Stability Analysis of a Discrete-Time Limit Cycle Model Predictive Controller

Recently, a novel discrete-time nonlinear limit cycle model predictive controller for harmonic compensation has been proposed. Its compensating action is achieved by using the dynamics of a supercritical Neimark–Sacker bifurcation normal form at the core of its cost function. This work aims to extend this approach's applicability by analyzing its stability. This is accomplished by identifying the normal form's region of attraction and final set, which enables the use of LaSalle's invariance principle. These results are then extended to the proposed controller under ideal conditions, i.e., zero-cost solutions with predictable disturbances. For nonideal scenarios, i.e., solutions with unpredictable disturbances and cost restrictions, conditions are developed to ensure that the closed-loop system remains inside the normal form's region of attraction. These findings are tested under nonideal conditions in a power systems application example. The results show successful power quality compensation and a satisfactory resilient behavior of the closed loop within the margins developed during this work.

in [2], which had a linear oscillator at the core of its cost function that could steer the system's frequency but not the amplitude of the oscillation directly, only constrain it [3].This motivates using nonlinear oscillators for control, particularly for applications where a fundamental harmonic behavior is desired.
Recently, in [4], a novel nonlinear MPC approach has been proposed, i.e., the limit cycle model predictive control (LCMPC).The LCMPC uses the dynamics of a nonlinear oscillator, i.e., a supercritical Neimark-Sacker normal form, as the basis for its cost function.The interest in using model predictive control (MPC) is the straightforward translation of the control objective into a cost function, which, for the LCMPC, is conveniently given by the chosen nonlinear oscillator.The supercritical Neimark-Sacker normal form (the discrete-time counterpart of the Hopf bifurcation) presents the desired attractor limit cycle dynamics ready for digital implementation on a discrete-time MPC.This motivated the inclusion of the normal form oscillator dynamics directly into the LCMPC cost function in a stepwise square error formulation.This approach differs from standard nonlinear MPC practice, where the reference system is typically unknown to the controller and is treated as a fixed external reference [5], [6].The stepwise inclusion of the reference system enables more flexible trajectories, particularly where tradeoffs are required due to disturbances or constraints, as a new compliant trajectory can be followed at any point.This allows the LCMPC, for instance, to adjust its amplitude after a disturbance while keeping the right frequency using a new compliant trajectory without retracing [4].This can be challenging for periodic MPC approaches, where transitionary trajectories are also restricted to be periodic, hindering amplitude correction (only at the limit cycle for LCMPC) [5].
When looking at relevant LCMPC applications, e.g., power systems harmonic compensation, most established approaches rely on finely tuned resonant controller chains [7].While their simpler implementation has proven successful in real time, in contrast to MPC, there is no explicit handling of constraints nor full exploitation of system and disturbance knowledge, e.g., look-ahead action, which further motivates the exploration of MPC options.
The previous LCMPC study in [4] focused on the cost function design from the Neimark-Sacker normal form, providing some first convexity results and an application example for power systems harmonic compensation with predictable disturbances.This article aims to expand this work by first identifying the normal form's region of attraction and final set based on its parameters, as presented in Section II.These findings are needed for the main contribution in Section III, where conditions are developed to ensure that the closed-loop system remains inside the normal form's region of attraction for sufficiently small disturbances.These results are showcased in Section IV, with a power system application example under unpredictable disturbances and cost restrictions.

II. NEIMARK-SACKER NORMAL FORM STABILITY
This section focuses on analyzing the stability of the Neimark-Sacker normal form.The aim is to characterize its region of attraction and final set, which are fundamental for the mitigation strategies developed in Section III.

A. Normal Form Parameters
The normal form used by the LCMPC is a supercritical Neimark-Sacker autonomous system given as in polar coordinates {r, θ} ∈ R and where the parameters {α, ρ, φ}∈R >0 (required for the attractor limit cycle supercritical case) stand for the scale factor, circular limit cycle radius, and phase step size correspondingly, in discrete time k ∈ Z, [4], [8], [9].Negative radii are considered to ease the analysis, but can be interpreted as phase jumps.
In Cartesian coordinates, for , the system is given as where Some critical radii that will prove helpful in the following analysis: ρ 0 = (1 + αρ 2 )/α is the starting radius magnitude that, after one step, leads to the unstable equilibrium at the origin; starting radius magnitudes larger than ρ ∞ = (2 + αρ 2 )/α will diverge toward ∞ [4].

B. Normal Form Phase Space Analysis
Next, the key phase space properties of (1) are identified, starting with its region of attraction.
Proof: Let V : D → R relate the distance of the radius r k in (1) to the circular limit cycle of radius ρ as then, starting at |r 0 | = ρ ∞ , the distances to the limit cycle of all following iterations of (1) are nonincreasing, i.e., 0 ≤ |r k | ≤ ρ ∞ , and thus, positively invariant in Ω.Looking at the roots of the r k polynomial in ( 5) , the roots r 7,k to r 10,k become complex.Evaluating the remaining double roots' second derivative all are negative for αρ 2 ∈ (0, 2 √ 2 − 2), thus, have negative "downwards" concavity toward −∞.This leaves the symmetrical single roots r 11,k and r 12,k , as only crossing points at V (f (x k )) − V (x k ) = 0. Since they are also the largest real roots in absolute magnitude, all other real roots lie between them.Therefore, the polynomial in ( 5) is always nonpositive for the interval bounded by the single roots at ±ρ ∞ , i.e., Ω.
Now that the region of attraction Ω is well defined for (1), next is the identification of the final set.
For further analysis, M can be divided into positively invariant subsets corresponding to each real root pair

C. Stability Analysis
Now that the region of attraction Ω and the final subsets of M have been characterized, their interactions with respect to (1) are described in the following.
Theorem 1: Every solution of the autonomous system in (1) starting in Ω approaches M as k → ∞.
Proof: Considering the compact set Ω ⊂ D, which is positively invariant with respect to (1), and choosing the continuous function V , where Applying LaSalle's invariance principle, then every solution of (1) starting in Ω approaches M as k → ∞ [10], [11].
While Theorem 1 establishes M as a final set for Ω, looking at the subsets of M in (7), their reachability differs within Ω. Starting with M ∞ , it can only be reached by solutions at r k = ±ρ ∞ , i.e., within M ∞ ; which is unlikely to be hit exactly in real applications.While M 0 is reached from any trajectory passing by r k = ±ρ 0 , it consists of a single unstable equilibrium point at the origin; thus, in practical applications with nonzero disturbances, it is not relevant.This leaves M ρ as the main final set for most trajectories in Ω.
Although these results apply only to the normal form in (1), some key connections exist to the nonlinear MPC approach proposed in this work.Assuming a second-order system (the simplest case for oscillation) and recalling from [4], let the LCMPC optimization problem be with nonlinear cost function Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. where tor containing an ordered stack of state vectors for the whole prediction horizon H p ∈ N, with parameter matrices , and is the Hadamard product.
The LCMPC cost function is, in essence, a stepwise squared error accumulation of the predicted state behavior versus the normal form in (2).This is further detailed in Section III-A, like in (18).Next, a connection between Theorem 1 findings and the LCMPC optimization in ( 8) can be given as follows.
Theorem 2: Let X = arg min J(X) be the solution of the optimization problem in (8), consisting of every consecutive state sequence x ∈ Ω from time step k + 1 up to k + H p , stacked in order into a column vector.If J(X ) = 0, then every consecutive state sequence x of the optimization problem solution X approaches M as k → ∞.
Proof: From [4, Th. 1], it that if the cost J(X ) = 0, then every consecutive state sequence x stacked in X obeys the dynamics of (2).Therefore, as every state sequence x of X is restricted to Ω, Theorem 1 is applicable, and thus, every consecutive state sequence x of X approaches M as k → ∞.

III. NONIDEAL SOLUTIONS
Theorem 2 assumes ideal conditions for a controllable system with predictable disturbances, where a zero-cost solution can be found.However, under practical assumptions, a zero-cost solution is typically not found even if it exists; this can be, among other reasons, due to constraints, numeric limitations, or convergence to a local minimum.Moreover, prediction errors, modeling limitations, measurement noise, and other unpredictable disturbances can lead the closed-loop system behavior away from ideal solutions.
This section analyzes these nonideal solutions, starting by defining error concepts that represent them.Then, critical error margins can be introduced, which will be fundamental for developing conditions to ensure that the closed-loop system remains in Ω.
To help distinguish these nonideal solutions from the ideal ones, from this section on, the solutions found by the optimization solver will be referred to as "computed," denoted by a tilde "∼" above, which can differ from the "true" optimum (ideal) solutions like X .

A. Nonzero Cost Solutions Error
The nonzero cost solutions error arises when the LCMPC solver cannot find a zero-cost solution, as discussed at the beginning of this section.Therefore, it relies on the solver's best solution, i.e., the computed solution Here, for ease of analysis, an underlying linear state-space model for the LCMPC is considered as with computed state vector x ∈ R n , computed control input ũ ∈ R m , computed measurable input disturbance prediction z ∈ R d , and parameter matrices A ∈ R n×n , B ∈ R n×m , and F ∈ R n×d .The system is assumed to be controllable, with direct access to the states for simplicity; thus, no output equations are given, though the approach can be extended to support them.Given an initial measured state x k and a prediction for the future measurable input disturbances Z, the computed state vector X lifted system can be expressed in terms of the computed input sequence vector Ũ where with parameter matrices Ψ ∈ R nHp×n , Θ ∈ R nHp×mHp , and Γ ∈ R nHp×dHp , resulting from (11), [12].From here on, the analysis is restricted to the relevant dynamics of the oscillation, which are captured by a second-order plant (n = 2).Using (12) leads to a reformulation of the LCMPC optimization problem as Next, an expression for nonzero cost errors can be developed.Let which is the normal form prediction in (2) at step k + 1 for xk .The nonzero cost step error is given as Expanding ε J,k+1 for the whole prediction H p leads to where is the stepwise normal form state prediction vector for the whole H p following (14).Notice that the normal form state prediction vector X will always depend on the computed state vector X at each step, i.e., it cannot be recursively calculated just from the starting state x 0 .
There is a clear connection between the nonzero cost error vector E J and the cost function J in (13).Calculating the cost function for the computed state vector X, it follows from [4, Proof of Th. 1] that This formulation shows that the cost error in (15) calculates the cost function J at each step, given by the error between the computed LCMPC state xk+1 and the "ideal" normal form prediction xk+1 , based on the previous step computed state xk .From (18), it is evident that there are sets of E J matrices leading to the same cost J X Ũ .

B. Unpredictable Disturbance Error
When comparing the computed state vector x with the measured state vector x from the plant, another source of error can arise from differences due to, e.g., measurement noise, prediction errors, model mismatches, or other unpredictable disturbances.For simplicity, these differences will be modeled as unpredictable state disturbances, extending (11) as with the vector w ∈ R 2 of unpredictable state disturbance.Notice that w is not included in the LCMPC optimization in (13), as it is assumed to be unpredictable.Similarly to ( 12), ( 19) can be lifted to the whole H p as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.where with parameter matrix Ξ ∈ R 2Hp×2Hp , which can be easily shown to be invertible; thus, ΞW can be treated as an output disturbance without loss of generality.
Considering this new formulation, the unpredictable disturbance error for a single step can be defined as which can be extended to the whole H p as

C. Combined Error and Critical Error Margins
Considering both error definitions in ( 16) and ( 22), an overall combined error can be defined as such that which stepwise is given as Notice that even though x is "canceled out" from a geometric perspective, it is always needed to calculate x at every step, as shown in (14).In summary, calculating the different components of the combined error E in (23) only requires the computed state vector X [from Ũ for a given x k and Z, see (12)] for the nonzero cost error E J and the unpredictable state disturbance W for the unpredictable disturbance error E W . Next, corresponding critical error margin concepts will be introduced, which are fundamental for the bounds proposed in Section III-D.Fig. 1 shows the phase space of the first two steps of an example nonideal solution starting at time k = 0. Here, the stepwise combined error ε from (25) for X and X is shown for a given nonzero cost computed state vector X and unpredictable state disturbance W.
The figure also shows critical error radius for each step k, which gives margins for the combined error ε, such that the next step remains within the attraction region Ω.Similarly, critical error radii can be defined for each combined error component such that at every step k,

D. Critical Combined Error
The critical margins introduced in the previous section serve as a reference to analyze if the combined error can lead the system outside the attraction region Ω.It is evident that whether the system actually goes outside Ω depends on the stepwise combined error dynamics.Therefore, this section will focus on analyzing the combined error dynamics that can lead the system outside Ω, starting with the following definition.
Definition 1 (Critical combined error): Let the magnitude of a combined error E be given as Given x k ∈ Ω, then, the combined error E with the smallest Λ(E ) that leads any state sequence x k+i outside Ω, i.e., x k+i 2 > ρ ∞ , where i ∈ [1, . . ., H p ], is the critical combined error.
With the main criteria defined, the critical combined error can be analyzed for initial conditions at different Ω subsets in the following.
Proof: From the proof of Theorem 1, it follows that state trajectories obeying the normal form are nonincreasing with respect to V from (4).Since V correlates with the distance to M ρ , then x k ∈ M ρ∞ is at the smallest distance to M ∞ for trajectories obeying the normal form.From (26), the critical radius ρ k+1 defines the distance from xk+1 (x k ) to M ∞ ; thus, if ε ρ∞,k+1 = ρ k+1 + , then x k+1 can be let outside Ω by just an arbitrarily small margin.By limiting the subsequent error steps in E ρ∞ to zero, the combined error magnitude is kept to a minimum, as required by Definition 1.
Proposition 4: Let Given x k ∈ M 0ρ , let X be a computed nonzero cost LCMPC state vector with unpredictable state disturbance W and predicted measurable input disturbances Z.Then, for an arbitrarily small > 0, the combined error with Λ(E 0ρ ) = ε 0ρ,k+Hp 2 = ρ k+Hp + , is the critical combined error E , as given by Definition 1.
Proof: Similarly to the proof of Proposition 3, as V correlates with the distance to M ρ for trajectories obeying the normal form starting at x k ∈ M 0ρ , x k+Hp is at the smallest distance to M ∞ .Thus, if ε 0ρ,k+Hp = ρ k+Hp + , then x k+Hp can be let outside Ω by .Limiting the previous error steps to zero, the combined error magnitude is kept to a minimum, as required by Definition 1.
Proposition 5: Given x k ∈ M 0 , let X be a computed nonzero cost LCMPC state vector with unpredictable state disturbance W and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
predicted measurable input disturbances Z.Then, for an arbitrarily small > 0, the combined error with ε 0,k+1 2 = and ε 0,k+Hp 2 = ρ k+Hp + , is the critical combined error E , as given by Definition 1.
Proof: As x k ∈ M 0 is in an unstable equilibrium, the smallest disturbance, i.e., ε 0,k+1 2 = , will lead to the case in Proposition 4. Thus, if ε 0,k+1 2 = and ε 0ρ,k+Hp = ρ k+Hp + , then x k+Hp can be let outside Ω.Limiting the in-between error steps to zero, the combined error magnitude is kept to a minimum, i.e., Λ(E 0 )=ρ k+Hp +2 , as required by Definition 1.
Having identified the LCMPC critical combined error E for different Ω subsets, a bound can be determined next for each case.
Theorem 3: Given x k ∈ Ω, assuming a computed nonzero cost LCMPC state vector X with unpredictable state disturbance W, predicted measurable input disturbances Z, and combined error configuration E. Let If Λ(E) < ρ up (x k ), then every consecutive state vector sequence x stacked in X remains in Ω.
Proof: For the first case in (34), where x k ∈ M ρ∞ , the conditions in x k are the same as in Proposition 3. Considering the structure of the critical combined error E ρ∞ , it follows that if Λ(E) < ρ k+1 , then ε k+1 2 < ρ k+1 , keeping x k+1 inside Ω.Since the subsequent combined error steps are zero, the remaining state vector steps x in X obey the normal form.Then, following Theorem 1, every state vector step x in X is nonincreasing with respect to V and, thus, remaining in Ω.Similarly, the second case in (34) includes the conditions for both Proposition 4 and Proposition 5 subsets.Both critical combined error structures, E 0ρ and E 0 , rely on the last step error step ε k+Hp to lead x k+Hp outside Ω, while keeping the previous combined error steps at zero (except for ε 0,k+1 2 = ); thus, obeying the normal form.Therefore, every state vector step x in X previous to x k+Hp remains inside M ρ since they are nonincreasing with respect to V .Assuming a big enough H p , then xk+Hp 2 = ρ, leading to the conservative critical margin ρ k+Hp = ρ ∞ − ρ.Since ε k+Hp 2 ≤ Λ(E) < ρ ∞ − ρ, then x k+Hp remains also inside Ω, along with every other state vector step x in X.Finally, as the claim holds for the critical combined error structures E of each Ω subset for both cases in (34), it holds by extension for any other error configuration that complies with the conditions given.
Theorem 3 critical radius ρ up serves as an upper bound of the combined error magnitude Λ(E), ensuring that every state vector x stacked in X stays in Ω.Moreover, this critical radius can be further distributed to bound each combined error component as in (27).
From geometry, it is clear that the critical combined error from Definition 1 arises when the combined error magnitude contribution has the same angle as the normal form prediction, i.e., pointing outward in a radial direction toward M ∞ (see the light gray dashed radial lines in Fig. 1).Thus, the error bounded by each critical radius component can be assumed to be aligned with the radial outward direction of xk , see Fig. 2.This implies, following the combined error composition (24), that Λ(E) = Λ(E J ) + Λ(E W ).
Therefore, following (18) and (28), part of Theorem 3 critical radius ρ up can be used to bound the cost function J through the nonzero cost error E J as Similarly, for the unpredictable disturbance error E W , following ( 19) and ( 21), the maximum unpredictable disturbance is bounded With ( 35) and (36), a tradeoff for the distribution of ρ up between cost and disturbance rejection can be established, leaving room for control objective tuning and design.Assuming critical error structures as in Definition 1 leads to very conservative bounds.Naturally, considerably less conservative formulations are possible, especially with more information about the plant and the disturbances.

IV. APPLICATION EXAMPLE
This section showcases the conditions developed over the past sections in an application example under nonideal conditions.The application field is harmonic compensation for power systems, where the attractor circular limit cycle dynamics of the normal form in (1) are desired.
As renewable energy share increases with the ongoing power system energy transition, a higher influx of harmonic distortion is expected due to the nonlinear loads involved [13].Active power filters (APFs) are the prime solution; their corrective action is driven by their reference compensation current [14].This is where the LCMPC's nonlinear oscillator dynamics can be used as a reference.

A. Grid Circuit Model
Fig. 3 shows the circuit diagram of the microgrid, taken from the test model from [4].The system's grid connection operating voltage v s is 400 V at 50 Hz of fundamental frequency f , with a feeder line resistance of R 1 = 100 Ω to the point of common coupling.The LCMPC goal is to steer the APF compensation current i c to mitigate the harmonic distortion introduced by i d while ensuring a fundamental harmonic supply for the load.The load parameters are: voltage v l , current i l , charge q l , resistance R 2 = 10 Ω, inductance L 2 = 100 mH, and capacitance C 2 = 10 mF.
The microgrid is modeled as a linear state-space model normalized so that its states x = [0.3i l 89.8 q l ] T oscillate at a radius of ρ = 1 when undisturbed.The system was discretized via zero-order hold at sampling time τ = 200 μs.The control input is u = i c , and the measurable input disturbance vector is z = [i d v s ] T .The output targets the load variables y = [v c i l ] T , with capacitor voltage v c .

B. Controller Setup
To reduce the LCMPC search space, the control input Ũ in (13a) can be restricted to a harmonic order h suitable for the application, [4], reformulating the optimization as min P J X P (37) with Kronecker product ⊗, identity matrix I, and where contains the Fourier coefficients f ∈ R m and g ∈ R m up to order h, with base function matrix M ∈ R Hp×2h .
For ease, direct access to the states is assumed.In addition, the future measurable disturbance prediction sequence vector Z is considered the same as the previous fundamental period measurement.This is a fair assumption when considering steady oscillations for Z, typical for the grid and nonlinear loads like rectifiers.Moreover, due to the periodic nature of the plant and disturbances, a periodic receding horizon is chosen, as introduced in [2].
In contrast to the classical MPC receding horizon approach, Ũ( P) is computed per period and given to the plant in a feedforward manner for an entire fundamental period.Then, a new Ũ( P) is computed in the next period, using new x and Z feedback measurements.This strategy reduces the computational burden, as the optimization is performed once per fundamental period.
To simplify implementation, the unpredictable state disturbance w is assumed to act as an output disturbance y k = Cx k + w k .This follows (20), as Ξ is invertible.Therefore, w will only influence the LCMPC at the end of each period due to the periodic receding horizon strategy.

C. Simulation Setup
The simulations are done for a fundamental frequency of f = 50 Hz and a sampling time of τ = 200 μs.For the LCMPC prediction horizon, two fundamental periods are considered, i.e., H p = 200.Finally, for the normal form parameterization, the target radius is kept at ρ = 1 due to the system normalization, with a phase step size corresponding to the target frequency, φ = 2πf τ , and scaling factor α = 0.1.Recalling Proposition 1, this implies that the controller will be operating at the "slower" end of the parameter range αρ 2 ∈ (0, 2 √ 2 − 2) but providing a wide attraction region Ω, as seen from ρ ∞ = 4.5826.
While a wide region of attraction Ω is important, operating across all Ω is not always convenient for every application.Thus, a new metric is needed.Solving (1a) for ρ, leads to three solutions: ±ρ and which can be interpreted as the "dead-beat" radius before the system reaches the limit cycle at ρ = 1.If ρ db > ρ, this new metric delimits an operational region within Ω, i.e., ρ ≤ r k ≤ ρ db , with few undershoots and phase jumps.This setup requires a scaling factor of 0 < α ≤ 0.4.
The chosen α = 0.1 leads to ρ db = 2.7016, providing a wide operation region.
A harmonic order of 3 and 5 is considered for the measurable input disturbance i d , typical of nonlinear loads like rectifiers.Therefore, the harmonic band of the controller is tightened to h = 5 for the search space of P.
The optimizations were solved with fminunc from MATLAB by embedding the equality constraints in (37a) directly into the cost function.The trust-region was used as an optimization algorithm with  optimality and step tolerances of 10 −6 and 10 −10 , respectively [15].The analytical gradient and Hessian were calculated using [16], [17].
Each iteration of the LCMPC starts with the same guess of P as a vector of zeros.The initial conditions of the state are x 0 = [0.1412− 1.1241] T , calculated after 500 periods of uncontrolled disturbed operation starting from 0 2×1 .

D. Base Scenario
Fig. 4 shows the phase portrait for the first 50 periods of the base scenario, with the controlled response as the blue-solid line and the uncontrolled free response as the red-dashed line.The LCMPC harmonic compensating action is immediately evident.Under this parameterization, the polar plot in Fig. 5 shows slow dynamics leading to a steady response with no severe radius undershoots, thanks to the LCMPC's control action i c and the design of ρ db .

E. Nonideal Conditions Scenario
To test the robustness of the LCMPC, first, the solver of the optimization algorithm is set to stop as soon as a solution with a cost value Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. of J ≤ 0.5 is found, more than an order of magnitude higher than typical solutions.In addition, uniformly distributed white noise between −0.5 and 0.5 is applied as unpredictable output disturbance w.
The control action and the system response in polar coordinates can be seen in Fig. 6.The setup manages to remain in Ω while presenting minor phase jumps and maintaining its radius close to ρ = 1.This can be explained by the conditions developed in Theorem 3. Recalling (34), the critical radius bound ρ up can range from 0 up to ρ ∞ − ρ = 3.5826 for this setup, depending on where x lands in Ω.The cost restriction only amounts to ρ J,up = 0.25, leaving ample margin for the disturbance ρ W,up , even when only considering the operation region ρ db − ρ = 1.7016.During simulation, the average length of w was 0.3805 from a maximum possible 0.7071, which makes it unlikely to hit the bounds with a random orientation at every step.Naturally, total certainty of not hitting the bounds will require a much smaller maximum unpredictable disturbance, as seen from (36).

V. CONCLUSION
This work focused on further developing a novel nonlinear MPC approach based on limit cycles, i.e., the LCMPC.The analysis started by identifying the supercritical Neimark-Sacker normal form's attraction region and final set based on its parameters.This allowed the characterization of its stability via LaSalle's invariance principle.
These findings connected the normal form dynamics and the LCMPC cost function for zero-cost solutions with predictable disturbances.In the case of nonideal solutions, i.e., unpredictable disturbances and cost restrictions, conditions were developed to ensure that the closed-loop system remains inside the normal form region of attraction for sufficiently small disturbances.Furthermore, a power systems application example for the LCMPC was showcased.This simulation study analyzed the systems' response under ideal and nonideal conditions, testing the concepts developed in this work.The compensation results were successful, and the resilience of the approach was satisfactorily within the margins established for nonideal cases.While the developed margins are very conservative, they still offer enough room for practical applications, as showcased here.
Future work will focus on analyzing the phase dynamics and its application in discrete-time nonlinear oscillator synchronization to enable power system ancillary services like grid forming.

Fig. 1 .
Fig. 1.Phase portrait of the measured state vector sequence x k with normal form state prediction xk and combined error ε k .

Fig. 2 .Fig. 3 .
Fig. 2. Phase portrait the measured state vector sequence x k showing the critical radius components of ρ k aligned in outwards radial direction.

Fig. 4 .
Fig.4.Base scenario's phase portrait of the measured state vector x controlled response and uncontrolled free response.

Fig. 5 .
Fig. 5. Base scenario's polar coordinates of the measured state vector x controlled response and control input i c .

Fig. 6 .
Fig. 6.Polar coordinates of the measured state vector x controlled response under cost restriction and noise, and control input i c .