case

• As t→∞, the result bounds the limsup of (ζ(t), ξ(t), ε) and of V(t)(ζ(t), ξ(t), z(t)).These are the estimates stated in (3.17).Source: §3 · p. 18
• The result also gives trajectory bounds in (3.18) and (3.19) in terms of initial values and system quantities.The displayed bounds include w(t), z(t), disturbance and coefficient quantities; the expressions are partly corrupted in the supplied text.Source: §3 · p. 19
• Under the stated additional conditions, either ζ(t), ξ(t), and w(t) all converge to zero, or the alternative asymptotic comparison in (3.19) holds.The conditions include γ(0)=Λ=0, γ(s)≡0, d(t)→0, and δ(t)→0.Source: §3 · p. 19

Example 1

• The cancellation-type example controller is simpler and avoids conservativeness, while differing from the general recursion, which dominates rather than cancels.The departure from the general design procedure is presented for pedagogical reasons as well as simplicity.Source: §4 · p. 21
• DADS achieves the residual-output objective for time-varying θ without knowing the disturbance, deadzone, or parameter bounds; the earlier controller requires constant θ and a known bound.The DADS guarantee requires a specified inequality to hold for some r ∈ (0,4), involving known controller parameters.Source: §4 · p. 22
• For constant θ, the estimate gives exponential convergence of the output to a set, followed by convergence toward the residual objective and possibly toward zero plant state.The third phase is guaranteed if disturbance and deadzone tend to zero and the stated condition on θ holds.Source: §4 · p. 22
• For the original disturbance, both controllers meet the 0.25 objective, but DADS is more aggressive initially, with maximum input 1651.94 versus 152.57.These maxima are over t ∈ [0,1]; for t ≥ 1, the control inputs are very similar.Source: §4 · p. 23
• With the disturbance changed, the conventional controller misses the 0.25 output objective, while DADS achieves it without retuning.The changed case uses δ(t) ≡ 2 and d(t) = 0.6 sin(2t); the conventional controller must be tuned differently to meet the objective.Source: §4 · p. 23, Figure 3 · p. 24

Example 2

• The PDE example gives an explicit partial-state DADS feedback for robust practical output regulation, extending the approach beyond finite-dimensional systems to a parabolic PDE.The analysis focuses on control and does not study existence or uniqueness; the authors note that other PDE types can also be handled.Source: §4 · pp. 24–26

Example 1

• Theorem 4’s conclusions follow from differential inequalities (4.19) and (4.30), together with inequalities (4.20) and (4.26).Source: §4 · p. 28

Proofs of Main Results

• Lemma 1 states conditions on smooth functions and system terms under which the parameterized control system satisfies specified Lyapunov inequalities.The lemma introduces positive constants and functions and requires the stated inequalities to hold over the parameter, disturbance, and state domains.Source: §5 · pp. 28–29
• The proof defines an augmented function V(z,xi,chi) from V(z,xi) and k(z) to establish a corresponding bound involving xi and chi.The construction is given in (5.7), and the proof derives the associated inequality (5.9).Source: §5 · pp. 29, 31
• The proof then derives the time-derivative inequality (5.10) using the constructed function and auxiliary facts introduced for the derivation.It introduces the notation zeta=k(z)−xi and applies the assumptions in (5.4) during the derivation.Source: §5 · pp. 31–32

5. Proofs of Main Results

• The proof uses the non-increasing property of a map to obtain a further estimate.The following passage connects that estimate to inequality (5.10), derived from definition (5.13), inequality (5.35), and additional bounds.Source: §5 · p. 37
• The proof establishes inequality (5.17) using bounds (5.36) and (5.37), together with stated facts about the parameters.It then uses inequalities (5.38) and (5.39) to derive a further inequality and estimate.Source: §5 · pp. 37–39
• The authors finish the proof by establishing the validity of inequalities (5.17) and (5.18), with the cited derivation invoking (5.6), (5.38), and (5.39).They obtain an inequality from (5.6), while (5.38) and (5.39) are used in subsequent steps.Source: §5 · pp. 37, 39

Part I

• Part I derives an inequality from (2.9) and (2.10), for arbitrary positive a and κ.The derivation is stated to hold for all (ξ, w) in the specified domain.Source: §5 · pp. 40, 42

Part II

• Part II constructs a smooth function k1, vanishing at z = 0, to satisfy the required inequality for arbitrary positive a, c, and κ.The construction defines an auxiliary function using ρ = exp(z) + κ and the smooth function from (2.6).Source: §5 · pp. 41, 43

Part III

• Part III applies the lemma m − 1 times with Assumption (A) to construct smooth k and V, with V positive when ξ ≠ 0.The constructed functions are stated to satisfy the proof’s listed bounds and inequalities.Source: §5 · p. 41

Part IV

• Part IV uses the constructed functions to analyze stability of the closed-loop system (2.1), (2.2) with (2.17), (2.18).The analysis is set up for arbitrary positive κ, ε, and Γ, with parameters specified in the proof.Source: §5 · pp. 42, 47

5. Proofs of Main Results

• Assuming (2.1), Lemma 4.1 on page 63 in and differential inequality (5.77) imply the ensuing estimate for every t in [0,t_max).Source: §5 · p. 50
• Under the stated boundedness and regularity conditions, Barbălat’s Lemma establishes estimate (2.19).The argument shows the relevant integral is finite and applies Barbălat’s Lemma to obtain the required limiting estimate.Source: §5 · p. 51
• With the additional limiting assumptions in the final case, U, w, and V converge to zero as time tends to infinity.The proof obtains v(t) → 0, then U(t) → 0; the definition of U and the stated inequalities yield convergence of w and V.Source: §5 · pp. 51–52

Proofs of Main Results

• A subsequence limit with ξ* ≥ ε would force V(ξ*, z*) = 0, contradicting positive definiteness and thus completing the convergence argument.The argument uses continuity of V and the limiting condition V(ξ(t), z(t)) → 0.Source: §5 · p. 52
• Theorem 2’s proof has three parts, beginning with construction of a positive-definite quadratic Lyapunov function using stabilizability of (A, b).The construction establishes a quadratic lower bound for the Lyapunov function.Source: §5 · pp. 52, 54
• Part II applies the lemma m − 1 times under Assumption (B) to construct a Lyapunov function satisfying positivity, bounded-sublevel-set, and derivative inequalities.The construction is stated for arbitrary positive constants κ, ε, and Γ.Source: §5 · pp. 52–53
• The function was designed in Part II for arbitrary positive constants κ, ε, and Γ.Source: §5 · p. 53

5. Proofs of Main Results

• Part III begins by fixing arbitrary positive κ, ε, and Γ and constructing functions used in the stability estimates.The ensuing estimates are stated for the closed-loop system with the specified controller.Source: §5 · pp. 62–63
• The closed-loop stability analysis uses the Lyapunov function U(w,z,ζ,ξ)=Φ(w)+V(z,ζ,ξ).The proof introduces U before considering solutions of the closed-loop system.Source: §5 · p. 64
• The proof establishes that the auxiliary function v belongs to L∞ under the stated regularity and positive lower-bound assumptions on the input coefficients.The passage states that the definition of v implies v ∈ L∞; the neighboring assumptions require b_j to be essentially bounded with a positive infimum.Source: §5 · p. 64

Proof of main results

• The estimates (5.129) imply global existence and estimates (2.27)–(2.28) for appropriate constants.In particular, the maximal existence time is infinite.Source: §5 · p. 66
• The proof concludes that the right estimate (2.26) holds, while the left estimate (2.26) follows from the right estimate and inequality (5.87).Source: §5 · p. 66
• When the specified disturbance terms vanish asymptotically and the input-gain condition holds, v(t) and the output w(t) converge to zero.The proof uses the resulting bound on U and the definition of the Lyapunov function.Source: §5 · p. 67
• Under those conditions, the state variables ζ(t) and ξ(t) also converge to zero, as shown by contradiction using the Lyapunov function’s positivity.A bounded subsequence with nonzero limiting state would contradict that the limiting Lyapunov value is zero.Source: §5 · p. 67

• DADS can provide robust, partial-state adaptive feedback laws.Source: §6 · p. 67
• With only one measured output, extending DADS is not trivial and requires an observer-based control structure.Source: §6 · p. 67
• A detailed comparison of DADS with other adaptive feedback design methodologies is still needed.Source: §6 · p. 67