Stability Guarantees for Data-Driven Predictive Control of Nonlinear Systems via Approximate Koopman Embeddings

Stability Guarantees for Data-Dri ven Predicti v e Control of Nonlinear Systems via Approximate K oopman Embeddings Amin T aghieh and SangW oo Park Abstract — Data-driven model predicti ve control based on Willems’ fundamental lemma has proven effective f or linear systems, but extending stability guarantees to nonlinear systems remains an open challenge. In this paper , we establish conditions under which data-driven MPC, applied directly to input-output data from a nonlinear system, yields practical exponential stability . The key insight is that the existence of an approximate Koopman linear embedding certifies that the nonlinear data can be interpreted as noisy data from a linear time-inv ariant system, enabling the application of existing rob ust stability theories. Crucially , the Koopman embedding serves only as a theoretical certificate; the controller itself operates on raw nonlinear data without knowledge of the lifting functions. W e further show that the proportional structure of the embedding residual can be exploited to obtain an ultimate bound that depends only on the irreducible offset, rather than the worst-case embedding error . The framework is demonstrated on a synchronous generator connected to an infinite bus, f or which we construct an explicit physics-inf ormed embedding with error bounds. I . I N T RO D U C T I O N A. Motivation and Backgr ound Data-driv en control methods that bypass explicit system identification hav e become a central topic in modern control theory , building on W illems’ fundamental lemma [1]. For linear time-in variant (L TI) systems, this lemma provides a complete, non-parametric characterization of the system’ s trajectory space using a single persistently exciting input- output trajectory . When combined with model predictive con- trol (MPC), this yields the Data-Enabled Predictive Control (DeePC) algorithm [2] and its variants with rigorous closed- loop stability and robustness guarantees [3]. Beyond MPC, W illems’ fundamental lemma has led to a broad range of data-dri ven controller designs for L TI systems. These include computing stabilizing feedback controllers and optimal LQR gains directly from data [4], characterizing which control-theoretic properties can be inferred from a giv en dataset through a “data informativity” frame work [5], and unifying direct data-driv en and indirect identification- based formulations via regularization [6]. These develop- ments pro vide a mature theoretical foundation for L TI sys- tems, but all rely fundamentally on the linearity assumption: the data must come from a linear system, possibly corrupted by noise. Sev eral approaches have been proposed to extend data- driv en methods to specific classes of nonlinear systems, including L TI embeddings for bilinear systems [7], sum- of-squares programs for polynomial systems [8], coordinate transformations for feedback-linearizable systems [9], and linearization-based tracking MPC [10]. A fundamentally dif ferent perspectiv e is offered by K oop- man operator theory [11], which provides a pathway toward applying linear methods to broad classes of nonlinear systems by lifting the dynamics into a higher-dimensional (potentially infinite) functional space where they e volve linearly . This K oopman-based idea has been pursued along two distinct lines. The first is identification-based : one learns a finite- dimensional K oopman model from data via extended dy- namic mode decomposition (EDMD) [12] and then applies standard MPC using the identified model. Recent work has established stability guarantees for this approach under pro- portional approximation error bounds [13]–[15]. The second line is dir ectly data-driven : the extended W illems’ funda- mental lemma of [16] sho ws that the trajectory space of a nonlinear system admitting an exact Koopman embedding can be represented directly from nonlinear data without knowledge of the lifting functions. Howe ver , a critical subtlety is that e xact finite-dimensional K oopman embeddings rarely exist for physical systems. For example, systems with trigonometric nonlinearities or bilinear terms generically require infinite-dimensional embeddings to achieve exact closure of the observables. Therefore, a fundamental question remains: under what conditions is the resulting closed loop stable, and how does the stability margin depend on the system’ s deviation from linearity? This makes the approximate embedding framew ork not merely a theoretical conv enience, but a practical necessity . In this paper , we establish rigorous conditions under which data-driv en MPC, applied directly to input-output data from a nonlinear system, yields practical exponential stability of the closed loop. The ke y insight is that the existence of an approximate K oopman linear embedding certifies that the nonlinear data can be interpreted as noisy data from a linear time-in variant system, enabling the application of existing robust stability theories. Our contributions are as follows: (i) W e formalize the notion of an appr oximate K oopman linear embedding with pr oportional-with-offset err or bounds and construct an explicit physics-informed embedding for the third-order synchronous generator; (ii) W e prove that the robust data-driv en MPC scheme of [3], applied directly to input-output data from a nonlinear system, yields practical exponential stability whenever the system admits an approx- imate K oopman embedding with sufficiently small error; (iii) W e provide a rigorous bridge between the Koopman approximation error and the bounded noise framework of [3], showing how the nonlinearity gap maps to an effecti ve noise lev el ¯ ϵ that determines the stability margin. B. Notation For a vector x and positiv e definite matrix P , a weighted norm is defined as ∥ x ∥ P := √ x ⊤ P x . W e write col( · ) for column stacking and x [ a,b ] := col( x a , . . . , x b ) . For a compact operating region X with equilibrium x s and input constraint set U with equilibrium input u s , we de- fine diam z := max x ∈X ∥ Φ( x ) − Φ( x s ) ∥ 2 and diam u := max u ∈U ∥ u − u s ∥ 2 . Finally , a sequence { x k } N − 1 k =0 induces the Hankel matrix H L ( x ) :=      x 0 x 1 · · · x N − L x 1 x 2 · · · x N − L +1 . . . . . . . . . x L − 1 x L · · · x N − 1      . I I . S Y N C H R O N O U S G E N E R ATO R M O D E L In this section, we present a third-order synchronous generator model, which is nonlinear and fails to admit an exact finite-dimensional Koopman embedding. Although the theoretical results of this paper are generally applicable, we will use the generator model as a guiding example. A. Continuous-T ime Dynamics In this paper , we consider the dynamics of a single syn- chronous generator connected to an infinite bus through a transmission line. The generator is modeled by the classical flux-decay model without exciter or gov ernor dynamics. The state is denoted by x = ( δ , ω , E ′ q ) ⊤ ∈ R 3 , where δ is the rotor angle, ω is the rotor angular velocity , and E ′ q is the q -axis transient internal voltage. The control input is u = T M ∈ R (mechanical torque). ˙ δ = ω − ω s (1a) M ˙ ω = h T M − D ( ω − ω s ) − E ′ q V ∞ X Σ sin δ i (1b) T ′ d 0 ˙ E ′ q = h E f d − X d + X e X Σ E ′ q + ( X d − X ′ d ) V ∞ X Σ cos δ i (1c) Here, H is the inertia constant, ω s the synchronous speed, M := 2 H ω s the normalized inertia, D the damping coefficient, T ′ d 0 the d -axis open-circuit transient time constant, X d the synchronous d -axis reactance, and E f d is the (constant) field voltage. In (1b), the term E ′ q V ∞ X Σ sin δ is the electrical power P e , representing the electromagnetic torque. The symbol V ∞ is the infinite b us voltage and X Σ := X ′ d + X e is the total reactance (transient d -axis reactance X ′ d plus external line reactance X e ). In (1c), the cos δ term arises from the d -axis stator current I d = E ′ q − V ∞ cos δ X Σ after substitution into the flux-decay equation. The nonlinearities are: (i) sin δ and cos δ (trigonometric); (ii) E ′ q sin δ (bilinear). The input u = T M enters affinely through (1b) only . B. Discr ete-T ime F ormulation W ith sampling period ∆ t > 0 and Euler discretization, define the discrete-time system: x k +1 = f ( x k , u k ) , y k = g ( x k , u k ) (2) where f : R 3 × R → R 3 is given component-wise by: δ k +1 = δ k + ∆ t ( ω k − ω s ) (3) ω k +1 = ω k + ∆ t M h u k − D ( ω k − ω s ) − E ′ q ,k V ∞ X Σ sin δ k i (4) E ′ q ,k +1 = E ′ q ,k + ∆ t T ′ d 0 h E f d − X d + X e X Σ E ′ q ,k + ( X d − X ′ d ) V ∞ X Σ cos δ k i . (5) The system has n = 3 states, m = 1 input, and p = 2 outputs y k = ( ˜ ω k , P e,k ) ⊤ ∈ R 2 , where ˜ ω k := ω k − ω s is the speed deviation and P e,k is the electrical power . Assumption 1 (Operating Regime) . The generator operates in a compact domain X := { ( δ, ω , E ′ q ) : | δ − δ s | ≤ δ max , | ω − ω s | ≤ ω max , E ′ q , min ≤ E ′ q ≤ E ′ q , max } where δ s is the pre-fault equilibrium angle, δ max < π / 2 (within the stability boundary), and ω max ≪ ω s (small speed deviations relativ e to synchronous speed). The input satisfies u ∈ U := [ T M , min , T M , max ] . Under Assumption 1, the per-step angle change satisfies | ∆ t ( ω k − ω s ) | ≤ ∆ t ω max ≪ 1 for typical sampling rates ( ∆ t ≤ 0 . 005 s), which will be exploited in the K oopman embedding analysis. I I I . A N A L Y S I S O F A P P RO X I M A T E K O O P M A N E M B E D D I N G F O R T H E G E N E R A T O R M O D E L A. K oopman Linear Embedding: Definition and Existence Definition 1 (K oopman Linear Embedding [16]) . The nonlin- ear system (2) admits a K oopman linear embedding of dimen- sion n z if there exist linearly independent lifting functions Φ = (Φ 1 , . . . , Φ n z ) ⊤ : R n → R n z such that z k := Φ( x k ) satisfies z k +1 = Az k + B u k , y k = C z k + D u k (6) along all trajectories of (2) within the operating domain X , with A ∈ R n z × n z , B ∈ R n z × m , C ∈ R p × n z , D ∈ R p × m . Definition 2 (Approximate Koopman Embedding) . The sys- tem (2) admits an appr oximate Koopman linear embedding centered at an equilibrium ( x s , u s ) with pr oportional err or bound ( ϵ A , ϵ B , ϵ C ) and offset c 0 ≥ 0 if there exist Φ , A , B , C , D such that, defining the deviation variables ¯ z k := Φ( x k ) − Φ( x s ) and ¯ u k := u k − u s , the dynamics satisfy ¯ z k +1 = A ¯ z k + B ¯ u k + e k , ¯ y k = C ¯ z k + D ¯ u k + η k (7) along all trajectories within X , where ¯ y k := y k − y s , and the residuals satisfy: ∥ e k ∥ 2 ≤ ϵ A ∥ ¯ z k ∥ 2 + ϵ B ∥ ¯ u k ∥ 2 + c 0 (8) ∥ η k ∥ 2 ≤ ϵ C ∥ ¯ z k ∥ 2 (9) for constants ϵ A , ϵ B , ϵ C ≥ 0 . When c 0 = 0 , the bound is pur ely proportional ; when c 0 > 0 , it is proportional-with- offset . The constant affine terms arising from the equilibrium (e.g., from constant inputs like E f d ) cancel exactly by centering at the equilibrium, since Φ( f ( x s , u s )) = Φ( x s ) . The concept of proportional error bounds for data-dri ven K oopman surrogates originates in the work of [13], which established pointwise error bounds for kernel EDMD that are proportional in the state and input. [14] used the same structure to prov e asymptotic stability of K oopman MPC with terminal conditions. The centering at equilibrium in (8) is essential: a purely proportional bound ( c 0 = 0 ) guarantees exponential stability (the error vanishes at the setpoint at an exponential rate), while a proportional-with- offset bound ( c 0 > 0 ) yields practical exponential stability (exponential con v ergence to a neighborhood of radius O ( c 0 ) ). Definition 2 combines the exact embedding concept of [16] with the proportional error framew ork of [13], [14], extended to include the of fset term that arises naturally in finite- dimensional approximations. Remark 1 (Interpretation of Approximate Embedding) . In Definition 2, the lifting Φ is an explicitly chosen, exactly computed function; it is not learned or approximated. The matrices A , B , C , D are the best linear description of how Φ( x k ) ev olves under the nonlinear dynamics. The residual e k arises because the nonlinear dynamics do not propagate Φ in a perfectly linear fashion: Φ( f ( x k , u k ))  = A Φ( x k ) + B u k exactly . Neither Φ nor A, B are “wrong”; the error is structural, reflecting the fact that the observable algebra does not close in finite dimensions. The L TI system obtained by propagating ( A, B , C , D ) without the residuals ( e k = 0 , η k = 0 ) is referred to as the nominal Koopman system . It is neither the true nonlinear dynamics nor a “true” (infinite- dimensional) Koopman operator; it is a finite-dimensional linear model whose de viation from the plant is quantified by ( ϵ A , ϵ B , ϵ C , c 0 ) . This terminology aligns with the robust control conv ention used in [3], where the nominal system is the model around which the robust MPC is designed, and the residuals play the role of plant-model mismatch. B. K oopman Embedding for the Synchr onous Generator Theorem 1. Consider the discr etized g enerator (3) – (5) under Assumption 1. Define the lifting: Φ( x ) = col( δ, ˜ ω , E ′ q , sin δ, cos δ, E ′ q sin δ, E ′ q cos δ ) . (10) Then z k = Φ( x k ) satisfies an approximate K oopman linear embedding centered at the equilibrium ( x s , u s ) with ϵ A = 2∆ t + c 6 + c 7 = O (∆ t ) , ϵ B = 0 , ϵ C = 0 , (11) c 0 = (2 + 2 E ′ q , max ) (∆ t ω max ) 2 = O ((∆ t ω max ) 2 ) , (12) wher e c 6 , c 7 depend on ∆ t/T ′ d 0 , µ , E ′ q , max , and E f d (see Appendix I for details). The bound is pr oportional with offset: ∥ e k ∥ 2 ≤ ϵ A ∥ Φ( x k ) − Φ( x s ) ∥ 2 + c 0 . (13) The pr oportional-with-offset structure of (13) is the form r equir ed by the stability analysis in Section IV -B. The offset c 0 > 0 limits any contr oller based on this embedding to practical exponential stability . Pr oof: See Appendix I. As we can see in Appendix I, the proportional constant ϵ A = O (∆ t ) recei ves contributions from the trigonometric propagation in components 4–5 ( 2∆ t ) and the bilinear prod- ucts in components 6–7 ( c 6 + c 7 ), with the latter dominating due to the cross-term z 3 ,k +1 · e 4 ,k . The of fset c 0 = (2 + 2 E ′ q , max )(∆ t ω max ) 2 arises from the T aylor remainders in the small-angle approximation. For typical parameters ( ∆ t = 0 . 0025 s, T ′ d 0 ≈ 6 s, ω max = 0 . 02 pu): ϵ A ≈ 0 . 015 and c 0 ≈ 10 − 8 . The offset is negligible relati ve to ϵ A , with c 0 /ϵ A ≈ 7 × 10 − 7 pu. Howe ver , as sho wn in Section IV -B, the dominant source of conserv atism in the stability bound is ϵ A itself, not c 0 ; the proportional error structure is exploited in Theorem 7 to recov er meaningful bounds. Remark 2 (Non-Controllability and Non-Observability of the Embedding) . The pair ( A, B ) in (6) with B = (0 , ∆ t M , 0 , 0 , 0 , 0 , 0) ⊤ is not controllable: the controllabil- ity matrix [ B , AB , . . . , A 6 B ] has rank 2, spanning only { z 1 , z 2 } = { δ, ˜ ω } . Components 3–7 are unreachable because their coupling to z 1 , z 2 (e.g., ˜ ω dri ving sin δ via ∆ t ˜ ω k cos δ k ) resides in the residual e k , not in A . Similarly , ( C, A ) with C ∈ R 2 × 7 is not observable: the observ ability matrix has rank 2, spanning { z 2 , z 6 } = { ˜ ω , E ′ q sin δ } . Consequently , the I/O transfer function G ( z ) = C ( z I − A ) − 1 B + D admits a minimal realization of order n eff ≤ n z , which is controllable and observable by construction. Neither controllability nor observability of the full n z -dimensional system is required: [16] shows that T ini ≥ n z suffices for the data-driven representation (Theorem 2), and [3] note e xplicitly that n z may be used as an “upper bound” on the true system order n eff , with all stability results remaining valid. I V . S T A B I L I T Y O F D AT A - D R I V E N M P C U N D E R A P P RO X I M A T E K O O P M A N E M B E D D I N G S A. Data-Driven Repr esentation W e now apply the extended W illems’ fundamental lemma [16] to the generator . Definition 3 (Lifted Excitation [16]) . Consider a nonlin- ear system (2) with a K oopman linear embedding (Defini- tion 1). W e say l trajectories of length L from (2), denoted col( u d,i , y d,i ) , i = 1 , . . . , l , with l ≥ mL + n z , provide lifted excitation of order L if the matrix H K :=  u d, 1 · · · u d,l Φ( x 1 0 ) · · · Φ( x l 0 )  ∈ R ( mL + n z ) × l (14) has full row rank, where x i 0 ∈ R n is the initial state of the i -th trajectory . Theorem 2 (Data-Dri ven Representation for the Generator (Exact Case)) . Consider the generator (2) with an exact K oopman embedding ( ϵ A = ϵ B = ϵ C = 0 ) of dimension n z = 7 . Collect a trajectory library of l ≥ L + 7 trajectories, each of length L := T ini + N with T ini ≥ 7 , that pr ovide lifted excitation of or der L (Definition 3). Arrange these trajectories into the data matrix H d :=  U ⊤ P Y ⊤ P U ⊤ F Y ⊤ F  ⊤ ∈ R ( m + p ) L × l (15) wher e U P , Y P ∈ R mT ini × l , R pT ini × l contain the first T ini steps (past) of the input and output trajectories, and U F , Y F ∈ R mN × l , R pN × l contain the remaining N steps (futur e). At time k , let u ini := col( u k − T ini , . . . , u k − 1 ) and y ini := col( y k − T ini , . . . , y k − 1 ) be the most r ecent T ini input- output measurements. Then, for any futur e input u F := col( u k , . . . , u k + N − 1 ) , the sequence col( u ini , y ini , u F , y F ) is a valid length- L trajectory of the generator if and only if ther e exists g ∈ R l such that col( U P , Y P , U F , Y F ) g = col( u ini , y ini , u F , y F ) . (16) Pr oof: This is a direct application of [16, Theorem 3] with n z = 7 . The conditions are verified as follows: (i) the K oopman embedding exists by assumption; (ii) the trajectory library pro vides lifted excitation by hypothesis; (iii) T ini ≥ n z = 7 ensures uniqueness of the initial lifted state by [16, Theorem 1], ev en without observability of ( C , A ) . B. F r om Exact to Appr oximate: Impact of K oopman Err or on Data-Driven Contr ol The exact case is idealised. In practice, the Koopman em- bedding is approximate (Theorem 1), and we must quantify how the approximation error propagates through the data- driv en control framework. The central question is the following. The earlier work [3] prov ed stability of data-driven MPC for L TI systems with bounded output noise. Our system is nonlinear . Why should their guarantees transfer? The existence of an approximate K oopman embedding provides the answer: it certifies that the nonlinear system’ s input-output data can be decomposed as y d = y d, nom + ϵ d , where y d, nom is what the nominal K oopman system (Remark 1) would produce, and ϵ d is bounded by an effecti ve noise lev el ¯ ϵ . The Koopman analysis quantifies ¯ ϵ in terms of the embedding quality ( ϵ A , ϵ B , ϵ C , c 0 ) , which in turn depends on physical parameters of the system (such as sampling period and operating regime). Thus, Theorem 4 be- low provides a stability guarantee for applying an LTI data- driven controller to a nonlinear system , with the K oopman embedding serving as the theoretical bridge. W e employ the robust data-dri ven MPC of [3, Prob- lem (6), Algorithm 2], which is summarized below . Consider a nonlinear system (2) admitting an approximate K oopman embedding (Definition 2) with parameters ( ϵ A , ϵ B , ϵ C , c 0 ) and lifted dimension n z . Collect a trajectory library as in Theorem 2. The robust MPC solves at each time t : min α ( t ) , σ ( t ) , ˆ u ( t ) , ˆ y ( t ) L − 1 X k =0 ℓ ( ˆ u k ( t ) , ˆ y k ( t )) + λ α ∥ α ( t ) ∥ 2 2 + λ σ ∥ σ ( t ) ∥ 2 2 (17a) s.t.  ˆ u [ − n z ,L − 1] ( t ) ˆ y [ − n z ,L − 1] ( t )  =  H L + n z ( u d ) H L + n z ( y d )  α ( t ) +  0 σ ( t )  (17b) ˆ u [ − n z , − 1] ( t ) = u [ t − n z ,t − 1] , ˆ y [ − n z , − 1] ( t ) = y [ t − n z ,t − 1] (17c) ˆ u [ L − n z ,L − 1] ( t ) = u s n z , ˆ y [ L − n z ,L − 1] ( t ) = y s n z (17d) ˆ u k ( t ) ∈ U , ∥ σ k ( t ) ∥ ∞ ≤ ¯ ϵ (1 + ∥ α ( t ) ∥ 1 ) (17e) where ℓ ( ˆ u, ˆ y ) = ∥ ˆ u − u s ∥ 2 R + ∥ ˆ y − y s ∥ 2 Q with Q, R ≻ 0 , and λ α , λ σ > 0 are regularization parameters. The Hankel matrices H L + n z ( u d ) and H L + n z ( y d ) are built directly from the measured nonlinear data (i.e., no model identification or noise decomposition is needed at the implementation level). The slack v ariable σ and its constraint (17e) absorb the mismatch between the nonlinear data and any L TI model. The scheme is applied in an n z -step fashion: solv e (17), apply u [ t,t + n z − 1] = ˆ u ∗ [0 ,n z − 1] ( t ) , then set t ← t + n z and repeat. Note that the MPC scheme (17) operates directly on input- output data from the nonlinear system; the Koopman embed- ding matrices A , B , C , D do not appear in the controller and need not be computed. The role of the Koopman analysis is purely certifying: the existence of an approximate embedding (Definition 2) guarantees that the nonlinear data can be interpreted as noisy data from an L TI system, enabling the stability theory of [3] to be applied. The effecti ve noise le vel ¯ ϵ , defined precisely in (21), quantifies the degree to which the nonlinear system deviates from L TI behavior . T o state the stability result, we require the following quantity from [3, eq. (8)]. Let H ux be the stacked input- state data matrix constructed from the offline data (where the state trajectory corresponds to some minimal realization of ( u d , y d ) ), and let H † ux := H ⊤ ux ( H ux H ⊤ ux ) − 1 be its right- in verse. Define: c pe :=   H † ux   2 2 . (18) This is a quantitativ e measure of the persistence of excitation: smaller c pe indicates better excitation, and c pe decreases with increasing data richness or larger amplitude of the e xciting input u d [3]. Lemma 3 (K oopman Error as Bounded Output Noise) . Consider a nonlinear system (2) admitting an approximate K oopman embedding (Definition 2) of dimension n z with parameters ( ϵ A , ϵ B , ϵ C , c 0 ) , operating within X under inputs u ∈ U (Assumption 1). Let ( u d,i , y d,i ) be any tr ajectory of the nonlinear system of length L . Let ( A, B , C, D ) be the matrices fr om Definition 2 and define the nominal K oop- man trajectory (cf. Remark 1): ˆ z nom ,i 0 = Φ( x i 0 ) , ˆ z nom ,i k +1 = A ˆ z nom ,i k + B u d,i k , and y nom ,i k = C ˆ z nom ,i k + D u d,i k . Then the output discrepancy at step k satisfies:    y d,i k − y nom ,i k    2 ≤ ¯ ϵ k , ∀ k = 0 , . . . , L − 1 (19) Defining ¯ e := ϵ A diam z + ϵ B diam u + c 0 , the per-step bound is: ¯ ϵ k := ∥ C ∥ 2 ¯ e k − 1 X l =0 ∥ A ∥ l 2 + ϵ C diam z . (20) A uniform bound over all k ≤ L , as requir ed by [3, Pr oblem (6)], is: ¯ ϵ := max 0 ≤ k ≤ L − 1 ¯ ϵ k = ∥ C ∥ 2 ¯ e L − 2 X l =0 ∥ A ∥ l 2 + ϵ C diam z . (21) Pr oof: The nominal K oopman trajectory ( ˆ z nom ,i k , y nom ,i k ) is the output of the L TI system ( A, B , C , D ) from Defi- nition 2, driv en by the same input u d,i and initialized at ˆ z nom ,i 0 = Φ( x i 0 ) , but propagated without the residuals e k and η k . It represents what the approximate embedding would predict if the linear relationship were exact. The actual output satisfies y d,i k = C Φ( x i k ) + D u d,i k + η i k with   η i k   2 ≤ ϵ C   ¯ z i k   2 (Definition 2). The output noise is: ϵ d,i k := y d,i k − y nom ,i k = C  Φ( x i k ) − ˆ z nom ,i k  | {z } =: ∆ i k + η i k . (22) W e bound ∆ i k . The actual lifted state satisfies Φ( x i k +1 ) = A Φ( x i k ) + B u d,i k + e i k , while the nominal satisfies ˆ z nom ,i k +1 = A ˆ z nom ,i k + B u d,i k . Subtracting giv es ∆ i k +1 = A ∆ i k + e i k with ∆ i 0 = 0 , which unrolls to ∆ i k = P k − 1 j =0 A k − 1 − j e i j . T aking norms:   ∆ i k   2 ≤ k − 1 X j =0 ∥ A ∥ k − 1 − j 2   e i j   2 . (23) From Definition 2,   e i j   2 ≤ ϵ A   ¯ z i j   2 + ϵ B   ¯ u i j   2 + c 0 . Since x i j ∈ X and u d,i j ∈ U by Assumption 1, each residual satisfies   e i j   2 ≤ ¯ e where ¯ e := ϵ A diam z + ϵ B diam u + c 0 , so   ∆ i k   2 ≤ ¯ e P k − 1 l =0 ∥ A ∥ l 2 . Combining with (22) via the triangle inequality:    ϵ d,i k    2 ≤ ∥ C ∥ 2 ¯ e k − 1 X l =0 ∥ A ∥ l 2 + ϵ C diam z = ¯ ϵ k . (24) Since the sum increases in k , the maximum is at k = L − 1 , giving (21). Remark 3. Lemma 3 is the k ey bridge between the K oopman theory and the data-dri ven MPC framework. It sho ws that the offline data ( u d , y d ) from the nonlinear system can be decomposed as ( u d , y d, nom + ϵ d ) with   ϵ d k   ∞ ≤ ¯ ϵ , where y d, nom are the outputs of the nominal K oopman system. This decomposition is purely analytical—the controller nev er computes y d, nom —but it enables the stability machinery of [3] to be applied. The same bound applies to online initial conditions: if ( u [ t − n z ,t − 1] , y [ t − n z ,t − 1] ) are the most recent n z ≤ L measurements from the nonlinear system in closed loop, the discrepancy from the nominal Koopman output also satisfies ∥ ϵ t ∥ ∞ ≤ ¯ ϵ . When ∥ A ∥ 2 < 1 , the geometric sum in (21) saturates: P L − 2 l =0 ∥ A ∥ l 2 ≤ 1 1 −∥ A ∥ 2 , so ¯ ϵ is bounded independently of the horizon L . Theorem 4 (Stability of Data-Dri ven MPC under Approxi- mate K oopman Embedding) . Consider a nonlinear system (2) admitting an appr oximate K oopman embedding (Definition 2) of dimension n z with parameters ( ϵ A , ϵ B , ϵ C , c 0 ) , and let ¯ ϵ be the effective noise level fr om Lemma 3. Suppose: (a) The offline input u d is persistently exciting of order L + 2 n z , (b) The pr ediction horizon satisfies L ≥ 2 n z , (c) The r e gularization parameters satisfy λ α > 0 , λ σ > 0 , (d) The quantity c pe ¯ ϵ is sufficiently small. Then the n z -step MPC scheme (17) is practically exponen- tially stable : there exist constants c > 0 and ρ ∈ (0 , 1) such that, for all initial states in a r e gion of attraction that gr ows as ¯ ϵ → 0 : ∥ x k − x s ∥ 2 ≤ c ρ k ∥ x 0 − x s ∥ 2 + β (¯ ϵ ) , ∀ k ≥ 0 (25) wher e β : R ≥ 0 → R ≥ 0 is continuous with β (0) = 0 . The constants c , ρ , and β depend on the cost matrices Q , R , the r e gularization parameter s λ α , λ σ , and the system dimensions. If ¯ ϵ = 0 (e xact K oopman embedding), then β (¯ ϵ ) = 0 and (25) r educes to exponential stability . Pr oof: By Lemma 3, the nonlinear I/O data decompose as ( u d , y d, nom + ϵ d ) with   ϵ d k   ∞ ≤ ¯ ϵ , where y d, nom are trajectories of the nominal K oopman L TI system G defined by ( A, B , C, D ) . As noted in Remark 2, the n z -dimensional re- alization ( A, B , C , D ) is neither controllable nor observable; howe ver , the I/O transfer function G ( z ) = C ( z I − A ) − 1 B + D admits a minimal realization of order n eff ≤ n z that is controllable and observable by construction. Since n z is a valid upper bound on n eff , all results of [3] hold with n replaced by n z [3, p. 1703]. Conditions (a)–(d) then imply [3, Assumptions 2 and 4], placing the problem in the exact setting of [3, Theorem 3]. That theorem yields a L yapunov function V t := J ∗ L ( t ) + γ W ( ξ t ) satisfying c 1 ∥ x t − x s ∥ 2 2 ≤ V t ≤ c 2 ∥ x t − x s ∥ 2 2 [3, Lemma 1] that conv erges exponen- tially to V t ≤ β ( ¯ ϵ ) . Con verting: ∥ x k − x s ∥ 2 2 ≤ V k /c 1 ≤ ( c 2 /c 1 ) ρ k ∥ x 0 − x s ∥ 2 2 + β (¯ ϵ ) /c 1 , yielding (25) with c = p c 2 /c 1 and β rescaled by 1 / √ c 1 . Application to the Synchr onous Generator: Theorem 4 applies to the generator (2) with n z = 7 , ϵ B = 0 , ϵ C = 0 , ϵ A = O (∆ t ) , and c 0 = O ((∆ t ω max ) 2 ) from Theorem 1. W ith ¯ e = ϵ A diam z + c 0 , the effecti ve noise level (21) becomes ¯ ϵ = ∥ C ∥ 2 ¯ e P L − 2 l =0 ∥ A ∥ l 2 . While each factor is individually moderate, the product can be lar ge. For the minimum horizon L = 14 with typical parameters ( ∆ t = 0 . 0025 s, ω max = 0 . 02 pu, diam z ≈ 1 ): ¯ ϵ ≈ 2 . 5 |{z} ∥ C ∥ 2 × 0 . 015 | {z } ¯ e × 43 |{z} P ∥ A ∥ l 2 ≈ 1 . 6 . This effecti ve noise le vel, while O (∆ t ) in scaling, has a large proportionality constant ( ∥ C ∥ 2 P ∥ A ∥ l 2 ≈ 107 ) due to the accumulation of Koopman errors ov er L steps and the non-normality of A ( ∥ A ∥ 2 = 1 . 18 while ρ ( A ) = 1 ). When ¯ ϵ ≈ 1 . 6 is substituted into the stability constants from [3, The- orem 3]—which in volv e the L yapuno v sandwich ratio c 2 /c 1 , the PE constant c pe , and the regularization parameters—the resulting β (¯ ϵ ) yields a state bound of O (10 3 ) pu. This is ph ys- ically meaningless, despite the theorem being qualitati vely correct ( β ( ¯ ϵ ) → 0 as ¯ ϵ → 0 ). The conservatism has two sources. First, the stability proof in [3] is designed for generic L TI systems with worst- case noise; it makes no use of the specific structure of the K oopman approximation error . Second, and more funda- mentally , the uniform bound ¯ ϵ replaces the state-dependent residual   e i j   2 ≤ ϵ A   ¯ z i j   2 + c 0 by its worst-case value ¯ e = ϵ A diam z + c 0 . This discards the proportional structure that is the hallmark of the Koopman embedding: near the equilibrium, the actual residual is O ( c 0 ) ≈ 10 − 8 , not O ( ¯ e ) ≈ 10 − 2 . The follo wing subsection dev elops tighter bounds that exploit this structure. C. T ighter Bounds via Pr oportional Err or Structur e The uniform bound ¯ ϵ in Lemma 3 is conservativ e in two ways: (i) it uses ∥ A ∥ l 2 (spectral norm raised to a po wer) rather than the tighter   A l   2 (norm of the matrix power), and (ii) it replaces the state-dependent residual   ¯ z i j   2 by the worst-case diam z . W e no w address both. Lemma 5 (Tighter Uniform Bound) . Under the same condi- tions as Lemma 3, the per-step output discrepancy satisfies: ¯ ϵ tight k := ¯ e k − 1 X l =0   C A l   2 + ϵ C diam z (26) with uniform bound ¯ ϵ tight := max 0 ≤ k ≤ L − 1 ¯ ϵ tight k = ¯ e P L − 2 l =0   C A l   2 + ϵ C diam z . This satisfies ¯ ϵ tight ≤ ¯ ϵ , with strict inequality whenever A is non-normal. Pr oof: From (23) and (22):   C ∆ i k   2 =       k − 1 X j =0 C A k − 1 − j e i j       2 ≤ k − 1 X j =0   C A k − 1 − j   2   e i j   2 ≤ ¯ e k − 1 X l =0   C A l   2 . The improvement over Lemma 3 comes from bound- ing   C A l   2 directly rather than splitting ∥ C ∥ 2 ∥ A ∥ l 2 ≥ ∥ C ∥ 2   A l   2 ≥   C A l   2 . For the generator, A is non-normal ( ∥ A ∥ 2 = 1 . 18 while ρ ( A ) = 1 ). The actual   A k   2 grows polynomially (linearly in k ), not exponentially as ∥ A ∥ k 2 would suggest. For L = 14 , ∥ C ∥ 2 P 12 l =0 ∥ A ∥ l 2 ≈ 107 while P 12 l =0   C A l   2 ≈ 49 , a factor of ∼ 2 ; but for L = 50 , the ratio exceeds 130 × , making Lemma 5 essential for longer horizons. The more significant improv ement comes from preserving the proportional error structure through the entire bound. Lemma 6 (State-Dependent Error Bound) . Under the same conditions as Lemma 3, the output discr epancy along the i -th trajectory satisfies the state-dependent bound:    ϵ d,i k    2 ≤ k − 1 X l =0   C A l   2  ϵ A   ¯ z i k − 1 − l   2 + ϵ B   ¯ u i k − 1 − l   2 + c 0  + ϵ C   ¯ z i k   2 . (27) In particular , if ϵ B = ϵ C = 0 (as for the generator), this simplifies to:    ϵ d,i k    2 ≤ ϵ A k − 1 X l =0   C A l   2   ¯ z i k − 1 − l   2 | {z } pr oportional: vanishes at equilibrium + c 0 k − 1 X l =0   C A l   2 | {z } offset: persists . (28) The pr oportional term depends on the actual trajectory deviation   ¯ z i j   2 , not the worst-case diam z as in Lemma 3. Near equilibrium (   ¯ z i j   2 → 0 ), the effective noise r educes to c 0 P k − 1 l =0   C A l   2 , which is O ( c 0 ) —or ders of magnitude smaller than the uniform bound ¯ ϵ = O ( ϵ A diam z ) . Pr oof: The proof follows Lemma 3 up to (23), b ut retains the state-dependent residual bound. From (22):    ϵ d,i k    2 ≤   C ∆ i k   2 +   η i k   2 =       k − 1 X j =0 C A k − 1 − j e i j       2 + ϵ C   ¯ z i k   2 ≤ k − 1 X j =0   C A k − 1 − j   2   e i j   2 + ϵ C   ¯ z i k   2 . Substituting   e i j   2 ≤ ϵ A   ¯ z i j   2 + ϵ B   ¯ u i j   2 + c 0 from Defi- nition 2 and re-indexing with l = k − 1 − j giv es (27). Theorem 7 (T ighter Stability via Proportional Error) . Con- sider the setting of Theorem 4, and suppose additionally that ϵ B = ϵ C = 0 (as holds for the generator by Theor em 1). Define: S L := L − 2 X l =0   C A l   2 (29) and the offset-only noise level ¯ ϵ 0 := c 0 S L . Then: (a) (Self-consistent bound) F or any r > 0 , if the closed-loop state satisfies ∥ ¯ z k ∥ 2 ≤ r for all k in the offline and online trajectories, the effective noise level can be tightened fr om ¯ ϵ to: ¯ ϵ ( r ) := ϵ A S L r + ¯ ϵ 0 . (30) (b) (F ixed-point argument) Pr ovided ϵ A S L is sufficiently small, the bound (25) fr om Theor em 4 holds with ¯ ϵ r eplaced by ¯ ϵ ( r ∗ ) , wher e r ∗ is the smallest positive solution of the fixed-point equation: r = c ∥ x 0 − x s ∥ 2 + β (¯ ϵ ( r )) , (31) with c, β from Theor em 4. Since β is continuous with β (0) = 0 and ¯ ϵ ( r ) → ¯ ϵ 0 = O ( c 0 ) as r → 0 , the fixed- point r ∗ satisfies r ∗ ≤ c ∥ x 0 − x s ∥ 2 + β (¯ ϵ 0 ) , yielding: lim sup k →∞ ∥ x k − x s ∥ 2 ≤ β  c 0 S L  (32) which depends on c 0 (the offset) rather than ϵ A diam z (the worst-case pr oportional err or). Pr oof: Part (a). If   ¯ z i j   2 ≤ r for all trajectory steps, Lemma 6 with ϵ B = ϵ C = 0 giv es    ϵ d,i k    2 ≤ ϵ A S L r + c 0 S L = ¯ ϵ ( r ) . This is a valid uniform noise bound, so Theorem 4 applies with ¯ ϵ replaced by ¯ ϵ ( r ) . Part (b). The bound (25) with noise lev el ¯ ϵ ( r ) gi ves ∥ x k − x s ∥ 2 ≤ c ρ k ∥ x 0 − x s ∥ 2 + β (¯ ϵ ( r )) . In particular , lim sup k →∞ ∥ x k − x s ∥ 2 ≤ β (¯ ϵ ( r )) . The self-consistency requirement is that r must be large enough to contain the entire trajectory , i.e., r ≥ c ∥ x 0 − x s ∥ 2 + β (¯ ϵ ( r )) . The smallest such r is the fixed point r ∗ of (31). Since β (¯ ϵ ( r )) is continuous and increasing in r with β (¯ ϵ (0)) = β ( ¯ ϵ 0 ) finite, and the left side r grows linearly , a fixed point e xists provided the slope β ′ (¯ ϵ ( r )) · ϵ A S L < 1 for large r , which holds when ϵ A S L is sufficiently small. The asymptotic tracking error is lim sup ∥ x k − x s ∥ 2 ≤ β (¯ ϵ ( r ∗ )) ≤ β (¯ ϵ 0 ) = β ( c 0 S L ) . Remark 4 (Improvement over Theorem 4) . Theorem 7 giv es an asymptotic tracking error of β ( c 0 S L ) compared to β (¯ ϵ ) from Theorem 4, where ¯ ϵ = ∥ C ∥ 2 ¯ e P L − 2 l =0 ∥ A ∥ l 2 from (21). The improv ement has two sources: (i) the sum S L = P   C A l   2 is tighter than ∥ C ∥ 2 P ∥ A ∥ l 2 (Lemma 5), and (ii) the proportional error structure replaces ¯ e = ϵ A diam z + c 0 by c 0 alone. The second effect dominates: for the generator with ϵ A ≈ 0 . 015 , diam z ≈ 1 , c 0 ≈ 10 − 8 : c 0 ¯ e = c 0 ϵ A diam z + c 0 ≈ 10 − 8 0 . 015 ≈ 7 × 10 − 7 . The effecti ve noise at the equilibrium is six orders of mag- nitude smaller than the worst-case noise. Since β scales as O (¯ ϵ 2 ) for small ¯ ϵ [3], the tracking error improv es by a factor of O (10 − 12 ) . The proportional error structure effecti vely eliminates ϵ A from the asymptotic bound, leaving only the irreducible offset c 0 . V . C O N C L U S I O N W e have established rigorous conditions under which data- driv en MPC, applied directly to input-output data from a nonlinear system, yields practical exponential stability . The key insight is that an approximate K oopman linear embed- ding serves as a purely analytical certificate: the controller itself nev er uses the Koopman model, but the existence of the embedding guarantees that the nonlinear data can be interpreted as noisy L TI data, enabling the robust stability theory of [3] to be applied. While the uniform noise bound ¯ ϵ can be conservati ve, the proportional error structure of the K oopman embedding (Theorem 7) recovers an asymptotic tracking error that depends only on the irreducible offset c 0 , which is negligible for the synchronous generator at practical sampling rates. A P P E N D I X I P RO O F O F T H E O R E M 1 W e denote the equilibrium lifted state by z s = Φ( x s ) and the deviation ¯ z k := z k − z s , ¯ u k := u k − u s . W e construct a matrix A ∈ R 7 × 7 and vector B ∈ R 7 × 1 such that ¯ z k +1 = A ¯ z k + B ¯ u k + e k , where e k is the residual satisfying the proportional bound. The constant affine terms (such as α q E f d in the flux-decay equation) cancel by subtraction of the equilibrium relation. Define the shorthand θ k := ∆ t ˜ ω k , α d := ∆ t/ M , α q := ∆ t/T ′ d 0 , γ := V ∞ /X Σ , κ := ( X d + X e ) /X Σ , µ := ( X d − X ′ d ) V ∞ /X Σ . Components 1–3: Exact Linear Propagation. From the Euler discretization (3)–(5), the absolute dynamics are: z 1 ,k +1 = z 1 ,k + ∆ t z 2 ,k (33) z 2 ,k +1 = (1 − α d D ) z 2 ,k − α d γ z 6 ,k + α d u k (34) z 3 ,k +1 = (1 − α q κ ) z 3 ,k + α q µ z 7 ,k + α q E f d . (35) At equilibrium, the same equations hold with z k = z s , u k = u s , and z k +1 = z s . Subtracting: ¯ z 1 ,k +1 = ¯ z 1 ,k + ∆ t ¯ z 2 ,k (36) ¯ z 2 ,k +1 = (1 − α d D ) ¯ z 2 ,k − α d γ ¯ z 6 ,k + α d ¯ u k (37) ¯ z 3 ,k +1 = (1 − α q κ ) ¯ z 3 ,k + α q µ ¯ z 7 ,k . (38) Note that the constant α q E f d in (35) cancels exactly in (38). All three deviation equations are exactly linear in ¯ z k and ¯ u k with zero residual: e i,k = 0 for i = 1 , 2 , 3 . Component 4 ( sin δ ). By the angle addition formula: z 4 ,k +1 = sin δ k +1 = sin( δ k + θ k ) = z 4 ,k cos θ k + z 5 ,k sin θ k . (39) Since | θ k | ≤ ∆ t ω max =: ¯ θ ≪ 1 by Assumption 1, we apply T aylor’ s theorem with explicit Lagrange remainders: cos θ k = 1 − θ 2 k 2 ξ c,k , sin θ k = θ k − θ 3 k 6 ξ s,k (40) where ξ c,k , ξ s,k ∈ [0 , 1] . Substituting (40) into (39): z 4 ,k +1 = z 4 ,k + ∆ t z 2 ,k z 5 ,k + r 4 ,k (41) where the higher-order remainder is r 4 ,k = − θ 2 k 2 ξ c,k z 4 ,k − θ 3 k 6 ξ s,k z 5 ,k . (42) The bilinear term ∆ t z 2 ,k z 5 ,k is not linear in z k . W e absorb it into the residual by writing z 4 ,k +1 = z 4 ,k + e 4 ,k with e 4 ,k := ∆ t z 2 ,k z 5 ,k + r 4 ,k . (43) Bounding each term: since | z 5 ,k | = | cos δ k | ≤ 1 and z 2 ,s = ˜ ω s = 0 , | ∆ t z 2 ,k z 5 ,k | ≤ ∆ t | z 2 ,k − z 2 ,s | ≤ ∆ t ∥ ¯ z k ∥ 2 . (44) For the remainder (42), using | θ k | 2 ≤ ¯ θ 2 and | z 4 ,k | , | z 5 ,k | ≤ 1 : | r 4 ,k | ≤ ¯ θ 2 2 + ¯ θ 3 6 ≤ ¯ θ 2 . (45) Combining (44) and (45): | e 4 ,k | ≤ ∆ t ∥ ¯ z k ∥ 2 + ¯ θ 2 . (46) Component 5 ( cos δ ). By the angle addition expansion: z 5 ,k +1 = cos( δ k + θ k ) = z 5 ,k cos θ k − z 4 ,k sin θ k = z 5 ,k − ∆ t z 2 ,k z 4 ,k + r 5 ,k (47) where r 5 ,k = − θ 2 k 2 ξ c,k z 5 ,k + θ 3 k 6 ξ s,k z 4 ,k . Defining e 5 ,k := − ∆ t z 2 ,k z 4 ,k + r 5 ,k and bounding identically to component 4: | e 5 ,k | ≤ ∆ t ∥ ¯ z k ∥ 2 + ¯ θ 2 . (48) Component 6 ( E ′ q sin δ ). By the product rule: z 6 ,k +1 = E ′ q ,k +1 sin δ k +1 = z 3 ,k +1 · z 4 ,k +1 . (49) Substituting z 3 ,k +1 from (35) and z 4 ,k +1 = z 4 ,k + e 4 ,k : z 6 ,k +1 =  (1 − α q κ ) z 3 ,k + α q µ z 7 ,k + α q E f d  ×  z 4 ,k + e 4 ,k  = (1 − α q κ ) z 6 ,k + α q µ z 7 ,k z 4 ,k + α q E f d z 4 ,k + z 3 ,k +1 e 4 ,k . (50) The first term (1 − α q κ ) z 6 ,k is linear in z k . The remaining terms are nonlinear . For the linear embedding, we assign the linear part to the A -matrix and collect all nonlinear and constant deviations into the residual: e 6 ,k := α q µ  z 7 ,k z 4 ,k − z 7 ,s z 4 ,s  + α q E f d ( z 4 ,k − z 4 ,s ) + z 3 ,k +1 e 4 ,k . (51) T o bound the bilinear term, we add and subtract z 7 ,s z 4 ,k :   z 7 ,k z 4 ,k − z 7 ,s z 4 ,s   =   ( z 7 ,k − z 7 ,s ) z 4 ,k + z 7 ,s ( z 4 ,k − z 4 ,s )   ≤ | z 4 ,k | | ¯ z 7 ,k | + | z 7 ,s | | ¯ z 4 ,k | ≤ (1 + E ′ q , max ) ∥ ¯ z k ∥ 2 (52) where we used | z 4 ,k | = | sin δ k | ≤ 1 , | z 7 ,s | = | E ′ q ,s cos δ s | ≤ E ′ q , max , and | ¯ z i,k | ≤ ∥ ¯ z k ∥ 2 . The remaining terms sat- isfy | α q E f d ¯ z 4 ,k | ≤ α q E f d ∥ ¯ z k ∥ 2 and | z 3 ,k +1 e 4 ,k | ≤ E ′ q , max (∆ t ∥ ¯ z k ∥ 2 + ¯ θ 2 ) , where we bounded | z 3 ,k +1 | ≤ E ′ q , max . Combining: | e 6 ,k | ≤ c 6 ∥ ¯ z k ∥ 2 + c ′ 6 ¯ θ 2 (53) with c 6 := α q µ (1+ E ′ q , max ) + α q E f d + E ′ q , max ∆ t and c ′ 6 := E ′ q , max . Component 7 ( E ′ q cos δ ). By the identical product-rule ar- gument applied to z 7 ,k +1 = z 3 ,k +1 · z 5 ,k +1 , with z 5 ,k +1 = z 5 ,k + e 5 ,k : z 7 ,k +1 = (1 − α q κ ) z 7 ,k + α q µ z 7 ,k z 5 ,k + α q E f d z 5 ,k + z 3 ,k +1 e 5 ,k . (54) The residual satisfies the analogous bound: | e 7 ,k | ≤ c 7 ∥ ¯ z k ∥ 2 + c ′ 7 ¯ θ 2 (55) with c 7 := α q µ (1+ E ′ q , max ) + α q E f d + E ′ q , max ∆ t and c ′ 7 := E ′ q , max . System Matrices and Aggregate Error Bound. Collecting results, the system matrices in the deviation- coordinate embedding ¯ z k +1 = A ¯ z k + B ¯ u k + e k are: A =           1 ∆ t 0 0 0 0 0 0 1 − α d D 0 0 0 − α d γ 0 0 0 1 − α q κ 0 0 0 α q µ 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 − α q κ 0 0 0 0 0 0 0 1 − α q κ           , (56) B =  0 α d 0 0 0 0 0  ⊤ . (57) Since y k = ( z 2 ,k , γ z 6 ,k ) ⊤ = ( ˜ ω k , P e,k ) ⊤ : C =  0 1 0 0 0 0 0 0 0 0 0 0 γ 0  , D = 0 . (58) Rows 1–3 follow from (36)–(38) (exact, no residual). Rows 4–7 have the identity as their linear part (since the leading term in each is z i,k ), with all remaining terms collected into the residuals e 4 ,k , . . . , e 7 ,k . The error vector e k = (0 , 0 , 0 , e 4 ,k , e 5 ,k , e 6 ,k , e 7 ,k ) ⊤ satisfies: ∥ e k ∥ 2 ≤ 7 X i =4 | e i,k | ≤ ( c 4 + c 5 + c 6 + c 7 ) ∥ ¯ z k ∥ 2 + ( c ′ 4 + c ′ 5 + c ′ 6 + c ′ 7 ) ¯ θ 2 (59) where c 4 = c 5 = ∆ t , c ′ 4 = c ′ 5 = 1 , and c 6 , c 7 , c ′ 6 , c ′ 7 are defined above. Since ¯ θ = ∆ t ω max , the aggregate bound is: ∥ e k ∥ 2 ≤ ϵ A ∥ ¯ z k ∥ 2 + c 0 (60) with ϵ A := 2∆ t + c 6 + c 7 = O (∆ t ) (61) and offset c 0 := ( c ′ 4 + c ′ 5 + c ′ 6 + c ′ 7 ) ¯ θ 2 = (2 + 2 E ′ q , max )(∆ t ω max ) 2 . This is precisely the proportional-with- offset bound (13) stated in Theorem 1. 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