Passivity and Practical Stability of Fast-Sampled Quantized Discrete-Time Nonlinear Systems with Finite-Register Effects

2.1 Notation

Let ℝⁿ denotes by the n-dimensional Euclidean space, and ℤ ≥ 0 the set of nonnegative integers, and for the vector x $\in$ ℝⁿ, ||x|| denotes the Euclidean norm. in addition to a symmetric matrix P = P^T $\in$ℝ{ⁿ^×ⁿ}, the notation P > 0 and P ≥ 0 indicate positive definiteness and positive semi definiteness respectively, and λmin(P) and λmax(P) were denote its smallest, and largest eigenvalues.

$\alpha_1, \alpha_2, \alpha_3$ class $K \infty$ definitions

2.2 Continuous-time plant and fast-sampled discrete-time model

Consider the nonlinear continuous-time plant:

$d x / d t=f_{-} c(x(t), u(t), w(t))$                     (1)

where, $x(t) \in R^n$ is the state, $\mathrm{u}(\mathrm{t}) \in R^m$ is the control input, and $\mathrm{w}(\mathrm{t}) \in R^{\mathrm{n}}$ is an exogenous disturbance. The function $f c: \mathbb{R}^{\mathrm{n}} \times \mathbb{R}^{\mathrm{m}} \times \mathbb{R}^{\mathrm{p}} \rightarrow \mathbb{R}^{\mathrm{n}}$ represents the vector field of the continuoustime plant, which is assumed to be locally Lipschitz continuous in all arguments on the operating region of interest. No specific parametric form is imposed on $f_c$, so that the framework applies to any nonlinear system satisfying the stated Lipschitz and dissipation conditions. The discrete-time representation obtained under zero-order hold and sampling period $T_s>0$ is:

$x_{\{k+1\}}=f\left(x_k, u_k, w_k ; T_s\right)$                   (2)

where, f (·,·,·; Tₛ) is the Tₛ-flow map of f_c, retaining explicit dependence on Tₛ to enableS fast-sampling analysis. For the scalar and two-dimensional examples in Section 5, f_c is given explicitly by the forward-Euler discretization stated there. Under sampling period Tₛ > 0 and zero-order hold, the discrete-time representation is:

$x_{\{k+1\}}=f\left(x_k, u_k, w_k ; T_s\right), k \in Z_{\geq 0}$

where, the explicit dependence on Tₛ is remined to capture fast-sampling effects in the analysis.

Assumption 1. There exists $\bar{T}>0$ such that for all $T_s \in(0$, $\bar{T}$ ), the map $f\left(T_s\right)$ is locally Lipschitz in ( $x, u, w$ ). Moreover, there exist nonnegative functions $L_x\left(T_s\right), L_u\left(T_s\right), L_w\left(T_s\right)$ satisfying:

$\begin{aligned} \| f\left(x_1, u_1, w_1 ; T_s\right) & -f\left(x_2, u_2, w_2 ; T_s\right) \|  \leq L_{x| | x_1-x_2| |}+L_{u| | u_1-u_2| |} +L_{w| | w_1-w_2| |}\end{aligned}$                   (3)

For all admissible arguments. This assumption is standard for fast-sampled nonlinear systems and is consistent with recent implementation-aware analyses in the discrete-time control literature [18-20]. The Lipschitz constants L_x (Tₛ), L_u (Tₛ), L_w (Tₛ) can be computed analytically for polynomial or rational vector fields using the mean-value theorem, or estimated numerically via finite-difference Jacobian bounds over the operating region. For many sampled-data systems, these constants grow with Tₛ, which is why fast sampling reduces implementation error propagation as shown in Lemma 3.

2.3 Nominal feedback law and storage property

Let the nominal control law be:

$u_k=\kappa\left(x_k\right)$                             (4)

where, κ from Rⁿto R^m is locally Lipschitz. The nominal closed-loop dynamics are:

$x_{\{k+1\}}=f\left(x_k, \kappa\left(x_k\right), w_k ; T_s\right)$                            (5)

Assumption 2. There exists a continuous storage function $V$ from $R^{\mathrm{n}}$ to $R(\geq 0)$, functions $\alpha_1, \alpha_2, \alpha_3$ in $\mathcal{K} \infty$, and $\sigma$ in $\mathcal{K}$ such that:

$\alpha_1\left(\left\|x_k\right\|\right) \leq V\left(x_k\right) \leq \alpha_2\left(\left\|x_k\right\|\right)$                           (6)

and

$V\left(x_{\{k+1\}}{ }^{\text {nom}}\right)-V\left(x_k\right) \leq-\alpha_3\left(\left\|x_k\right\|\right)+\sigma\left(\left\|w_k\right\|\right)$                          (7)

where, $x_{\{k+1\}}{ }^{\text {nom }}=f\left(x_k, \kappa\left(x_k\right), w_k ; T_s\right)$. This captures a discretetime dissipation or ISS-type property for the nominal closed loop and forms the baseline for later perturbation analysis [21, 22]. The storage function V and the comparison functions $\alpha_1$, $\alpha_{2,} \alpha_3 \in \mathcal{K} \infty$ can be obtained from a nominal Lyapunov or dissipative analysis of the continuous-time plant before discretization. For ISS systems, V is typically a Lyapunov function satisfying the ISS decay condition, and $\alpha_3$ is derived from the ISS gain. For the examples in Section 5, $V\left(x_{\mathrm{k}}\right)= 1 / 2 x \_k^2$ is used, which is a standard quadratic storage function that satisfies Assumption 2 with $\alpha_1(r)=\alpha_2(r)=1 / 2 r^2$ and $\alpha_3(r)=c \cdot r^2$ for appropriate constants $\mathrm{c}>0$.

2.4 Quantized measurement and actuation

In digital implementation, the controller operates on quantized signals. Define:

x^k = Q_x(xₖ), u^k=Q_u(uk)                              (8)

where, Q_x  from Rⁿ to Rⁿ and Q_u  from R^m  to R^m are static quantizers with steps Δx and Δu, respectively. Define the quantization errors:

e_k^x = Q_x(xₖ) - xₖ, e_k^u=Q_u(uk) - uk                              (9)

Assumption 3. There exist constants c_x > 0 and c_u > 0 such that:

$\left\|e_k^x\right\| \leq c_x \Delta x,\left\|e_k^u\right\| \leq c_u \Delta u$                        (10)

For all k. This is standard for finite-level uniform quantizes [1, 7].

Figure 1 is a block diagram of the implemented closed-loop system showing state quantizer Q_x, controller κ, input quantizer Q_u, interference block Φ, summing junction, and plant.

Figure 1. Block diagram of the digitally implemented nonlinear closed-loop system

With quantized sensing, the implemented command becomes:

u_k = κ(Q_x(xₖ)), ū_k=Q_u(κ(Q_x(xₖ)))                              (11)

2.5 Finite-register effects and interference operator

Finite-word-length arithmetic introduces further distortions including saturation, truncation, and overflow. These are modeled through an interference operator Φ from R^m to R^m, with the actually applied plant input:

ũ_k = Φ(ū_k)=Φ(Q_u(κ(Q_x(xₖ))))                              (12)

Assumption 4. There exist ρ₀, ρ₁ ≥ 0 such that:

$\|\Phi(v)-v\| \leq \rho_1^*\|v\|+\rho_0, \forall \,v \in R^{m_e}$                         (13)

This compactly captures a broad family of finite-precision effects through a sector-bounded or incrementally bounded interference model [8, 23].

2.6 Implemented closed loop and aggregate error

The implemented closed-loop dynamics are:

x_{k+1} = f(xₖ, Φ(Q_u(κ(Q_x(xₖ)))), wₖ; Tₛ)                              (14)

Define the aggregate implementation error:

ε = Φ(Q_u(κ(Q_x(xₖ))))-κ(xₖ)                              (15)

Then:

x_{k+1} = f(xₖ, κ(xₖ)+εₖ, wₖ; Tₛ)                              (16)

V(x_{k+1})−V(xₖ)≤s (v_k, z_k)                              (17)

This shows all implementation nonlinearities as a structured perturbation around the nominal closed-loop input. This, for emulation, is the central device of the paper and enables the subsequent passivity-based analysis.

2.7 Passivity, semi-passivity, and practical stability

Here, $v_k$$\in$ℝ^m denotes the external input (port variable) and $z_k$ $\in$ℝ^m denotes the output port variable of the system at time step k, forming the supply rate pair used in the passivity definition. For the standard supply rate s = z_k^T× v_k, strict passivity requires an additional negative definite term $\in$xₖ. Semi-passivity relaxes this by permitting the presence of dissipation outside a compact set, or by permitting a residual positive term because of implementation effects, which is valid when quantization prevents the asymptotic convergence, and instead results practical stability with a residual set.

A system is practically stable, having there exist a $\mathcal{K} \mathscr{L}$ function β and a constant r ≥ 0, that [10, 24]:

$\left\|x_k\right\| \leq \beta\left(\left\|x_0\right\|, k\right)+r, \quad \forall k \geq 0$                         (18)

In cases where r is dependent on the implementation parameters (Δx, Δu, ρ₀, ρ₁ and Tₛ) the outcome is a direct measure of the digital realization degradation.

2.8 Problem statement

The target is to obtain implementation-sensitive sufficient conditions that guarantee semi-passivity and practical stability of the system in Eq. (14), under constraints on the disturbance wₖ, quantization thresholds Δx and Δu, implementation parameters ρ₀ and ρ₁, and sampling period Tₛ. To be more precise, the objectives are to:

(i) Demonstrates that the applied closed-loop system is, in an appropriate sense, passive or semi-passive even when quantized and on finite-register models;

(ii) Establish that state paths are uniformly bounded with bounded disturbances;

(iii) Express the final set of bounds explicitly in terms of Tₛ, Δx, Δu, ρ₀, and ρ₁;

(iv) Transform those resulting conditions into the practical design guidelines of digital controller implementation.

2.9 Additional assumption on the controller

Assumption 5. Lipschitz κ is the nominal feedback map of the operating region of interest. That is, there exists Lκ > 0 such that:

$\left\|\kappa\left(x_1\right)-\kappa\left(x_2\right)\right\| \leq L \kappa \times\left\|x_1-x_2\right\|$                       (19)

For all admissible x₁, x₂. In addition, there exists a_κ ≥ zero such that:

$\|\kappa(x)\| \leq a_\kappa \times\|x\|$                     (20)

This assumption is mild, on the region of interest, linear, and most nonlinear (standard) feedback laws used in practice satisfy it.

4.1 Theorem 1: Semi-passivity preservation

Theorem 1. Suppose Assumptions 1 through 6 hold. Assume there exists a radius r* > 0 such that:

$\tilde{\alpha}(r)>0, \forall \, r>r^*$                  (43)

Then the implemented closed-loop system is semi-passive with respect to storage V. Specifically:

$V\left(x_{\{k+1\}}\right)-V\left(x_k\right) \leq \sigma\left(\left\|w_k\right\|\right)+\delta_0$                       (44)

for all states, and

$V\left(x_{\{k+1\}}\right)-V\left(x_k\right) \leq-\eta\left(\left\|x_k\right\|\right)+\sigma\left(\left\|w_k\right\|\right)+\delta_0, \forall\left\|x_k\right\|>r^*$            (45)

where, ƞ(r) = α̃(r) for r > r*. Thus, the system is strictly dissipative outside the compact set {x: ||x|| ≤ r*} and hence semi-passive.

Proof. From Proposition 1, the implemented system satisfies:

$V\left(x_{\{k+1\}}\right)-V\left(x_k\right) \leq-\tilde{\alpha}\left(| | x_k| |\right)+\sigma\left(\| w_k \mid\right)+\delta_0$                    (46)

When ||xₖ|| > r*, the assumption α̃(||xₖ||) > 0 ensures the right-hand side has a strictly negative state-dependent term, establishing strict dissipation outside {x: ||x|| ≤ r*}. In every state, the first inequality is obtained by dropping the negative term. By the definition of semi-passivity, the system is semi-passive. This is the end of the evidence.

Remark 5. Explains what is physically meant by implementation degradation. The nominal control law can be strictly dissipative; however, quantization and finite-precision implementation decrease the storage decay rate by the term L_V  × Lᵤ(Tₛ) × c̄₁ × r and introduce a residual leakage term δ₀ [11, 15]. Semi-passivity is maintained as long as this degradation is weaker than the nominal dissipation outside a small radius r*. The radius r* in Theorem 1 is the smallest r for which α̃(r) > 0, i.e., r* satisfies $\alpha_3(r *)=L_1 \cdot L_u\left(T_s\right) \cdot \bar{c}_1 \cdot r *$. For the scalar quadratic case where α₃(r) = c₃r², this gives $r *=c_3 /\left(L_V \cdot L_u\left(T_s\right) \cdot \bar{c}_1\right)$, which decreases as implementation quality improves.

4.2 Theorem 2: Practical stability and ultimate boundedness

Theorem 2. In the hypothesis of Theorem 1, suppose that the disturbance is bounded i.e. namely ||wₖ|| ≤ w̄, all k greater than or equal to 0. Then there is a fixed r_f ≥ r* which ensures that all trajectories of the closed-loop system that is implemented satisfy:

$\left\|x_k\right\| \leq \beta\left(\left\|x \_0\right\|, k\right)+r_{\mathrm{f}},  \forall \, k \geq 0$           (47)

For some $\mathcal{K L}$ function β. The system is virtually steady with final bound r_f a function of δ₀, w̄ and the dissipation margin of α̃.

Proof. From Theorem 1, whenever ||xₖ|| > r*, the storage satisfies:

$V\left(x_{\{k+1\}}\right)-V\left(x_k\right) \leq-\tilde{\alpha}\left(\left\|x_k\right\|\right)+\sigma(\bar{w})+\delta_0$              (48)

Define r_f ≥ r* such that:

$\tilde{\alpha}(r)>\sigma(\bar{w})+\delta_0, \forall \,  r>r_{\mathrm{f}}$                       (49)

Such r_f exists because α̃(r) = α₃(r)-L_V  × Lᵤ(Tₛ) × c̄₁ × r grows without bound when α₃ is of $\mathcal{K} \infty$ and the degradation term remains sublinear relative to α₃. Then for all ||xₖ||>r_f:

V(x_{k+1})-V(xₖ)<0                    (50)

That is why V is reduced where the state is beyond the ball of radius r_f. To apply the sandwich bounds α₁(||xₖ||) ≤ V(xₖ) ≤ α₂(||xₖ||) and standard discrete-time Lyapunov proofs of ultimate boundedness [25], the trajectories approach the non-large set {x: ||xₖ|| ≥ r_f} in finite time and stay thereafter. So there is a KL function β so that, in all ||xₖ|| ≤ β(||x_0||, k) + r_ffor all k ≥ 0. This is where the evidence is completed.

Remark 6. Theorem 2 clarifies the general impossibility of exact asymptotic convergence in finite precision digital control [7, 11]. The values δ₀ and σ (w̄) leave a residual floor that is nonzero, hence the state approaches a neighborhood instead of the origin. Under the ideal condition of Δx = Δu = ρ₀ = ρ₁ = 0 and wₖ = 0 (all k), the residual set reduces to a point and nominal asymptotic behavior is regained.

4.3 Corollary 1: Implementation-oriented admissibility conditions

Corollary 1. Under the conditions of Theorem 2, with ||κ(x)|| ≤ a_κ × ||x||, define:

c̄₀ = (1+ρ₁) × c_u × Δu+(1+ρ₁) × Lκ × c_x× Δx+ρ₀                    (51)

c̄₁ = ρ₁ × a_κ                  (52)

δ₀ = L_V× Lᵤ(Tₛ) × c̄₀                        (53)

If there exists a target residual radius r_d > 0 such that:

$\begin{gathered}\alpha_3(r)-L_V \times L_u\left(T_s\right) \times \rho_1 \times a_\kappa \times r>\sigma(\bar{w})+L_V \times L_u\left(T_s\right) \times \bar{c}_0, \\ \forall r>r_d\end{gathered}$                 (54)

Then every trajectory ultimately enters and remains in the ball {x: ||x|| ≤ r_d}. The above inequality allows any choice of Tₛ, Δx, Δu, ρ₀, ρ₁ that will be admissible with the desired practical-stability specification.

Proof. The assertion is a direct consequence of Theorem 2 that equates r_f with the design radius r_d. The specified inequality is the guarantee that the dissipative term is the leading one in the overall residual beyond r_d, and it means that the entrance will eventually occur, and the whole process will become invariant. This finishes the evidence.

Remark 7. Corollary 1 makes the proof an engineering rule. An engineer can calculate the largest possible number of quantization steps, the smallest possible register length or the largest possible sampling period [17], or any combination of these that will produce practical stability within a desired bound, r_d.

4.4 Corollary 2: Linear state-feedback specialization

Corollary 2. Consider the special case κ(x) = K × x where K $\in$ R^{(m × n)} is a constant gain matrix. Then Assumption 5 holds with Lκ = ||K|| and aκ = ||K||. Consequently:

$\bar{c}_0=\left(1+\rho_1\right) \times c_u \times \Delta u+\left(1+\rho_1\right) \times\|K\| \times c_x \times \Delta x+\rho_0$      (55)

$\bar{c}_1=\rho_1 \times\|K\|$            (56)

If:

$\begin{gathered}\alpha_3(r)-L_V \times L_u\left(T_s\right) \times \rho_1 \times\|K\| \times r>\sigma(\bar{w})+L_V \times L_u\left(T_s\right) \times \bar{c} 0, \forall r>r_d\end{gathered}$              (57)

Then, the practical stability of the implemented closed-loop system is guaranteed with an upper bound on the ultimate bound of rd.

Proof. This is the result of Corollary 1 by replacing Lκ = ||K|| and aκ = ||K||. This is the end of the evidence.

Remark 8. Corollary 2 shows that a large gain matrix K can indeed speed up the rate of nominal convergence at the expense of an increase in the sensitivity of measurement quantization, c̄₀, and finite-register sensitivity, c̄₁. This corollary quantifies this trade-off and can be used as a gain tuning rule for a digital design.

4.5 Design procedure for digital controller implementation

The following procedure takes the theoretical results of this section and converts them into an implementation procedure for the design of a digital controller.

Step 1: Specify the target residual bound r_d

Choose the maximum allowable steady-state error r_d > 0 based on the application requirement. For example, r_d = 0.05 means the state must remain within a ball of radius 0.05 around the origin in steady state.

Step 2: Obtain the nominal dissipation function α₃

From the nominal continuous-time design, identify the storage function V and the dissipation function $\alpha_3 \in \mathcal{K} \infty$ satisfying Assumption 2. For quadratic $V(x)=1 / 2 x^{\wedge} T P x$, compute $\alpha_3(r)=\lambda \min (Q) \cdot r^2$ where Q comes from the nominal Lyapunov equation. Record the controller Lipschitz constants Lκ and a_κ from Assumption 5.

Step 3: Select the sampling period T_s

Choose T_s small enough that:

• The Lipschitz constant L_u (T_s) satisfies L_v · L_u(T_s) · ρ₁ · a_κ < α₃( r_d)/ r_d
• This ensures α̃(r_d) > 0 as required by Theorem 1

A conservative starting point is T_s ≤ 0.1/L_u where L_u is estimated from the nominal plant Jacobian.

Step 4: Select quantization steps Δx and Δu

From Corollary 1, compute the maximum allowable quantization steps by solving:  

$\begin{aligned} \alpha_3\left(r_d\right)-L_V \cdot L_u\left(T_s\right) \cdot \rho_1 \cdot a_k \cdot r_d & \geq L_V \cdot L_u\left(T_s\right) \cdot\left[\left(1+\rho_1\right) c u \Delta u+(1\right. \left.\left.+\rho_1\right) L_k c x \Delta x+\rho_0\right]+\sigma(\bar{w})\end{aligned}$

Choose Δx and Δu (and hence the ADC/DAC resolution in bits: N_bits = log₂(Range/Δ)) to satisfy this inequality. A symmetric choice Δx = Δu = Δ is often convenient.

Step 5: Select finite-register parameters ρ₀ and ρ₁

The parameters ρ₀ and ρ₁ are determined by the word-length W of the processor: typically, ρ₀ ≈ 2^{-W} and ρ₁ ≈ 2^{-W} for fixed-point arithmetic. Verify that the chosen W satisfies the inequality from Step 4 with the selected Δx, Δu.

Step 6: Verify the design inequality

With all parameters selected, compute:

• c̄₀ = (1+ρ₁)c_u Δu+(1+ρ₁)Lκ c_x Δx+ρ₀

• c̄₁ = ρ₁ · a_κ

• δ₀ = L_V · L_u(T_s) · c̄₀

• α̃(r_d) = α₃(r_d)-L_V · L_u(T_s) · c̄₁ · r_d

If α̃(r_d) > σ(w̄) + δ₀, the design is admissible and practical stability with bound r_d is guaranteed by Theorem 2.

Step 7: Iterate if necessary

If the inequality in Step 6 is not satisfied, either increase word-length W (reduce ρ₀, ρ₁), decrease Δx or Δu (increase ADC/DAC resolution), or decrease T_s (faster sampling), and repeat from Step 3.

4.6 Theorem 3: Strengthened form for polynomial dissipation

Theorem 3. Suppose the conditions of Theorem 2 hold with $w_k=0 \,\forall \,\mathrm{k}$, and assume $\alpha_3(r) \geq c_3 \times r^p$ for constants $c_3>0$ and $p \geq 1$. If:

$c_3 \times r^p>L_V \times L_u\left(T_s\right) \times\left(\bar{c}_0+\bar{c}_i \times r\right), \forall \, r>r_d$                 (58)

Then the ball {x: ||x|| ≤ r_d} is globally attractive and positively invariant for the implemented closed-loop system.

Proof. Under the stated conditions, the storage inequality from Proposition 1 gives V(x_{k+1}) - V(xₖ) ≤ -( c₃ × r^p-L_V× Lᵤ(Tₛ) × c̄₁ × r) + L_V× Lᵤ(Tₛ) × c̄₀. Hypothesis the bracket is positive at all r bigger than r_d; therefore, V is decreasing outside the ball. The K∞ bounds of V and the fact that the decrease is true to all initial conditions result in global attraction. This fulfils the evidence.

Remark 9. Theorem 3 gives a more prescriptive scaling law between implementation-induced degradation and nominal dissipation growth. In the quadratic case, p = 2, the condition can be simplified to a mere quadratic versus linear comparison, which can be proved analytically in certain plant models.